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Anthropomorphic Image Reconstruction via Hypoelliptic Diffusion
 SIAM Journal on Control and Optimization, Volume 50, Issue
, 2012
"... Abstract. In this paper we study a model of geometry of vision due to Petitot, Citti and Sarti. One of the main features of this model is that the primary visual cortex V1 lifts an image from R2 to the bundle of directions of the plane. Neurons are grouped into orientation columns, each of them corr ..."
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Abstract. In this paper we study a model of geometry of vision due to Petitot, Citti and Sarti. One of the main features of this model is that the primary visual cortex V1 lifts an image from R2 to the bundle of directions of the plane. Neurons are grouped into orientation columns, each of them corresponding to a point of this bundle. In this model a corrupted image is reconstructed by minimizing the energy necessary for the activation of the orientation columns corresponding to regions in which the image is corrupted. The minimization process gives rise to an hypoelliptic heat equation on the bundle of directions of the plane. In the original model directions are considered both with and without orientation giving rise respectively to a problem on the group of rototranslations of the plane SE(2) or on the projective tangent bundle of the plane PTR2. We provide a mathematical proof of several important facts for this model. We rst prove that the model is mathematically consistent only if direction are considered without orientation. We then prove that generically the convolution of a L2(R2;R) function (e.g. an image) with a 2D Gaussian is generically a Morse function. This fact is important since the lift of Morse functions to PTR2 is dened on a smooth manifold. We then provide the explicit expression of the hypoelliptic heat kernel on PTR2 in terms of Mathieu functions. Finally we present the main ideas of an algorithm which allows to perform image reconstruction on real nonacademic images. A very interesting point is that this algorithm is massively parallelizable and needs no information on where the image is corrupted.
Exemplar based Image Inpainting with Reduced Search Region
 No.12
, 2014
"... Image inpainting is a technique to recover lost data or manually remove data from an image. This process is carried out in such a way that it seems reasonable to human eyes. The lost part of the image is filled with the remaining available data of the image. Inpainting can be done using various tech ..."
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Image inpainting is a technique to recover lost data or manually remove data from an image. This process is carried out in such a way that it seems reasonable to human eyes. The lost part of the image is filled with the remaining available data of the image. Inpainting can be done using various techniques, mainly variation based, exemplar based and wavelet based. In this paper, exemplar based image inpainting techniques is used and an algorithm which improves and extends the previously proposed algorithm is presented. Instead of searching the entire image, the proposed approach has reduced the search region by searching the nearby pixels since most of the relevant information lies in neighbouring pixels which reduces time to complete inpainting. This technique is mainly used for removing objects and cracks from the image.
Parallel Algorithm and Software for Image Inpainting via SubRiemannian Minimizers on the Group of Rototranslations
, 2012
"... Abstract. The paper is devoted to an approach for image inpainting developed on the basis of neurogeometry of vision and subRiemannian geometry. Inpainting is realized by completing damaged isophotes (level lines of brightness) by optimal curves for the leftinvariant subRiemannian problem on the ..."
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Abstract. The paper is devoted to an approach for image inpainting developed on the basis of neurogeometry of vision and subRiemannian geometry. Inpainting is realized by completing damaged isophotes (level lines of brightness) by optimal curves for the leftinvariant subRiemannian problem on the group of rototranslations (motions) of a plane SE(2). The approach is considered as anthropomorphic inpainting since these curves satisfy the variational principle discovered by neurogeometry of vision. A parallel algorithm and software to restore monochrome binary or halftone images represented as series of isophotes were developed. The approach and the algorithm for computation of completing arcs are presented in detail. AMS subject classifications: 65M10, 78A48 Key words: Image inpainting, subRiemannian geometry, neurogeometry of vision, group of rototranslations
Multiscale analysis of images on Riemannian manifolds
"... Abstract. In this paper we study multiscale analyses for images defined on Riemannian manifolds and extend the axiomatic approach proposed by Álvarez, Guichard, Lions, and Morel to this general case. This covers the case of two and threedimensional images and video sequences. After obtaining the ..."
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Abstract. In this paper we study multiscale analyses for images defined on Riemannian manifolds and extend the axiomatic approach proposed by Álvarez, Guichard, Lions, and Morel to this general case. This covers the case of two and threedimensional images and video sequences. After obtaining the general classification, we consider the case of morphological scale spaces, which are given in terms of geometric equations, and the linear case given by the Laplace–Beltrami flow. We consider in some detail the case of image metrics given in terms of the structure tensor and compute some cases of such a tensor for video. Then we comment on the connections with variational formulations of image diffusion comparing the anisotropies that appear. Finally, we include numerical experiments illustrating some of the models. Namely, we compare some examples for still images using the Laplace–Beltrami flow and some variational models. We also consider several examples in video: the mean curvature motion and the extension of the morphological and Galilean invariant scale spaces to the video manifold case, and the Laplace–Beltrami flow. We point out that the number of models that appear is huge, and we have restricted ourselves to such cases for the sake of brevity and illustration.
PRIOR AND MACROFILLING ORDER FOR IMAGE COMPLETION
"... This paper introduces an algorithm to build priors that help image completion to produce better and visually plausible results. The goal of the prior is to seamlessly reconstruct long edges, that exemplarbased inpainting fails to restore. The prior consists in locating the edges across the missing r ..."
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This paper introduces an algorithm to build priors that help image completion to produce better and visually plausible results. The goal of the prior is to seamlessly reconstruct long edges, that exemplarbased inpainting fails to restore. The prior consists in locating the edges across the missing region, and separating the image in regions of relative similar textures. Unlike other techniques, our proposal does not use segmentation or inpainting to obtain the prior, providing a fast technique. Moreover, the technique does not rely on image dependent threshold. Finally, we propose a scheduling for the synthesis of the missing region based on the prior, and Criminisi’s algorithm. The scheduling may be thought as a novel macrofilling order for exemplarbased synthesis. The results are comparable with recent and more complex proposals based on superresolution or Photoshop. Finally, we emphasize that, our prior is not dedicated to our image completion algorithm, and can be used to drive other image completion or inpainting algorithms. Index Terms — prior, image completion, inpainting, filling priority 1.
∇u(α + βdiv
, 2011
"... coareatype formula for the relaxation of a generalized ..."
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"... coareatype formula for the relaxation of a generalized elastica functional ..."
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coareatype formula for the relaxation of a generalized elastica functional