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
This report introduces a geometrical constraint kernel for handling the location in space and time of polymorphic k-dimensional objects subject to various geometrical and time constraints. The constraint kernel is generic in the sense that one of its parameters is a set of constraints on subsets of the objects. These constraints are handled globally by the kernel. We first illustrate how to model several placement problems with the constraint kernel. We then explain how new constraints can be introduced and plugged into the kernel. Based on these interfaces, we develop a generic k-dimensional lexicographic sweep algorithm for filtering the attributes of an object (i.e., its shape and the coordinates of its origin as well as its start, duration and end in time) according to all constraints where the object occurs. Experiments involving up to hundreds of thousands of objects and 1 million integer variables are provided in 2, 3 and 4 dimensions, both for simple shapes (i.e., rectangles, parallelepipeds) and for more complex shapes.
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.
In the context of video editing, enforcing spatio-temporal consistency is an important issue. With that purpose, the current variational models for gradient domain video editing include space and time regularization terms. The spatial terms are based on the usual space derivatives, the temporal ones are based on the convective derivative, and both are balanced by a parameter β. However, the usual discretizations of the convective derivative limit the value of β to a certain range, thus limiting these models from achieving their full potential. In this paper, we propose a new numerical scheme to compute the convective derivative, the deblurring convective derivative, which allows us to lift this constraint. Moreover, the proposed scheme introduces less errors than other discretization schemes without adding computational complexity. We use this scheme in the implementation of two gradient domain models for temporally consistent video editing, based on Poisson and total variation type formulations, respectively. We apply these models to three video editing tasks: inpainting correction, object insertion and object removal.
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