Abstract
In volumetric medical imaging the boundaries of structures are frequently blurred due to insufficient resolution. This artefact is particularly serious in structures such as articular joints, where different cartilage surfaces appear to be linked at the contact regions. Traditional image segmentation techniques fail to separate such erroneously linked structures, and a sensible approach has been the introduction of prior-knowledge to the segmentation process. Although several 3D prior-knowledge based techniques that could successfully segment these structures have been published, most of them are pixel-labelling schemes that generate pixellated images with serious geometric distortions. The Simplex Mesh Diffusion Snakes segmentation technique presented here is an extension of the two dimensional Diffusion Snakes, but without any restriction on the number of dimensions of the data set. This technique integrates a Simplex Mesh, a region-based deformable model and Statistical Shape Knowledge into a single energy functional, so that it takes into account both the image information available directly from the data set, and the shape statistics obtained from a training process. The resulting segmentations converge correctly to well defined boundaries and provide a feasible location for those removed boundaries. The algorithm has been evaluated using 2D and 3D data sets obtained with Magnetic Resonance Imaging (MRI) and has proved to be robust to most of the MRI artefacts, providing continuous and smooth curves or surfaces with sub-pixel resolution. Additionally, this novel technique opens a wide range of opportunities for segmentation and tracking time-dependent 3D structures or data sets with more than three dimensions, due to its non-restrictive mathematical formulation.
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References
Bajcsy, R., & Kovačič, S. (1989). Multiresolution elastic matching. Computer Vision, Graphics, and Image Processing, 46(1), 1–21.
Blake, A., & Isard, M. (2000). Active contours. London: Springer.
Blake, A., & Zisserman, A. (1987). Visual reconstruction. Cambridge: MIT Press.
Caselles, V., Catté, F., Coll, T., & Dibos, F. (1993). A geometric model for Active Contours in image processing. Numerische Mathematik, 66(1), 1–31.
Caselles, V., Kimmel, R., & Sapiro, G. (1995). Geodesic Active Contours. In Proceedings of the IEEE international conference on computer vision (pp. 694–699) 1995.
Caselles, V., Kimmel, R., & Sapiro, G. (1997). Geodesic Active Contours. International Journal of Computer Vision, 22(1), 61–79.
Chakraborty, A., Staib, L. H., & Duncan, J. S. (1996). Deformable boundary finding in medical images by integrating gradient and region information. IEEE Transactions on Medical Imaging, 15(6), 859–870.
Chan, T. F., & Vese, L. A. (2001). Active Contours without edges. IEEE Transactions on Image Processing, 10(2), 266–277.
Chen, Y., Thiruvenkadam, S., Tagare, H. D., Huang, F., Wilson, D., & Geiser, E. A. (2001). On the incorporation of shape priors into geometric active contours. In Proceedings of the IEEE workshop on variational and level set methods in computer vision, Vancouver (pp. 145–152) 2001.
Cohen, L. D. (1991). Note on Active Contour models and balloons. CVGIP: Image Understanding, 53(2), 211–218.
Cohen, L. D., & Cohen, I. (1993). Finite-element methods for Active Contour models and balloons for 2-D and 3-D images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 15(11), 1131–1147.
Cootes, T. F., Taylor, C. J., Cooper, D., & Graham, J. (1992). Training Models of shape from sets of examples. In Proceedings of the third British machine vision conference (pp. 9–18) 1992.
Cootes, T. F., Cooper, D., Taylor, C. J., & Graham, J. (1995). Active shape models—their training and application. Computer Vision and Image Understanding, 61(1), 38–59.
Cremers, D. (2002). Statistical Shape Knowledge in variational image segmentation. PhD Thesis, Universität Mannheim, Mannheim.
Cremers, D., Kohlberger, T., & Schnörr, C. (2003). Shape statistics in kernel space for variational image segmentation. Pattern Recognition, 36(9), 1929–1943.
Cremers, D., & Soatto, S. (2003). A pseudo distance for shape priors in level set segmentation. In O. Faugeras & N. Paragios (Eds.), Proceedings of the IEEE international workshop on variational, geometric and level set methods in computer vision, Nice (pp. 169–176) 2003.
Cremers, D., Tischhäuser, F., Weickert, J., & Schnör, C. (2002). Diffusion Snakes: introducing Statistical Shape Knowledge into the Mumford-Shah functional. International Journal of Computer Vision, 50(3), 295–313.
Dam, E., Fletcher, P.T., Pizer, S. M., Tracton, G., & Rosenman, J. (2004). Prostate shape modeling based on principal geodesic analysis bootstrapping. In Lecture notes in computer science: Vol. 3217. Medical image computing and computer-assisted intervention MICCAI’04, Saint-Malo (pp. 1008–1016). Berlin: Springer.
Davies, R. H. (2002). Learning shape: optimal models for analysing natural variability. PhD Thesis. University of Manchester, Manchester.
Delingette, H. (2004). General object reconstruction based on simplex mesh (Research Report). INRIA, Sophia Antipolis.
Delingette, H., & Montagnat, J. (2000). New algorithms for controlling active contours shape and topology. In Lecture notes in computer science: Vol. 1843. Proceedings of the 6th European conference on computer vision (pp. 381–395). Berlin: Springer.
Dryden, I. L., & Mardia, K. V. (2002). Statistical shape analysis. Chichester: Wiley.
Kass, M., Witkin, A., & Terzopoulos, D. (1988). Snakes: Active Contour models. International Journal of Computer Vision, 1(4), 321–331.
Kichenassamy, S., Kumar, A., Olver, P. J., Tannenbaum, A., & Yezzi, A. J. (1995). Gradient flows and geometric active contour models. In Proceedings of the IEEE international conference on computer vision (pp. 810–815) 1995.
Klein, P. (1999). Calculus with vectors and matrices. Stockholm: Stockholms Universitet.
Lachaud, J. O., & Montanvert, A. (1999). Deformable meshes with automated topology changes for coarse-to-fine three-dimensional surface extraction. Medical Image Analysis, 3(2), 187–207.
Leitner, F., & Cinquin, P. (1991). Complex topology 3D objects segmentation. In Proceedings of the SPIE conference on advances in intelligent robotics systems (Vol. 1609) 1991.
Leroy, B., Herlin, I., & Cohen, L. D. (1996). Multi-resolution algorithms for Active Contour models. In Proceedings of the 12th international conference on analysis and optimization of systems, images, wavelets and PDE’S, Rocquencourt, France, 1996.
Leventon, M. E., Grimson, W. E. L., & Faugeras, O. (2000). Statistical shape influence in Geodesic Active Contours. In Proceedings on the conference on computer vision and pattern recognition, Hilton Head Islands (Vol. 1, pp. 316–323) 2000.
Malladi, R., Sethian, J. A., & Vemuri, B. C. (1995). Shape modeling with front propagation: a level set approach. IEEE Transactions on Pattern Analysis and Machine Intelligence, 17(2), 158–175.
McInerney, T., & Terzopoulos, D. (1996). Deformable Models in medical image analysis: a survey. Medical Image Analysis, 1(2), 91–108.
McInerney, T., & Terzopoulos, D. (1999). Topology adaptive deformable surfaces for medical image volume segmentation. IEEE Transactions on Medical Imaging, 18(10), 840–850.
Mumford, D., & Shah, J. (1989). Optimal approximations by piecewise smooth functions and associated variational problems. Communications on Pure and Applied Mathematics, 42(5), 577–685.
Osher, S., & Sethain, J. A. (1988). Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulation. Journal of Computational Physics, 79(1), 12–49.
Papadopoulo, T., & Lourakis, M. I. A. (2004). Estimating the Jacobian of the singular value decomposition: theory and applications (Research Report). INRIA, Sophia Antipolis.
Paragios, N., & Deriche, R. (1999). Geodesic active regions for supervised texture segmentation. In Proceedings of IEEE international conference on computer vision (Vol. 2, pp. 926–932) 1999.
Paragios, N., & Deriche, R. (2002). Geodesic active regions and level set methods for supervised texture segmentation. International Journal of Computer Vision, 46(3), 223–247.
Paragios, N., & Rousson, M. (2002). Shape prior for level set representations. In Proceedings of the European conference on computer vision (Vol. 2, pp. 78–92) 2002.
Pentland, A., & Sclaroff, S. (1991). Closed-form solutions for physically based shape modeling and recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 13(7), 715–729.
Pitiot, A., Delingette, H., Ayache, N., & Thompson, P. M. (2003). Expert-knowledge-guided segmentation system for brain MRI. In R. E. Ellis & T. M. Peters (Eds.), Lecture notes in computer science: Vol. 2879. Medical image computing and computer-assisted intervention MICCAI’03, Montreal (pp. 644–652). Berlin: Springer.
Pizer, S. M., Fletcher, P. T., Joshi, S., Thall, A., Chen, J. Z., Fridman, Y. (2003). Deformable M-reps for 3D medical image segmentation. International Journal of Computer Vision, 55(2), 85–106.
Romdhani, S., Gong, S., & Psarrou, A. (1999). A multi-view nonlinear active shape model using kernel PCA. In Proceedings of the British machine vision conference, Nottingham (pp. 483–492) 1999.
Ronfard, R. (1994). Region-based strategies for Active Contour models. International Journal of Computer Vision, 13(2), 229–251.
Rousson, M., & Paragios, N. (2002). Shape priors for level set representations. In Lecture notes in computer science: Vol. 2351. Proceedings of the European conference in computer vision (pp. 78–93). Berlin: Springer.
Samson, C., Blanc-Féraud, L., Aubert, G., & Zerubia, J. (1999). A level set model for image classification. In Proceedings of the international conference of scale-space theories in computer vision (pp. 306–317) 1999.
Singh, A., Goldgof, D., & Terzopoulos, D. (Eds.) (1998). Deformable Models in medical image analysis. Los Alamitos: IEEE Press.
Staib, L.H., & Duncan, J. (1989). Parametrically Deformable Contour models. In Proceedings of IEEE conference on computer vision and pattern recognition (pp. 98–103) 1989.
Tejos, C., Hall, L. D., & Cárdenas-Blanco, A. (2004). Segmentation of articular cartilage using active contours and prior knowledge. In Proceedings of the annual international conference of the IEEE engineering in medicine and biology society, San Francisco (pp. 1648–1651).
Tejos, C., Cárdenas-Blanco, A., & Hall, L. D. (2005). Active Contours and Statistical Shape Knowledge: an automatic solution for segmentation of articular cartilage from MR images which have a partial loss of boundaries. In Proceedings 13th international conference of international society of magnetic resonance in medicine, Miami, 2004.
Tsai, A., Yezzi, A., Wells, W., Tempany, C., Tucker, D., Fan, A., et al. (2001). Model-based curve evolution technique for image segmentation. In Proceedings of the IEEE conference on computer vision and pattern recognition (pp. 463–468) 2001.
Tsai, A., Yezzi, A., Wells, W., Tempany, C., Tucker, D., Fan, A., (2003). A shape-based approach to the segmentation of medical imagery using level sets. IEEE Transactions on Medical Imaging, 22(2), 137–154.
Twining, C. J., & Taylor, C. J. (2001). Kernel principal component analysis and the construction of non-linear active shape models. In Proceedings of the British machine vision conference, Manchester (pp. 23–32) 2001.
Xu, C., & Prince, J. L. (1998). Snakes, shapes, and gradient vector flow. IEEE Transactions on Image Processing, 7(3), 359–369.
Yezzi, A., Tsai, A., & Willsky, A. (1999). A statistical approach to snakes for bimodal and trimodal imagery. In Proceedings of the IEEE international conference on computer vision (Vol. 2, pp. 898–903) 1999.
Yuille, A. L., Hallinan, P. W., & Cohen, D. S. (1992). Feature extraction from faces using deformable templates. International Journal of Computer Vision, 8(2), 99–111.
Zhu, S. C., & Yuille, A. (1996). Region competition: unifying snakes, region growing, and Bayes/MDL for multi-band image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 18(9), 884–900.
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Tejos, C., Irarrazaval, P. & Cárdenas-Blanco, A. Simplex Mesh Diffusion Snakes: Integrating 2D and 3D Deformable Models and Statistical Shape Knowledge in a Variational Framework. Int J Comput Vis 85, 19–34 (2009). https://doi.org/10.1007/s11263-009-0241-1
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DOI: https://doi.org/10.1007/s11263-009-0241-1