Multiscale Analysis of Similarities between Images on Riemannian Manifolds

Abstract

Abstract. 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 byÁlvarez 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. 1. Introduction. Our purpose in this paper is to compare 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 in [1].Let us review the fundamentals of that approach. A multiscale analysis represents a given image at different scales of smoothing, the scale being related to the size of the neighborhood which is used to give an estimate of the brightness of the picture at a given point. It is a basic preprocessing step for shape recognition [27] (see [16,6] and references therein).The systematic study of multiscale analyses for images was the purpose of the axiomatic approach proposed in [1]. Based on a series of axioms which define the structure of the multiscale space and a set of geometric and photometric invariants, multiscale analyses were defined in terms of (viscosity) solutions of a parabolic equation. In the case of linear multiscale analysis they obtained the Gaussian scale space (already proposed and studied in [28,24,25,49,18,17,26,48], using also an axiomatic approach in some of those papers). In addition to the Gaussian scale space, classification covers many of the classical models that were proposed in the literature, such as the Perona-Malik equation [34] (see also [9]), the Rudin-Osher-Fatemi model

Affine Invariant Self-similarity for Exemplar-based Inpainting

“…We first briefly discuss the concept of affine covariant tensors. Then we show how they are used to define the affine invariant similarity measure and establish the relation between our derivation and the theory of (Ballester et al, 2014;Fedorov et al, 2015). Finally, we describe an algorithm to compute the affine convariant tensors.…”

“…It was derived as an approximation to a more general framework introduced in (Ballester et al, 2014), where similarity measures between images on Riemannian manifolds are studied.…”

“…Let us also remark that formula (7) has the same complexity of the patch comparison formula (1). The similarity measure corresponding to (7) was derived in (Fedorov et al, 2015) as a computationally tractable approximation of the linear case of the multiscale similarity measures introduced in (Ballester et al, 2014). There, the authors show that all scale spaces of similarity measures D(t,x,y) satisfying a set of appropriate axioms are solutions of a family of degenerate elliptic partial differential equations (PDE).…”

“…In this Riemannian framework P R defines an isometry between the tangent spaces in two manifolds. The authors refer to it as the a priori connection, since it is related to the notion of connection appearing in parallel transport (see (Ballester et al, 2014) for details).…”

Multiscale Analysis for Images on Riemannian Manifolds

“…It is interesting to note at this point that in the case where the image domain is R N the manifold point of view permits us to construct some examples of affine invariant multiscale spaces (even new linear and affine invariant multiscale spaces). This deserves a more detailed study (its performance and applications), which we postpone to a subsequent paper to avoid a longer extension of the present paper [7]. Second, in the context of manifolds, multiscale analyses for video fall into the same context as multiscale analyses of 3D images, the only difference being due to the specific metric for video.…”