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 classicalÁlvarez-Guichard-Lions-Morel work to the nonlocal case. Namely, we study a linear model to compare patches defined on two images in 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. Introduction.Image comparison is a topic that has received a lot of attention in the image processing and computer vision communities since it is a main ingredient in many applications, such as object recognition, stereo vision, image interpolation, image denoising, and exemplar-based image inpainting, among others. A common way to define a nonlocal similarity measure between two images is to compare the patches (local neighborhoods) around each pair of points formed by taking one point from each image. We consider a general setting in which images are defined on Riemannian manifolds. Such manifolds arise, for instance, for images defined on R N , endowed with a suitable metric depending on the image.In [3] it was shown that multiscale analyses of similarities between images on Riemannian manifolds, satisfying a certain set of axioms, are (viscosity) solutions of a family of degenerate PDEs. Our goal in this paper is to study one particular instance of the set of models derived in [3], namely a linear model to compare patches defined on two images in R N endowed

Image inpainting aims to obtain a visually plausible image interpolation in a region of the image in which data is missing due to damage or occlusion. Usually, the only available information is the portion of the image outside the inpainting domain. Besides its numerous applications, the inpainting problem is of theoretical interest since its analysis involves an understanding of the self-similarity present in natural images. In this work, we present a detailed description and implementation of three exemplar-based inpainting methods derived from the variational framework introduced by Arias et al. Source CodeThe implementation written in C++ for this algorithm is available in the IPOL web page of this article 1 . Compilation and usage instructions are included in the README.md file of the archive.

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.

Abstract:This paper presents a new method for exemplar-based image inpainting using transformed patches. We build upon a recent affine invariant self-similarity measure which automatically transforms patches to compare them in an appropriate manner. As a consequence, it intrinsically extends the set of available source patches to copy information from. When comparing two patches, instead of searching for the appropriate patch transformation in a highly dimensional parameter space, our approach allows us to determine a single transformation from the texture content in both patches. We incorporate the affine invariant similarity measure in a variational formulation for inpainting and present an algorithm together with experimental results illustrating this approach.

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 L 1-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.