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

Using a classical result on algebraic invariants of the unimodular group, we present in this paper some basic geometric affine invariant quantities, and we use them to construct some distinctive descriptors for object detection. Although full affine invariance cannot be guaranteed due to noncommutativity of camera blur with affine maps and the domain problem (that is, the difficulty of finding an affine covariant domain), the proposed descriptors behave more robustly than SIFT with respect to affine deformations. This is supported by our comparisons both with the version of SIFT computed on an affine normalized neighborhood, and with ASIFT, which solves both the previously mentioned camera blur and domain problems by cleverly sampling the orbit of affine transformations of the images.