Linear multiscale analysis of similarities between images on Riemannian manifolds: practical formula and affine covariant metrics

Abstract

In this paper we study the problem of comparing two patches of images defined on Riemannian/nmanifolds which in turn can be defined by each image domain with a suitable metric depending on/nthe image. For that we single out one particular instance of a set of models defining image similarities/nthat was earlier studied in [C. Ballester et al., Multiscale Model. Simul., 12 (2014), pp. 616–649],/nusing an axiomatic approach that extended the classical Alvarez–Guichard–Lions–Morel work to the ´/nnonlocal case. Namely, we study a linear model to compare patches defined on two images in RN/nendowed with some metric. Besides its genericity, this linear model is selected by its computational/nfeasibility since it can be approximated leading to an algorithm that has the complexity of the/nusual patch comparison using a weighted Euclidean distance. Moreover, we propose and study some/nintrinsic metrics which we define in terms of affine covariant structure tensors and we discuss their/nproperties. These tensors are defined for any point in the image and are intrinsically endowed with/naffine covariant neighborhoods. We also discuss the effect of discretization over the affine covariance/nproperties of the tensors. We illustrate our theoretical results with numerical experiments.