We propose and analyze a nonparametric region-based active contour model for segmenting cluttered scenes. The proposed model is unsupervised and assumes pixel intensity is independently identically distributed. Our proposed energy functional consists of a geometric regularization term that penalizes the length of the partition boundaries and a region-based image term that uses histograms of pixel intensity to distinguish different regions. More specifically, the region data encourages segmentation so that local histograms within each region are approximately homogeneous. An advantage of using local histograms in the data term is that histogram differentiation is not required to solve the energy minimization problem. We use Wasserstein distance with exponent 1 to determine the dissimilarity between two histograms. The Wasserstein distance is a metric and is able to faithfully measure the distance between two histograms, compared to many pointwise distances. Moreover, it is insensitive to oscillations, and therefore our model is robust to noise. A fast global minimization method based on (Chan et al. in SIAM J. Appl. Math. 66(5):1632–1648, 2006; Bresson et al. in J. Math. Imaging Vis. 28(2):151–167, 2007) is employed to solve the proposed model. The advantages of using this method are two-fold. First, the computational time is less than that of the method by gradient descent of the associated Euler-Lagrange equation (Chan et al. in Proc. of SSVM, pp. 697–708, 2007). Second, it is able to find a global minimizer. Finally, we propose a variant of our model that is able to properly segment a cluttered scene with local illumination changes.
Ambrosio, L., & Tortorelli, V. M. (1990). Approximation of functionals depending on jumps by elliptic functionals via Gamma convergence. Communications on Pure and Applied Mathematics, 43, 999–1036.
Aubert, G., Barlaud, M., Faugeras, O., & Jehan-Besson, S. (2005). Image segmentation using active contours: calculus of variations or shape gradients? SIAM Journal on Applied Mathematics, 1(2), 2128–2145.
Aujol, J. F., Gilboa, G., Chan, T., & Osher, S. (2006). Structure-texture image decomposition—modeling, algorithms, and parameter selection. International Journal of Computer Vision, 67(1).
Bresson, X., & Chan, T. (2007). Fast minimization of the vectorial total variation norm and applications to color image processing. UCLA CAM report 07-25.
Bresson, X., Esedoglu, S., Vandergheynst, P., Thiran, J. P., & Osher, S. (2007). Fast global minimization of the active contour/snake model. Journal of Mathematical Imaging and Vision, 28(2), 151–167.
Caselles, V., Kimmel, R., & Sapiro, G. (1997). Geodesic active contours. International Journal of Computer Vision, 22(1), 61–79.
Chambolle, A. (2004). An algorithm for total variation minimization and applications. Journal of Mathematical Imaging and Vision, 20(1–2), 89–97.
Chan, T. F., & Vese, L. A. (2001). Active contours without edges. IEEE Transactions on Image Processing, 10(2), 266–277.
Chan, T., Esedoglu, S., & Nikolova, M. (2006). Algorithms for finding global minimizers of image segmentation and denoising models. SIAM Journal on Applied Mathematics, 66(5), 1632–1648.
Chan, T., Esedoglu, S., & Ni, K. (2007). Histogram based segmentation using Wasserstein distances. In Proceedings of SSVM (pp. 697–708).
Chartrand, R., Vixie, K., Wohlber, B., & Bollt, E. (2005). A gradient descent solution to the Monge-Kantorovich problem. Preprint: LA-UR-04-6305.
Cohen, L. (1991). On active contour models and balloons. Computer Vision, Graphics, and Image Processing, 53, 211–218.
Georgiou, T., Michailovich, O., Rathi, Y., Malcolm, J., & Tannenbaum, A. (2007). Distribution metrics and image segmentation. Linear Algebra and its Applications, 405, 663–672.
Haker, S., Zhu, L., & Tannnenbaum, A. (2004). Optimal mass transport for registration and warping. International Journal of Computer Vision, 60(3), 225–240.
Herbulot, A., Jehan-Besson, S., Barlaud, M., & Aubert, G. (2004). Shape gradient for image segmentation using information theory. In ICASSP (Vol. 3, pp. 21–24).
Herbulot, A., Jehan-Besson, S., Duffner, S., Barlaud, M., & Aubert, G. (2006). Segmentation of vectorial image features using shape gradients and information measures. Journal of Mathematical Imaging and Vision, 25(3), 365–386.
Kantorovich, L. V. (1942). On the translocation of masses. Doklady Akademii Nauk SSSR, 37, 199–201.
Kass, M., Witkin, A., & Terzopoulos, D. (1991). Snakes: active contours model. International Journal of Computer Vision, 1, 1167–1186.
Kichenesamy, S., Kumar, A., Olver, P., Tannenbaum, A., & Yezzi, A. (1996). Conformal curvature flows: from phase transitions to active vision. Archive for Rational Mechanics and Analysis, 134, 275–301.
Kim, J., Fisher, J. W., Yezzi, A., Cetin, M., & Willsky, A. S. (2005). A nonparametric statistical method for image segmentation using information theory and curve evolution. IEEE Transactions on Image Processing, 14, 1486–1502.
Michailovich, O., Rathi, Y., & Tannenbaum, A. (2007). Image segmentation using active contours driven by the Bhattacharya gradient flow. IEEE Transactions on Image Processing, 16(11), 2787–2801.
Mory, B., & Ardon, R. (2007). Fuzzy region competition: a convex two-phase segmentation framework. In Proceedings of SSVM (pp. 214–226).
Mumford, D., & Shah, J. (1989). Optimal approximation by piecewise smooth functions and associated variational problems. Communications on Pure and Applied Mathematics, 42, 577–685.
Osher, S., & Fedkiw, R. (2002). Applied mathematical sciences: Vol. 153. Level set methods and dynamic implicit surfaces. New York: Springer.
Osher, S., & Sethian, J. A. (1988). Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulation. Journal of Computational Physics, 79, 12–49.
Paragios, N., & Deriche, R. (2002). Geodesic active regions: a new paradigm to deal with frame partition problems in computer vision. Journal of Visual Communication and Image Representation, 13(1–2), 249–268.
Parzen, E. (1962). On estimation of a probability density function and mode. Annals of Mathematical Statistics, 33, 1065–1076.
Rachev, S., & Rüschendorf, L. (1998). Mass transportation problems. Vol. I: Theory, Vol. II: Applications. Probability and its applications. New York: Springer.
Rubner, Y., Tomasi, C., & Guibas, L. J. (1998). A metric for distributions with applications to image databases. In: IEEE international conference on computer vision (pp. 59–66).
Sapiro, G., & Caselles, V. (1997). Histogram modification via differential equations. Journal of Differential Equations, 135(2), 238–268.
Tu, Z., & Zhu, S. (2002). Image segmentation by data-driven Markov chain Monte Carlo. IEEE Transactions on Pattern Analysis and Machine Intelligence, 24(5).
Vese, L. A., & Chan, T. F. (2002). A multiphase level set framework for image segmentation using the Mumford and Shah model. International Journal of Computer Vision, 50(3), 271–293.
Villani, C. (2003). Graduate studies in mathematics: Vol. 58. Topics in optimal transportation. Providence: American Mathematical Society.
Yezzi, A. Jr., Tsai, A., & Willsky, A. (1999). A statistical approach to snakes for bimodal and trimodal imagery. In International conference on computer vision (pp. 898–903).
Zhu, S. C., & Yuille, A. (1996). Region competition: unifying snakes, region growing, and Bayes/MDL for multiband image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 18(9), 884–900.
Zhu, W., Jiang, T., & Li, X. (2005). Local region based medical image segmentation using J-divergence measures. In Proceedings of EMBC.
This research is supported by ONR grant N00014-09-1-0105 and NSF grant DMS-0610079.
About this article
Cite this article
Ni, K., Bresson, X., Chan, T. et al. Local Histogram Based Segmentation Using the Wasserstein Distance. Int J Comput Vis 84, 97–111 (2009). https://doi.org/10.1007/s11263-009-0234-0
- Image segmentation
- Wasserstein distance
- Image processing
- Computer vision