A Joint Segmentation/Registration Model Based on a Nonlocal Characterization of Weighted Total Variation and Nonlocal Shape Descriptors

Segmentation and registration are cornerstone steps of many imaging situations: while segmentation aims to identify relevant constituents of an image for visualization or quantitative analysis, registration consists of mapping salient features of an image onto the corresponding ones in another. Instead of treating these tasks linearly one after another, so without correlating them, we propose a unified variational model, in a hyperelasticity setting, processing these two operations simultaneously. The dissimilarity measure relates local and global (or region-based) information, since it relies on weighted total variation and nonlocal shape descriptors inspired by the piecewise constant Mumford-Shah model. Theoretical results emphasizing the mathematical and practical soundness of the model are provided, including existence of minimizers, connection with the segmentation step, nonlocal characterization of weighted seminorms, asymptotic results, and-convergence properties. A preliminary version of this work appeared in [N. Debroux and C. Le Guyader, A unified hyperelastic joint segmentation/registration model based on weighted total variation and nonlocal shape descriptors, in Sixth International Conference on Scale Space and Variational Methods in Computer Vision, F. Lauze, Y. Dong, and A. B. Dahl, eds., Springer International, Cham, 2017, pp. 614-625], but it contains neither proofs of the proposed material nor details on the numerical treatment (developed nonlocal algorithm and extensive comparisons with related works).