Abstract
We present a modification of the Mumford-Shah functional and its cartoon limit which facilitates the incorporation of a statistical prior on the shape of the segmenting contour. By minimizing a single energy functional, we obtain a segmentation process which maximizes both the grey value homogeneity in the separated regions and the similarity of the contour with respect to a set of training shapes. We propose a closed-form, parameter-free solution for incorporating invariance with respect to similarity transformations in the variational framework. We show segmentation results on artificial and real-world images with and without prior shape information. In the cases of noise, occlusion or strongly cluttered background the shape prior significantly improves segmentation. Finally we compare our results to those obtained by a level set implementation of geodesic active contours.
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Alcouffe, R.E., Brandt, A., Dendy, Jr.,J.E., and Painter, J.W. 1981. The multi-grid method for the diffusion equation with strongly discontinuous coefficients. SIAM J.Sci.Stat.Comp., 2(4):430–454.
Blake, A. and Isard, M. 1998. Active Contours, Springer: London.
Briggs, W.L., Henson, V.E., and McCormick, S.F. 2000. A Multigrid Tutorial, 2nd edn. SIAM: Philadelphia.
Caselles, V., Kimmel, R., and Sapiro, G. 1995. Geodesic active contours. In Proc.IEEE Internat.Conf.on Comp.Vis., Boston, USA, pp. 694–699.
Chan, T. and Vese, L. 2001. A level set algorithm for minimizing the Mumford-Shah functional in image processing. In IEEE Workshop on Variational and Level Set Methods, Vancouver, CA, pp. 161–168.
Chen, Y., Thiruvenkadam, S., Tagare, H., Huang, F., Wilson, D., and Geiser, E. 2001. On the incorporation of shape priors into geometric active contours. In IEEE Workshop on Variational and Level Set Methods, Vancouver, CA, pp. 145–152.
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.
Cremers, D., Kohlberger, T., and Schnörr, C. 2002. Nonlinear shape statistics in Mumford-Shah based segmentation. In Proc.of the Europ.Conf.on Comp.Vis., Copenhagen, A. Heyden et al. (Eds.), vol. 2351 of LNCS. Springer: Berlin, pp. 93–108.
Cremers, D. and Schnärr, C. 2002. Motion competition: Variational integration of motion segmentation and shape regularization. In Pattern Recognition, L. van Gool (Ed.), Zürich, LNCS, Springer: Berlin.
Cremers, D., Schnörr, C., and Weickert, J. 2001. Diffusion snakes: Combining statistical shape knowledge and image information in a variational framework. In IEEE First Workshop on Variational and Level Set Methods, Vancouver, pp. 137–144.
Cremers, D., Schnörr, C., Weickert, J., and Schellewald, C. 2000. Diffusion snakes using statistical shape knowledge. In Alge-braic Frames for the Perception-Action Cycle, G. Sommer and Y.Y. Zeevi (Eds.), vol. 1888 of LNCS, pp. 164–174. Springer: Berlin.
Dendy, J.E. 1982. Black box multigrid. J.Comp.Phys., 48:366–386.
Dryden, I.L. and Mardia, K.V. 1998. Statistical Shape Analysis. Wiley: Chichester.
Farin, G. 1997. Curves and Surfaces for Computer–Aided Geometric Design. Academic Press: San Diego, CA.
Goodall, C. 1991. Procrustes methods in the statistical analysis of shape. J.Roy.Statist.Soc., Ser.B., 53(2):285–339.
Grenander, U., Chow, Y., and Keenan, D.M. 1991. Hands: A Pattern Theoretic Study of Biological Shapes. Springer: New York.
Kass, M., Witkin, A., and Terzopoulos, D. 1988. Snakes: Active contour models. Int.J.of Comp.Vis., 1(4):321–331.
Kervrann, C. 1995. Modèles statistiques pour la segmentation et le suivi de structures déformables bidimensionnelles dans une séquence d'images. Ph.D. Thesis, Université de Rennes I, France.
Kervrann, C. and Heitz, F. 1999. Statistical deformable model-based segmentation of image motion. IEEE Trans.on Image Processing, 8:583–588.
Kichenassamy, S., Kumar, A., Olver, P.J., Tannenbaum, A., and Yezzi, A.J. 1995. Gradient flows and geometric active contour models. In Proc.IEEE Internat.Conf.on Comp.Vis., Boston, USA, pp. 810–815.
Leventon, M.E., Grimson, W.E.L., and Faugeras, O. 2000. Statistical shape influence in geodesic active contours. In Proc.Conf.Computer Vis.and Pattern Recog., Hilton Head Island, SC, vol. 1, pp. 316–323. June 13–15.
Mantegazza, C. 1993. Su Alcune Definizioni Deboli di Curvatura per Insiemi Non Orientati. Ph.D. Thesis, Dept. of Mathematics, SNS Pisa, Italy.
Moghaddam, B. and Pentland, A. 1995. Probabilistic visual learning for object detection. In Proc.IEEE Internat.Conf.on Comp.Vis., pp. 786–793.
Morel, J.-M. and Solimini, S. 1988. Segmentation of images by variational methods: A constructive approach. Revista Matematica de la Universidad Complutense de Madrid, 1(1–3):169–182.
Mumford, D. and Shah, J. 1989. Optimal approximations by piecewise smooth functions and associated variational problems. Comm.Pure Appl.Math., 42:577–685.
Paragios, N. and Deriche, R. 2000. Coupled geodesic active regions for image segmentation: a level set approach. In ECCV, D. Vernon (Ed.), vol. 1843 of LNCS, Springer: Berlin, pp. 224–240.
Roweis., S. EM algorithms for PCA and SPCA. 1998. In Advances in Neural Information Processing Systems 10, M. Jordan, M. Kearns, and S. Solla (Eds.), MIT Press: Cambridge, MA, pp. 626–632.
Terzopoulos, D. 1983. Multilevel computational processes for visual surface reconstruction. Comp.Vis., Graph., and Imag.Proc., 24:52–96.
Tipping, M.E. and Bishop, C.M. 1997. Probabilistic principal component analysis. Neural Computing Research Group, Aston University, UK, Technical Report Woe-19.
Tischhäuser, F. 2001. Development of a multigrid algorithm for diffusion snakes. Diploma thesis Department of Mathematics and Computer Science, University of Mannheim, Mannheim, Germany (in German).
Wang, Y. and Staib, L.H. 1998. Boundary finding with correspondence using statistical shape models. In Proc.Conf.Computer Vis.and Pattern Recog., Santa Barbara, CA, pp. 338–345.
Weickert, J. 2001. Applications of nonlinear diffusion filtering in image processing and computer vision. Acta Mathematica Universitatis Comenianae, LXX(1):33–50.
Werman, M. and Weinshall, D. 1995. Similarity and affine invariant distances between 2d point sets. IEEE Trans.on Patt.Anal.and Mach.Intell., 17(8):810–814.
Wesseling, P. 1992. An Introduction to Multigrid Methods. John Wiley: Chichester.
Yezzi, A., Soatto, S., Tsai, A., and Willsky, A. 2002. The Mumford-Shah functional: From segmentation to stereo. Mathematics and Multimedia.
de Zeeuw, P.M. 1990. Matrix-dependent prolongations and restrictions in a blackbox multigrid solver. J.Comp.Appl.Math., 33:1–27.
Zhu, S.C. and Mumford, D. 1997. Prior learning and Gibbs reaction–diffusion. IEEE Trans.on Patt.Anal.and Mach.Intell., 19(11):1236–1250.
Zhu, S.C. and Yuille, A. 1996. Region competition: Unifying snakes, region growing, and Bayes/MDL for multiband image segmentation. IEEE Trans.on Patt.Anal.and Mach.Intell., 18(9):884–900.
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Cremers, D., Tischhäuser, F., Weickert, J. et al. Diffusion Snakes: Introducing Statistical Shape Knowledge into the Mumford-Shah Functional. International Journal of Computer Vision 50, 295–313 (2002). https://doi.org/10.1023/A:1020826424915
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DOI: https://doi.org/10.1023/A:1020826424915