Task-based measures of image quality and their relation to radiation dose and patient risk

Consider a binary task in which the goal is to use an image vector g to classify the object that produced it as belonging to either class C₁ (e.g. signal present) or C₀ (e.g. signal absent). The hypothesis that the object was drawn from class C₁ is denoted H₁, and the hypothesis that the object was drawn from class C₀ is denoted H₀.

For algorithmic observers (though not for humans), we can assume that the task is performed without any additional randomness beyond that inherent in the image data (no coin flips or random-number generators can be used) and that no equivocation is permitted (every image must be classified as either C₀ or C₁). Under these two assumptions, it can be shown (see Barrett et al (1998) or section 13.2 in Barrett and Myers (2004)) that any decision rule is equivalent to computing a scalar test statistic t (g) and comparing it to a threshold t_c. If t (g) > t_c, then the decision is that the object came from class C₁, or equivalently that hypothesis H₁ is true; we refer to this decision as ${{\mathcal{D}}_{1}}$. Conversely, if t (g) < t_c, then the decision is that the object came from class C₀, so H₀ is true, and in this case we refer to the decision as ${{\mathcal{D}}_{0}}$.

We can express this decision rule succinctly as

$$\begin{eqnarray}&&t\left(\mathbf{g}\right)\begin{array}{c} \begin{array}{c} {{\mathcal{D}}_{1}} \\ > \\ \end{array} \\ \begin{array}{c} < \\ {{\mathcal{D}}_{0}} \\ \end{array} \\ \end{array}{{t}_{c}},\end{eqnarray} \tag{ 22 }$$

which is read, 'choose ${{\mathcal{D}}_{1}}$ if > holds, and vice versa'. Specific choices for t (g), i.e. specific mathematical observers for the binary task, are discussed in section 3.3.

For each image vector g and any particular choice of t_c, there are four possible outcome scenarios, as summarized in table 1. The true-positive and true-negative outcomes are correct decisions and the false-positive and false-negative decisions are erroneous. In the terminology of hypothesis testing, a false positive is a Type I error and a false negative is Type II.

Table 1. Possible outcomes for a binary classification task.

1.	True positive (TP):	H1 is true; observer makes decision ${{\mathcal{D}}_{1}}$.
2.	False positive (FP):	H0 is true; observer makes decision ${{\mathcal{D}}_{1}}$.
3.	False negative (FN):	H1 is true; observer makes decision ${{\mathcal{D}}_{0}}$.
4.	True negative (TN):	H0 is true; observer makes decision ${{\mathcal{D}}_{0}}$.

Associated with each of these outcomes there is a fraction or probability of occurrence. For example, the true-positive fraction, denoted TPF, is the probability that the observer makes the correct decision ${{\mathcal{D}}_{1}}$ when the object that produced g was actually in class C₁, so that hypothesis H₁ is true. In signal-detection terminology, TPF is the probability of detecting a signal when it is really there. Conversely, the false-positive fraction, FPF, is the probability of deciding that a signal is present when it really is not, so it is the false-alarm probability.

In terms of conditional probabilities, the four fractions are defined as

$$\begin{eqnarray} &&\text{TPF}=\Pr \left({{\mathcal{D}}_{1}}\mid {{H}_{1}}\right),\quad \text{TNF}=\Pr \left({{\mathcal{D}}_{0}}\mid {{H}_{0}}\right),\\ &&\text{FPF}=\Pr \left({{\mathcal{D}}_{1}}\mid {{H}_{0}}\right)=1-\text{TNF},\quad \text{FNF}=\Pr \left({{\mathcal{D}}_{0}}\mid {{H}_{1}}\right)=1-\text{TPF}.\end{eqnarray} \tag{ 23 }$$

In the medical literature, the TPF is referred to as the sensitivity, since it is an indication of the sensitivity of the test to the presence of an abnormality. The TNF is commonly referred to as the specificity; it measures the proportion of the tests in which the subject really does not have the abnormality that are correctly identified as normal. Because FPF = 1 − TNF, a test with low specificity is one where there are many false positive decisions.

These fractions can be computed if we have knowledge of the probability density functions on the test statistic t. For the generic decision rule of (22) and a given threshold t_c, we can express the four fractions as

$$\begin{eqnarray}&&\text{TPF}=\Pr \left(t > {{t}_{c}}\mid {{H}_{1}}\right)={\int}_{{{t}_{c}}}^{\infty}\text{d}t\ \text{pr}\left(t\mid {{H}_{1}}\right),\end{eqnarray} \tag{ 24a }$$

$$\begin{eqnarray}&&\text{FPF}=\Pr \left(t > {{t}_{c}}\mid {{H}_{0}}\right)={\int}_{{{t}_{c}}}^{\infty}\text{d}t\ \text{pr}\left(t\mid {{H}_{0}}\right),\end{eqnarray} \tag{ 24b }$$

$$\begin{eqnarray}&&{\rm{TNF}} = 1 - {\rm{FPF}} = \int_{ - \infty }^{{t_c}} {{\rm{d}}t\;{\rm{pr}}\left( {t\mid {H_0}} \right),} \end{eqnarray} \tag{ 24c }$$

$$\begin{eqnarray}&&\text{FNF}=1-\text{TPF}=\int_{ - \infty }^{{t_c}} \text{d}t\ \text{pr}\left(t\mid {{H}_{1}}\right).\end{eqnarray} \tag{ 24d }$$

Recall that Pr denotes a probability and $\text{pr}$ denotes a probability density function; thus 0 ⩽ Pr(· ) ⩽ 1 and $\int_{ - \infty}^{\infty} \text{d}t\ \text{pr}\left(t\mid \cdot \right)=1$.

The integrals in (24) show that all four fractions depend on the chosen decision threshold t_c, and in particular that decreasing t_c increases both TPF and FPF. We can achieve any desired probability of detection, but only at the expense of an increased false-alarm probability. This trade-off is illustrated in a Receiver Operating Characteristic (ROC) curve (see figure 1), which is a plot of TPF versus FPF (or equivalently, sensitivity versus 1 − specificity) as t_c is varied.

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As seen from (24a) and (24b), an ROC curve is determined by the two PDFs on the test statistic, $\text{pr}\left(t\mid {{H}_{1}}\right)$ and $\text{pr}\left(t\mid {{H}_{0}}\right)$. If these PDFs are identical, then there is no way to discriminate between the two hypotheses, and the ROC curve is the diagonal line TPF = FPF as shown in figure 1. Conversely, if there is no overlap of $\text{pr}\left(t\mid {{H}_{1}}\right)$ and $\text{pr}\left(t\mid {{H}_{0}}\right)$, then perfect discrimination is possible and the ROC curve is a line vertically from the origin to the upper left corner, then horizontally to the upper right corner. Perfect separation of the two PDFs (hence perfect performance on this binary classification task) corresponds to an area under the curve (AUC) of 1.0. The line TPF = FPF, indicating no ability to perform the task, has AUC = 0.5. Real ROC curves will have AUCs intermediate between these extremes, as shown on the right in figure 1, and AUC is a commonly used figure of merit for binary tasks.

In an imaging context, AUC depends on the classes of objects considered (C₀ and C₁), on the form of the test statistic t (g), and on the statistical properties of the image vectors g. For imaging with ionizing radiation, the statistics of the images, and hence the AUC, depend on radiation dose.

An alternative FOM for separability of $\text{pr}\left(t\mid {{H}_{1}}\right)$ and $\text{pr}\left(t\mid {{H}_{0}}\right)$ is the signal-to-noise ratio of the test statistic, denoted SNR_t. With the notation developed in section 2.2, we can express the means and variances of the test statistic under the two hypotheses as

$$\begin{eqnarray}&&{{\left\langle t\right\rangle}_{j}}\equiv {{\left\langle t\left(\mathbf{g}\right)\right\rangle}_{\mathbf{g}\mid {{\mathbf{C}}_{j}}}},~~~\sigma _{j}^{2}\equiv {{\left\langle {{\left[t\left(\mathbf{g}\right)-{{\left\langle t\right\rangle}_{j}}\right]}^{2}}\right\rangle}_{\mathbf{g}\mid {{\mathbf{C}}_{j}}}},~~~\left(j=0,1\right).\end{eqnarray} \tag{ 25 }$$

Then SNR_t is defined as

$$\begin{eqnarray}&&\text{SN}{{\text{R}}_{t}}=\frac{{{\left\langle t\right\rangle}_{1}}-{{\left\langle t\right\rangle}_{0}}}{\sqrt{\frac{1}{2}\sigma _{0}^{2}+\frac{1}{2}\sigma _{1}^{2}}}.\end{eqnarray} \tag{ 26 }$$

When $\text{pr}\left(t\mid {{H}_{1}}\right)$ and $\text{pr}\left(t\mid {{H}_{0}}\right)$ are both univariate normal PDFs, the area under the ROC curve is given by Barrett et al (1998) as

$$\begin{eqnarray}&&\text{AUC}=\frac{1}{2}+\frac{1}{2}\text{erf} \left(\frac{1}{2}\text{SN}{{\text{R}}_{t}}\right),\end{eqnarray} \tag{ 27 }$$

where erf(· ) is the error function, defined as $\text{erf}(z)=\frac{2}{\sqrt{\pi}}{\int}_{0}^{z}\text{d}t\ \text{exp}\left(-{{t}^{2}}\right)$.

Even when $\text{pr}\left(t\mid {{H}_{1}}\right)$ and $\text{pr}\left(t\mid {{H}_{0}}\right)$ are not normals, AUC can be used to compute an effective signal-to-noise ratio simply by inverting (27). The resulting figure of merit is denoted d_A and defined as (ICRU 1997)

$$\begin{eqnarray}&&{{d}_{A}}=2\text{er}{{\text{f}}^{-1}}\left[2\left(\text{AUC}\right)-1\right].\end{eqnarray} \tag{ 28 }$$

This definition shows that d_A is a nonlinear but monotonic transformation of AUC, so these two figures of merit are equivalent.

There has been some effort to extend the ROC paradigm to classification problems with three or more classes, but the results are at best unwieldy. With N classes, N − 1 scalar variables are needed to make a decision and the ROC curve becomes an (N − 1)-dimensional surface.

It is not clear how to relate these generalized ROC concepts to Stage 2 efficacy, but as discussed below it is relatively straightforward in principle to use Stage 3 efficacy for any number of classes.

The FOMs used for binary classification tasks—AUC, d_A and SNR_t—are properties of the complete ROC curve, so they are independent of the operating point t_c. In practice, however, the choice of operating point itself depends on the clinical task and on the consequences of the outcome. In screening mammography, for example, a clinician might choose a low t_c (high TPF and FPF) in order to avoid missing any true malignancies, especially if the consequence of a false positive is just that a second test will be performed to confirm the result from mammography. If, on the other hand, the consequence of a false positive is that the patient will undergo an unnecessary surgery, then a higher t_c (lower TPF and FPF) is more appropriate.

In order to account for the consequences of clinical decisions, we must use Stage 3 efficacy. The key feature that sets Stage 3 apart from Stage 2 is the introduction of some measure of the clinical usefulness of the diagnostic information. One way to do this is to introduce the concept of utility.

The utility of a classification outcome has been defined, somewhat subjectively, as a measure of the desirability of the outcome relative to other outcomes (Patton and Woolfenden 1989). The utility can also be viewed as the negative of the cost of the outcome, especially if 'cost' is interpreted more broadly than simply monetary cost. The methodology of Fryback and Thornbury is to elicit estimates of utility from questionnaires submitted to expert clinicians.

The overall diagnostic utility of an imaging test can be evaluated as an expectation of the utility of the test over the population of patients and outcomes. For classification tasks with N classes and imaging system S_k, the overall utility U_k is defined as

$$\begin{eqnarray}&&{{U}_{k}}=\underset{i=1}{\overset{N}{\sum}} \underset{j=1}{\overset{N}{\sum}} {{u}_{ij}}\Pr \left({{\mathcal{D}}_{i}}\mid {{H}_{j}},{{S}_{k}}\right)\Pr \left({{H}_{j}}\right),\end{eqnarray} \tag{ 29 }$$

where u_ij is the utility assigned a priori to decision ${{\mathcal{D}}_{i}}$ when hypothesis H_j is true, and Pr(H_j) is the probability that H_j is true (i.e. the object being imaged came from class C_j) in the population under study. The conditional probability $\Pr \left({{\mathcal{D}}_{i}}\mid {{H}_{j}},{{S}_{k}}\right)$ is the generalization of TPF, FPF, etc to more than 2 classes; it depends on the decision rule and on the statistical properties of the image vectors g, hence on the imaging system and the radiation dose. Note that the other factors in the summand of (29), u_ij and Pr(H_j), are independent of the imaging system and the radiation dose.

Wunderlich and Abbey (2013) have shown that utility can also be used for choosing the performance-assessment paradigm itself.

The concept of expected utility, and the assessment of diagnostic image quality by Stage 3 efficacy in general, has been criticized on the grounds that it is difficult to know what utilities to assign to different outcomes and what probabilities of disease to use for computing the expected utility (Metz 2006). The increasing availability of Big Data should mitigate the latter problem, and the dependence of overall utility on the choice of the component utilities u_ij can be regarded as an opportunity for further research rather than a problem. We can in principle study the variation of $\Pr \left({{\mathcal{D}}_{i}}\mid {{H}_{j}},{{S}_{k}}\right)$ with radiation dose for different true hypotheses and decision outcomes and determine not only the overall effect of radiation dose but also its effect on various possible decision outcomes; if the outcomes for which large values of $\Pr \left({{\mathcal{D}}_{i}}\mid {{H}_{j}},{{S}_{k}}\right)$ require large dose are associated with relatively small utilities, then the dose can be reduced with little effect on overall utility.

Moreover, expected utility can be used to compare different imaging systems or to study the effect of radiation dose with a given system just by selecting reasonable values for u_ij and Pr(H_j) and holding them constant across the comparisons. The robustness of the conclusions can then be checked by varying u_ij and Pr(H_j) after the fact. In many cases it will suffice if the rank order of the comparisons is maintained as utilities and probabilities are varied over reasonable ranges.

On the other hand, defensible utilities are essential if we want to evaluate image quality not only for Stage 3 efficacy but also for Stages 5 and 6, and in particular if we want to study the impact of radiation dose on long-term patient outcomes and on society.

In image-quality analysis, the goal of an estimation task is to use image data to estimate one or more numerical parameters defined in terms of the object. This goal is distinct from the goals in other areas of imaging, such as data mining, pattern recognition, texture analysis and computer-aided diagnosis (CAD), which often use parameters of an image.

To illustrate this distinction, there is considerable current interest in the heterogeneity of the uptake of fluorodeoxyglucose (FDG) in tumors, as observed in PET scans, because heterogeneity can have prognostic value for radionuclide therapy. Various metrics from the literature on texture analysis (see, for example, Tixier et al (2012)) have been proposed for this purpose, but they describe the heterogeneity of the image, not the object. Noise in the image can introduce heterogeneity in the image where none exists in the object, and limited resolution of the imaging system can hide fine structures that are really present. The identification of image texture features with physical properties of the object, much less with biological significance, has been criticized by Brooks (2013). An approach to avoiding this problem and directly estimating object texture parameters is given in Kupinski et al (2003b).

A scalar parameter θ(f), which is defined in terms of the object f, is said to be linear if it can be written as a scalar product. If f represents a time-independent spatial function f(r), the scalar product has the form (see (23)):

$$\begin{eqnarray}&&\theta \left(\mathbf{f}\right)=\underset{\infty}{\int} {{\text{d}}^{3}}r\ \chi \left(\mathbf{r}\right)f\left(\mathbf{r}\right)\equiv {{\mathbf{\chi}}^{\dagger}}\mathbf{f},\end{eqnarray} \tag{ 30 }$$

where χ represents the spatial function χ(r) (strictly, as a vector in a Hilbert space), and χ^†f is a notation for the scalar product of functions. As an example, if f(r) is a radionuclide distribution (activity per unit volume) and χ(r) is an indicator function equal to 1 for r inside the boundary of some region of interest (ROI) and 0 outside, then the integral in (30) is the total amount of activity in the ROI.

If we have a raw image data vector g, we can use one of the estimation rules to be discussed in section 3.4 to form an estimate $\hat{\theta}\left(\mathbf{g}\right)$. On the other hand, if the data vector is a reconstructed image $\mathbf{\hat{f}}$, then the estimate is denoted $\hat{\theta}\left(\mathbf{\hat{f}}\right)$. For simplicity, we consider only $\hat{\theta}\left(\mathbf{g}\right)$ here.

Because g is random, $\hat{\theta}\left(\mathbf{g}\right)$ is also random. That is, if you repeatedly image the same object, hence keeping θ(f) constant, you will get a distribution of random images g and hence a distribution of values of $\hat{\theta}\left(\mathbf{g}\right)$.

The full statistics of $\hat{\theta}\left(\mathbf{g}\right)$ can be difficult to derive. For this reason, it is common practice to provide the bias and variance only. These quantities are defined, respectively, as

$$\begin{eqnarray}&&b\left(\mathbf{f}\right)\equiv {{\left\langle \hat{\theta}\left(\mathbf{g}\right)\right\rangle}_{\mathbf{g}\mid \mathbf{f}}}-\theta \left(\mathbf{f}\right)=\overline{{\hat{\theta}}}\left(\mathbf{f}\right)-\theta \left(\mathbf{f}\right),~~~\text{Var}\left\{\hat{\theta}\mid \mathbf{f}\right\}\equiv {{\left\langle {{\left[\hat{\theta}\left(\mathbf{g}\right)-\overline{{\hat{\theta}}}\left(\mathbf{f}\right)\right]}^{2}}\right\rangle}_{\mathbf{g}\mid \mathbf{f}}}.\end{eqnarray} \tag{ 31 }$$

Thus the bias is the average deviation of the estimate from the true θ (f), and the variance is the mean-square deviation of the estimate from the average of the estimates themselves. Bias is the systematic error and variance is the random error in the measurement of θ.

Bias and variance can be combined into a single FOM, the mean-square error (MSE), which is defined as

$$\begin{eqnarray}&&\text{MSE}\left(\mathbf{f}\right)\equiv {{\left\langle {{\left[\hat{\theta}\left(\mathbf{g}\right)-\theta \left(\mathbf{f}\right)\right]}^{2}}\right\rangle}_{\mathbf{g}\mid \mathbf{f}}}={{\left[b\left(\mathbf{f}\right)\right]}^{2}}+\text{Var}\left\{\hat{\theta}\mid \mathbf{f}\right\}.\end{eqnarray} \tag{ 32 }$$

Thus MSE accounts for the total deviation (systematic and random) between the estimate and the true θ.

As the notation implies, bias, variance and MSE all depend on the object f; it is not correct, however, to say that these metrics depend on the true value of the parameter. The problem is that a linear CD imaging operator $\mathcal{H}$ necessarily has null functions, for which $\mathcal{H}{{\mathbf{f}}_{\text{null}}}=0$. Moreover, any object vector can be decomposed uniquely into a null component and an orthogonal measurement component, f = f_meas + f_null. Objects that differ by a null function produce the same mean data $\mathbf{\bar{g}}$, and the same conditional PDF $\text{pr}\left(\mathbf{g}\mid \mathbf{f}\right)$, so they must produce the same conditional mean estimate, $\overline{{\hat{\theta}}}\left(\mathbf{f}\right)$, and the same conditional variance, $\text{Var}\left\{\hat{\theta}\mid \mathbf{f}\right\}$. Objects that differ by a null function do not, however correspond to the same true value of the parameter unless θ(f) is independent of the null functions. This condition can be stated as θ(f) = θ (f_meas). A parameter that satisfies this condition is said to be estimable, which means that there exists an unbiased estimate of it for all true values.

For linear parameters as defined in (30), we can state the estimability condition more directly by writing

$$\begin{eqnarray}&&\theta \left(\mathbf{f}\right)=\left(\mathbf{\chi},\mathbf{f}\right)=\left({{\mathbf{\chi}}_{\text{meas}}},{{\mathbf{f}}_{\text{meas}}}\right)+\left({{\mathbf{\chi}}_{\text{null}}},{{\mathbf{f}}_{\text{null}}}\right).\end{eqnarray} \tag{ 33 }$$

There are no cross terms (χ_meas, f_null) or (χ_null, f_meas) because null space and measurement space are orthogonal subspaces of object space (Barrett and Myers 2004). Thus, for a general object, a linear parameter is estimable only if the function that defines it, χ (r), contains no null functions, as determined, for example, by an SVD analysis. In particular, the integral of an object over a voxel is virtually never estimable. Intuitively, the sharp edges of the voxel function contain high spatial frequencies that are not captured by the imaging system; for more detail, see Whitaker et al (2008) and Kupinski et al (2013).

One way to account for null functions in the figure of merit for a scalar estimation task is to average the MSE over some class of objects, called an ensemble in this context. The ensemble mean-square error (EMSE) for object class C is defined as

$$\begin{eqnarray}&&\text{EMSE}\equiv {{\left\langle \text{MSE}\left(\mathbf{f}\right)\right\rangle}_{\mathbf{f}\mid \mathbf{C}}}={{\left\langle {{\left[b\left(\mathbf{f}\right)\right]}^{2}}+\text{Var}\left\{\hat{\theta}\mid \mathbf{f}\right\}\right\rangle}_{\mathbf{f}\mid \mathbf{C}}}={{\left\langle {{\left\langle {{\left[\hat{\theta}\left(\mathbf{g}\right)-\theta \left(\mathbf{f}\right)\right]}^{2}}\right\rangle}_{\mathbf{g}\mid \mathbf{f}}}\right\rangle}_{\mathbf{f}\mid \mathbf{C}}}.\end{eqnarray} \tag{ 34 }$$

The discussion above on estimation of a single scalar parameter is readily extended to a set of P parameters. The parameters {θ_p, p = 1, ..., P } can be organized into a P × 1 parameter vector θ, and the estimates can be similarly organized as a P × 1 vector $\mathbf{\hat{\theta}}$. Likewise the biases and variances are components of P × 1 vectors. A vector parameter θ(f) is estimable if and only if every component is estimable: θ_p(f) = θ_p(f_meas).

For P parameters, the EMSE is given by

$$\begin{eqnarray}&&{\rm{EMSE}} = \mathop \sum \limits_{p = 1}^P {\left\langle {{{\left[ {{b_p}({\bf{f}})} \right]}^2} + {\rm{Var}}\left\{ {{{\hat \theta }_p}\mid {\bf{f}}} \right\}} \right\rangle _{{\bf{f}}\mid {\bf{C}}}} = {\left\langle {{{\left\langle {{{\left\| {{\bf{\hat \theta }}({\bf{g}}) - {\bf{\theta }}({\bf{f}})} \right\|}^2}} \right\rangle }_{{\bf{g}}\mid {\bf{f}}}}} \right\rangle _{{\bf{f}}\mid {\bf{C}}}},\end{eqnarray} \tag{ 35 }$$

where ∣∣ · ∣∣ denotes the norm of the vector.

If the parameter values obtained in an estimation task are used to make diagnostic decisions, then Stage 2 efficacy (diagnostic accuracy) becomes relevant, and if we want to discuss the diagnostic impact of an estimation task, thereby moving on to Stage 3 efficacy, we must assign utilities to the outcomes. For classification tasks, we defined u_ij as the utility of making decision ${{\mathcal{D}}_{i}}$ when hypothesis H_j is true. A natural analog of this definition for a vector estimation task is the utility of obtaining estimate $\widehat{\boldsymbol{\theta }}$ when the true parameter is θ, denoted $u\left(\widehat{\boldsymbol{\theta }},\boldsymbol{\theta }\right)$.

It is common in the estimation literature to use a quadratic cost function $c(\hat \theta ,\theta ) = {\left\| {\hat \theta - \theta } \right\|^2}$. Since a cost can be viewed as a constant minus a utility, the negative of the EMSE in (35) is a possible expected utility associated with this quadratic cost.

EMSE is not directly related to diagnosis or Stage 3 efficacy, in part because it treats all components of the vector θ as equally important to the figure of merit. Some of the components may be nuisance parameters, defined as ones that influence the image data g but are of no direct clinical interest. It may be necessary to estimate the nuisance parameters in order to get good estimates of the parameters of interest, but they should be omitted from the EMSE if diagnostic relevance is to be ascribed to that metric. Conversely, high utilities should be assigned to accurate estimates of clinically relevant parameters.

One way to bring these judgments about clinical impact into the estimation figure of merit is to define a weighted EMSE by

$$\begin{eqnarray}&&\text{WEMSE}=\underset{p=1}{\overset{P}{\sum}} {{w}_{p}}{{\left\langle {{\left[{{b}_{p}}\left(\mathbf{f}\right)\right]}^{2}}+\text{Var}\left\{{{\hat{\theta}}_{p}}\mid \mathbf{f}\right\}\right\rangle}_{\mathbf{f}\mid \mathbf{C}}},\end{eqnarray} \tag{ 36 }$$

where all of the weights w_p are nonnegative. A component considered to be a nuisance parameter is assigned the weight w_p = 0, and components of high clinical relevance are assigned high weights. As a side benefit, if different components of θ(f) have different units, the weights can be used to normalize each component by either the variance or the square of the mean of that component, thereby making WEMSE dimensionless.

Even with these modifications, however, there can be objections to relating EMSE to efficacy. There is no reason other than mathematical convenience to assume that the cost increases quadratically with the discrepancy between an estimate of a parameter and its true value. We know that the bias term in MSE or EMSE is influenced by null functions of the imaging system, and this term can be arbitrarily large because a null function remains a null function when it is multiplied by a constant. Rather than accepting a large MSE as indicative of low clinical utility, it is better to construct utility functions more directly related to the clinical tasks.

Diagnosis is ultimately about classification. If one or more of the estimated parameters is used for classification, then the expected utility as defined in (29) is a relevant FOM for an estimation task.

In some clinical scenarios, one wants to detect a tumor or other lesion and, if it is present, estimate some numerical parameters such as the size or location of the lesion. These situations lead to several new kinds of ROC curve, including the localization ROC (LROC) and estimation ROC (EROC), both of which are illustrated in figure 2.

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The LROC curve plots the joint probability of detection and correct localization (within some tolerance) versus the false-positive fraction as the decision threshold for detection is varied. The EROC curve (Clarkson 2007), a generalization of LROC, plots the expected utility of a true positive decision against the false-positive fraction, again as the decision threshold is varied. An LROC curve is an EROC where the parameter being estimated is the lesion location and the utility function is unity if the estimate is within the set tolerance of the true location, and zero otherwise.

Popescu and Myers (2013) recently published a paper describing a signal-search paradigm, along with a phantom design, that enables the evaluation and comparison of iterative reconstruction algorithms in terms of signal detectability versus dose performance using a few tens of images. Standard commercial phantoms for CT quality assessment have limited utility for quantitative evaluation of image quality, because of the limited area available for deriving the statistics of a model observer in the signal-absent condition. Moreover, such phantoms yield only one signal size/contrast combination per image. Phantoms that allow for multiple realizations of background-only and signal-present regions in each image, along with the addition of search as a component of the detection task, enable the investigator to customize the signal detectability level through adjustment of the search region area so that meaningful image quality comparisons can be made across algorithms or imaging systems with a fairly small number of images.

Imaging with ionizing radiation is often used in the planning and monitoring of radiation therapy. For a discussion of anatomical imaging for radiotherapy applications, see Evans (2008), and for a discussion of the role of PET in radiotherapy, see Nestle et al (2009).

A recent paper in this journal (Barrett et al 2013) discussed how radiobiological models could be used to assess image quality in terms of therapeutic efficacy, defined by the tradeoff between the probability of tumor control and the probability of some critical adverse reaction in normal tissues. In essence, this paper defines image quality in terms of Stage 4 efficacy, and if we assign utilities, the theory is applicable to Stage 5.

The ideal observer (IO) for a binary classification task can be defined in several equivalent ways: It maximizes the TPF for any given FPF; it maximizes the area under the ROC curve (AUC); it maximizes the detectability index d_A, and it maximizes the overall utility. A well-known result from statistical decision theory is that the ideal observer implements the generic decision strategy (22) with the test statistic t (g) being the likelihood ratio (LR), defined as

$$\begin{eqnarray}&&\Lambda \left(\mathbf{g}\right)\equiv \frac{\text{pr}\left(\mathbf{g}\mid {{H}_{1}}\right)}{\text{pr}\left(\mathbf{g}\mid {{H}_{0}}\right)},\end{eqnarray} \tag{ 37 }$$

where $\text{pr}\left(\mathbf{g}\mid {{H}_{j}}\right)$ is the multivariate PDF for the data given that hypothesis H_j (j = 0, 1) is true.

An important, though little recognized, point is that the LR is invariant to any invertible transformation of the data (see Barrett and Myers (2004), pp 829–30). For example, if we base the classification decision on a reconstructed image $\mathbf{\hat{f}}$ instead of raw projection data g, and it is possible to go backwards and compute g from $\mathbf{\hat{f}}$, then $\Lambda \left(\mathbf{\hat{f}}\right)=\Lambda \left(\mathbf{g}\right)$, and hence the ROC and any FOM derived from it will be the same for the two data sets. If the reconstruction algorithm is not invertible in this sense, then the IO performance on the reconstruction will be less than or equal to its performance on the projection data; if this were not the case, the ideal observer, being ideal, would include the reconstruction algorithm in her decision process when given g.

An important example of an invertible transformation is the logarithm. The logarithm of the likelihood ratio (LLR),

$$\begin{eqnarray}&&\lambda \left(\mathbf{g}\right)\equiv \ln \Lambda \left(\mathbf{g}\right)= \ln\ \text{pr}\left(\mathbf{g}\mid {{H}_{1}}\right)- \ln\ \text{pr}\left(\mathbf{g}\mid {{H}_{0}}\right),\end{eqnarray} \tag{ 38 }$$

produces exactly the same ROC curve as the LR itself, hence the same value for any figure of merit derived from the ROC curve. Thus LR and LLR are equivalent test statistics for the ideal observer.

Analytical forms for the LR and LLR are available for several different families of data PDFs, $\text{pr}\left(\mathbf{g}\mid {{H}_{j}}\right)$, including multivariate normal (Gaussian), log-normal, exponential and Poisson distributions (see sections 13.2.8 and 13.2.9 in Barrett and Myers (2004)).

Here we focus on multivariate normal data, for which the PDF is given by

$$\begin{eqnarray}&&\text{pr}\left(\mathbf{g}\mid {{H}_{j}}\right)={{\left[{{\left(2\pi \right)}^{M}} \text{det}\left({{\mathbf{K}}_{j}}\right)\right]}^{-1/2}} \text{exp} \left[-\frac{1}{2}{{\left(\mathbf{g}-{{\overline{{\mathbf{\bar{g}}}}}_{j}}\right)}^{t}}\mathbf{K}_{j}^{-1}\left(\mathbf{g}-{{\overline{{\mathbf{\bar{g}}}}}_{j}}\right)\right],\end{eqnarray} \tag{ 39 }$$

where det(· ) denotes determinant, and ${{\overline{{\mathbf{\bar{g}}}}}_{j}}$ and K_j, (j = 0, 1) are shorthand forms for the mean data vector and covariance matrix, respectively, when the object class is C_j. We remind the reader that the means and covariances include averages over both the conditional noise for a given object, described by the PDF $\text{pr}\left(\mathbf{g}\mid \mathbf{f}\right)$, and the randomness of the object itself; this double averaging is indicated by the double overbar on ${{\overline{{\mathbf{\bar{g}}}}}_{j}}$.

With (38) and (39), we see that the LLR is given by

$$\begin{eqnarray}&&\lambda \left(\mathbf{g}\right)=-\frac{1}{2} \ln \left[\frac{ \text{det}\,\left({{\mathbf{K}}_{1}}\right)}{ \text{det}\,\left({{\mathbf{K}}_{0}}\right)}\right]-\frac{1}{2}{{\left(\mathbf{g}-{{\overline{{\mathbf{\bar{g}}}}}_{1}}\right)}^{t}}\mathbf{K}_{1}^{-1}\left(\mathbf{g}-{{\overline{{\mathbf{\bar{g}}}}}_{1}}\right)+\frac{1}{2}{{\left(\mathbf{g}-{{\overline{{\mathbf{\bar{g}}}}}_{0}}\right)}^{t}}\mathbf{K}_{0}^{-1}\left(\mathbf{g}-{{\overline{{\mathbf{\bar{g}}}}}_{0}}\right).\end{eqnarray} \tag{ 40 }$$

Thus the test statistic for a binary classification task with multivariate-normal data is, in general, a quadratic function of the data vector, referred to as a quadratic discriminant.

In x-ray and gamma-ray imaging, however, it is frequently an excellent approximation to say that the two covariance matrices, K₀ and K₁, are nearly equal. In transmission x-ray imaging, the signal to be detected results from a localized change in the x-ray attenuation coefficient, but it is observed by its effect on a line integral of the attenuation coefficient through the patient's body. Similarly, in radionuclide imaging a localized change in the activity per unit volume of the tracer is observed by its effect on a line integral through all of the background activity within the body. In neither case is a significant change in the value of a single line-integral projection to be expected; a large change in one projection would be too easily detected from the full set of projections and would be of little use in comparing imaging systems or studying the effect of increased radiation dose. Thus we are interested in detecting small signals in this sense, and the signal is in the mean data vector, so we can usually neglect its effect on the covariances.

If we assume K₀ = K₁ ≡ K_g, and take advantage of the fact that covariance matrices are symmetric, then (40) becomes

$$\begin{eqnarray}&&\lambda \left(\mathbf{g}\right)=\Delta {{\overline{{\mathbf{\bar{g}}}}}^{t}}\mathbf{K}_{\mathbf{g}}^{-1}\mathbf{g}-\frac{1}{2}\overline{{\mathbf{\bar{g}}}}_{1}^{t}\mathbf{K}_{\mathbf{g}}^{-1}{{\overline{{\mathbf{\bar{g}}}}}_{1}}+\frac{1}{2}\overline{{\mathbf{\bar{g}}}}_{0}^{t}\mathbf{K}_{\mathbf{g}}^{-1}\overline{{\mathbf{\bar{g}}}},\quad \text{where} \Delta \overline{{\mathbf{\bar{g}}}}\equiv {{\overline{{\mathbf{\bar{g}}}}}_{1}}-{{\overline{{\mathbf{\bar{g}}}}}_{0}}.\end{eqnarray} \tag{ 41 }$$

The last two terms in (41) are constants, independent of the data, so they can be lumped into the decision threshold. Thus the ideal observer for a binary classification task with multivariate normal data and equal covariance matrices is a linear discriminant, with test statistic given by

$$\begin{eqnarray}&&{{t}_{\text{IO}}}\left(\mathbf{g}\right)=\Delta {{\overline{{\mathbf{\bar{g}}}}}^{t}} \mathbf{K}_{\mathbf{g}}^{-1}\mathbf{g}.\end{eqnarray} \tag{ 42 }$$

Because a covariance matrix is symmetric and positive definite, we can define its square root $\mathbf{K}_{\mathbf{g}}^{1/2}$ and then rewrite (42) as

$$\begin{eqnarray}&&{{t}_{\text{IO}}}\left(\mathbf{g}\right)={{\left[\mathbf{K}_{\mathbf{g}}^{-1/2}\Delta \overline{{\mathbf{\bar{g}}}}\right]}^{t}}\left[\mathbf{K}_{\mathbf{g}}^{-1/2}\mathbf{g}\right].\end{eqnarray} \tag{ 43 }$$

The M × M matrix $\mathbf{K}_{\mathbf{g}}^{-1/2}$ is called a prewhitening operator because the transformed data, $\mathbf{K}_{\mathbf{g}}^{-1/2}\mathbf{g}$, exhibit uncorrelated or 'white' noise. More precisely, the covariance matrix of $\mathbf{K}_{\mathbf{g}}^{-1/2}\mathbf{g}$ is the M × M unit matrix . Thus the IO test statistic for the present problem is just the scalar product of the prewhitened data and the prewhitened mean difference signal.

Because any linear transformation of a normal random vector or random variable is also normal, it follows that $\text{pr}\left({{t}_{\text{IO}}}\mid {{H}_{1}}\right)$ and $\text{pr}\left({{t}_{\text{IO}}}\mid {{H}_{0}}\right)$ are both univariate normal PDFs. Thus AUC_IO is given by (27), with the SNR as defined in (26); it can be shown that

$$\begin{eqnarray}&&\text{SNR}_{\text{IO}}^{2}=\Delta {{\overline{{\mathbf{\bar{g}}}}}^{t}} \mathbf{K}_{\mathbf{g}}^{-1} \Delta \overline{{\mathbf{\bar{g}}}}.\end{eqnarray} \tag{ 44 }$$

Note that calculation of both the test statistic in (43) and its SNR in (44) requires prior knowledge of the mean data vector for both classes and their common covariance matrix.

Some alternative ways of writing this SNR are:

$$\begin{eqnarray}&&\text{SNR}_{\text{IO}}^{2}={{\left[\mathbf{K}_{\mathbf{g}}^{-1/2} \Delta \overline{{\mathbf{\bar{g}}}}\right]}^{t}}\left[\mathbf{K}_{\mathbf{g}}^{-1/2} \Delta \overline{{\mathbf{\bar{g}}}}\right]={\left\|\mathbf{K}_{\mathbf{g}}^{-1/2} \Delta \overline{{\mathbf{\bar{g}}}}\right\|}^{2}=\text{tr}\left\{\mathbf{K}_{\mathbf{g}}^{-1} \Delta \overline{{\mathbf{\bar{g}}}} \Delta {{\overline{{\mathbf{\bar{g}}}}}^{t}}\right\},\end{eqnarray} \tag{ 45 }$$

where tr(·) denotes the trace (sum of the diagonal elements) of a matrix.

In 1931, the American statistician Harold Hotelling devised a multivariate generalization of Student's t test for determining the separability of two groups of N-dimensional random vectors (Hotelling 1931). In essence, Hotelling's separability measure, which he called T², was a linear discriminant function (Fisher's linear discriminant (Fisher 1936), in fact, though Hotelling's work was five years earlier) in the ND space.

The concept we now refer to as the Hotelling observer was introduced into the image-quality literature by Smith and Barrett (1986) and Fiete et al (1987). Because T² was in the form of a trace of a matrix, Smith and Fiete referred to this proposed FOM for classification tasks as the Hotelling trace criterion. A key difference, however, is that T² is a random variable (a test statistic) but the Hotelling trace, as Smith and Fiete used the term, is an ensemble average separability (a figure of merit).

Current terminology in the image-quality literature is that the Hotelling observer is a linear discriminant in the form of a scalar product, $t\left(\mathbf{g}\right)={{\mathbf{w}}^{t}}\mathbf{g}=\underset{m=1}{\overset{M}{\sum}} {{w}_{m}}{{g}_{m}}$, where w is a vector of the same dimension as the data g. We refer to w as a template because the scalar product can be visualized by thinking of g as a transparency on a light box and w as a second transparency laid over the first one. The total light that emerges, in this metaphor, is proportional to the product of the two transmittances, integrated over position on the light box.

Specifically, the Hotelling observer uses the linear test statistic that maximizes SNR². It can be shown by a straightforward calculation (Barrett and Myers 2004) that the Hotelling test statistic is given by

$$\begin{eqnarray}&&{{t}_{\text{Hot}}}\left(\mathbf{g}\right)=\mathbf{w}_{\text{Hot}}^{t}\mathbf{g}=\Delta {{\overline{{\mathbf{\bar{g}}}}}^{t}}\mathbf{K}_{\mathbf{g}}^{-1}\mathbf{g},\end{eqnarray} \tag{ 46 }$$

where $\mathbf{w}_{\text{Hot}}^{t}=\Delta {{\overline{{\mathbf{\bar{g}}}}}^{t}} \mathbf{K}_{\mathbf{g}}^{-1}$, so that ${{\mathbf{w}}_{\text{Hot}}}=\mathbf{K}_{\mathbf{g}}^{-1} \Delta \overline{{\mathbf{\bar{g}}}}$ (because the inverse of a covariance matrix is symmetric), and now

$$\begin{eqnarray}&&{{\mathbf{K}}_{\mathbf{g}}}\equiv \frac{1}{2}{{\mathbf{K}}_{0}}+\frac{1}{2}{{\mathbf{K}}_{1}}.\end{eqnarray} \tag{ 47 }$$

The corresponding SNR² is found to be

$$\begin{eqnarray}&&\text{SNR}_{\text{Hot}}^{2}=\Delta {{\overline{{\mathbf{\bar{g}}}}}^{t}} \mathbf{K}_{\mathbf{g}}^{-1} \Delta \overline{{\mathbf{\bar{g}}}}.\end{eqnarray} \tag{ 48 }$$

Comparing these results to (42) and (44), respectively, we see that the Hotelling observer is also the ideal observer, in the sense of maximizing AUC, when the data PDFs are multivariate normal with equal covariances under the two hypotheses. Even when these conditions are not satisfied, however, the Hotelling observer is still optimal among all linear observers in the sense of maximizing SNR².

On the other hand, no linear observer performs well on a signal detection task when the signal is at a random location in the image. Recall that the double overbar on $\Delta \overline{{\mathbf{\bar{g}}}}$ indicates averages over both image noise (Poisson or electronic) and object randomness, and Δ indicates the difference in these averages between the signal-absent and signal-present classes. If the background (non-signal) part of the data has the same statistics in the two classes, then $\Delta \overline{{\mathbf{\bar{g}}}}$ reflects only the randomness of the signal. If the signal can appear randomly anywhere in the image, the components of the Hotelling template vector, [w_Hot]_m, will have very little variation with the pixel index m, and the test statistic t_Hot(g) is little more than a sum over the components of g, which conveys very little information about the presence or absence of the signal.

For this reason, the use of the Hotelling observer, as we have described it so far, is restricted to situations where the location of a potential signal is known, and the task is to determine whether or not it is present at that location. If we further restrict the signal to be completely nonrandom, then the task is said to be SKE (signal known exactly). These restrictions will be lifted below when we discuss scanning observers that can perform well even when the signal location and other parameters are random variables. The converse of SKE is SKS (signal known statistically).

We might also consider the case of BKE (background known exactly), which for example would describe the task of detecting a lesion in a water tank or other spatially uniform phantom. The much more realistic case is BKS (background known statistically), where there can be organs of unknown size, shape, location and radiographic appearance in the phantom or object being imaged, and there can be unknown fine structure within the organs. The SKE/BKS combination is often useful in image-quality studies, but SKE/BKE should be avoided because the BKE assumption eliminates the object-variability term in the covariance matrix. In particular, as we shall see in section 6, the noise and object-covariance terms depend differently on radiation dose, so conclusions about the effect of dose in a BKE task are not applicable in the real world of random, unknown anatomical backgrounds.

An important early example of a mathematically tractable random background is the lumpy background developed by Rolland and Barrett (1992) and used by Myers et al (1990) to study the optimization of pinhole apertures in nuclear medicine. A later variant called the clustered lumpy background provides an excellent model for the random background inhomogeneity seen in mammography (Bochud et al 1998).

An immediate difficulty with implementing the Hotelling observer is that the covariance matrix is enormous. As a simple example, a rotating-camera SPECT system with 64 2D projection images, each consisting of a grid of 128 × 128 pixels, acquires a data vector with approximately 10⁶ elements, so the covariance matrix of the data is 10⁶ × 10⁶, or 10¹² elements. Modern CT systems can deliver even larger data vectors and covariance matrices; a typical single-organ volumetric scan might consist of 2000 projection images, each 2500 × 320 pixels.

The daunting prospect of computing and storing such large matrices, much less inverting them, has led to a malady termed megalopinakophobia (fear of large matrices) (Barrett et al 2001). The key to the treatment of this syndrome is the covariance decomposition of (16) or (21), where the covariance matrix is split into two terms, one arising from Poisson and other noise sources, and one arising because the objects being imaged are unknown and unpredictable, hence best regarded as random processes.

Many studies of task-based image quality these days are performed with simulated objects. Simulations can be very realistic and flexible, and they avoid all issues related to knowing the true class of the object or the true values of parameters being estimated. Most importantly for the present paper, they permit arbitrary variation of the radiation dose delivered to the 'patient' and hence allow control of the relative importance of the two terms in the covariance decomposition. In particular, we can simulate images with no noise, and with modern computers we can simulate as many of them as we wish. Therefore we can generate image data or reconstructions related just to the second term in the decomposition, with no noise term. Analytic models such as (9) then permit construction of the first term at any desired level of radiation dose.

If we have simulated a set of N_s noise-free sample images, and we denote the kth such image as g_k, then we can form the M × N_s matrix R given by Smith and Barrett (1986) and Fiete et al (1987),

$$\begin{eqnarray}&&\mathbf{R}=\frac{1}{\sqrt{{{N}_{s}}-1}}\left[\delta {{\mathbf{g}}_{1}},\delta {{\mathbf{g}}_{2}},...,\delta {{\mathbf{g}}_{{{N}_{s}}}}\right],\end{eqnarray} \tag{ 49 }$$

where

$$\begin{eqnarray}&&\delta {{\mathbf{g}}_{k}}\equiv {{\mathbf{g}}_{k}}-\frac{1}{{{N}_{s}}}\underset{j=1}{\overset{{{N}_{s}}}{\sum}} {{\mathbf{g}}_{j}}.\end{eqnarray} \tag{ 50 }$$

Thus each column of R is $1/\sqrt{{{N}_{s}}-1}$ times the difference between a sample image and the arithmetic average of all of the sample images.

With these definitions, we can estimate the object covariance by a sample covariance matrix of the form,

$$\begin{eqnarray}&&\mathbf{\hat{K}}_{\mathbf{g}}^{\text{obj}}=\mathbf{R}{{\mathbf{R}}^{t}},\end{eqnarray} \tag{ 51 }$$

where the hat indicates an estimate. The overall covariance matrix can then be estimated via

$$\begin{eqnarray}&&{{\mathbf{\hat{K}}}_{\mathbf{g}}}=\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}+\mathbf{R}{{\mathbf{R}}^{t}}.\end{eqnarray} \tag{ 52 }$$

The noise covariance matrix $\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}$ will be diagonal if we use noise models (8) or (9) but it will still depend on the grand mean of the data, which for a linear system is $\overline{{\mathbf{\bar{g}}}}=\mathcal{H}\mathbf{\bar{f}}$, where $\mathbf{\bar{f}}$ can be estimated as the average of the noise-free images used to construct R. Logically, therefore, we should include a hat on $\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}$, but in practice the error in the estimate of this term is negligible because we are making use of the analytic noise models and we can treat $\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}$ as a known, full-rank, diagonal matrix.

An exact inverse of the estimated overall covariance matrix ${{\mathbf{\hat{K}}}_{\mathbf{g}}}$ can be obtained without onerous calculations or impossibly large matrix inversions by use of the Woodbury matrix inversion lemma (Tylavsky and Sohie 1986, Barrett and Myers 2004). From (52), the lemma allows us to write

$$\begin{eqnarray}&&{{\left[{{\mathbf{\hat{K}}}_{\mathbf{g}}}\right]}^{-1}}={{\left[\mathbf{K}_{\mathbf{g}}^{\text{noise}}\right]}^{-1}}-{{\left[\mathbf{K}_{\mathbf{g}}^{\text{noise}}\right]}^{-1}}\mathbf{R}{{\left\{\mathbf{I}+{{\mathbf{R}}^{t}} {{\left[\mathbf{K}_{\mathbf{g}}^{\text{noise}}\right]}^{-1}}\mathbf{R}\right\}}^{-1}}{{\mathbf{R}}^{t}} {{\left[\mathbf{K}_{\mathbf{g}}^{\text{noise}}\right]}^{-1}},\end{eqnarray} \tag{ 53 }$$

where I is the N_s × N_s identity matrix.

The advantage of this form is that $\mathbf{I}+{{\mathbf{R}}^{t}} {{\left[\mathbf{K}_{\mathbf{g}}^{\text{noise}}\right]}^{-1}} \mathbf{R}$ is an N_s × N_s matrix, readily inverted numerically for N_s up to 10⁵ or so. The matrix $\mathbf{K}_{\mathbf{g}}^{\text{noise}}$ is still M × M, but it is full rank and in fact diagonal if we work with the noise model of (9). Hence all of the inverses indicated in (53), including ${{\left[{{\mathbf{\hat{K}}}_{\mathbf{g}}}\right]}^{-1}}$ itself, will exist.

An alternative to inverting ${{\mathbf{\hat{K}}}_{\mathbf{g}}}$ is to use it in an iterative algorithm to find the Hotelling template. A simple manipulation of (46) shows that this template satisfies ${{\mathbf{K}}_{\mathbf{g}}}{{\mathbf{w}}_{\text{Hot}}}=\Delta \overline{{\mathbf{\bar{g}}}}$, so an estimate of the template can be found by solving (see (52))

$$\begin{eqnarray}&&\left[\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}+\mathbf{R}{{\mathbf{R}}^{t}}\right]{{\mathbf{\hat{w}}}_{\text{Hot}}}=\Delta \overline{{\mathbf{\bar{g}}}}.\end{eqnarray} \tag{ 54 }$$

A variant of the iterative Landweber algorithm for solving this equation is given in Barrett and Myers (2004); the iteration rule is

$$\begin{eqnarray}&&\mathbf{\hat{w}}_{\text{Hot}}^{(n+1)}=\mathbf{\hat{w}}_{\text{Hot}}^{(n)}+\alpha {{\left[\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}\right]}^{-1}}\left[\Delta \overline{{\mathbf{\bar{g}}}}-{{\mathbf{\hat{K}}}_{\mathbf{g}}}\mathbf{\hat{w}}_{\text{Hot}}^{(n)}\right],\end{eqnarray} \tag{ 55 }$$

where $\mathbf{\hat{w}}_{\text{Hot}}^{(n)}$ is the estimated template at the nth iteration and α is a positive constant that controls the convergence rate.

Once the template has been estimated, the SNR can be found by applying the template to a set of sample images, determining the mean and variance of the resulting scalar test statistic under each hypothesis, and estimating the observer's performance via (26). Alternatively, we can directly estimate $\text{SNR}_{\text{Hot}}^{2}$ by (48) as $\Delta {{\overline{{\mathbf{\bar{g}}}}}^{t}}{{\mathbf{\hat{w}}}_{\text{Hot}}}$. For a discussion of the errors in the estimate, see Kupinski et al (2007).

A channel vector is a relatively low-dimensional vector used in place of the original large data vector to perform the desired detection and/or estimation task. In almost all of the copious literature on this topic (which began with Myers and Barrett (1987)), the jth component of the channel vector is the scalar product of a channel template vector with a data vector of the same dimensionality:

$$\begin{eqnarray}&&{{v}_{j}}\equiv \mathbf{u}_{j}^{t}\mathbf{g}=\underset{m}{\sum} {{u}_{jm}}{{g}_{m}},~~~\left(j=1,...,{{N}_{c}}\right),\end{eqnarray} \tag{ 56 }$$

where N_c is the number of channels and u_jm is the mth component of template vector u_j. Equation (56) can also be written in matrix-vector form as

$$\begin{eqnarray}&&\mathbf{v}\equiv {{\mathbf{U}}^{t}}\mathbf{g},\end{eqnarray} \tag{ 57 }$$

where the jth column of the M × N_c matrix U is the template vector for the jth channel.

By analogy to (46), the weak-signal form of the Hotelling discriminant for operating on channelized data is

$$\begin{eqnarray}&&{{t}_{\text{CHO}}}\left(\mathbf{v}\right)=\Delta {{\overline{{\mathbf{\bar{v}}}}}^{t}}\mathbf{K}_{\mathbf{v}}^{-1}\mathbf{v},\end{eqnarray} \tag{ 58 }$$

where

$$\begin{eqnarray}&&\Delta \overline{{\mathbf{\bar{v}}}}\equiv {{\mathbf{U}}^{t}}\left[{{\overline{{\mathbf{\bar{g}}}}}_{{{H}_{1}}}}- {{\overline{{\mathbf{\bar{g}}}}}_{{{H}_{0}}}}\right],\end{eqnarray} \tag{ 59 }$$

$$\begin{eqnarray}&&{{\mathbf{K}}_{\mathbf{v}}}={{\mathbf{U}}^{t}}\left[\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}+\mathbf{K}_{{\mathbf{\bar{g}}}}^{\text{obj}}\right]\mathbf{U}.\end{eqnarray} \tag{ 60 }$$

Since K_v is an N_c × N_c matrix, computation of the CHO test statistic is much less onerous than for the full Hotelling observer.

The signal-to-noise ratio for the channelized Hotelling observer is given by (see (48))

$$\begin{eqnarray}&&\text{SNR}_{\text{CHO}}^{2}=\Delta {{\overline{{\mathbf{\bar{v}}}}}^{t}}\mathbf{K}_{\mathbf{v}}^{-1}\Delta \overline{{\mathbf{\bar{v}}}}.\end{eqnarray} \tag{ 61 }$$

One possible objective in designing a set of channels is to make $\text{SNR}_{\text{CHO}}^{2}$ as close to $\text{SNR}_{\text{Hot}}^{2}$ as possible. Channels that achieve this goal to a good approximation are said to be efficient, meaning that there is little loss of information (defined by the performance of a Hotelling observer) as a result of the dimensionality reduction inherent in projecting the full data vector onto the channels. The condition for approximate efficiency in this sense is

$$\begin{eqnarray}&&{{\left[{{\mathbf{U}}^{t}}\Delta \overline{{\mathbf{\bar{g}}}}\right]}^{t}}{{\left[{{\mathbf{U}}^{t}}{{\mathbf{K}}_{\mathbf{G}}}\mathbf{U}\right]}^{-1}}{{\mathbf{U}}^{t}}\Delta \overline{{\mathbf{\bar{g}}}}\approx {{\left[\Delta \overline{{\mathbf{\bar{g}}}}\right]}^{t}}\mathbf{K}_{\mathbf{g}}^{-1}\Delta \overline{{\mathbf{\bar{g}}}},\end{eqnarray} \tag{ 62 }$$

for all signals of interest. Channel vectors that satisfy this condition can be regarded as approximate sufficient statistics for the detection problem.

One way to construct efficient channels is to take advantage of the symmetries of the signal and noise in the images under study. For example, if we are dealing with 2D images and we know that the signal is rotationally symmetric and that the noise covariance is isotropic (independent of direction in the plane), then we can expand the channel template in rotationally symmetric functions such as Laguerre–Gauss polynomials (Barrett et al 1998b). For a detailed study of Laguerre–Gauss (LG) channels and the errors in estimating the CHO SNR from sample images, see Gallas and Barrett (2003).

Much of the early development of statistical decision theory was aimed at analyzing the results of psychophysical studies with human observers (Tanner et al 1954, Peterson et al 1954, Swets 1964, Green and Swets 1966, Chesters 1992), and some of the early papers on the Hotelling observer showed that $\text{SNR}_{\text{Hot}}^{2}$ was a good predictor of human performance on SKE/BKS tasks with uncorrelated noise (Fiete et al 1987, Rolland and Barrett 1992). The work of Myers on channels (Myers and Barrett 1987), however, was motivated by the observation that the Hotelling observer was not a good predictor of human performance when the noise was correlated as a result of the high-pass filter required in tomographic reconstruction. At the time, available psychophysical data on perception in tomographic images (e.g. Judy and Swensson 1981, Burgess 1984, Myers et al 1985) were generally summarized by saying that humans were unable to prewhiten the noise. Myers, however, was able to build on many previous experiments in human vision (e.g. Campbell and Robson 1968, Sachs et al 1971, Graham and Nachmias 1971, Strohmeyer and Julesz 1972, Mostafavi and Sakrison 1976) which showed that humans process visual information through spatial-frequency-selective channels. Typically, the channels had about one octave of bandwidth in the magnitude of the spatial-frequency vector and ∼± 45° selectivity in the orientation of the spatial frequency vector. By incorporating such channels in a channelized Hotelling model, Myers was able to account for the extant data on human detection performance in tomographic images.

Many subsequent studies (Barrett et al 1992, Barrett et al 1993, Abbey and Barrett 2001) confirmed the broad applicability of the CHO as a predictor of human task performance and refined the model by including an additional source of noise internal to the visual system; no adjustable parameters were needed in the resultant model (Burgess and Colborne 1988, Eckstein et al 1997, Abbey and Barrett 2001).

Of particular importance to the present paper, Abbey and Barrett (2001) showed that the CHO correctly predicted the minimum contrast required for a human observer to detect a weak signal as a function of radiation dose, over a range of several decades in dose.

This same paper (Abbey and Barrett 2001) showed that the CHO performance and its correlation with human performance were very insensitive to the spatial-frequency profiles of the channels, so long as all channels had a bandpass structure with zero response at zero spatial frequency. Interestingly, the response of Laguerre–Gauss channels is maximum, not zero, at zero frequency, so LG channels should not be used as models for human observers.

For a comprehensive review of research through 2000 on visual detection tasks in correlated image noise with linear model observers, see Abbey and Bochud (2000), and for a practical guide to model observers for visual detection in synthetic and natural noisy images, see Eckstein et al (2000). A more recent review of these topics is given in Burgess (2011).

Yu et al (2013) investigated how well a channelized Hotelling observer predicts human observer performance to validate their use for the evaluation of CT reconstruction algorithms. Their work made use of a 2-alternative forced choice (2AFC) lesion-detection paradigm. They considered two different reconstruction algorithms: a filtered-backprojection (FBP) and an iterative reconstruction (IR) method. Images of a torso-shaped phantom containing three rods of different diameters that simulated small, medium and large lesions were reconstructed. Because the goal of the work was to show the predictive ability of the model observer as a surrogate for human observers, the observer model incorporated Gabor channels intended to model the human visual system. Sixty channels in all were utilized, including six frequency-selective passbands, five orientations, and two phases. With this model the investigators found that the performance of the human and model observers were highly correlated. Investigations of this type provide strong evidence that the channelized Hotelling observer with anthropomorphic channels can be used to predict human performance for the evaluation of CT image reconstruction algorithms.

The use of a CHO for the assessment of volumetric imaging data sets is presented in Platisa et al (2011).

Linear observers are computationally convenient, and in section 6.3 they will prove to be useful vehicles for the study of task performance as a function of radiation dose. If the background statistics are not Gaussian, however, the performance of the ideal observer can significantly exceed that of the ideal linear observer, especially at high dose where object variability can dominate over measurement noise.

One fruitful approach to ideal-observer computation in medical imaging is the use of Markov-chain Monte Carlo (MCMC) techniques (Kupinski et al 2003c, Barrett and Myers 2004, He et al 2008, Park and Clarkson 2009), in which one generates a sequence of likelihood ratios drawn from the posterior density of the background given a measured image g or $\mathbf{\hat{f}}$. The average of this sequence is then an estimate of the LR for that image, and repeating the estimation for many images leads to an ROC curve for the ideal observer. This method can then be used for optimizing imaging hardware (Gross et al 2003) or for determining the efficiency of a CHO (Park and Clarkson 2009) or a human observer (Park et al 2005) relative to the ideal observer.

Another approach is to use surrogate figures of merit based on Fisher information (see section 3.4), which accurately approximate the performance of the ideal observer (Shen and Clarkson 2006, Clarkson and Shen 2010, Clarkson 2012).

Much of the early research on task-based assessment of image quality in radiology used continuous Fourier transforms to define quantities such as TF (transfer function), MTF (modulation transfer function), NPS (noise power spectrum), NEQ (noise equivalent quanta) and DQE (detective quantum efficiency), all of which are functions of the spatial frequency vector ρ. One of the first papers to make use of these quantities was the treatment of image quality in CT presented by Hanson (1979). In that paper Hanson lays out the various contributors to image quality in CT, including the noise power spectrum, the MTF, and the NEQ. Further, Hanson presents signal detection theory and the concept of an optimum receiver that makes use of knowledge of the noise correlations in the image as part of its test statistic, first in tutorial/general form and then as it applies to the particular case of CT, including the concept of dose efficiency in the comparison of systems. For a history of the use of Fourier methods in the evaluation of image quality, see Barrett (2009), and for a detailed critique of its applicability to task-based assessment, see section 16.1.6 of Barrett and Myers (2004).

Continuous Fourier transforms are useful when the image is a function of continuous spatial variables instead of discrete pixel indices; when the imaging system is linear and shift invariant, so that the operator $\mathcal{H}$ can be modeled as a convolution; and when both the noise and object randomness are statistically stationary continuous random processes, so that the autocovariance function of the continuous data depends only on the vector distance between two points of interest. When all of these conditions are satisfied for 2D continuous data, the Hotelling test statistic and SNR in the spatial-frequency domain are given, respectively, by (see (46) and (48))

$$\begin{eqnarray}&&{t_{{\rm{Hot}}}}({\bf{G}}) = \int_\infty {{{\rm{d}}^2}\rho \frac{{{{[\Delta \overline {\bar G} ({\bf{\rho }})]}^ * }}}{{{\rm{NPS}}({\bf{\rho }})}}} G({\bf{\rho }}),\quad {\rm{SNR}}_{{\rm{Hot}}}^2 = \int_\infty {{{\rm{d}}^2}\rho \frac{{|\Delta \overline {\bar G} ({\bf{\rho }}){|^2}}}{{{\rm{NPS}}({\bf{\rho }})}},}\end{eqnarray} \tag{ 63 }$$

where ρ is a 2D spatial-frequency vector, G (ρ) is the 2D transform of the continuous data g (r), the asterisk denotes complex conjugate, and NPS(ρ) is the noise power spectrum, defined as the 2D Fourier transform of the autocovariance function of the data.

One thing to note about (63), in contrast to all of the other Hotelling expressions in this section, is that there is no inverse covariance present in either t_Hot or $\text{SNR}_{\text{Hot}}^{2}$. In essence, under the stated assumptions, the Fourier transform simultaneously diagonalizes both the system operator $\mathcal{H}$ and the covariance operator $\mathcal{K}$. The system operator (now a convolution) is realizable as a product in the Fourier domain and the inverse of the covariance becomes a simple division by the NPS.

As in the covariance decomposition of (16), the NPS can be expressed as a sum of two terms, one arising from the noise in the data conditional on the object, and one arising from object randomness, provided that both of these effects are described by stationary random processes. Even with the simplest of noise models, (8) or (9), however, the noise in imaging with ionizing radiation is not stationary because it depends on the mean image, which must necessarily vary from point to point in any interesting image. Similarly, the object variability, described in continuous terms by (19), cannot be expected to depend on only r − r'.

Another conclusion to be drawn from (63) is that this form of $\text{SNR}_{\text{Hot}}^{2}$ is independent of signal location; a shift in position of a signal in the space domain corresponds to a phase change in its Fourier transform, so the square modulus of $\Delta \overline{{\bar{G}}}\left(\mathbf{\rho}\right)$ is invariant to signal location.

Further problems arise when one replaces the Fourier integrals required in (63) with discrete Fourier transforms (DFTs) for purposes of computer implementation. The correct discretization of a linear, shift-invariant operator or a stationary autocovariance function is a Toeplitz matrix, but DFTs diagonalize only circulant matrices; for a graphical depiction of the difference between a Toeplitz matrix and its circulant approximation, see figure 7.15 in Barrett and Myers (2004). A DFT will actually undiagonalize a noise covariance described by (8) or (9).

In spite of these difficulties, there is considerable current interest in Fourier-based approaches to task-based assessment, with their simple, intuitive interpretations (Siewerdsen et al 2002, Richard et al 2005, Tward et al 2007, Richard and Siewerdsen 2008, Pineda et al 2008, Pineda et al 2012, Tward and Siewerdsen 2008, Tward and Siewerdsen 2009, Gang et al 2010). In many cases deviations from strict shift-invariance or strict stationarity can be accommodated by considering detection of a signal at a specific location r₀ (in 2D or 3D as appropriate), then defining a local point spread function and a local autocovariance function, from which we can use a Fourier transform to get a local modulation transfer function LMTF(ρ; r₀), and a local noise power spectrum LNPS(ρ; r₀), respectively. There are many options as to how the intermediate functions are defined and how one handles the discretization, but the end result will be a Fourier-based formula for a Hotelling-like figure of merit appropriate to detection of a signal localized at r₀.

For more on the mathematics of quasistationary noise and local Fourier metrics for image quality, see sections 13.2.3 and 13.2.4 in Barrett and Myers (2004). Fourier-based approaches to task-based assessment will be discussed further in section 7 of this paper.

The likelihood function for estimating a P × 1 parameter vector θ from an M × 1 data vector g is defined as the conditional probability density function (or probability, for discrete-valued data), $\text{pr}\left(\mathbf{g}\mid \boldsymbol{\theta }\right)$. For a particular data vector, $\text{pr}\left(\mathbf{g}\mid \boldsymbol{\theta }\right)$ can be regarded as a function in a P-dimensional parameter space.

The score s(g) is a P × 1 vector of partial derivatives of the log of the likelihood function, with components given by

$$\begin{eqnarray}&&{{s}_{p}}\left(\mathbf{g}\right)\equiv \frac{\partial}{\partial {{\theta}_{p}}} \text{log}\ \text{pr}\left(\mathbf{g}\mid \boldsymbol{\theta }\right),~~~\left(p=1,...,P\right).\end{eqnarray} \tag{ 64 }$$

In terms of repeated data acquisitions with θ fixed, the score is a random vector, and its mean is readily shown to be zero, i.e. 〈s(g)〉_g∣θ = 0. The P × P covariance matrix of the score, which is called the Fisher Information Matrix (FIM) and denoted F(θ), is defined in component form by

$$\begin{eqnarray}&&{{F}_{pp\prime}}\left(\boldsymbol{\theta }\right)\equiv {{\left\langle \frac{\partial}{\partial {{\theta}_{p}}} \text{log}\ \text{pr}\left(\mathbf{g}\right)\cdot \frac{\partial}{\partial {{\theta}_{p\prime}}} \text{log}\ \text{pr}\left(\mathbf{g}\right)\right\rangle}_{\mathbf{g}\mid \boldsymbol{\theta }}}.\end{eqnarray} \tag{ 65 }$$

The FIM sets a fundamental lower bound, called the Cramér-Rao Bound (CRB), on the conditional variance of any unbiased estimator. If the bias of $\widehat{\boldsymbol{\theta }}\left(\mathbf{g}\right)$ is zero, then it can be shown that the variance of the pth component must satisfy

$$\begin{eqnarray}&&\text{Var}\left\{{{\hat{\theta}}_{p}}\left(\mathbf{g}\right)\mid \boldsymbol{\theta }\right\}\geqslant {{\left[{{\mathbf{F}}^{-1}}\left(\boldsymbol{\theta }\right)\right]}_{pp}}.\end{eqnarray} \tag{ 66 }$$

An unbiased estimator for which this inequality becomes an equality is said to be efficient, which means that it achieves the best possible variance among all unbiased estimators for a specific estimation task.

It is always possible to find an estimator with zero bias just by ignoring the data and setting $\widehat{\boldsymbol{\theta }}$ to a constant. More generally, variance can always be made arbitrarily small just by accepting as much bias as needed. The tradeoff between variance and bias in an estimation task is analogous to the tradeoff between sensitivity and specificity in a detection task; in both cases, the tradeoff should be made in terms of clinical efficacy.

The form of the CRB in (66) immediately suggests that one or more of the diagonal elements of the inverse of the FIM can be used as FOMs for estimation tasks (Kupinski et al 2003a).

The maximum-likelihood estimate (MLE) is defined as the point in the P-dimensional parameter space where the likelihood function, $\text{pr}\left(\mathbf{g}\mid \mathbf{\theta}\right)$ for a given, observed g, is maximal. This condition is expressed mathematically as

$$\begin{eqnarray}&&{{\widehat{\boldsymbol{\theta }}}_{\text{ML}}}\left(\mathbf{g}\right)\equiv \underset{\theta}{{\text{argmax}}} \left\{\text{pr}\left(\mathbf{g}\mid \boldsymbol{\theta }\right)\right\}=\underset{\theta}{{\text{argmax}}} \left\{ \ln\ \left[\text{pr}\left(\mathbf{g}\mid \boldsymbol{\theta }\right)\right]\right\},\end{eqnarray} \tag{ 67 }$$

where the second form yields the same MLE as the first because the maximum of the logarithm of a function occurs at the same point as the maximum of the function itself.

The MLE is efficient (unbiased and best possible variance) in any problem where an efficient estimator exists. Further properties of the MLE are discussed in Barrett and Myers (2004), section 13.3.6.

In Bayesian estimation, the parameter of interest is treated as a random variable, described a priori (i.e. before any data are acquired) by a PDF $\text{pr}\left(\boldsymbol{\theta }\right)$. The estimation rule is then based on the posterior density, $\text{pr}\left(\boldsymbol{\theta }\mid \mathbf{g}\right)$, which is interpreted as the probability of the parameter after a data set is acquired. By Bayes' rule, this posterior can be written as $\text{pr}\left(\mathbf{g}\mid \boldsymbol{\theta }\right)\text{pr}\left(\boldsymbol{\theta }\right)/\text{pr}\left(\mathbf{g}\right)$, and because $\text{pr}\left(\mathbf{g}\right)$ is independent of θ, the maximum a posteriori (MAP) estimate is given by

$$\begin{eqnarray}&&{{\widehat{\theta}}_{\text{MAP}}}\left(\mathbf{g}\right)\equiv \underset{\theta}{{\text{argmax}}}\ \left\{\text{pr}\left(\mathbf{g}\mid \theta \right)\ \text{pr}\left(\theta \right)\right\}=\underset{\theta}{{\text{argmax}}}\ \left\{\ln \left[\text{pr}\left(\mathbf{g}\mid \theta \right)\right]-\ln \left[\text{pr}\left(\theta \right)\right]\right\}.\end{eqnarray} \tag{ 68 }$$

Thus the MAP estimator maximizes the product of the likelihood and the prior. Equivalently, it is the mode of the posterior density.

Another important Bayesian estimator is the posterior mean,

$$\begin{eqnarray}&&{{\widehat{\boldsymbol{\theta }}}_{\text{PM}}}\left(\mathbf{g}\right)\equiv {{\int}_{\infty}}{{\text{d}}^{P}}\theta \boldsymbol{\theta }\ \text{pr}\ \left(\boldsymbol{\theta }\mid \mathbf{g}\right),\end{eqnarray} \tag{ 69 }$$

which can be shown to minimize the EMSE defined in (35).

As we saw in (30), a linear, scalar parameter has the form of a scalar product, θ (f) = χ^†f. The general linear estimator of this parameter has the same structure as a general linear discriminant, $\widehat{\boldsymbol{\theta }}={{\mathbf{w}}^{t}}\mathbf{g}$, and design of a linear estimator is equivalent to choosing the template w. The template can range from a simple indicator function that defines a region of interest to ones that account for the statistics of the object and image noise; whatever template is chosen, however, the task performance will depend on these statistics.

The Gauss–Markov Estimator (GME) accounts for the noise statistics but not the object variability. It applies to data obtained with a linear system operator $\mathcal{H}$, and it minimizes the conditional MSE for a given object f by choosing (Barrett 1990)

$$\begin{eqnarray}&&{{\widehat{\boldsymbol{\theta }}}_{\text{GM}}}\left(\mathbf{g}\right)\equiv {{\mathbf{\chi}}^{\dagger}}{{\left[\mathbf{K}_{\mathbf{g}\mid \mathbf{f}}^{-1/2}\mathcal{H}\right]}^{+}}\mathbf{K}_{\mathbf{g}\mid \mathbf{f}}^{-1/2}\mathbf{g},\end{eqnarray} \tag{ 70 }$$

where $\mathbf{K}_{\mathbf{g}\mid \mathbf{f}}^{-1/2}$ is recognized from section 3.3 (see (43)) as a prewhitening operator and [ · ]⁺ denotes a Moore–Penrose pseudoinverse; for many properties of pseudoinverses and algorithms for computing them, see chapter 1 in Barrett and Myers (2004).

For completeness, the GME for estimation from a reconstructed image with a linear reconstruction operator $\mathcal{O}$ is given by

$$\begin{eqnarray}&&{{\widehat{\boldsymbol{\theta }}}_{\text{GM}}}\left(\mathbf{\hat{f}}\right)\equiv {{\mathbf{\chi}}^{\dagger}}{{\left[\mathbf{K}_{\mathbf{\hat{f}}\mid \mathbf{f}}^{-1/2}\mathcal{O}\mathcal{H}\right]}^{+}}\mathbf{K}_{\mathbf{\hat{f}}\mid \mathbf{f}}^{-1/2}\mathbf{\hat{f}}.\end{eqnarray} \tag{ 71 }$$

Even though the GME was derived by minimizing MSE for a chosen object, we can still get the EMSE by averaging over an ensemble of objects. An expression for the resulting EMSE is given in Barrett (1990). This averaging step does not make the GME a Bayesian estimator because the statistics of the object ensemble are not used in defining the estimator.

We have noted above that the Bayesian estimator that minimizes EMSE among all estimators is the posterior mean estimator defined in (69), but the integral in that expression usually requires Markov-chain Monte Carlo techniques as in Kupinski et al (2003c). A more tractable approach is to use the Wiener Estimator (WE) which minimizes the EMSE among all linear estimators.

The most compact way of writing the WE is to assume that the prior means of the parameter and the data are zero, in which case the best linear estimate of a P × 1 parameter vector is given by

$$\begin{eqnarray}&&{{\widehat{\boldsymbol{\theta }}}_{\text{W}}}\left(\mathbf{g}\right)\equiv {{\mathbf{K}}_{\boldsymbol{\theta },\mathbf{g}}}\mathbf{K}_{\mathbf{g}}^{-1}\mathbf{g},\end{eqnarray} \tag{ 72 }$$

where K_θ,g is the P × M cross-covariance matrix of the random parameter and the random data. When the prior means of the parameter and the data are not zero but known, the WE becomes

$$\begin{eqnarray}&&{{\widehat{\boldsymbol{\theta }}}_{\text{W}}}\left(\mathbf{g}\right)=\overline{\boldsymbol{\theta }}+{{\mathbf{K}}_{\boldsymbol{\theta },\mathbf{g}}}\mathbf{K}_{\mathbf{g}}^{-1}\left[\mathbf{g}-\overline{{\mathbf{\bar{g}}}}\right].\end{eqnarray} \tag{ 73 }$$

In this form, the WE is an affine transformation of the data, not a strictly linear one.

To apply the estimators discussed above to imaging, it is useful to decompose the object being imaged into signal and background components. The distinction is that the signal contributes to task performance and hence to efficacy and the background does not; therefore we treat the background as a nuisance parameter, albeit a very high-dimensional one.

For imaging with ionizing radiation, where no part of the object is opaque to the radiation, we can write the signal/background decomposition as a sum, and for estimation tasks we can assume that the parameters of interest are associated only with the signal. Thus we can write the object function as

$$\begin{eqnarray}&&f\left(\mathbf{r};\boldsymbol{\theta }\right)={{f}_{bg}}\left(\mathbf{r}\right)+{{f}_{\text{sig}}}\left(\mathbf{r};\boldsymbol{\theta }\right),\end{eqnarray} \tag{ 74 }$$

or, in our usual shorthand, f = f_bg + f_sig(θ). Common choices for the components of θ might include its spatial coordinates, some measures of its shape and size, and a measure of the strength of the signal, such as its mean x-ray attenuation coefficient or the uptake of a radiotracer.

It may be a useful approximation to treat the likelihood for the signal parameters as a multivariate normal of the form

$$\begin{eqnarray}&&\text{pr}\left(\mathbf{g}\mid \boldsymbol{\theta }\right)=\frac{1}{\sqrt{{{\left(2\pi \right)}^{M}} \text{det}\left({{\mathbf{K}}_{\mathbf{g}\mid \boldsymbol{\theta }}}\right)}} \text{exp} \left[-\frac{1}{2}{{\left[\mathbf{g}-\overline{{\mathbf{\bar{g}}}}\left(\boldsymbol{\theta }\right)\right]}^{t}}\mathbf{K}_{\mathbf{g}\mid \boldsymbol{\theta }}^{-1}\left[\mathbf{g}-\overline{{\mathbf{\bar{g}}}}\left(\boldsymbol{\theta }\right)\right]\right].\end{eqnarray} \tag{ 75 }$$

This PDF is a complicated function of the parameter vector through the covariance matrix, K_g∣θ, but there are many cases where this dependence can be neglected. The easiest approach is to consider signals that are very weak compared to the background. If the signal is not weak, we may still be able to approximate K_g∣θ by either ${{\mathbf{K}}_{\mathbf{g}\mid \overline{\boldsymbol{\theta }}}}$ or 〈K_g∣θ〉_θ (Whitaker et al 2008, Kupinski et al 2013).

If the θ dependence of the covariance matrix can be neglected or approximated away, then a Gaussian log-likelihood can be written as

$$\begin{eqnarray} &&\ln\ \text{pr}\left(\mathbf{g}\mid \boldsymbol{\theta }\right)={{\left[\overline{{\mathbf{\bar{g}}}}\left(\boldsymbol{\theta }\right)\right]}^{t}}\mathbf{K}_{\mathbf{g}}^{-1}\mathbf{g}-\frac{1}{2}{{\left[\overline{{\mathbf{\bar{g}}}}\left(\boldsymbol{\theta }\right)\right]}^{t}}\mathbf{K}_{\mathbf{g}}^{-1}\left[\overline{{\mathbf{\bar{g}}}}\left(\boldsymbol{\theta }\right)\right]+\text{terms}~\text{independent}~\text{of}~\boldsymbol{\theta },\end{eqnarray} \tag{ 76 }$$

which is a linear function of g. Under the same assumptions, the components of the Fisher information matrix can be shown to be:

$$\begin{eqnarray}&&{{F}_{pp\prime}}={{\left[\frac{\partial \overline{{\mathbf{\bar{g}}}}\,\left(\boldsymbol{\theta }\right)}{\partial {{\theta}_{p}}}\right]}^{t}}\mathbf{K}_{\mathbf{g}}^{-1}\left[\frac{\partial \overline{{\mathbf{\bar{g}}}}\,\left(\boldsymbol{\theta }\right)}{\partial {{\theta}_{p\prime}}}\right].\end{eqnarray} \tag{ 77 }$$

As we mentioned in section 3.3, linear discriminant functions do not perform well on detection tasks when the signal, if present, can be at a random location in the image. Similarly when the task is to estimate the location, size or other parameters of a lesion at a random location, then linear estimators do not perform well either (Whitaker et al 2008).

A Scanning Linear Estimator (SLE) is an estimator that computes a scalar product of the data vector with a template w(θ) and then searches for the point in parameter space where the scalar product is maximum. Formally,

$$\begin{eqnarray}&&{\widehat {\boldsymbol{\theta }} _{{\rm{SL}}}}\left( {\bf{g}} \right) \equiv \mathop {{\rm{argmax}}}\limits_\theta {\left[ {{\bf{w}}\left( {\boldsymbol{\theta }} \right)} \right]^t}{\bf{g}}.\end{eqnarray} \tag{ 78 }$$

If the parameter vector consists only of spatial coordinates of a tumor or other signal, then ${{\widehat{\boldsymbol{\theta }}}_{\text{SL}}}\left(\mathbf{g}\right)$ has a natural interpretation as spatial scanning of a template; more generally, (75) describes scanning in parameter space, not just spatially. The argmax operator is a nonlinear selection of the point in parameter space where the maximum occurs, so the SLE is nonlinear even though it performs only linear operations on the data.

Moreover, if the task requires detection of a signal before estimating its parameters, the same search in parameter space yields the test statistic for detection:

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} t\left(\mathbf{g}\right)=\underset{\theta}{{\text{max}}} {{\left[\mathbf{w}\left(\boldsymbol{\theta }\right)\right]}^{t}}\mathbf{g}. \\ \end{array}\end{eqnarray} \tag{ 79 }$$

As in any detection problem, we make the decision that the signal is present if t (g) exceeds the decision threshold (see (22)).

Conversely, any of the linear observers for detection discussed in section 3.3 can be converted to an SLE simply by showing its dependence on the signal parameters explicitly. For example, the Hotelling observer of (46) leads to

$$\begin{eqnarray}&&{{\left[\mathbf{w}\left(\boldsymbol{\theta }\right)\right]}^{t}}={{\left[\Delta \overline{{\mathbf{\bar{g}}}}\left(\boldsymbol{\theta }\right)\right]}^{t}}\mathbf{K}_{\mathbf{g}}^{-1}.\end{eqnarray} \tag{ 80 }$$

If the data covariance arises entirely from the background, as in the case of a weak signal, then K_g is independent of θ and w(θ) can be computed with methods from section 3.3 without performing a matrix inverse at each point in parameter space.

The performance of an SLE on a joint detection/estimation task can be specified by the area under an LROC or EROC curve (see figure 2). If detection is not at issue, the SLE performance can be specified by some variant of EMSE or by an expected utility. All of these FOMs will vary with radiation dose (see section 6.4).

Khurd and Gindi presented ideal-observer decision strategies that maximize the area under the LROC curve (Khurd and Gindi 2005), and Clarkson extended that work to the area under the EROC curve (Clarkson 2007). For a general treatment of methods for performing, analyzing and validating observer studies involving detection and localization, see Chakraborty and Berbaum (2004).

An interesting practical application is the work of Yendiki and Fessler (2007), who considered model observers for which the decision variable was the maximum value of a local test statistic within a search area. Their goal was to optimize regularized image reconstruction for emission tomography with respect to lesion detectability. They made use of approximations of tail probabilities for the maximum of correlated Gaussian random fields to facilitate the analytical evaluation of detection performance. They illustrated the reasonable accuracy of these approximations in the regime of low probability of false alarms and their potential use in the optimization of regularization functions.

Radiation treatment planning almost always starts by segmenting a CT image to identify the boundaries of the tumors and surrounding normal tissues. It is usually assumed for purposes of treatment planning that the tissue distribution within an organ is homogeneous and that the boundaries returned by the segmentation algorithm are exact. In some case substructures within an organ, such as cortical bone and marrow, are considered separately, and there are some current investigations into how much the substructure in organs like the lung would change the results of the dose calculations if they were not considered separately.

Even with the assumption that the tissues in an organ are homogeneous, the simulation tools used in therapy planning are able to provide some information about the distribution of doses within an organ. However, at the current state, this information is not used; the absorbed doses predicted by the plan are averaged over the organs, and often even these average values are then summarized by a single value, typically the effective dose.

If one wants to use the information available from the planning algorithm for more in-depth evaluation one could use the so-called dose–volume histograms (DVHs), preferably for each organ. This concept was introduced in 1979 in the context of radiation oncology (Shipley et al 1979, Drzymala et al 1991), where it is used to predict probability of tumor control and/or probability of healthy tissue detriment.

One might expect the DVH for an organ to be a histogram of the dose values in the volume elements (voxels) in that organ, hence a discretized approximation of the probability density function for occurrence of different values of the absorbed dose in a voxel, D_v. Mathematically, this PDF could be denoted $\text{pr}\left({{D}_{v}}\right)$. In the radiation-oncology literature, however, the term DVH corresponds to a cumulative probability distribution function rather than a PDF. Specifically, the DVH is defined as the fraction of voxels that receive a dose greater than some amount D₀, plotted as a function of D₀. Thus it is a discretized approximation of the probability (not PDF) Pr(D_v > D₀), which is expressed mathematically as $1-\underset{0}{\overset{{{D}_{0}}}{\int}} pr\left({{D}_{v}}\right)\text{d}{{D}_{v}}$.

The equivalent uniform dose (EUD), a concept similar to that of DVH but more related to possible biological effects, was established by Niemierko (1997) in the context of radiation oncology. This concept can be used for any organ specifically or for the whole body. It is designed in a way that the current dose distribution is compared to another one which is assumed to have the same radiobiological effect. If the biological effect follows the linear-no-threshold theory, and if only photons and electrons are involved, the EUD is equivalent to average of the dose distribution. If so-called deterministic effects (see section 5) might occur, for example due to large dose variations inside an organ like the skin, the EUD can be much higher than the average dose.

The breast is not a homogeneous tissue. Rather, it consists of glandular tissue, fibrous tissue, fatty tissue and the skin. Mainly depending on the size of the breast, the fractional content of the various tissues is different and hence the optimum imaging conditions are different (Thilander-Klang et al 1997). Besides the different contrast appearances of the different tissues, it is also important to note that the glandular (and the fibrous) tissues are much more sensitive to ionizing radiation than the fatty tissue and the skin. To compare mammographic imaging systems on the basis of radiation risk, it is therefore more useful to evaluate the average dose to the glandular and fibrous tissue. This can be done by measuring the air kerma at the entrance of a breast phantom, multiplying this value by a conversion coefficient to get the entrance dose for the real breast and then multiplying by a factor relating this entrance dose to the mean glandular dose. These factors have been determined by Dance et al, first for 50% glandularity and then for various different glandularities (Dance 1990, Dance et al 2000). While Dance was assuming a homogeneous distribution in the main area of the breast between fatty and glandular tissue, this is obviously not the case. Therefore there are approaches that use more realistic phantoms of the breast either by evaluating large number of mammograms and finding mathematical representations resulting in 3D tissue distributions (Lau et al 2012, Bakic et al 2011, Bakic et al 2002) or by high-resolution 3D-scanning of specially fixed breast specimens from corpses (Hoeschen et al 2005). The segmented high-resolution phantoms need to be extended to whole-body phantoms in order to gain additional dosimetric information for comparison of different imaging modalities.

The effective dose is intended to be a way to compare systems regarding their radiation burden for a large group of patients undergoing similar examinations, but it is not a directly measurable unit. In addition, for CT applications, which generate the highest per capita radiation burden in most of the developed countries today, there is no available measurement system for really measuring radiation dose to the patient directly. Instead, various dose indicators have been proposed. They are typically based on measuring absorbed dose within polymethyl methacrylcate (PMMA) phantoms during standard dose protocols. With these values and the known system parameters for a particular patient examination, dose indicators such as the so-called dose–length product (DLP) and the computed tomography dose index (CTDI) can be calculated and displayed. The DLP in general is the dose in a certain area multiplied by the length of the investigated area; in a CT scanner this would mean the scan length (typically calculated as the slice thickness multiplied by the number of slices). However, there is not a measured or calculated value for the absorbed dose within the body of the patient. Therefore the CTDI_vol is used as the multiplier.

The CTDI as it was originally defined by the Food and Drug Administration (FDA) in 1981 was introduced for axial scanners (Shope et al 1981). It was meant to characterize the dose from the primary beam and that resulting from scattered radiation emitted from surrounding slices in such a scanning geometry and for standard conditions. The CTDI is thus defined as the equivalent value of absorbed dose in a central slice assuming that the total radiation is concentrated in that specific slice. Normalization of this value by the current output of the tube gives a hint for the dose profile which can be reached by a scanner but not about the dose of the patient or the dose efficiency of the scanner. This basic concept has been modified in various approaches to make CTDI meaningful for spiral and multi-detector-row CT. CTDI₁₀₀ is assumed to represent the dose contribution from a 100 mm measurement length along the axis with 100 mm dosimetric chambers measured in standard conditions for a single collimation scan still in a phantom. Doing such a measurement in the center of the acrylic phantom typically 32 cm or 16 cm diameter and on various positions at the periphery permits calculation of the weighted CTDI (CTDI_w). For this purpose, one weights the central measurement by 1/3 and the average of the peripheral measurements by 2/3 as expressed in the following equation:

$$\begin{eqnarray}&&\text{CTD}{{\text{I}}_{w}}=\frac{1}{3}\text{CTD}{{\text{I}}_{c}}+\frac{2}{3}\frac{{{\sum}_{n=1}^{N}} \text{CTD}{{\text{I}}_{\text{per},n}}}{N},\end{eqnarray} \tag{ 83 }$$

From CTDI_w one can calculate the volume CTDI (CTDI_vol), which is the most commonly used CT dose index today, by dividing the weighted CTDI by the pitch factor. This dose index is somehow related to an absorbed dose of the patient in a specific volume, but only indirectly because it does not refer to either a phantom measurement or to measurements in the center and the periphery with a fixed weighting. Instead, CTDI refers to a standard operation protocol for the scanner. Using the CTDI_vol, the scan length, the body part investigated and conversion coefficients resulting from calculation programs based on results from Monte Carlo simulations seems to give results consistent with thermoluminescent dosimeter (TLD) measurements in anthropomorphic phantoms (Lechel et al 2007) for calculated effective doses on standard patients as indicators for dose comparisons of systems. Using these values for risk estimates on single patients is probably not useful for various reasons.

One approach to getting a better patient-specific dose estimate is the size-specific dose estimate as introduced by the AAPM (AAPM 2011), which tries to find a way to calculate dose values dependent on the size of the patient. However, again this value does not take into account the patient-specific radiation sensitivity and it therefore does not give a true patient-specific risk value, but only an indicator.

Li et al (2011b) developed a method to estimate patient-specific dose and cancer risk from CT examinations by combining a validated Monte Carlo program with patient-specific anatomical models. The organ dose, effective dose, and risk index (a surrogate of cancer risk) was estimated for clinical chest and abdominopelvic protocols. In Li et al (2011a), this same group evaluated the dependence of dose and risk on patient size and scanning parameters. The reported relationships can be used to estimate patient-specific dose and risk in clinical practice for a given pediatric patient.

Schmidt and Kalender (2002) describe a voxel-based Monte Carlo method for scanner- and patient-specific dose calculations in computed tomography.

Dose values cannot usually be measured inside a patient. It is possible to measure the skin dose directly on the patient or to calculate it from dose measurements (mostly air kerma measurements) in the x-ray beam, but this measurement does not yield the dose distributions in specific organs throughout the body. Therefore, ICRP and ICRU as well as many research groups have been using simulation tools in combination with models of the human body and biokinetic models to determine absorbed doses to organs. The simulation tools that are most commonly used today are Monte Carlo simulations (e.g. EGS, MCNP, Fluka, Penelope and GEANT (Bielajew et al 1994, Baró et al 1995, Agostinelli et al 2003, Ferrari et al 2005, Allison et al 2006, Reed 2007, Ljungberg and King 2012, Pozuelo et al 2012)). The models range from pure mathematical models (ICRP 1979, Zankl et al 1988)) to various anthropomorphic voxel phantoms (Zubal et al 1994, Xu et al 2000, Petoussi-Henss et al 2002, Kramer et al 2004, Lee et al 2006, Menzel et al 2009, Xu and Eckerman 2010).

The voxel models for average adults in ICRP 110 (Menzel et al 2009) are based on real patient images that were scanned and segmented. These models were then modified to fit the conditions of the organs of average human beings. ICRP and ICRU are currently working on models for children of different ages. There are also other phantoms available (Zhang et al 2007, Becker et al 2007, Segars et al 2010). For all these phantoms and all these kinds of simulation tools, there are many publications investigating the variation of dose with different imaging parameters e.g. for angiography or CT scanning in Schlattl et al (2007, 2012), Li et al (2012).

Many details on computational phantoms were presented at the 4th International Workshop on Computational Phantoms for Radiation Protection, Imaging, and Radiotherapy in 2013, and selected papers from this workshop were published in a special section of Physics in Medicine and Biology in September, 2014; included is a topical review on computational phantoms (Xu 2014).

Two separate steps are required for rigorous dose calculation in emission imaging and radionuclide therapy. First, the spatiotemporal radiopharmaceutical distribution should be determined. Then, in principle, the distribution of absorbed dose, d (r, t), should be calculated, but in practice usually only time-integrated organ doses are considered.

Time-dependent radiopharmaceutical distributions for nuclear medical imaging have been investigated in Zankl et al (2012) and Giussani et al (2012). A detailed review of dosimetry in nuclear medicine is given by Stabin (2006).

The heritable genetic risks associated with radiation exposure are estimated directly from animal studies in combination with data on baseline incidence of disease in human populations (UNSCEAR 2001). There are no usable human data on radiation-induced germ cell mutations, let alone induced genetic diseases, and the results of the largest and most comprehensive of human epidemiological studies, namely that carried out on the children of the Japanese atomic bomb survivors, are negative; there are no statistically significant radiation-associated adverse hereditary effects in this cohort (Neel et al 1990). The data on the induction of germline mutations at human minisatellite loci (Dubrova et al 1996), although of importance from the standpoint of direct demonstration of radiation-induced heritable genetic changes in humans, are probably not suitable for risk estimation; they occur in non-coding DNA and are thought not to be associated with heritable disease (UNSCEAR 2001, Little et al 2013).

Simple linear models of dose are generally employed to model genetic effects. For example, the preferred risk model used in the 2001 report of the United Nations Scientific Committee on the Effects of Atomic Radiation (UNSCEAR 2001) assumes that the excess heritable genetic risk for disease class i associated with a dose D of radiation to the parental gonads is given by:

$$\begin{eqnarray}&&{{R}_{i}}={{P}_{i}} \cdot \left[\frac{1}{D{{D}_{i}}}\right]\cdot M{{C}_{i}} \cdot PCR{{F}_{i}} \cdot D,\end{eqnarray} \tag{ 86 }$$

where P_i is the baseline incidence of that disease class, MC_i is the mutational component of the disease class (defined as the relative increase in disease frequency (relative to the baseline) per unit relative increase in mutation rate (relative to the spontaneous rate)), PCRF_i is the potential recoverability correction factor for the disease class (the fraction of induced mutations compatible with live births) and DD_i is the mutational doubling dose (i.e. the dose required to double the mutational load associated with the disease). Table 3 summarizes estimates of risk for the first- and second-generation progeny of an irradiated population that has sustained radiation exposure in the parental generation and no radiation subsequently. These risk estimates assume a mutational doubling dose, DD_i, of 1 Gy, that is based on human data on spontaneous mutation rates and mouse data on induced mutation rates (UNSCEAR 2001). Neel et al (1990) derived an estimate of the doubling dose in the atomic bomb survivors of 3.4–4.5 Sv, based on examination of a combination of five endpoints. However, as discussed in UNSCEAR (2001), in view of the differences between the endpoints considered by Neel et al (1990) and those employed by UNSCEAR, and the uncertainties in the estimates of Neel et al, their estimate of the doubling dose is entirely compatible with that used by UNSCEAR.

Most of the information on radiation-induced cancer risk comes from (a) the Japanese atomic bomb survivors, (b) medically exposed populations and (c) occupationally exposed groups (UNSCEAR 2008b). The Life Span Study (LSS) cohort of Japanese atomic bomb survivors is unusual among exposed populations in that both genders and a wide range of ages were exposed, comparable with those of a general population (Ozasa et al 2012, Preston et al 2007). Most medically treated groups are more restricted in the age and gender mix. For example, the International Radiation Study of Cervical Cancer patients (IRSCC), a cohort of women followed up after treatment for cancer of the cervix, were all treated as adults, most above the age of 40 (Boice et al 1987, Boice et al 1988). Organ doses among those treated with radiotherapy tend to be higher than those received by the Japanese atomic bomb survivors, although there are some exceptions, e.g. breast doses in the IRSCC patients (Boice et al 1987, Boice et al 1988).

Occupationally exposed groups are also more restricted in their age and gender mix. For example, the cohorts of workers exposed in the nuclear industry (Cardis et al 2007, Muirhead et al 2009) are overwhelmingly male and exposed in adulthood, as are the groups of underground miners (National Research Council 1999). For these reasons, most standard-setting bodies (National Research Council 1999, National Research Council 2006, ICRP 2007) use the LSS as the basis for estimates of population cancer risk associated with exposure to low-LET radiation. For certain cancer sites and types of radiation exposures, other groups are occasionally used; in particular for high-LET (α-particle) exposure to the lung, underground miners are used (National Research Council 1999). However, lung-cancer risks estimated for the miners by applying those estimated in the LSS in combination with the current ICRP dosimetric model (ICRP 1994) are close to those that can be estimated directly (Little 2002).

One of the principal uncertainties that surround the calculation of population cancer risks from epidemiological data results from the fact that few radiation-exposed cohorts have been followed up to extinction. For example, 58 years after the atomic bombings of Hiroshima and Nagasaki, about 42% of the survivors were still alive (Ozasa et al 2012). In attempting to calculate lifetime population cancer risks, it is therefore important to predict how risks might vary as a function of time after radiation exposure, in particular for that group for whom the uncertainties in projection of risk to the end of life are most uncertain, namely those who were exposed in childhood.

Analyses of solid cancers in the LSS and other exposed groups have found that the radiation-induced excess risk can be approximately described by a constant-relative-risk model (UNSCEAR 2008b). The time-constant excess relative risk (ERR) model assumes that if a dose of radiation is administered to a population, then, after some latent period, there is an increase in the cancer rate, the excess rate being proportional to the underlying cancer rate in an unirradiated population.

It is well known that for all cancer subtypes (including leukemia) the ERR diminishes with increasing age at exposure (UNSCEAR 2008b). For those irradiated in childhood there is some evidence of a reduction in the ERR of solid cancer 25 or more years after exposure. Therefore, even for solid cancers various factors have to be employed to modify the ERR. For many solid cancers a generalized relative risk model is commonly used, in which the cancer rate t years after exposure for gender s following exposure at age e to a dose D of radiation is given by

$$\begin{eqnarray}&&R\left(a,s,e,D\right)={{r}_{0}}(a,s)\left[1+F(D)\phi (t,e,s)\right],\end{eqnarray} \tag{ 87 }$$

where r₀(a, s) is the cancer rate in the absence of irradiation, i.e. the baseline cancer rate, a = t + e is the age at observation (attained age) of the person and F(D) is the function determining the dose dependency of the cancer risk, to be discussed below. The expression ϕ(t, e, s) describes the modification to the ERR, F(D), as a function of time since exposure t, age at exposure e and gender s. Similar types of model have been fitted to leukemia (Hsu et al 2013). However, generalized absolute risk models can also be used to model both solid cancers and leukemia, in which the cancer rate t years after exposure for gender s following exposure at age e to a dose D of radiation is given by:

$$\begin{eqnarray}&&R\left(a,s,e,D\right)={{R}_{0}}(a,s)+F(D)\psi (t,e,s).\end{eqnarray} \tag{ 88 }$$

The expression ψ(t, e, s) describes the modification to the EAR, F(D), as a function of time since exposure t, age at exposure e and gender s.

Given appropriate forms of the modifying functions ϕ(t, e, s) and ψ(t, e, s) of the relative and absolute risk respectively, equivalently good fits to the solid cancer and leukemia mortality dataset were achieved using both generalized ERR and generalized EAR models (Little et al 2008a). It is to some extent arbitrary which of these two models one uses. However, as can be seen from table 3, models with equivalent fit can yield slightly different lifetime population risks. The reason for this is that, as noted above, about 42% of the LSS cohort are still alive (Ozasa et al 2012), so that population risk calculations based on this dataset (used by many scientific committees (National Research Council 2006, ICRP 2007)) crucially depend on extrapolating the current mortality and incidence follow-up of this group to the end of life.

Uncertainties due to risk projection are greatest for solid cancers, because the radiation-associated excess risk in the LSS is still increasing (Ozasa et al 2012, Preston et al 2007). For leukemia the excess risk is reducing over time (Hsu et al 2013), and most models used predict very few radiation-associated leukemia deaths or cases from the current follow-up point in the LSS to extinction.

Use can also be made of so-called biologically based cancer risk models, as summarized by Little (2010). However, these would not be expected to yield materially different population cancer risks to those given by purely empirical models of the form discussed above, although they may be parametrically more parsimonious than purely empirical models.

It has been customary to model the dose-response function F(D) that appears in (87) and (88) in fits to biological (UNSCEAR 1993) and epidemiological data (UNSCEAR 2008b) by the linear-quadratic expression:

$$\begin{eqnarray}&&F(D)=\alpha D+\beta {{D}^{2}}.\end{eqnarray} \tag{ 89 }$$

There is significant curvilinearity in the dose response for leukemia in the LSS (Hsu et al 2013), although for solid cancers, apart from non-melanoma skin cancer (Ron et al 1998, Preston et al 2007), there is little evidence for anything other than a linear dose response in the Japanese cohort or in any other group (UNSCEAR 2008b). It should be noted that as well as modifications in effectiveness (per unit dose) relating to alterations in the total dose there are also possible variations of effectiveness as a result of dose fractionation (the process of splitting a given dose into a number of smaller doses suitably separated in time) and dose rate (UNSCEAR 1993). This is not surprising radiobiologically; by administering a given dose at progressively lower dose rates (i.e. giving the same total dose over longer periods of time), or by splitting it into many fractions, the biological system has time to repair the damage, so that the total damage induced will be less (UNSCEAR 1993). Therefore, although for cancers other than leukemia there is generally little justification for assuming anything other than a linear dose response, i.e. β = 0, it may nevertheless be justifiable to employ a dose and dose-rate effectiveness factor (DDREF) other than 1. The DDREF is the factor by which one divides risks for high-dose and high-dose-rate exposure to obtain risks for low doses and low dose rates. The ICRP (2007) recommended that a DDREF of 2 be used together with models linear in dose for all cancer sites, on the basis largely of the observations in various epidemiological datasets. UNSCEAR (2008b) used linear-quadratic models to derive cancer risks (as shown in table 3), so that a DDREF should not be applied.

Another form of dose response, perhaps less commonly used, slightly generalizes (89):

$$\begin{eqnarray}&&F(D)=\left(\alpha D+\beta {{D}^{2}}\right) \text{exp} \left(\gamma D\right),\end{eqnarray} \tag{ 90 }$$

and this form has been employed in fits to biological data (UNSCEAR 1993) and epidemiological data (Boice et al 1987, Thomas et al 1992, Little et al 1999). The αD + βD² component represents the effect of (carcinogenic) mutation induction, while the exp(γD) term represents the effect of cell sterilization or killing. In general the cell sterilization coefficient γ is negative. Essentially this implies that there is a competing effect due to cell killing which is greater at higher radiation doses. A dead cell cannot proliferate and become the focus of a malignant clone. Variant forms of the cell-sterilization term exp(γD), incorporating higher powers of dose D, i.e. exp(γD^k) for k > 1, are sometimes employed (Little and Charles 1997).

In SPECT and PET, the object of interest is the spatiotemporal distribution of a radiopharmaceutical, described in detail by the function f (r, t), which describes the mean number of emitted gamma-ray photons per second per unit volume in the patient's body. The mean number of photons incident (from within) on the patient's body, following an injection at t = 0 is given by

$$\begin{eqnarray}&&{\bar N^{{\rm{inc}}}}({\bf{f}}) = \int_0^\infty {{\rm{d}}t} \,\int_{{\rm{body}}} {{{\rm{d}}^3}r} \,f({\bf{r}},t),\end{eqnarray} \tag{ 96 }$$

where the single overbar indicates that ${{\bar{N}}^{\text{inc}}}\left(\mathbf{f}\right)$ is averaged only over Poisson statistics, not over object variability, so it remains a function of f. We can define a normalized object function f₀(r, t) by

$$\begin{eqnarray}&&{{f}_{0}}\left(\mathbf{r},t\right)=\frac{f\left(\mathbf{r},t\right)}{{{{\bar{N}}}^{\text{inc}}}\left(\mathbf{f}\right)},\end{eqnarray} \tag{ 97 }$$

so that $f\left(\mathbf{r},t\right)={{\bar{N}}^{\text{inc}}}\left(\mathbf{f}\right){{f}_{0}}\left(\mathbf{r},t\right)$.

Because SPECT and PET systems are linear continuous-to-discrete mappings, the mean detector output (averaged only over Poisson statistics) is given by (see (2))

$$\begin{eqnarray}&&{{\bar g}_m}({\bf{f}}) = \int_{{t_1}}^{{t_2}} {{\rm{d}}t} \,\int_{{\rm{body}}} {{{\rm{d}}^3}r\,{h_m}({\bf{r}},t)f({\bf{r}},t)} = {{\bar N}^{{\rm{inc}}}}({\bf{f}})\int_{{t_1}}^{{t_2}} {{\rm{d}}t} \,\int_{{\rm{body}}} {{{\rm{d}}^3}r\,{h_m}({\bf{r}},t){f_0}({\bf{r}},t),}\end{eqnarray} \tag{ 98 }$$

where the image acquisition takes place over t₁ < t < t₂ and h_m(r, t) is the probability (not PDF) that a gamma-ray photon emitted at time t and location r will be detected in detector bin m. The mean number of detected photons, in all bins, for a given object is

$$\begin{eqnarray}&&{{\bar{N}}^{\text{det}}} \left(\mathbf{f}\right)=\underset{m=1}{\overset{M}{\sum}} {{\bar{g}}_{m}}\left(\mathbf{f}\right)={\int}_{{t_1}}^{{t_2}}\text{d}t {{\int}_{\text{body}}}{{\text{d}}^{3}}r\,s\left(\mathbf{r},t\right)f\left(\mathbf{r},t\right)={{\bar{N}}^{\text{inc}}}\left(\mathbf{f}\right){\int}_{{{t}_{1}}}^{{{t}_{2}}}\text{d}t {{\int}_{\text{body}}}{{\text{d}}^{3}}r\ s\left(\mathbf{r},t\right){{f}_{0}}\left(\mathbf{r},t\right),\end{eqnarray} \tag{ 99 }$$

where s (r, t), defined as $\underset{m=1}{\overset{M}{\sum}} {{h}_{m}}\left(\mathbf{r},t\right)$ is called the system sensitivity function; in SPECT it can vary with time as the detector rotates around the patient's body.

The mean number of incident photons, ${{\bar{N}}^{\text{inc}}}\left(\mathbf{f}\right)$, varies from patient to patient because the total injected activity is not always the same and because different patients may have different biological elimination rates. It is reasonable to assume that these effects are statistically independent of the more complicated patient-to-patient variations captured in the normalized distribution f₀(r, t). Thus the efficiency defined in (94) is given, for patients in group C, by

$$\begin{eqnarray}&&\bar \eta = \frac{{{{\left\langle {{{\bar N}^{{\rm{det}}}}\left( {\bf{f}} \right)} \right\rangle }_{{\bf{f}}\mid {\bf{C}}}}}}{{{{\left\langle {{{\bar N}^{{\rm{inc}}}}\left( {\bf{f}} \right)} \right\rangle }_{{\bf{f}}\mid {\bf{C}}}}}} = \int_{{t_1}}^{{t_2}} {{\rm{d}}t} \,\int_{{\rm{body}}} {{{\rm{d}}^3}r{{\left\langle {s\left( {{\bf{r}},t} \right){f_0}\left( {{\bf{r}},t} \right)} \right\rangle }_{{\bf{f}}\mid {\bf{C}}}},}\end{eqnarray} \tag{ 100 }$$

where the expectation within the integral is the average of the normalized object distribution weighted by the system sensitivity. We see immediately that the efficiency can be increased just by increasing the acquisition time t₂ −t₁, though practical issues of patient motion and patient throughput may intrude.

As we saw in section 3, many figures of merit for image quality depend on some form of data covariance matrix. By (16) or (21), these matrices can always be decomposed into two terms, one arising from measurement noise and one from the randomness in the object being imaged. As noted in section 2.2, these terms have different dependences on ${{\overline{{\bar{N}}}}^{\text{det}}}$ and hence on any measure of the radiation dose administered during the imaging procedure.

For SPECT and PET, the noise term in the covariance expansion is obtained from (8) by averaging over object variability, which requires only adding an overbar to f. From this result and (97) we obtain

$$\begin{eqnarray}&&{{\left[\mathbf{\bar{K}}_{\mathbf{g}\mid \mathbf{C}}^{\text{noise}}\right]}_{mm\prime}}={{\left(\mathcal{H}\mathbf{\bar{f}}\right)}_{m}}{{\delta}_{mm\prime}}={{\overline{{\bar{N}}}}^{\text{inc}}}{{\left(\mathcal{H}{{\mathbf{\bar{f}}}_{0}}\right)}_{m}}{{\delta}_{mm\prime}},\end{eqnarray} \tag{ 101 }$$

where the last step has used the fact that $\mathcal{H}$ is linear for SPECT and PET. We can express (101) in vector form, without explicit components, as

$$\begin{eqnarray}&&\mathbf{\bar{K}}_{\mathbf{g}\mid \mathbf{C}}^{\text{noise}}=\text{Diag}\left(\mathcal{H}\mathbf{\bar{f}}\right)={{\overline{{\bar{N}}}}^{\text{inc}}}\text{Diag}\left(\mathcal{H}{{\mathbf{\bar{f}}}_{0}}\right)\equiv {{\overline{{\bar{N}}}}^{\text{inc}}}\mathbf{\bar{K}}_{{{\mathbf{g}}_{0}}\mid \mathbf{C}}^{\text{noise}},\end{eqnarray} \tag{ 102 }$$

where Diag indicates a diagonal matrix and the final covariance on the right is the noise covariance for the normalized object f₀. Similar manipulations show that

$$\begin{eqnarray}&&\mathbf{K}_{\mathbf{\bar{g}}\mid \mathbf{C}}^{\text{obj}}={{\left[{{\overline{{\bar{N}}}}^{\text{inc}}}\right]}^{2}}\mathbf{K}_{{{{\mathbf{\bar{g}}}}_{0}}\mid \mathbf{C}}^{\text{obj}}.\end{eqnarray} \tag{ 103 }$$

Thus, for the linear system operators in SPECT and PET, the noise covariance term for the raw projection data varies linearly with the mean number of incident photons, and hence linearly with any measure of radiation dose, and the object-variability term varies quadratically. The same conclusions hold if we consider SPECT and PET images reconstructed with linear or nonlinear algorithms.

The situation is more complicated for x-ray modalities where the system operator is nonlinear and integrating detectors are usually used rather than photon-counting detectors as in SPECT and PET. We begin by examining the statistics of the raw detector outputs, and how they vary with dose, and then we derive expressions for the mean vectors and covariance matrices after logarithmic mapping and also after image reconstruction.

The object of interest in CT is the 3D distribution of x-ray attenuation coefficient in some portion of the patient's body. Though the attenuation coefficient is conventionally called μ, we refer to it as f in order to maintain a common notation across modalities. As noted in section 2, we can continue to write $\mathbf{\bar{g}}\left(\mathbf{f}\right)=\mathcal{H}\mathbf{f}$, but the system operator is now nonlinear because f appears in an exponent. Moreover, in a complete treatment, the energy spectrum of the source, the energy dependence of the attenuation coefficient and the variation of the detector efficiency with photon energy would all have to be taken into account, yielding a very complicated nonlinear expression for the mean data (see section 16.1.4 in Barrett and Myers (2004), especially equation (16.109)).

For the purposes of this paper, however, we approximate the mean number of primary (unscattered) x-ray photons reaching the mth detector element in some exposure time by

$$\begin{eqnarray}&&\bar{g}_{m}^{\text{pri}}\left(\mathbf{f}\right)=\bar{N}_{m}^{\text{inc}} \text{exp} \left[-{{\left(\mathcal{R}\mathbf{f}\right)}_{m}}\right],\end{eqnarray} \tag{ 104 }$$

where $\mathcal{R}$ is a linear operator that integrates the object over a thin tube of directions from the x-ray source to the detector element and $\bar{N}_{m}^{\text{inc}}$ is the mean number of photons that would be incident on the detector element in the absence of the object, and hence it is the number incident on the object in the tube of integration when the object is present. The steps involved in getting to (104) are detailed in section 16.1.7 in Barrett and Myers (2004).

In addition to the primary photons, there are also some photons scattered in the patient's body that reach the detector element, so we can write

$$\begin{eqnarray}&&{{\bar{g}}_{m}}\left(\mathbf{f}\right)=\bar{g}_{m}^{\text{pri}}\left(\mathbf{f}\right)+\bar{g}_{m}^{\text{sc}}\left(\mathbf{f}\right).\end{eqnarray} \tag{ 105 }$$

Detected scattered photons convey very little information about the task, but it is important that we retain them in the noise analysis.

An initial step in the processing of CT data is usually to normalize each detector output to the number of incident photons (presumably known by careful scanner calibration) and take the negative logarithm of the ratio:

$$\begin{eqnarray}&&{{y}_{m}}\equiv - \ln \left(\frac{{{g}_{m}}}{\bar{N}_{m}^{\text{inc}}}\right).\end{eqnarray} \tag{ 106 }$$

The conditional mean of y_m is

$$\begin{eqnarray}&&{{\bar{y}}_{m}}\left(\mathbf{f}\right)= \ln \left(\bar{N}_{m}^{\text{inc}}\right)-{{\left\langle \ln \left({{g}_{m}}\right)\right\rangle}_{\mathbf{g}\mid \mathbf{f}}}= \ln \left(\bar{N}_{m}^{\text{inc}}\right)-{{\left\langle \ln \left[{{\bar{g}}_{m}}\left(\mathbf{f}\right)+{{g}_{m}}-{{\bar{g}}_{m}}\left(\mathbf{f}\right)\right]\right\rangle}_{\mathbf{g}\mid \mathbf{f}}}\end{eqnarray} \tag{ 107 }$$

$$\begin{eqnarray}&&\approx - \ln \left(\frac{{{{\bar{g}}}_{m}}\left(\mathbf{f}\right)}{\bar{N}_{m}^{\text{inc}}}\right)+\frac{1}{2}\frac{\alpha {{{\bar{g}}}_{m}}\left(\mathbf{f}\right)+\sigma _{m}^{2}}{{{\left[{{{\bar{g}}}_{m}}\left(\mathbf{f}\right)\right]}^{2}}},\end{eqnarray} \tag{ 108 }$$

where we have used a truncated Taylor expansion, $\ln (1+x)\approx x-\frac{1}{2}{{x}^{2}}$, and the last step has used the noise model of (9). Notice that the variance of g_m affects the mean of y_m because of the logarithmic nonlinearity.

The usual assumption in CT is that ${{\bar{y}}_{m}}\left(\mathbf{f}\right)$ is equal to the tube integral ${{\left(\mathcal{R}\mathbf{f}\right)}_{m}}$, but it isn't because of the last term in (108) and the scattered radiation. Instead we get

$$\begin{eqnarray}&&{{\bar{y}}_{m}}\left(\mathbf{f}\right)={{\left(\mathcal{R}\mathbf{f}\right)}_{m}}+{{b}_{m}}\left(\mathbf{f}\right),\end{eqnarray} \tag{ 109 }$$

where

$$\begin{eqnarray}&&{{b}_{m}}\left(\mathbf{f}\right)=\frac{1}{2}\frac{\alpha {{{\bar{g}}}_{m}}\left(\mathbf{f}\right)+\sigma _{m}^{2}}{{{\left[{{{\bar{g}}}_{m}}\left(\mathbf{f}\right)\right]}^{2}}}- \ln \left[1+\frac{\bar{g}_{m}^{\text{sc}}\left(\mathbf{f}\right)}{\bar{g}_{m}^{\text{pri}}\left(\mathbf{f}\right)}\right].\end{eqnarray} \tag{ 110 }$$

The second term in b_m is negligible if the scatter-to-primary ratio is small, and the first term is small if the mean number of detected photons for one detector element is large; neither assumption is guaranteed to be valid in practice.

Calculation of the conditional covariance matrix, K_y∣f, follows the same pattern that led to (108), but with a different truncated Taylor series: [ln(1 + x)]² ≈ x². The result is

$$\begin{eqnarray}&&{{\left[{{\mathbf{K}}_{\mathbf{y}\mid \mathbf{f}}}\right]}_{mm\prime}}=\frac{\alpha {{{\bar{g}}}_{m}}\left(\mathbf{f}\right)+\sigma _{m}^{2}}{{{\left[{{{\bar{g}}}_{m}}\left(\mathbf{f}\right)\right]}^{2}}}{{\delta}_{mm\prime}}.\end{eqnarray} \tag{ 111 }$$

Averaging over all objects f in set C yields the average noise covariance matrix for data vector y:

$$\begin{eqnarray}&&{{\left[\mathbf{\bar{K}}_{\mathbf{y}\mid \mathbf{C}}^{\text{noise}}\right]}_{mm\prime}}={{\left\langle \frac{\alpha {{{\bar{g}}}_{m}}\left(\mathbf{f}\right)+\sigma _{m}^{2}}{{{\left[{{{\bar{g}}}_{m}}\left(\mathbf{f}\right)\right]}^{2}}}\right\rangle}_{\mathbf{f}\mid \mathbf{C}}}{{\delta}_{mm\prime}}.\end{eqnarray} \tag{ 112 }$$

In principle, this average can be estimated by collecting raw projection data for a large number of real or accurately simulated patients in some defined set C and estimating the ensemble average with an arithmetic average, separately for each index m.

The object term in the covariance matrix for y is complicated by the object-dependent bias b_m. We should write (see (18))

$$\begin{eqnarray}&&{\bf{K}}_{\overline {\bf{y}} \mid {\bf{C}}}^{{\rm{obj}}} \equiv {\left\langle {\left[ {\overline {\bf{y}} ({\bf{f}}) - \overline {\overline {\bf{y}} } } \right]{{\left[ {(\overline {\bf{y}} ({\bf{f}}) - \overline {\overline {\bf{y}} } } \right]}^t}} \right\rangle _{{\bf{f}}\mid {\bf{C}}}} = {\left\langle {\left[ {{\cal R}({\bf{f}} - \overline {\bf{f}} ) + ({\bf{b}} - \overline {\bf{b}} )} \right]{{\left[ {{\cal R}({\bf{f}} - \overline {\bf{f}} ) + ({\bf{b}} - \overline {\bf{b}} )} \right]}^t}} \right\rangle _{{\bf{f}}\mid {\bf{C}}}},\end{eqnarray} \tag{ 113 }$$

but for simplicity we assume that ${{b}_{m}}-{{\bar{b}}_{m}}$ is small, and we find

$$\begin{eqnarray}&&\mathbf{K}_{\mathbf{\bar{y}}\mid \mathbf{C}}^{\text{obj}}={{\left\langle \left[\mathcal{R}\left(\mathbf{f}-\mathbf{\bar{f}}\right)\right]{{\left[\mathcal{R}\left(\mathbf{f}-\mathbf{\bar{f}}\right)\right]}^{t}}\right\rangle}_{\mathbf{f}\mid \mathbf{C}}}=\mathcal{R}{{\mathcal{K}}_{\mathbf{f}\mid \mathbf{C}}}{{\mathcal{R}}^{\dagger}},\end{eqnarray} \tag{ 114 }$$

where ${{\mathcal{K}}_{\mathbf{f}\mid \mathbf{C}}}$ was introduced in (19) and (20).

By analogy to (21), we can express the covariance matrix for a CT image reconstructed by a linear algorithm $\mathcal{O}$, such that $\mathbf{\hat{f}}=\mathcal{O}\mathbf{y}$, as

$$\begin{eqnarray}&&{{\mathbf{K}}_{\mathbf{\hat{f}}\mid \mathbf{C}}}=\mathcal{O}\mathbf{\bar{K}}_{\mathbf{y}\mid \mathbf{C}}^{\text{noise}}{{\mathcal{O}}^{t}}+\mathcal{O}\mathbf{K}_{\mathbf{\bar{y}}\mid \mathbf{C}}^{\text{obj}}{{\mathcal{O}}^{t}}=\mathcal{O}\mathbf{\bar{K}}_{\mathbf{y}\mid \mathbf{C}}^{\text{noise}}{{\mathcal{O}}^{t}}+\mathcal{O}\mathcal{H}{{\mathcal{K}}_{\mathbf{f}}}{{\mathcal{H}}^{\dagger}}{{\mathcal{O}}^{t}}.\end{eqnarray} \tag{ 115 }$$

We are now able to discuss the relationship of these various mean vectors and covariance matrices to radiation dose and to compare the results for transmission computed tomography (TCT, or CT for short) to what we found earlier for ECT (SPECT and PET). One obvious difference is that the object in ECT is the source of the ionizing radiation, so we immediately have that f_ECT is linearly related to the absorbed dose at any point or to any organ dose. In CT, on the other hand, f_CT is an x-ray attenuation coefficient, hence unrelated to dose. In both cases, however, $\mathcal{H}\mathbf{f}$ is linearly related to dose; the proportionality to dose comes from f in ECT and from $\mathcal{H}$ in TCT.

These and other dose dependences are summarized in table 4, and their relation to figures of merit for image quality are discussed in sections 6.3 and 6.4.

Table 4. Dose dependences of mean vectors and covariance matrices for emission computed tomography (ECT) and transmission computed tomography (TCT). Photon-counting detectors are assumed for ECT and integrating detectors are assumed for TCT. Symbol D indicates a generic dose quantity (such as absorbed dose at a point, organ dose or effective dose), and D², for example, indicates that the quantity in the left column varies quadratically with dose; D⁰ is used for a quantity independent of dose. The notation ≈ D⁰ in the TCT column means that the quantity in the first column would be independent of dose if we neglected the noise-induced bias resulting from the logarithmic step in CT reconstruction. Angle brackets, included only when they affect the dose dependence, denote an average over object variability. Constant c relates to the dose-independent electronic noise in integrating detectors, and constant a serves to scale the dose-dependent part of the noise covariance to the dose-independent part. Double right arrow indicates asymptotic dependence for large dose. All other symbols are defined in the text.

Quantity	ECT	TCT
f, $\mathbf{\bar{f}}$	D	D0
$\mathcal{H}$	D0	D
$\mathbf{\bar{g}}$, $\overline{{\mathbf{\bar{g}}}}$	D	D
$\mathbf{\bar{y}}$, $\overline{{\mathbf{\bar{y}}}}$	NA	≈ D0
$\overline{{\mathbf{\hat{f}}}}$, $\overline{\overline{{\mathbf{\hat{f}}}}}$	D	≈ D0
$\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}$	D	〈bD + c〉 ⇒ D
$\mathbf{K}_{{\mathbf{\bar{g}}}}^{\text{obj}}$	D2	D2
$\mathbf{\bar{K}}_{\mathbf{y}}^{\text{noise}}$	NA	$\left\langle \frac{bD+c}{{{D}^{2}}}\right\rangle \Rightarrow {{D}^{-1}}$
$\mathbf{K}_{{\mathbf{\bar{y}}}}^{\text{obj}}$	NA	D0
$\mathbf{\bar{K}}_{{\mathbf{\hat{f}}}}^{\text{noise}}$	D	$\left\langle \frac{bD+c}{{{D}^{2}}}\right\rangle \Rightarrow {{D}^{-1}}$
$\mathbf{K}_{\overline{{\mathbf{\hat{f}}}}}^{\text{obj}}$	D2	D0

Consider first a simple linear discriminant for the task of detecting the presence of a lesion at a single, specified location. If the task is performed with raw projection data g (as opposed to a reconstructed image $\mathbf{\hat{f}})$, then the linear discriminant is defined by a template w_ld, and the test statistic is a simple scalar product:

$$\begin{eqnarray}&&{{t}_{ld}}\left(\mathbf{g}\right)=\mathbf{w}_{ld}^{t}\mathbf{g}.\end{eqnarray} \tag{ 116 }$$

We saw in (46) that the Hotelling observer uses a test statistic of this form, but there the template depends on the data mean vectors and covariance matrices, hence on the radiation dose. Here we consider templates that do not depend on the dose.

The template might have been designed for a certain signal s₀ and without regard to the background in which it is immersed, but if it is applied in a real clinical setting (as opposed to a simulation study or a simplified psychophysical study with synthetic data), then the inevitable random variations in signal and background determine the statistics of g and hence of t_ld(g).

These statistics can be different for images that do not contain the signal and those that do, so we define two subsets of objects C₀ and C₁, respectively, and we denote averages over data vectors in the subsets with the corresponding subscripts. Thus

$$\begin{eqnarray}&&{{\overline{{\bar{t}}}}_{0}}\equiv {{\left\langle {{\left\langle \mathbf{w}_{ld}^{t}\mathbf{g}\right\rangle}_{\mathbf{g}\mid \mathbf{f}}}\right\rangle}_{\mathbf{f}\mid {{\mathbf{C}}_{0}}}}=\mathbf{w}_{ld}^{t}{{\overline{{\mathbf{\bar{g}}}}}_{0}},~ {{\overline{{\bar{t}}}}_{1}}\equiv {{\left\langle {{\left\langle \mathbf{w}_{ld}^{t}\mathbf{g}\right\rangle}_{\mathbf{g}\mid \mathbf{f}}}\right\rangle}_{\mathbf{f}\mid {{\mathbf{C}}_{1}}}}=\mathbf{w}_{ld}^{t}{{\overline{{\mathbf{\bar{g}}}}}_{1}}.\end{eqnarray} \tag{ 117 }$$

Because t_ld(g) is a scalar, it does not have a covariance matrix, but its variance can still be decomposed into terms arising from noise and object variability. From (16), it is easy to show that

$$\begin{eqnarray}&&\text{Var}\left\{{{t}_{ld}}\left(\mathbf{g}\right)\mid {{\mathbf{C}}_{j}}\right\}=\mathbf{w}_{ld}^{t}{{\mathbf{K}}_{\mathbf{g}\mid {{\mathbf{C}}_{j}}}}{{\mathbf{w}}_{ld}}=\mathbf{w}_{ld}^{t}\mathbf{\bar{K}}_{\mathbf{g}\mid {{\mathbf{C}}_{j}}}^{\text{noise}}{{\mathbf{w}}_{ld}}+\mathbf{w}_{ld}^{t}\mathbf{K}_{\mathbf{\bar{g}}\mid {{\mathbf{C}}_{j}}}^{\text{obj}}{{\mathbf{w}}_{ld}}, \left(j=0,1\right).\end{eqnarray} \tag{ 118 }$$

The signal-to-noise ratio associated with t_ld(g) is defined by (see (26))

$$\begin{eqnarray}&&\text{SNR}_{ld}^{2}=\frac{{{\left[{{\overline{{\bar{t}}}}_{1}}-{{\overline{{\bar{t}}}}_{0}}\right]}^{2}}}{\frac{1}{2}\left[\text{Var}\left\{{{t}_{ld}}\left(\mathbf{g}\right)\mid {{\mathbf{C}}_{0}}\right\}+\text{Var}\left\{{{t}_{ld}}\left(\mathbf{g}\right)\mid {{\mathbf{C}}_{1}}\right\}\right]}.\end{eqnarray} \tag{ 119 }$$

For a weak signal, it will usually be a valid approximation to replace the average variance in the denominator here with the signal-absent variance (see section 3.3).

The template w_ld is independent of dose, and the dose dependence of the terms in (117) and (118) can be found in table 4. For ECT, we find that

$$\begin{eqnarray}&&\text{SNR}_{ld}^{2}\propto \frac{{{D}^{2}}}{a{{D}^{2}}+bD},\end{eqnarray} \tag{ 120 }$$

where a and b are constants that depend not only on the task, system operator and object and image statistics, but also on the particular measure of dose used. If D is an organ dose, D_k, the constants depend on which organ is chosen.

For TCT with integrating detectors and the task performed on raw data, we find a different dependence,

$$\begin{eqnarray}&&\text{SNR}_{ld}^{2}\propto \frac{{{D}^{2}}}{a{{D}^{2}}+bD+c},\end{eqnarray} \tag{ 121 }$$

where c arises from the electronic readout noise, introduced in (9). The discussion below that equation shows that the readout noise is negligible if ${{\bar{g}}_{m}}\left(\mathbf{f}\right)>>\sigma _{m}^{2}$. where σ_m is the standard deviation of the noise in detector m in units of the mean output for one x-ray photon, but this condition might be difficult to achieve for small detector pixels and low-dose imaging.

In TCT, the situation is more complicated when the detection task is performed on reconstructed images rather than raw projections, as it almost always will be in clinical practice. In that case w is a vector with the same dimension as $\mathbf{\hat{f}}$, and ${{t}_{ld}}\left(\mathbf{\hat{f}}\right)=\mathbf{w}_{ld}^{t}\mathbf{\hat{f}}$. Equations (117)–(119) still hold if we replace g with $\mathbf{\hat{f}}$ everywhere, but when we consult the TCT column of table 4 we find

$$\begin{eqnarray}&&\text{SNR}_{ld}^{2}\propto \frac{1}{a+{{\left\langle \frac{bD+c}{{{D}^{2}}}\right\rangle}_{\mathbf{f}\mid {{\mathbf{C}}_{0}}}}},\end{eqnarray} \tag{ 122 }$$

where a, b and c depend on the object statistics but are independent of D. The necessity for averaging the noise term over objects in this manner arises from the logarithmic nonlinearity used in CT reconstruction.

It follows from (121) and (122) that the SNR² for any linear discriminant with a dose-independent template must asymptotically approach an upper limit as dose is increased.

As we saw in section 3.3, the Hotelling observer is also a linear discriminant of the form (116), but now the observer has full knowledge of the mean and covariance matrix of the data and uses it optimally to construct the template w; as a result the template depends on the dose.

A convenient way to study the dose-dependence of the performance of the Hotelling observer is to use the Karhunen-Loève (KL) expansion of the estimated covariance matrix ${{\mathbf{\hat{K}}}_{\mathbf{g}}}$, defined in (52). In brief, KL is an expansion of a covariance matrix in terms of its eigenvectors, with its eigenvalues as coefficients.

Following Kupinski et al (2007), we rewrite (52) as

$$\begin{eqnarray}&&{{\mathbf{\hat{K}}}_{\mathbf{g}}}={{\left[\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}\right]}^{1/2}}\left\{\mathbf{I}+{{\left[\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}\right]}^{-1/2}}\mathbf{R}{{\mathbf{R}}^{t}} {{\left[\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}\right]}^{-1/2}}\right\}{{\left[\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}\right]}^{1/2}},\end{eqnarray} \tag{ 123 }$$

where I is the M × M unit matrix and R is the M × N_s matrix of noise-free sample images as defined by (49) and (50). We recognize that ${{\left[\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}\right]}^{-1/2}}$ has the same form as the prewhitening operator, introduced in (43), but it prewhitens with respect to the average noise covariance only. Because $\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}$ is often a diagonal matrix, as in (9), it can be very simple in practice to perform this kind of prewitening operation.

After prewhitening in this sense, the data vector g and the matrix R become, respectively,

$$\begin{eqnarray}&&{{\mathbf{\tilde g}}} \equiv {{\left[\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}\right]}^{-1/2}}\mathbf{g},~~~{{\mathbf{\widetilde R}}} \equiv {{\left[\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}\right]}^{-1/2}}\mathbf{R}.\end{eqnarray} \tag{ 124 }$$

Thus (123) becomes

$$\begin{eqnarray}&&{{\mathbf{\hat{K}}}_{\mathbf{g}}}={{\left[\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}\right]}^{1/2}}\left[\mathbf{I}+{{\mathbf{\widetilde R}}} {{{{\mathbf{\widetilde R}}}}^{t}}\right]{{\left[\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}\right]}^{1/2}}.\end{eqnarray} \tag{ 125 }$$

The next step is to solve the eigenvalue problem for the M × M prewhitened sample covariance matrix ${{\mathbf{\widetilde R}}} {{{{\mathbf{\widetilde R}}}}^{t}}$:

$$\begin{eqnarray}&&\mathbf{\tilde{R}}{{\mathbf{\tilde{R}}}^{t}}\boldsymbol{\phi }_{{j}}={{\mu}_{j}}\boldsymbol{\phi }_{{j}}.\end{eqnarray} \tag{ 126 }$$

If the complete set of M × 1 eigenvectors, {ϕ_j , j = 1, ..., M}, is found, it will form a complete orthonormal basis for the M-dimensional data space.

However, because a sample covariance matrix with N_s < M (where N_s is the number of samples) has rank N_s − 1, only N_s − 1 of the eigenvalues will be nonzero. We can find these nonzero eigenvalues and the corresponding eigenvectors by solving the eigenvalue problem for the much smaller N_s × N_s matrix ${{{{\mathbf{\widetilde R}}}}^{t}}{{\mathbf{\widetilde R}}}$:

$$\begin{eqnarray}&&{{\mathbf{\tilde{R}}}^{t}}\mathbf{\tilde{R}}\boldsymbol{\psi }_{{j}}={{\mu}_{j}}\boldsymbol{\psi }_{{j}}.\end{eqnarray} \tag{ 127 }$$

It can be shown that all of these eigenvalues are nonnegative, and it is convenient to order them from largest to smallest, i.e. μ₁ ⩾ μ₂ ⩾ μ₃ ⩾ .... ⩾ μ_Ns − 1 > 0.

Equation (127) can be solved by standard linear-algebra programs such as Matlab or Lapack for N_s up to 10⁴ or so, and once the N_s × 1 eigenvectors ψ_j and the corresponding eigenvalues μ_j are determined, the requisite M × 1 eigenvectors can be found from (Barrett and Myers 2004)

$$\begin{eqnarray}&&\boldsymbol{\phi }_{{j}}=\frac{1}{\sqrt{{{\mu}_{j}}}}\mathbf{\tilde{R}}\boldsymbol{\psi }_{{j}},~~~ \text{for}\text{}j < {{N}_{s}}\left(\text{i}.\text{e}.~{{\mu}_{j}}\ne 0\right).\end{eqnarray} \tag{ 128 }$$

With these eigenvectors and eigenvalues, we can express ${{\mathbf{\widetilde R}}} {{{{\mathbf{\widetilde R}}}}^{t}}$ exactly by its spectral representation,

$$\begin{eqnarray}&&\mathbf{\tilde{R}}{{\mathbf{\tilde{R}}}^{t}}=\sum\limits_{j=1}^{{{N}_{s}}-1} {{\mu}_{j}}\boldsymbol{\phi }_{{j}}\boldsymbol{\phi }_{j}^{t}.\end{eqnarray} \tag{ 129 }$$

If we choose M − N_s + 1 additional vectors that form an orthonormal set and satisfy $\mathbf{\tilde{R}}{{\mathbf{\tilde{R}}}^{t}}\boldsymbol{\phi }_{{j}}=0,{{N}_{s}}<j\leqslant M$, then we can express the M × M unit matrix as

$$\begin{eqnarray}&&\mathbf{I}=\underset{j=1}{\overset{M}{\sum}} \boldsymbol{\phi }_{{j}}\boldsymbol{\phi }_{j}^{t},.\end{eqnarray} \tag{ 130 }$$

It turns out that we will not need any particular eigenvectors for j ⩾ N_s; it will suffice to know that they exist.

The estimated covariance matrix in the KL domain is now

$$\begin{eqnarray}&&{{\mathbf{\hat{K}}}_{\mathbf{g}}}={{\left[\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}\right]}^{1/2}}\left[\sum\limits_{j=1}^{M} \left(1+{{\mu}_{j}}\right)\boldsymbol{\phi }_{{j}}\boldsymbol{\phi }_{j}^{t}\right]{{\left[\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}\right]}^{1/2}},\end{eqnarray} \tag{ 131 }$$

and by the orthonormality of the eigenfunctions, the inverse of the estimated covariance is

$$\begin{eqnarray}&&\mathbf{\hat{K}}_{\mathbf{g}}^{-1}={{\left[\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}\right]}^{-1/2}}\left[\sum\limits_{j=1}^{M} \frac{1}{1+{{\mu}_{j}}}\boldsymbol{\phi }_{{j}}\boldsymbol{\phi }_{j}^{t}\right]{{\left[\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}\right]}^{-1/2}}.\end{eqnarray} \tag{ 132 }$$

If we use $\frac{1}{1+{{\mu}_{j}}}=1-\frac{{{\mu}_{j}}}{1+{{\mu}_{j}}}$, we find

$$\begin{eqnarray}&&\mathbf{\hat{K}}_{\mathbf{g}}^{-1}={{\left[\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}\right]}^{-1}}-{{\left[\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}\right]}^{-1/2}}\left[\sum\limits_{j=1}^{{{N}_{s}}-1} \frac{{{\mu}_{j}}}{1+{{\mu}_{j}}}\boldsymbol{\phi }_{{j}}\boldsymbol{\phi }_{j}^{t}\right]{{\left[\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}\right]}^{-1/2}}.\end{eqnarray} \tag{ 133 }$$

This form of the inverse is equivalent to (53) but now in terms of KL eigenfunctions.

The estimated Hotelling FOM, $\widehat{\text{SNR}_{\text{Hot}}^{2}}\equiv \Delta {{\overline{{\mathbf{\bar{g}}}}}^{t}}\mathbf{\hat{K}}_{\mathbf{g}}^{-1}\Delta \overline{{\mathbf{\bar{g}}}}$, is given in terms of these eigenfunctions by

$$\begin{eqnarray}&&\widehat{\text{SNR}_{\text{Hot}}^{2}}={{\overline{{\mathbf{\bar{g}}}}}^{t}}{{\left[\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}\right]}^{-1}}\Delta \overline{{\mathbf{\bar{g}}}}-\Delta {{\overline{{\mathbf{\bar{g}}}}}^{t}}{{\left[\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}\right]}^{-1/2}}\left[\sum\limits_{j=1}^{{{N}_{s}}-1} \frac{{{\mu}_{j}}}{1+{{\mu}_{j}}}\boldsymbol{\phi }_{{j}}\boldsymbol{\phi }_{j}^{t}\right]{{\left[\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}\right]}^{-1/2}}\Delta \overline{{\mathbf{\bar{g}}}}.\end{eqnarray} \tag{ 134 }$$

The first term in this expression is exactly what we would have for an SKE/BKE problem, and the second term (with its minus sign) is the estimated reduction in the FOM resulting from the object randomness. Because $\frac{{{\mu}_{j}}}{1+{{\mu}_{j}}}<1$, the overall estimated SNR² cannot go negative.

From table 4, we see that $\mathbf{\bar{g}}$, $\overline{{\mathbf{\bar{g}}}}$ and hence $\Delta \overline{{\mathbf{\bar{g}}}}$ are ∝ D and that ${{\left[\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}\right]}^{-1/2}}\propto {{D}^{-1/2}}$ in ECT. It follows that ${{\mathbf{\widetilde R}}} {{{{\mathbf{\widetilde R}}}}^{t}}\propto D$. Moreover, because the eigenvectors ϕ_j always have unit norm, μ_j, ∝ D. Thus we can write μ_j = μ_j1D, where μ_j1 is the eigenvalue found by solving (127) for D = 1 in whatever units we wish (e.g. Gy or mGY) and for whatever particular dose quantity we are discussing (e.g. organ dose, ROI dose or even effective dose.)

Assembling these results, we find that the estimated FOM scales with dose as

$$\begin{eqnarray}&&\widehat{\text{SNR}_{\text{Hot}}^{2}}=aD-\sum\limits_{j=1}^{{{N}_{s}}-1} \mid {{\gamma}_{j}}{{\mid}^{2}}\frac{{{\mu}_{j1}}{{D}^{2}}}{1+{{\mu}_{j1}}D},\end{eqnarray} \tag{ 135 }$$

where a is $\Delta {{\overline{{\mathbf{\bar{g}}}}}^{t}}{{\left[\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}\right]}^{-1}}\Delta \overline{{\mathbf{\bar{g}}}}$ as computed for D = 1 and γ_j is the difference signal after prewhitening for the noise covariance and projecting it onto the jth basis vector in the KL expansion of the object covariance:

$$\begin{eqnarray}&&{{\gamma}_{j}}=\boldsymbol{\phi }_{j}^{t}{{\left[\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}\right]}^{-1/2}}\Delta \overline{{\mathbf{\bar{g}}}}.\end{eqnarray} \tag{ 136 }$$

The result in (134) holds for TCT as well as ECT, in spite of the fact that the system operator $\mathcal{H}$ is nonlinear for TCT; the exponential nonlinearity is included in the simulation code that generates noise-free sample images. From results in table 4, we can also show that (135) holds for TCT provided the excess detector noise (c in the table) can be neglected.

So far we have discussed application of the Hotelling observer to raw projection data g, but the expression in (134) still applies formally if we substitute the reconstruction $\mathbf{\hat{f}}$ for g everywhere. The practical difference is that the prewhitening operator is more complicated with reconstructions.

For linear algorithms in ECT, we know from (21) that $\mathbf{\bar{K}}_{{\mathbf{\hat{f}}}}^{\text{noise}}=\mathcal{O}\mathbf{\bar{K}}_{\mathbf{g}}^{\text{noise}}{{\mathcal{O}}^{t}}$, where the reconstruction operator $\mathcal{O}$ is an N × M matrix with elements O_nm, with index n specifying a voxel located at position r_n in the reconstructed image. With the pure Poisson noise model of (8), we find (see Barrett and Myers (2004), section 15.2.6)

$$\begin{eqnarray}&&{{\left[\mathbf{\bar{K}}_{{\mathbf{\hat{f}}}}^{\text{noise}}\right]}_{nn\prime}}=\sum\limits_{m=1}^{M} {{O}_{\text{nm}}}{{O}_{n\prime m}}{{\left[\mathcal{H}\mathbf{\bar{f}}\right]}_{m}}.\end{eqnarray} \tag{ 137 }$$

To understand this expression and its implications for prewhitening, it is useful to look at the elements of the conditional covariance matrix for a single object, ${{\left[\mathbf{K}_{\mathbf{\hat{f}}\mid \mathbf{f}}^{\text{noise}}\right]}_{nn\prime}}$, obtained from (137) just by omitting the overbars on K and f. With (2), we find

$$\begin{eqnarray}&&{{\left[\mathbf{K}_{\mathbf{\hat{f}}\mid \mathbf{f}}^{\text{noise}}\right]}_{nn\prime}}=\sum\limits_{m=1}^{M} {{O}_{\text{nm}}}{{O}_{n\prime m}}{{{\int}^{}}_{\infty}}{{\text{d}}^{3}}r{{h}_{m}}\left(\mathbf{r}\right)f\left(\mathbf{r}\right).\end{eqnarray} \tag{ 138 }$$

The conditional variance of the reconstruction at voxel n is the nn diagonal element of this covariance matrix, which can be written as

$$\begin{eqnarray}&&\text{Var}\left({{\hat{f}}_{n}}\right)={{\left[\mathbf{K}_{\mathbf{\hat{f}}\mid \mathbf{f}}^{\text{noise}}\right]}_{nn}}={{{\int}^{}}_{\infty}}{{\text{d}}^{3}}r{{\aleph}_{n}}\left(\mathbf{r}\right)f\left(\mathbf{r}\right),\end{eqnarray} \tag{ 139 }$$

where ℵ_n(r), called the noise kernel (see Barrett and Myers (2004), section 15.2.6), is defined by

$$\begin{eqnarray}&&{{\aleph}_{n}}\left(\mathbf{r}\right)\equiv \sum\limits_{m=1}^{M} {{\left[{{O}_{\text{nm}}}\right]}^{2}}{{h}_{m}}\left(\mathbf{r}\right).\end{eqnarray} \tag{ 140 }$$

In 2D filtered backprojection (FBP), ℵ_n(r) varies approximately as ∣r − r_n∣⁻¹, with the disconcerting consequence that the variance at the center of the image of a disk object increases linearly, without bound, as the diameter of the disk increases (see Barrett and Myers (2004), section 17.3.2). A variance map (image of $\text{Var}({{\hat{f}}_{n}})$ versus r_n) looks like a highly blurred version of the object f (r). Moreover, the correlations predicted by (138) have very long range, especially if a line drawn through voxels n and n' passes through the center of rotation in a tomographic imaging system; low-dose FBP images exhibit radial noise pattern extending across the whole reconstruction.

In short, the noise in tomographic reconstructions with linear algorithms is very nonlocal, with strong, long-range correlations. For the present discussion, this means that $\mathbf{\bar{K}}_{{\mathbf{\hat{f}}}}^{\text{noise}}$, though in principle full rank, is nowhere close to being diagonal, and computation of the prewhitening operator ${{\left[\mathbf{\bar{K}}_{{\mathbf{\hat{f}}}}^{\text{noise}}\right]}^{-1/2}}$ is far from trivial.

The noise covariance matrix behaves quite differently when nonlinear reconstruction algorithms such as MLEM (maximum-likelihood expectation maximization) are used. The noise is much more local, the variance map looks very much like the object, and ${{\left[\mathbf{K}_{\mathbf{\hat{f}}\mid \mathbf{f}}^{\text{noise}}\right]}_{nn\prime}}$ falls off rapidly as ∣r_n − r_n∣ increases (Barrett et al 1994, Wilson et al 1994). The average noise covariance matrix $\mathbf{\bar{K}}_{{\mathbf{\hat{f}}}}^{\text{noise}}$ is sparse, and there are efficient algorithms for prewhitening.

All of the computational difficulties associated with estimating SNR² for the Hotelling observer, and its dependence on radiation dose, disappear if we use the channelized Hotelling observer. If the number of sample images is much greater than the number of channels (N_s >> N_c), then the sample covariance matrix for the channel outputs, ${{\mathbf{\hat{K}}}_{\mathbf{v}}}$, will be invertible and well conditioned, for both raw data and for reconstructed images. There is no need to use the covariance decomposition, and there may not be much advantage to simulating noise-free images.

The estimators used for estimating radiotracer uptake in SPECT or PET are usually simple templates applied to the reconstructed images. The templates do not account for the spatial resolution of the imaging system, the background statistics, the noise correlations introduced by the reconstruction algorithm or the effects of radiation dose. They do not acknowledge the presence of null functions in the system operator or the effect of nuisance parameters. As a result, the estimates can be highly biased and suboptimal in terms of variance. Various correction methods referred to as contrast recovery or partial-volume correction have been invented, but they usually assume that the background is spatially uniform and nonrandom. For a recent review of partial-volume-correction techniques for ECT, see Erlandsson et al (2012).

The activity of a SPECT or PET tracer inside a specified ROI is a scalar defined as θ ≡ χ^†f, where χ represents an indicator function χ (r) which equals 1 inside the ROI and 0 outside. The expression χ^†f denotes a scalar product in object space as defined by the integral in (30).

A linear estimate of the uptake from a reconstructed image is $\hat{\theta}\left(\mathbf{\hat{f}}\right)\equiv {{\mathbf{w}}^{t}}\mathbf{\hat{f}}$, where w is some template in reconstruction space, perhaps just a voxelized version of χ. For simplicity we assume that the reconstruction is linear so the operator $\mathcal{O}$ is the N × M matrix O.

It is straightforward to show that the bias and variance of this estimate, conditioned on a particular object, are given, respectively, by

$$\begin{eqnarray}&&{{b}_{{\hat{\theta}}}}\left(\mathbf{f}\right)={{\mathbf{w}}^{t}}\mathbf{O}\mathcal{H}\mathbf{f}-{{\mathbf{\chi^\dagger}}^{\text{}}}\mathbf{f};~~~\text{Var}\left(\hat{\theta}\mid \mathbf{f}\right)={{\mathbf{w}}^{t}}{{\mathbf{K}}_{\mathbf{\hat{f}}\mid \mathbf{f}}}\mathbf{w}={{\mathbf{w}}^{t}}\mathbf{O}{{\mathbf{K}}_{\mathbf{g}\mid \mathbf{f}}}{{\mathbf{O}}^{t}}\mathbf{w}.\end{eqnarray} \tag{ 141 }$$

The conditional mean-square error, MSE(f), is related to conditional bias and variance by (32), and a further average over objects, followed by some algebra, yields

$$\begin{eqnarray}&&{\rm{EMSE}} = {{\mathbf{w}}^t}{\mathbf{O\bar K}}_{\mathbf{g}}^{{\rm{noise}}}{{\mathbf{O}}^t}{\mathbf{w}} + \left[ {{{\mathbf{w}}^t}{\mathbf{O}}{\cal H} - {\chi ^\dagger }} \right]{{\cal K}_{\mathbf{f}}}{\left[ {{{\mathbf{w}}^t}{\mathbf{O}}{\cal H} - {\chi ^\dagger }} \right]^\dagger } + {\left[ {{{\mathbf{w}}^t}{\mathbf{O}}{\cal H}{\mathbf{f}} - {\chi ^\dagger }{\mathbf{f}}} \right]^2}.\end{eqnarray} \tag{ 142 }$$

The first term in this equation is the average variance in the estimate arising from noise in the image; the second term is the variance arising from object randomness, and the third term is the square of the ensemble-average bias. Both the second and third terms would be zero if we could choose the template so that ${{\mathbf{w}}^{t}}\mathbf{O}\mathcal{H}={\mathbf{\chi^\dagger}}$, or equivalently ${{(\mathbf{O}\mathcal{H})}^\dagger} \mathbf{w}=\mathbf{\chi}$, but this is not possible if the indicator function has null components, where $\mathbf{O}\mathcal{H}{{\mathbf{\chi}}_{\text{null}}}=0$. In essence, the last two terms in (142) arise because we are trying to estimate a parameter that is not estimable.

With the dose dependences from table 4, we find that the first term in (142) varies linearly with any measure of dose, D, and the second and third terms go as D². Of course, the parameter being estimated—activity in an ROI—is also linearly related to dose, and it is reasonable to normalize the estimate to the total injected activity. The normalized EMSE (denoted NEMSE) then has a noise term that varies as 1/D plus two dose-independent terms that come from object randomness and from the null functions. Smaller NEMSE is better, so its reciprocal can be used as a figure of merit, and we see that

$$\begin{eqnarray}&&\frac{1}{\text{NEMSE}}\propto \frac{{{D}^{2}}}{a{{D}^{2}}+bD},\end{eqnarray} \tag{ 143 }$$

which is the same dependence as seen in (120) for a detection task with a dose-independent template.

We cannot eliminate the bias term in (142), but we can minimize it by finding a least-squares solution to ${{(\mathbf{O}\mathcal{H})}^\dagger}\mathbf{w}=\mathbf{\chi}$; that is, we seek a template whose backprojection is, as closely as possible, the indicator function that defines the parameter being estimated. From Barrett and Myers (2004), section 1.7.4, the solution is

$$\begin{eqnarray}&&{{\mathbf{w}}_{\text{LS}}}={{\left(\mathbf{O}\mathcal{H}{{\mathcal{H}^\dagger}^{\text{}}}{{\mathbf{O}}^{t}}\right)}^{+}}\mathbf{O}\mathcal{H}\mathbf{\chi},\end{eqnarray} \tag{ 144 }$$

where superscript + denotes a Moore–Penrose pseudoinverse. The operator $\mathbf{O}\mathcal{H}{{\mathcal{H}}^\dagger}{{\mathbf{O}}^{t}}$ is an N × N matrix, and in many cases the pseudoinverse becomes an ordinary matrix inverse.

As we saw in section 3.4, the Gauss–Markov estimator (GME) minimizes the first term in (143). From (70), as transcribed to apply to reconstructed images, its template satisfies

$$\begin{eqnarray}&&\mathbf{w}_{\text{GM}}^{t}={{\mathbf{\chi}}^\dagger}{{\left[\mathbf{K}_{\mathbf{\hat{f}}\mid \mathbf{f}}^{-1/2}\mathbf{O}\mathcal{H}\right]}^{+}}\mathbf{K}_{\mathbf{\hat{f}}\mid \mathbf{f}}^{-1/2}.\end{eqnarray} \tag{ 145 }$$

The dose dependence of the FOM given in (143) still holds for the LS and GM templates, though of course the constants are different.

If we wish to minimize the sum of the three terms in (142), we should use the Wiener estimator from (73). For the special case of estimating a scalar linear parameter such as uptake in an ROI from a reconstructed image, the Wiener estimator becomes (Whitaker et al 2008)

$$\begin{eqnarray}&&{{\hat{\theta}}_{W}}\left(\mathbf{\hat{f}}\right)=\bar{\theta}-{{\mathbf{\chi}}^\dagger}{{\mathcal{K}}_{\mathbf{f}}}{{\mathcal{H}}^\dagger}{{\mathbf{O}}^{t}}\mathbf{K}_{{\mathbf{\hat{f}}}}^{-1}(\mathbf{\hat{f}}-\overline{\overline{{\mathbf{\hat{f}}}}}),\end{eqnarray} \tag{ 146 }$$

where $\bar{\theta}$ is the prior mean of θ. The corresponding EMSE is

$$\begin{eqnarray}&&\text{EMSE}=\sigma _{\theta}^{2}-{{\mathbf{\chi}}^{\dagger}}{{\mathcal{K}}_{\mathbf{f}}}{{\mathcal{H}}^{\dagger}}{{\mathbf{O}}^{t}}\mathbf{K}_{\widehat{\mathbf{f}}}^{-1}\mathbf{O}\mathcal{H}\mathcal{K}_{\mathbf{f}}^{\dagger}\mathbf{\chi}=\sigma _{\theta}^{2}-{{\mathbf{\chi}}^{\dagger}}{{\mathcal{K}}_{\mathbf{f}}}{{\mathcal{H}}^{\dagger}}{{\mathbf{O}}^{t}}{{\left[\overline{\mathbf{K}}_{\widehat{\mathbf{f}}}^{\text{noise}}+\mathbf{K}_{\overline{\widehat{\mathbf{f}}}}^{\text{obj}}\right]}^{-1}}\mathbf{O}\mathcal{H}\mathcal{K}_{\mathbf{f}}^{\dagger}\mathbf{\chi},\end{eqnarray} \tag{ 147 }$$

where $\sigma _{\theta}^{2}$ is the prior variance of θ. From the ECT column of table 4, $\sigma _{\theta}^{2}\sim {{D}^{2}}$, ${{\mathcal{K}}_{\mathbf{f}}}\sim {{D}^{2}}$, $\mathbf{\bar{K}}_{{\mathbf{\hat{f}}}}^{\text{noise}}\sim D$ and $\mathbf{K}_{\overline{{\mathbf{\hat{f}}}}}^{\text{obj}}\sim {{D}^{2}}$. To get the overall dose dependence, we can approximate $\mathbf{K}_{{\mathbf{\hat{f}}}}^{-1}$ as in (133), but with eigenvalues and eigenvectors determined by use of simulated noise-free reconstructions $\overline{{\mathbf{\hat{f}}}}$ rather than simulated, noise-free g.

Now suppose we separate the object into background and signal parts as in (74), f = f_bg + f_sig(θ), and our task is to estimate the P × 1 signal parameter vector θ. For example, the components of θ might be the 3D coordinates, diameter and tracer uptake for a tumor modeled as a sphere, or they might include other parameters to incorporate departures from sphericity or spatial variations of uptake within the boundaries. Some of these components might be null functions (e.g. fine structures in the signal beyond the resolution limit of the system) and some may be nuisance parameters (e.g. precise location of a tumor when the task is to estimate the total tracer uptake).

The use of models specified by a small number of parameters usually means that all of the parameters are estimable—provided the model is correct. On the other hand, if the model is wrong (as all models are), then there is a new form of bias arising from modeling error. There is no reason to think that bias from modeling error will depend on dose in the same way as bias arising from properties of the estimator or bias stemming from null functions in the template that defines the parameter, but this point does not seem to have been studied in the imaging literature.

We will proceed as if the model used in defining f_sig(θ) is an accurate description of the real object and θ is estimable. We assume also that the set of P parameters (components of θ) is adequate to describe the signal completely, with no remaining randomness.

In section 3.4 we introduced a multivariate normal data model and some reasonable approximations under which the log-likelihood, $\ln\ \text{pr}\left(\mathbf{g}\mid \boldsymbol{\theta }\right)$, reduces to a linear function of g, albeit a nonlinear function of θ (see (76)). The corresponding P × P Fisher information matrix is given in (77).

This data model was also considered by Abbey et al (1998), who used earlier work by Barndorff-Nielsen (1983) and Fessler (1996) to develop methods for approximating the PDF of maximum-likelihood and maximum a posteriori estimates. For low noise and ML estimation, Abbey et al showed that

$$\begin{eqnarray}&&\text{pr}\left(\widehat{\boldsymbol{\theta }}\mid \boldsymbol{\theta }\right)\approx {{\left[\frac{ \text{det} \left[\mathbf{F}\left(\boldsymbol{\theta }\right)\right]}{{{\left(2\pi \right)}^{P}}}\right]}^{1/2}} \text{exp} \left[-\frac{1}{2}{{\left(\widehat{\boldsymbol{\theta }}-\boldsymbol{\theta }\right)}^{t}}\left[\mathbf{F}\left(\boldsymbol{\theta }\right)\right]\left(\widehat{\boldsymbol{\theta }}-\boldsymbol{\theta }\right)\right],\end{eqnarray} \tag{ 148 }$$

which has the same structure as the generic multivariate normal PDF (see (39)) except that the Fisher information matrix F (θ) has replaced the inverse of the covariance matrix, ${{\left[{{\mathbf{K}}_{\widehat{\boldsymbol{\theta }}\mid \boldsymbol{\theta }}}\right]}^{-1}}$, and the true parameter vector θ has replaced the mean of the estimate.

The PDF in (148) could have been obtained by citing a theorem that says that in the asymptotic (low-noise) limit, a maximum-likelihood estimate is unbiased, efficient and normally distributed, but Abbey et al derived it from a different viewpoint that facilitated its extension to higher noise where the PDF is not multivariate normal and to MAP estimates which are necessarily biased. The approximations derived by Abbey et al were in excellent agreement with Monte Carlo simulations.

The FIM that appears in (148), defined in (77), involves ${\bf{K}}_{\bf{g}}^{ - 1}$, which can be approximated by (133) if we can simulate realistic noise-free images. If all components of θ are defined to be independent of dose, then the derivatives in the FIM, $\partial {{\overline{{\bar{g}}}}_{m}}/\partial {{\theta}_{p}}$, have the same dependence on dose as ${{\overline{{\bar{g}}}}_{m}}$ itself, and the dose dependence of any component of the FIM has the same form as the detection SNR² in (135). The variance of any component of an ML estimate as a function of dose can then easily be found numerically by performing a P-dimensional inverse of the FIM, and from there we can get the dose dependence of the EMSE (35), the WEMSE (36), or any other desired measure of efficacy for the estimation task.

The detectors commonly used in SPECT and planar nuclear medicine are scintillation cameras, modern variants of the venerable Anger camera (Peterson and Furenlid 2011), but there is also considerable current interest in semiconductor detector arrays. The important characteristics of these detectors that can affect image quality, and eventually radiation dose, are detection efficiency, energy resolution, spatial resolution and overall detector area. The detector area divided by the area of one resolution cell is referred to as the space-bandwidth product.

Of these detector properties, perhaps the most important in terms of dose and image quality is the space-bandwidth product, which is basically the dimension M of the data vector g in the preceding sections. All else being equal, increasing M will always increase the task performance at any given dose, or reduce the dose required for a specified task performance for any task.

Moreover, increasing the space-bandwidth product of the detector also opens up the opportunity for new imaging configurations that improve task performance. For example, in a smulation of new pinhole imaging systems for brain SPECT, Rogulski et al (1999) demonstrate that use of a detector with a very large number of resolvable elements permits the use of a large number of pinholes, but without overlap of the images on the detector. The result is that the system can collect more photons without multiplexing or loss of spatial resolution than would be possible with a detector having smaller space-bandwidth product.

Improvements in energy resolution in SPECT detectors result immediately in improvements in task performance by reducing the number of scattered photons that contribute to the data. In addition, better energy resolution facilitates imaging studies with multiple radioisotopes or multiple emission lines from a single isotope. Improved energy resolution is the main motivation for developing semiconductor detector arrays.

One way to improve the spatial and energy resolution of either scintillation cameras or semiconductor detector arrays is by using maximum-likelihood estimation to determine the position and energy of the gamma-ray interaction event. For a review of ML estimation applied to scintillation cameras, see Barrett et al (2009), and for a discussion of real-time implementations, see Hesterman et al (2010). For more on this topic. see section 7.4.

For a review of the basic principles of PET imaging, see Muehllehner and Karp (2006), and for a tutorial on recent developments in PET detector technology, see Lewellen (2008).

Current PET systems usually use discrete arrays of scintillation detectors. Space-bandwidth product in these detectors can be interpreted simply as the number of individual scintillation crystals. In an attempt to get high absorption efficiency for 511 keV gamma rays and good spatial resolution, these crystals are usually several cm long and a few mm in cross section.

Early PET systems used a single ring of scintillation detector to image a single 2D slice of the patient at a time, but recent trends are in the direction of multiple contiguous rings for imaging a significant volume of the patient with one position of the detector array. This benefits the photon collection efficiency, and it facilitates dynamic studies which are an important application of PET.

Advances in scintillation crystals, photodetector technology and readout electronics have led to improvements in coincidence timing resolution, thereby improving the rejection of accidental coincidences and allowing some degree of localization of a positron annihilation event in three dimensions by estimating the difference in time of flight (TOF) of the two gamma rays to their respective detectors.

These technological advances should lead directly to improvements in image quality or allow a reduction in the administered activity (hence reducing the absorbed dose) at the same image quality. A few groups are now using task-based assessment methods to investigate these gains quantitatively.

El Fakhri, Surti and coworkers have used model and human observers to demonstrate improvements in lesion detection in oncological PET both by volumetric imaging and with TOF-PET (El Fakhri et al 2007, El Fakhri et al 2011, Surti et al 2011). Lartizien et al (2004) compared the impact of 2-dimensional (2D) and fully 3-dimensional (3D) acquisition modes on the performance of human observers in detecting and localizing tumors in whole-body 18F-FDG images.