A Multiscale Deformation Representation

Motivated by Tadmor, Nezzar, and Vese`s work [30] dedicated to multiscale image representation using hierarchical (BV, L2) decompositions, we propose transposing their approach to the case of reg-istration, a task which consists in determining a smooth deformation aligning the salient constituents visible in an image into their counterpart in another. The underlying goal is to obtain a hierarchical decomposition of the deformation in the form of a composition of intermediate deformations: the coarser one, computed from versions of the two images capturing the essential features, encodes the main structural/geometrical deformation, while iterating the procedure and refining the versions of the two images yields more accurate deformations that faithfully map small-scale features. The proposed model falls within the framework of variational methods and hyperelasticity by viewing the shapes to be matched as Ogden materials. The material behavior is described by means of a specif-ically tailored strain energy density function, complemented by L\infty-penalizations ensuring that the computed deformation is a bi-Lipschitz homeomorphism. Theoretical results emphasizing the math-ematical soundness of the model are provided, among which the existence of minimizers/asymptotic results, and a suitable numerical algorithm is supplied, along with numerical simulations demon-strating the ability of the model to produce accurate hierarchical representations of deformations. A very preliminary version of this work has been accepted for publication in the Eighth International Conference on Scale Space and Variational Methods in Computer Vision [Springer, Cham, Switzer-land, 2021] but it does not include all the theoretical results, nor the detailed related proofs. A more complete and detailed analysis of the numerical experiments is also provided. The theoretical analysis of the numerical algorithm (introduced in section 3 and which is a result in itself) will be the subject of a separate article in preparation.