Multiscale analysis of similarities between images on Riemannian manifolds

Ballester, Coloma; Calderero, Felipe; Caselles, Vicente; Facciolo Furlan, Gabriele

doi:http://dx.doi.org/10.1137/130926833

Multiscale analysis of similarities between images on Riemannian manifolds

Citació

Ballester C, Calderero F, Caselles V, Facciolo G. Multiscale analysis of similarities between images on Riemannian manifolds. Multiscale Model Simul. 2014 May 1;12(2):616-49. DOI: 10.1137/130926833

Resum

In this paper we study the problem of comparing two patches of an image defined on a Riemannian manifold, which can be defined by the image domain with a suitable metric depending on the image. The size of the patch will not be determined a priori, and we identify it with a variable scale. Our approach can be considered as a nonlocal extension (comparing two points) of the multiscale analyses defined using the axiomatic approach by Álvarez et al. [Arch. Ration. Mech. Anal., 123 (1993), pp. 199--257]. Following this axiomatic approach, we can define a set of similarity measures that appear as solutions of a degenerate partial differential equation. This equation can be further specified in the linear case, and we observe that it contains as a particular instance the case of using weighted Euclidean distances as comparison measures. Finally, we discuss the case of some morphological scale spaces that exhibit a higher complexity.