In this paper, we make two contributions to the field of level set based image segmentation. Firstly, we propose shape dissimilarity measures on the space of level set functions which are analytically invariant under the action of certain transformation groups. The invariance is obtained by an intrinsic registration of the evolving level set function. In contrast to existing approaches to invariance in the level set framework, this closed-form solution removes the need to iteratively optimize explicit pose parameters. The resulting shape gradient is more accurate in that it takes into account the effect of boundary variation on the object’s pose.
Secondly, based on these invariant shape dissimilarity measures, we propose a statistical shape prior which allows to accurately encode multiple fairly distinct training shapes. This prior constitutes an extension of kernel density estimators to the level set domain. In contrast to the commonly employed Gaussian distribution, such nonparametric density estimators are suited to model aribtrary distributions.
We demonstrate the advantages of this multi-modal shape prior applied to the segmentation and tracking of a partially occluded walking person in a video sequence, and on the segmentation of the left ventricle in cardiac ultrasound images. We give quantitative results on segmentation accuracy and on the dependency of segmentation results on the number of training shapes.
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Basri, R., Costa, L., Geiger, D., and Jacobs D. 1998. Determining the similarity of deformable shapes. Vision Research, 38:2365–2385.
Bresson, X., Vandergheynst, P., and Thirau, J.P. 2003. A priori information in image segmentation: Energy functional based on shape statistical model and image information. In Proc. IEEE Int. Conf. Image Processing, pp. 425–428.
Brox, T. and Weickert, J. 2004. A TV flow based local scale measure for texture discrimination. In (eds.), T.Pajdla and V.Hlavac European Conf. on Computer Vision, volume 3022 of Lect. Not. Comp. Sci., Prague, Springer.
Caselles, V., Kimmel, R., and Sapiro, G. 1995. Geodesic active contours. In Proc. IEEE Intl. Conf. on Comp. Vis., pages 694–699, Boston, USA pp. 694–699..
Chan, T. and Zhu, W. 2003. Level set based shape prior segmentation. Technical Report 03-66, Computational Applied Mathematics, UCLA, Los Angeles.
Chan, T.F. and Vese, L.A. 2001. Active contours without edges. IEEE Trans. Image Processing, 10(2):266–277.
Charpiat, G., Faugeras, O., and Keriven, R. 2005. Approximations of shape metrics and application to shape warping and empirical shape statistics. Journal of Foundations Of Computational Mathematics, 5(1):1–58.
Chen, Y., Tagare, H., Thiruvenkadam, S., Huang, F., Wilson, D., Gopinath, K.S., Briggs, R.W., and Geiser, E. 2002. Using shape priors in geometric active contours in a variational framework. Int.J.ofComputer Vision, 50(3):315–328.
Chow, Y.S., Geman, S., and Wu, L.D. 1983. Consistent cross-validated density estimation. Annals of Statistics, 11:25–38.
Cootes, T.F., Taylor, C.J., Cooper, D.M., and Graham, J. 1995. Active shape models–their training and application. Comp.Vision Image Underst., 61(1):38–59.
Cootes T.F. and Taylor C.J. 1999. A mixture model for representing shape variation. Image and Vision Computing, 17(8):567–574.
Cremers, D. 2006. Dynamical statistical shape priors for level set based tracking. IEEE Trans. on Patt. Anal. and Mach. Intell.. To appear.
Cremers, D., Kohlberger, T., and Schnörr, C. 2003. Shape statistics in kernel space for variational image segmentation. Pattern Recognition, 36(9):1929–1943.
Cremers, D., Osher, S., and Soatto, S. 2004. Kernel density estimation and intrinsic alignment for knowledge-driven segmentation: Teaching level sets to walk. In Pattern Recogn., volume 3175 of Lect. Not. Comp. Sci., pp. 36–44. Springer, C.E. Rasmussen (ed.).
Cremers, D. and Soatto, S. 2003. A pseudo-distance for shape priors in level set segmentation. In, IEEE 2nd Int. Workshop on Variational, Geometric and Level Set Methods, N.Paragios (eds.), Nice, pp. 169–176.
Cremers, D. and Soatto, S. May 2005. Motion Competition: A variational framework for piecewise parametric motion segmentation. Int.J.ofComputer Vision, 62(3):249–265.
Cremers, D., Sochen, N., and Schnörr, C. 2003. Towards recognition-based variational segmentation using shape priors and dynamic labeling. In, Int. Conf. on Scale Space Theories in Computer Vision, volume 2695 of Lect. Not. Comp. Sci., L.Griffith, editor, pp. 388–400, Isle of Skye, Springer.
Cremers, D., Sochen, N., and Schnörr, C. 2006. A multiphase dynamic labeling model for variational recognition-driven image segmentation. Int.J.ofComputer Vision, 66(1):67–81.
Cremers, D., Tischhäuser, F., Weickert, J., and Schnörr, C. 2002. Diffusion Snakes: Introducing statistical shape knowledge into the Mumford—Shah functional. Int.J.ofComputer Vision, 50(3):295–313.
Deheuvels, P. 1977. Estimation non paramétrique de la densité par histogrammes généralisés. Revue de Statistique Appliquée, 25:5–42.
Dervieux, A. and Thomasset, F. 1979. A finite element method for the simulation of Raleigh-Taylor instability. Springer Lect. Notes in Math., 771:145–158.
Devroye, L. and Györfi, L. 1985. Nonparametric Density Estimation. The L1 View. John Wiley, New York.
Dryden, I.L. and Mardia, K.V. 1998. Statistical Shape Analysis. Wiley, Chichester.
Duin, R.P.W. 1976. On the choice of smoothing parameters for Parzen estimators of probability density functions. IEEE Trans. on Computers, 25:1175–1179.
Fréchet, M. 1961. Les courbes aléatoires. Bull. Inst. Internat. Stat., 38:499–504.
Gdalyahu, Y. and Weinshall, D. 1999. Flexible syntactic matching of curves and its application to automatic hierarchical classication of silhouettes. IEEE Trans. on Patt. Anal. and Mach. Intell., 21(12):1312–1328.
Grenander, U. 1976. Lectures in Pattern Theory. Springer, Berlin.
Grenander, U., Chow, Y., and Keenan, D.M. 1991. Hands: A Pattern Theoretic Study of Biological Shapes. Springer, New York.
Heiler, M. and Schnörr, C. 2003. Natural image statistics for natural image segmentation. In IEEE Int. Conf. on Computer Vision, pp. 1259–1266.
Kendall, D.G. 1977. The diffusion of shape. Advances in Applied Probability, 9:428–430.
Kichenassamy, S., Kumar, A., Olver, P.J., Tannenbaum, A., and Yezzi, A.J. 1995. Gradient flows and geometric active contour models. In IEEE Int. Conf. on Computer Vision, pp. 810–815.
Klassen, E., Srivastava, A., Mio, W., and Joshi, S.H. 2004. Analysis of planar shapes using geodesic paths on shape spaces. IEEE Trans. on Patt. Anal. and Mach. Intell., 26(3):372–383.
Leventon, M., Grimson, W., and Faugeras, O. 2000. Statistical shape influence in geodesic active contours. In CVPR, Hilton Head Island, SC.1:316–323.
Malladi, R., Sethian, J.A., and Vemuri, B.C. 1995. Shape modeling with front propagation: A level set approach. IEEE Trans. on Patt. Anal. and Mach. Intell., 17(2):158–175.
Matheron, G. 1975. Random Sets and Integral Geometry. Wiley & Sons.
Mio, W., Srivastava, A., and Liu, X. 2004. Learning and Bayesian shape extraction for object recognition. In European Conf. on Computer Vision, volume 3024 of Lect. Not. Comp. Sci., Prague, Springer, pp. 62–73.
Moelich, M. and Chan, T. 2003. Tracking objects with the Chan-Vese algorithm. Technical Report 03-14, Computational Applied Mathematics, UCLA, Los Angeles.
Mumford, D., and Shah, J. 1989. Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math., 42:577–685.
Osher, S.J. and Fedkiw, R.P. 2002. Level Set Methods and Dynamic Implicit Surfaces. Springer, New York.
Osher, S.J. and Paragios, N. 2003. Geometric Level Set Methods in Imaging, Vision and Graphics. Springer, Telos.
Osher, S.J. and Sethian, J.A. 1988. Fronts propagation with curvature dependent speed: Algorithms based on Hamilton–Jacobi formulations. J. of Comp. Phys., 79:12–49.
Paragios, N. and Deriche, R. 2002. Geodesic active regions and level set methods for supervised texture segmentation. Int.J.ofComputer Vision, 46(3):223–247.
Parzen, E. On the estimation of a probability density function and the mode. Annals of Mathematical Statistics, 33:1065–1076.
Pons, J.-P., Hermosillo, G., Keriven, R., and Faugeras, O. 2003. How to deal with point correspondences and tangential velocities in the level set framework. In IEEE Int. Conf. on Computer Vision, pages 894–899.
Rasmussen, C.-E. and Williams, C.K.I. 2006. Gaussian Processes for Machine Learning, MIT Press.
Rathi, Y., Vaswani, N., Tannenbaum, A., and Yezzi, A. 2005. Particle filtering for geometric active contours and application to tracking deforming objects. In IEEE Int. Conf. on Computer Vision and Pattern Recognition, volume2, pp.2–9.
Riklin-Raviv, T., Kiryati, N. and Sochen, N. 2004. Unlevel sets: Geometry and prior-based segmentation. In T.Pajdla and V.Hlavac, editors, European Conf. on Computer Vision, volume 3024 of Lect. Not. Comp. Sci., pp.50–61, Prague. Springer.
Rosenblatt, F. 1956. Remarks on some nonparametric estimates of a density function. Annals of Mathematical Statistics, 27:832–837.
Rousson, M., Brox, T., and Deriche, R. 2003. Active unsupervised texture segmentation on a diffusion based feature space. In Proc.IEEE Conf.on Comp.Vision Patt.Recog., pp. 699–704, Madison, WI.
Rousson, M. and Cremers, D. 2005. Efficient kernel density estimation of shape and intensity priors for level set segmentation. In Intl.Conf.on Medical Image Computing and Comp.Ass.Intervention (MICCAI), 1:757–764.
Rousson, M. and Paragios, N. 2002. Shape priors for level set representations. In A. Heyden etal., editors, Europ. Conf. on Comp. Vis., volume 2351 of Lect. Not. Comp. Sci., pages 78–92. Springer.
Rousson, M., Paragios, N., and Deriche, R. 2004. Implicit active shape models for 3d segmentation in MRI imaging. In Intl.Conf.on Medical Image Computing and Comp.Ass.Intervention (MICCAI), volume 2217 of Lect. Not. Comp. Sci., Springer, pp. 209–216
Silverman, B.W. 1978. Choosing the window width when estimating a density. Biometrika, 65:1–11.
Silverman, B.W. 1992. Density estimation for statistics and data analysis. Chapman and Hall, London.
Sussman, M., Smereka P., and Osher S.J. 1994. A level set approach for computing solutions to incompressible twophase flow. J. of Comp. Phys., 94:146–159.
Trouvé, A. 1998. Diffeomorphisms, groups and pattern matching in image analysis. Int.J.ofComputer Vision, 28(3):213–21.
Tsai, A., Yezzi, A., Wells, W., Tempany, C., Tucker, D., Fan, A., Grimson, E., and Willsky A. 2001. Model–based curve evolution technique for image segmentation. In Comp.Vision Patt.Recog., Kauai, Hawaii, pp. 463–468.
Tsai, A., Yezzi, A.J., and Willsky, A.S. 2001. Curve evolution implementation of the Mumford-Shah functional for image segmentation, denoising, interpolation, and magnification. IEEE Trans. on Image Processing, 10(8):1169–1186.
Wagner, T.J. 1975. Nonparametric estimates of probability densities. IEEE Trans. on Inform. Theory, 21:438–440.
Yezzi, A. and Soatto, S. 2003. Deformotion: Deforming motions and shape averages. Int.J.ofComputer Vision, 53(2):153–167.
Younes, L. 1998. Computable elastic distances between shapes. SIAM J. Appl. Math., 58(2):565–586.
Zhao, H.-K., Chan, T., Merriman, B. and Osher, S. 1996. A variational level set approach to multiphase motion. J. of Comp. Phys., 127:179–195.
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Cremers, D., Osher, S.J. & Soatto, S. Kernel Density Estimation and Intrinsic Alignment for Shape Priors in Level Set Segmentation. Int J Comput Vision 69, 335–351 (2006). https://doi.org/10.1007/s11263-006-7533-5