Globally Optimal Joint Image Segmentation and Shape Matching Based on Wasserstein Modes

Agarwal, S., Awan, A., Roth, D.: Learning to detect objects in images via a sparse, part-based representation. IEEE Trans. Pattern Anal. Mach. Intell. 26(11), 1475–1490 (2004)

Agueh, M., Carlier, G.: Barycenters in the wasserstein space. SIAM J. Math. Anal. 43(2), 904–924 (2011)

Ambrosio, L., Gigli, N.: A user’s guide to optimal transport. In: Modelling and Optimisation of Flows on Networks. Lect. Not. Math., vol. 2062, pp. 1–155. Springer, Heidelberg (2013)

Beg, M.F., Miller, M.I., Trouvé, A., Younes, L.: Computing large deformation metric mappings via geodesic flows of diffeomorphisms. Int. J. Comput. Vis. 61(2), 139–157 (2005)

Berkels, B., Fletcher, T., Heeren, B., Rumpf, M., Wirth, B.: Discrete geodesic regression in shape space. In: Energy Minimization Methods in Computer Vision and Pattern Recognition (EMMCVPR 2013), pp. 108–122 (2013)

Boykov, Y., Kolmogorov, V.: An experimental comparison of min-cut/max-flow algorithms for energy minimization in computer vision. IEEE Trans. Pattern. Anal. Mach. Intell. 26(9), 1124–1137 (2004)

Bronstein, A.M., Bronstein, M.M., Kimmel, R.: Efficient computation of isometry-invariant distances between surfaces. SIAM J. Sci. Comput. 28, 1812–1836 (2006)

Chan, T.F., Esedoglu, S., Nikolova, M.: Algorithms for finding global minimizers of image segmentation and denoising models. SIAM J. Appl. Math. 66(5), 1632–1648 (2006)

Charon, N., Trouvé, A.: Functional currents: a new mathematical tool to model and analyse functional shapes. J. Math. Imaging Vis. 48(3), 413–431 (2014)

Charpiat, G., Faugeras, O., Keriven, R.: Approximations of shape metrics and application to shape warping and empirical shape statistics. Found. Comput. Math. 5(1), 1–58 (2005)

Cohen, S.D., Guibas, L.J.: The earth mover’s distance under transformation sets. In: International Conference on Computer Vision (ICCV 1999), pp. 1076–1083 (1999)

Cremers, D., Kohlberger, T., Schnörr, C.: Shape statistics in kernel space for variational image segmentation. Pattern Recognit. 36(9), 1929–1943 (2003)

Cuturi, M., Doucet, A.: Fast computation of wasserstein barycenters. In: International Conference on Machine Learning (2014)

Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7(3), 1005–1028 (2008)

Glaunes, J., Trouve, A., Younes, L.: Diffeomorphic matching of distributions: a new approach for unlabelled point-sets and sub-manifolds matching. In Computer Vision and Pattern Recognition (CVPR 2004), vol. 2, pp. 712–718 (2004)

Haker, S., Zhu, L., Tannenbaum, A., Angenent, S.: Optimal mass transport for registration and warping. Int. J. Comput. Vis. 60, 225–240 (2004)

Heeren, B., Rumpf, M., Wardetzky, M., Wirth, B.: Time-discrete geodesics in the space of shells. Comput. Gr. Forum 31(5), 1755–1764 (2012)

Klodt, M., Cremers, D.: A convex framework for image segmentation with moment constraints. In: International Conference on Computer Vision (ICCV 2011), pp. 2236–2243 (2011)

Kriegl, A., Michor, P.W.: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, vol. 53. American Mathematical Society, Providence (1997)

Kuhn, H.W.: The Hungarian method for the assignment problem. Naval Res. Logist. 2, 83–97 (1955)

Kumar, M.P., Torr, P.H.S., Zisserman, A.: OBJ CUT. In Computer Vision and Pattern Recognition (CVPR 2005), vol. 1, pp. 18–25 (2005)

Lellmann, J., Schnörr, C.: Continuous multiclass labeling approaches and algorithms. SIAM J. Imaging Sci. 4(4), 1049–1096 (2011)

Lempitsky, V., Blake, A., Rother, C.: Image segmentation by branch-and-mincut. In: European Conference on Computer Vision (ECCV 2008), pp. 15–29 (2008)

Lott, J.: Some geometric calculations on Wasserstein space. Commun. Math. Phys. 277, 423–437 (2008)

Mémoli, F.: Gromov-Wasserstein distances and the metric approach to object matching. Found. Comput. Math. 11, 417–487 (2011)

Michor, P.W., Mumford, D.: Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc. 8(1), 1–48 (2006)

Mio, W., Srivastava, A., Joshi, S.: On shape of plane elastic curves. Int. J. Comput. Vis. 73(3), 307–324 (2007)

Pele, O., Werman, W.: Fast and robust Earth Mover’s Distances. In: International Conference on Computer Vision (ICCV 2009) (2009)

Pock, T., Chambolle, A., Cremers, D., Bischof, H.: A convex relaxation approach for computing minimal partitions. In: Computer Vision and Pattern Recognition (CVPR 2009), pp. 810–817 (2009)

Schmitzer, B., Schnörr, C.: Weakly convex coupling continuous cuts and shape priors. In: Scale Space and Variational Methods (SSVM 2011), pp. 423–434 (2012)

Schmitzer, B., Schnörr, C.: Contour manifolds and optimal transport. http://arxiv.org/abs/1309.2240 (2013). preprint

Schmitzer, B., Schnörr, C.: Modelling convex shape priors and matching based on the Gromov-Wasserstein distance. J. Math. Imaging Vis. 46(1), 143–159 (2013)

Schmitzer, B., Schnörr, C.: Object segmentation by shape matching with Wasserstein modes. In: Energy Minimization Methods in Computer Vision and Pattern Recognition (EMMCVPR 2013), pp. 123–136 (2013)

Sundaramoorthi, G., Mennucci, A., Soatto, S., Yezzi, A.: A new geometric metric in the space of curves, and applications to tracking deforming objects by prediction and filtering. SIAM J. Imaging Sci. 4(1), 109–145 (2011)

Villani, C.: Optimal Transport: Old and New, vol. 338. Grundlehren der mathematischen Wissenschaften. Springer, Berlin (2009)

Wang, W., Slepčev, D., Basu, S., Ozolek, J.A., Rohde, G.K.: A linear optimal transportation framework for quantifying and visualizing variations in sets of images. Int. J. Comput. Vis. 101, 254–269 (2012)

Yangel, B., Vetrov, D.: Learning a model for shape-constrained image segmentation from weakly labeled data. In: Energy Minimization Methods in Computer Vision and Pattern Recognition (EMMCVPR 2013), pp. 137–150 (2013)

Younes, L., Michor, P.W., Shah, J., Mumford, D.: A metric on shape space with explicit geodesics. Rend. Lincei Mat. Appl. 9, 25–57 (2008)

Younes, L.: Shapes and Diffeomorphisms. Applied Mathematical Sciences, vol. 171. Springer (2010)