This paper proposes a novel patch-based variational segmentation method that considers adaptive patches to characterize, in an affine invariant way, the local structure of each homogeneous texture region of the image and thus being capable of grouping the same kind of texture regardless of differences in the point of view or suffered perspective distortion. The patches are computed using an affine covariant structure tensor defined at every pixel of the image domain, so that they can automatically adapt its shape and size. They are used in a segmentation model that uses an L1-norm fidelity term and fuzzy membership functions, which is solved by an alternating scheme. The output of the method is a partition of the image in regions with homogeneous texture together with a patch representative of the texture of each region.

This paper presents an extension of the Non-Local Means denoising method, that effectively exploits the affine invariant self-similarities present in the images of real scenes. Our method provides a better image denoising result by grounding on the fact that in many occasions similar patches exist in the image but have undergone a transformation. The proposal uses an affine invariant patch similarity measure that performs an appropriate patch comparison by automatically and intrinsically adapting the size and shape of the patches. As a result, more similar patches are found and appropriately used. We show that this image denoising method achieves top-tier performance in terms of PSNR, outperforming consistently the results of the regular Non-Local Means, and that it provides state-of-the-art qualitative results.

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.