An analysis of scatter characteristics in x-ray CT spectral correction

In x-ray CT, the detected x-ray intensity I_t after passing through an object, in total, is the summation of primary after object attenuation, I_p , and scatter, I_s , i.e.

$$\begin{eqnarray}&&{I}_{t}={I}_{p}+{I}_{s}={I}_{0}\int S(E){e}^{-\mu (E)L}{\rm{d}}E+{I}_{s},\end{eqnarray} \tag{ 1 }$$

where, I₀ is the initial x-ray intensity without object in the beam; S(E) is a normalized effective spectrum of a polychromatic CT system (∫S(E)dE = 1); and μ(E) and L are the linear attenuation coefficients of the object and its path-length, respectively. Here, for simplicity, a single-material object is assumed. Consequently, the total projection with scatter contamination can be written as:

$$\begin{eqnarray}&&{p}_{t}=-\mathrm{ln}\left(\displaystyle \frac{{I}_{p}+{I}_{s}}{{I}_{0}}\right)=-\mathrm{ln}\left(\displaystyle \frac{\int S(E){e}^{-\mu (E)L}{\rm{d}}E}{\int S(E){\rm{d}}E}\right)-\mathrm{ln}(1+\mathrm{SPR}).\end{eqnarray} \tag{ 2 }$$

Here, SPR = I_s /I_p denotes the SPR. According to equation (2), a polychromatic projection free from scatter (SPR = 0) can be defined by,

$$\begin{eqnarray}&&p=-\mathrm{ln}\left(\displaystyle \frac{\int S(E){e}^{-{\mu }^{{\prime} }(E)\cdot m}{\rm{d}}E}{\int S(E){\rm{d}}E}\right),\end{eqnarray} \tag{ 3 }$$

with ${\mu }^{{\prime} }(E)=\mu (E)/\bar{\mu }$ being the normalized attenuation coefficients, and $m=\bar{\mu }L$ being the corresponding monochromatic projection whose effective attenuation coefficient is set at $\bar{\mu }$.

Spectral correction for a polychromatic CT system, in essence, is to find a linearization mapping function, f, such that

$$\begin{eqnarray}&&m=f(p),\end{eqnarray} \tag{ 4 }$$

for each and every CT detector pixel in the scan field of view. The mapping, f, is usually called the beam hardening correction (BHC) function. For a single-material object, equation (3) gives the inverse function of BHC function, i.e. p = f⁻¹(m). By simulating or measuring polychromatic projections at a range of object thicknesses, one can build a linearization mapping function from polychromatic projections to their corresponding monochromatic ones.

However, with scatter uncorrected, the outcome of the linearization mapping is to applying BHC function, f, to the total projection, p_t , rather than the primary projection, p, causing errors in the corrected projection and CT image after reconstruction.

Referring to equation (2), by using the Taylor expansion, the linearization mapping of total projection p_t can be approximated as,

$$\begin{eqnarray}&&f({p}_{t})=f(p-{\rm{\Delta }}p)=f(p)-{f}^{{\prime} }(p){\rm{\Delta }}p+\displaystyle \frac{1}{2}{f}^{{\prime\prime} }(p){\left({\rm{\Delta }}p\right)}^{2}\cdots ,\end{eqnarray} \tag{ 5 }$$

with ${\rm{\Delta }}p=\mathrm{ln}(1+\mathrm{SPR})$. With equation (3), one can easily derive the derivative of f(p) for arbitrary orders. Here, we explicitly give out the first three orders as follows (Gao et al 2006), with the first derivative being

$$\begin{eqnarray}&&{f}^{{\prime} }(p)=\displaystyle \frac{\int {S}^{{\prime} }(E){\rm{d}}E}{\int {S}^{{\prime} }(E){\mu }^{{\prime} }(E){\rm{d}}E},\end{eqnarray} \tag{ 6 }$$

where, ${S}^{{\prime} }(E)=S(E){e}^{-{\mu }^{{\prime} }(E)\cdot f(p)};$ the second derivative being

$$\begin{eqnarray}&&{f}^{{\prime\prime} }(p)=\left(\displaystyle \frac{\int {S}^{{\prime} }(E){[{\mu }^{{\prime} }(E)]}^{2}{\rm{d}}E\times \int {S}^{{\prime} }(E){\rm{d}}E}{{\left[\int {S}^{{\prime} }(E){\mu }^{{\prime} }(E){\rm{d}}E\right]}^{2}}-1\right){f}^{{\prime} }(p);\end{eqnarray} \tag{ 7 }$$

and the third derivative being

$$\begin{eqnarray}&&\begin{array}{l}{f}^{\prime\prime\prime }(p)=\left(3\displaystyle \frac{\displaystyle \int {S}^{{\prime} }(E){[{\mu }^{{\prime} }(E)]}^{2}{\rm{d}}E\times \displaystyle \int {S}^{{\prime} }(E){\rm{d}}E}{{\left[\displaystyle \int {S}^{{\prime} }(E){\mu }^{{\prime} }(E){\rm{d}}E\right]}^{2}}-1\right){f}^{{\prime\prime} }(p)\\ \ \ +\,\left(\displaystyle \frac{\displaystyle \int {S}^{{\prime} }(E){[{\mu }^{{\prime} }(E)]}^{2}{\rm{d}}E}{\displaystyle \int {S}^{{\prime} }(E){\rm{d}}E}-\displaystyle \frac{\displaystyle \int {S}^{{\prime} }(E){[{\mu }^{{\prime} }(E)]}^{3}{\rm{d}}E}{\displaystyle \int {S}^{{\prime} }(E){\mu }^{{\prime} }(E){\rm{d}}E}\right){[{f}^{{\prime} }(p)]}^{3}.\end{array}\end{eqnarray} \tag{ 8 }$$

Due to the subtractions within equations (7) and (8), the second and third order derivatives of BHC function are both relatively small. For a better understanding, we plot the BHC curves of water at 80 kVp and 120 kVp spectra and their corresponding derivatives, respectively. To illustrate the influence of the effective water attenuation coefficient, $\bar{\mu }$, on the BHC curves, we select $\bar{\mu }$ to be the attenuation coefficient of water at 70 keV (0.1929 cm⁻¹), and the spectrum weighted attenuation coefficients [∫S(E)μ(E)dE, i.e. 0.2479 cm⁻¹ at 80 kVp, and 0.2228 cm⁻¹ at 120 kVp]. As shown in figure 1, the second and third derivative values are both relatively small for all cases.

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In addition, in practice most SPR (or the residual SPR after first-pass correction) for medical CT is less than 200%, making ${\rm{\Delta }}p=\mathrm{ln}(1+{\rm{SPR}})\lt 1$. Therefore, spectral correction for total projections with scatter contamination can be reasonably approximated by,

$$\begin{eqnarray}\begin{array}{rcl}f({p}_{t}) & \approx & f(p)-{f}^{{\prime} }(p)\mathrm{ln}(1+\mathrm{SPR})\\ & \approx & f(p)-{f}^{{\prime} }({p}_{t})\mathrm{ln}(1+\mathrm{SPR}).\end{array}\end{eqnarray} \tag{ 9 }$$

It is seen that the scatter caused discrepancy between primary projection p and total projection p_t is $\mathrm{ln}(1+\mathrm{SPR})$, which after spectral correction will be scaled approximately by the first derivative of the mapping function, ${f}^{{\prime} }(p)$. Given that the first derivative changes relatively slowly, ${f}^{{\prime} }(p)$ can be replaced by ${f}^{{\prime} }({p}_{t})$, which is more convenient to obtain in practice. So far, we have obtained the scatter property in the projection domain.

By going back to the attenuation domain, one can also figure out how the scatter distribution will change. According to equation (9), after spectral correction the total x-ray intensity can be written as,

$$\begin{eqnarray}\begin{array}{rcl}{\hat{I}}_{t} & = & {I}_{0}{e}^{-f({p}_{t})}\\ & \approx & {I}_{0}{e}^{-f\left(p\right)}\cdot {\left(1+\mathrm{SPR}\right)}^{{f}^{{\prime} }({p}_{t})}.\end{array}\end{eqnarray} \tag{ 10 }$$

When taking ${f}^{{\prime} }({p}_{t})\approx 1$ into equation (10) or SPR is a small number, ${\hat{I}}_{t}$ can be further simplified as,

$$\begin{eqnarray}\begin{array}{rcl}{\hat{I}}_{t} & \approx & {I}_{0}{e}^{-f\left(p\right)}+{I}_{0}{e}^{-f\left(p\right)}\cdot {f}^{{\prime} }({p}_{t})\mathrm{SPR}\\ & = & {I}_{0}{e}^{-f\left(p\right)}+{I}_{s}{f}^{{\prime} }({p}_{t}){e}^{-f(p)+p}\\ & \approx & {I}_{0}{e}^{-f\left(p\right)}+{I}_{s}{f}^{{\prime} }({p}_{t}){e}^{-f({p}_{t})+{p}_{t}}.\end{array}\end{eqnarray} \tag{ 11 }$$

Here, we use the facts that SPR = I_s /I_p and I_p = I₀ e^−p . It is also assumed that the projection difference before and after spectral correction, p − f(p), can be approximated by, p_t − f(p_t ), which is again more convenient to obtain in practice. Therefore, the new scatter distribution after spectral correction can be estimated by

$$\begin{eqnarray}&&{\hat{I}}_{s}\approx {I}_{s}{f}^{{\prime} }({p}_{t}){e}^{[{p}_{t}-f({p}_{t})]}.\end{eqnarray} \tag{ 12 }$$

It is seen that the scatter distribution after spectral correction is scaled approximately by a factor of ${f}^{{\prime} }({p}_{t}){e}^{{p}_{t}-f({p}_{t})}$, which is the product of the first derivative of the linearization mapping function and a natural exponential of the projection difference before and after the mapping. Therefore, we have obtained the scatter property in the attenuation domain.

We note that in case the SPR is somehow available, after applying the spectral correction before scatter being removed, an SPRC can be done to recover f(p) from f(p_t ), in the following two ways.

• (1)  
  SPRC-I: $f(p)\approx f({p}_{t})+\mathrm{ln}(1+\mathrm{SPR})$, which totally ignores the impact of spectral correction.
• (2)  
  SPRC-II: $f(p)\approx f({p}_{t})+{f}^{{\prime} }({p}_{t})\mathrm{ln}(1+\mathrm{SPR})$, which uses ${f}^{{\prime} }({p}_{t})$ to adjust the SPR change due to spectral correction.

According to our proposed scatter property, SPRC-II will be more accurate. In figure 2, we further illustrate the process of recovering f(p) from f(p_t ) using the proposed SPRC methods, where the case with ${f}^{{\prime} }({p}_{t})\gt 1$ is for a greater effective water attenuation coefficient ($\bar{\mu }$) selected for the BHC curve, and the case with ${f}^{{\prime} }({p}_{t})\lt 1$ is for a smaller $\bar{\mu }$. As a result, such an SPRC can be used to validate our proposed new scatter property indirectly.

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In this section, we want to apply the proposed new scatter property to CT imaging using a SMFFS as an example to demonstrate its potential benefits in practical applications. SMFFS is a promising low-cost approach to accurately solving the x-ray scattering problem and physically enabling multi-energy imaging in a unified framework, with no significant sparsity or misalignment in data sampling (Gao et al 2020). However, due to the x-ray spectral change from spectral modulation, correcting for scatter using SMFFS needs careful consideration and its related algorithm highly depends on the design and implementation of SMFFS.

For SMFFS, when a focal spot FS_k whose position is labeled as k fires, the measured x-ray signal at a detector element (i, j) can be expressed as:

$$\begin{eqnarray}&&{I}_{{t}_{k}}(i,j)={I}_{{p}_{k}}(i,j)+{I}_{{s}_{k}}(i,j),\end{eqnarray} \tag{ 13 }$$

where, ${I}_{{p}_{k}}(i,j)$ and ${I}_{{s}_{k}}(i,j)$ denote the intensities of the primary beam and the scatter, respectively, at source position k. And ${I}_{{p}_{k}}(i,j)$, can be further written as:

$$\begin{eqnarray}&&{I}_{{p}_{k}}(i,j)={I}_{{m}_{k}}\int {S}_{k}(E,i,j){e}^{-{u}_{o}(E){L}_{k}(i,j)}{\rm{d}}E.\end{eqnarray} \tag{ 14 }$$

Here, ${I}_{{m}_{k}}$ is the intensity of x-ray after beam passing through the modulator; S_k (E, i, j) is the normalized spectrum with the modulator in the beam; u_o(E) and L_k (i, j) are the linear attenuation coefficients and the path-length of projection ray of the scanned object.

As investigated in the literature, scatter is mostly dominated by low-frequency components (Zhu et al 2006, Gao et al 2010). In SMFFS, it is further assumed that there is a strong correlation between scatter distributions at varying focal spot positions when the flying distance is relatively small (Gao et al 2020). Based on our experience, it is also safe to assume that the low- and high-energy projection rays are aligned quite well (i.e. L₁(i, j) = L₂(i, j) =L(i, j)), and their scatter share the same distribution (i.e. ${I}_{{s}_{1}}(i,j)={I}_{{s}_{2}}(i,j)={I}_{s}(i,j)$). For simplicity, we omit the expression (i, j) from the variables, e.g. ${I}_{{t}_{k}}$ represents ${I}_{{t}_{k}}(i,j)$.

For the two-point flying focal spot scenario in SMFFS, using the projection ray alignment and the scatter similarity, equations (13) and (14) can be written as,

$$\begin{eqnarray}&&{I}_{{t}_{1}}={I}_{{m}_{1}}\int {S}_{1}(E){e}^{-{u}_{o}(E)L}{\rm{d}}E+{I}_{s},\end{eqnarray} \tag{ 15a }$$

$$\begin{eqnarray}&&{I}_{{t}_{2}}={I}_{{m}_{2}}\int {S}_{2}(E){e}^{-{u}_{o}(E)L}{\rm{d}}E+{I}_{s}.\end{eqnarray} \tag{ 15b }$$

To estimate the scatter distributions for SMFFS, we need to the solve the unknown variable I_s in equation (15). A scatter estimation method for SMFFS based on our proposed scatter property is given in the following.

For the data acquired from SMFFS, after spectral correction one can write the total x-ray intensity as:

$$\begin{eqnarray}&&{\hat{I}}_{{t}_{1}}={I}_{{m}_{1}}{e}^{-{f}_{1}({p}_{{t}_{1}})}={I}_{{m}_{1}}{e}^{-{f}_{1}({p}_{1})}+{\hat{I}}_{{s}_{1}},\end{eqnarray} \tag{ 16a }$$

$$\begin{eqnarray}&&{\hat{I}}_{{t}_{2}}={I}_{{m}_{2}}{e}^{-{f}_{2}({p}_{{t}_{2}})}={I}_{{m}_{2}}{e}^{-{f}_{2}({p}_{2})}+{\hat{I}}_{{s}_{2}},\end{eqnarray} \tag{ 16b }$$

where, ${p}_{{t}_{k}}=-\mathrm{ln}({I}_{{t}_{k}}/{I}_{{m}_{k}})$ is the total projection with scatter; ${p}_{k}=-\mathrm{ln}({I}_{{p}_{k}}/{I}_{{m}_{k}})$ is the projection without scatter, f_k is the spectral correction linearization mapping function and $f{{\prime} }_{k}$ is the first order derivative of f_k .

Due to the assumed projection alignment from different focal spot positions, we can have f₁(p₁) ≈ f₂(p₂). If one simply ignores the impact of spectral correction on the distribution of scatter (i.e. ${\hat{I}}_{s}={I}_{s}$), then I_s can be easily computed using equation (16) as

$$\begin{eqnarray}&&{I}_{s}\approx \frac{{I}_{{m}_{1}}{I}_{{m}_{2}}\left[{e}^{-{f}_{1}({p}_{{t}_{1}})}-{e}^{-{f}_{2}({p}_{{t}_{2}})}\right]}{{I}_{{m}_{2}}-{I}_{{m}_{1}}}.\end{eqnarray} \tag{ 17 }$$

This oversimplified scatter estimate is usually biased and less accurate in practice, which is denoted as SE_d.

According to our proposed scatter property, however, we know that the scatter distribution after spectral correction will be scaled by the product of the first derivative of the linearization mapping function and a natural exponential of the projection difference before and after the mapping. So, by substituting ${\hat{I}}_{{s}_{k}}$ in equation (12) into equation (16), equation (17) should be revised as

$$\begin{eqnarray}&&{I}_{s}\approx \displaystyle \frac{{I}_{{m}_{1}}{I}_{{m}_{2}}\left[{e}^{-{f}_{1}({p}_{{t}_{1}})}-{e}^{-{f}_{2}({p}_{{t}_{2}}})\right]}{{I}_{{m}_{2}}{f}_{1}^{{\prime} }\left({p}_{{t}_{1}}\right){e}^{[{p}_{{t}_{1}}-{f}_{1}({p}_{{t}_{1}})]}-{I}_{{m}_{1}}{f}_{2}^{{\prime} }\left({p}_{{t}_{2}}\right){e}^{[{p}_{{t}_{2}}-{f}_{2}({p}_{{t}_{2}})]}}.\end{eqnarray} \tag{ 18 }$$

Hence, an analytic form of an accurate scatter estimate in this paper has been obtained with no iteration needed, which is denoted as SE_sps.

For the SMFFS and other similar scenarios, it is worth noting that due to the spatially-varying modulator in the beam, the effective x-ray spectrum at each detector pixel could be quite different from one to another. In other words, the spectral correction mapping functions, f, across the entire scan field of view vary spatially too (Gao et al 2019). As a result, after spectral correction the new scatter distribution will be significantly distorted (according to our proposed scatter property) and the assumption of a low-frequency distribution may no longer be appropriate.

There exist other ways to estimate the scatter for SMFFS using relations in equation (15). For comparison with the proposed method SE_sps , we also give another scatter estimation method for SMFFS based on residual spectral linearization (denoted as SE_rsl). Using the subtraction between equations (15a ) and (15b ), the scatter can be eliminated,

$$\begin{eqnarray}&&{I}_{{t}_{1}}-{I}_{{t}_{2}}=\int {S}_{{res}}(E){e}^{-{\mu }_{o}(E)L}{\rm{d}}E,\end{eqnarray} \tag{ 19 }$$

here, ${S}_{{res}}(E)={I}_{{m}_{1}}{S}_{1}(E)-{I}_{{m}_{2}}{S}_{2}(E)$, which is the residual spectrum between the modulated spectra from flying focal spot positions 1 and 2. Based on the single-material assumption, the path-length L in equation (19) can be computed by the conventional linearization mapping based on the residual spectrum S_res (E). Once path-length L is obtained, one can easily estimate the scatter using equations (15a ) or (15b ),

$$\begin{eqnarray}\begin{array}{rcl}{I}_{s} & = & {I}_{{t}_{1}}-{I}_{{m}_{1}}\displaystyle \int {S}_{1}(E){e}^{-{\mu }_{o}(E)L}{\rm{d}}E\\ & = & {I}_{{t}_{2}}-{I}_{{m}_{2}}\displaystyle \int {S}_{2}(E){e}^{-{\mu }_{o}(E)L}{\rm{d}}E.\end{array}\end{eqnarray} \tag{ 20 }$$

Equation (20) is the scatter estimation formula for the SE_rsl method.

For the SPRC, evaluations were performed using a Catphan phantom and an anthropomorphic chest phantom. As illustrated in figure 3(a), we took a fan-beam (FB) scan and a cone-beam (CB) scan of the phantoms on a tabletop CBCT system with some major imaging parameters listed in table 1. The x-ray tube was operated at 80 kVp and 120 kVp, and the effective spectra were obtained by an expectation method (Duan et al 2011) as shown in figures 1(a) and (b). The collimator used in the fan-beam scans limited the coverage of the x-ray beam on the detector along the longitudinal (z-) direction to about 10 mm; for the cone-beam scans the collimator was opened up to allow full x-ray illumination on the entire detector of about 300 mm along the z-direction. Along the lateral (x-) direction, for both fan-beam and cone-beam scans, the x-ray beam illuminated the entire detector of about 400 mm. With an assumption that the fan-beam scan is scatter free, scatter distributions can be obtained through subtracting the fan-beam transmission image from the cone-beam transmission image, and then SPR was computed by the ratio of the measured scatter to the fan-beam transmission image. Finally, projections for reconstructing CT images were recovered using the SPRC method as described in section 2.2.1.

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For the simulation study of estimated scatter error, the low- and high-energy x-rays passing through the same path of a object were simulated. As shown in figure 3(b), the low-energy x-ray was generated by the simulated 120 kVp spectrum while the high-energy was obtained after filtering the 120 kVp spectrum with molybdenum (thickness denoted by T_Mo) or copper (thickness denoted by T_Cu). And the object was simulated by a combination of water and bone, with the length being L_water and L_bone for water and bone, respectively. We define the water fraction as,

$$\begin{eqnarray}&&\beta =\displaystyle \frac{{\overline{\mu }}_{{water}}{L}_{{water}}}{{\overline{\mu }}_{{water}}{L}_{{water}}+{\overline{\mu }}_{{bone}}{L}_{{bone}}},\end{eqnarray} \tag{ 21 }$$

here, ${\overline{\mu }}_{{water}}$ and ${\overline{\mu }}_{{bone}}$ are the effective linear attenuation coefficients of water and bone for the spectrum at 120 kVp, which are 0.2228 cm⁻¹ and 0.7649 cm⁻¹, respectively. A larger β indicates that the relative contribution from bone is less in the corresponding object path-length. The primaries after the object attenuation were denoted as ${I}_{{p}_{1}}$ and ${I}_{{p}_{2}}$ for the low- and high-energy projection rays. With simulating a single ray at a time, the scatter is calculated by the product of the primary after the object attenuation and the SPR value with the nominal SPR value being set to 0.5. Considering that the assumption of scatter similarity for low- and high-energy projection rays may not hold strictly (i.e. ${I}_{{s}_{1}}\ne {I}_{{s}_{2}}$), the simulated scatter consisted of nominal part (I_s ) and variation part (α I_s , with α being the scatter deviation). The detected x-ray total intensity was given by summing up the scatter and primary after the object attenuation. As a result, the detected x-ray total intensity of low- and high-energy projection rays are ${I}_{{t}_{1}}={I}_{{p}_{1}}+\left(1+\tfrac{1}{2}\alpha \right){I}_{s}$ and ${I}_{{t}_{2}}={I}_{{p}_{2}}+\left(1-\tfrac{1}{2}\alpha \right){I}_{s}$, respectively. The parameters T_Mo, T_Cu, β and α can be adjusted in order to simulate a range of imaging cases, with parameters selection given in table 2. Finally, a comparison analysis between the methods SE_rsl and SE_sps was carried out on the error of the estimated scatter of the simulated data.

For the scatter correction from CT scans using the SMFFS, data acquisition and processing are illustrated in figure 3(c). To conduct CT scans with SMFFS on the tabletop CBCT system, a stationary copper filter in a 2D checkboard pattern with 0.42 mm thickness and 0.889 mm spacing was used as the spectral modulator and the x-ray tube was moved by 1.24 mm in the X direction to mimic the flying focal spot. An adaptive spectral estimation and compensation algorithm in Gao et al (2019) was adopted to capture the overall detected spectrum of each and every detector pixels in the presence of the spectral modulator across the scan field of view at each of the focal spot positions, respectively. Cone-beam scans with the modulator (CB_M) scans were taken with and without the Catphan phantom, with the x-ray source operated at 120 kVp. Scatter distributions are estimated using equations (17), (20) and (18), respectively. And projections after scatter correction and spectral correction are obtained and ready for reconstruction.