Gabriele Facciolo

TV-L1 Optical Flow Estimation

This article describes an implementation of the optical flow estimation method introduced by Zach, Pock and Bischof in 2007. This method is based on the minimization of a functional containing a data term using the L 1 norm and a regularization term using the total variation of the flow. The main feature of this formulation is that it allows discontinuities in the flow field, while being more robust to noise than the classical approach by Horn and Schunck. The algorithm is an efficient numerical scheme, which solves a relaxed version of the problem by alternate minimization. Source Code A C implementation of this algorithm is provided. The source code and an online demo are accessible at the web page of this article 1 .

Anisotropic Cheeger Sets and Applications

The main purpose of this paper is to develop the mathematical analysis of anisotropic total variation problems with a degenerate metric and the computation of the associated Cheeger sets. We illustrate our analysis with the computation of Cheeger sets with respect to different anisotropic norms of relevance in applications to image processing. In particular, we describe the computation of global minima of geodesic active contour models, and we illustrate the use of Cheeger sets for the problem of edge linking.

Analysis of a Variational Framework for Exemplar-Based Image Inpainting

In this paper we study some variational models for exemplar-based image inpainting, namely the patch nonlocal means and the patch nonlocal Poisson methods, previously studied by the authors from the experimental point of view. In both cases, the unknowns are the image u to be reconstructed and a weight function w expressing the similarity of patches. As a limit case of the studied framework, the weight function reduces to a correspondence map from the inpainting domain to the known part of the image. We prove the existence and regularity of minima for both functionals. In particular, we prove the existence of optimal correspondence maps which are uniform limits of maps of bounded variation with finitely many values. Then we prove the convergence of an alternating optimization scheme for the variables

Non-local dual image denoising

(u,w)

Constrained Anisotropic Diffusion and some Applications.

. We also prove the convergence in probability of the PatchMatch method, a recently introduced and efficient algorithm for computing optimal correspondence maps. Finally, we display some numerical expe...

MGM: A Significantly More Global Matching for Stereovision

The current state-of-the-art non-local algorithms for image denoising have the tendency to remove many low contrast details. Frequency-based algorithms keep these details, but on the other hand many artifacts are introduced. Recently, the Dual Domain Image Denoising (DDID) method has been proposed to address this issue. While beating the state-of-the-art, this algorithm still causes strong frequency domain artifacts. This paper reviews DDID under a different light, allowing to understand their origin. The analysis leads to the development of NLDD, a new denoising algorithm that outperforms DDID, BM3D and other state-of-the-art algorithms. NLDD is also three times faster than DDID and easily parallelizable.

NONLOCAL VARIATIONAL MODELS FOR INPAINTING AND INTERPOLATION

Minimal surface regularization has been used in several applications ranging from stereo to image segmentation, sometimes hidden as a graph-cut discrete formulation, or as a strictly convex approximation to TV minimization. In this paper we consider a modified version of minimal surfac e regularization coupled with a robust data fitting term for interpolatio n purposes, where the corresponding evolution equation is constrained to diffuse only along the isophotes of a given image u and we design a convergent numerical scheme to accomplish this. To illustrate the usefulness of our appr oach, we apply this framework to the digital elevation model interpolatio n and to constrained vector probability diffusion.

Variational Framework for Non-Local Inpainting

Semi-global matching (SGM) is among the top-ranked stereovision algorithms. SGM is an efficient strategy for approximately minimizing a global energy that comprises a pixel-wise matching cost and pair-wise smoothness terms. In SGM the two-dimensional smoothness constraint is approximated as the average of one-dimensional line optimization problems. The accuracy and speed of SGM are the main reasons for its widespread adoption, even when applied to generic problems beyond stereovision. This approximate minimization, however, also produces characteristic low amplitude streaks in the final disparity image, and is clearly suboptimal with respect to more comprehensive minimization strategies. Based on a recently proposed interpretation of SGM as a min-sum Belief Propagation algorithm, we propose a new algorithm that allows to reduce by a factor five the energy gap of SGM with respect to reference algorithms for MRFs with truncated smoothness terms. The proposed method comes with no compromises with respect to the baseline SGM, no parameters and virtually no computational overhead. At the same time it attains higher quality results by removing the characteristic streaking artifacts of SGM.

Linear Multiscale Analysis of Similarities between Images on Riemannian Manifolds: Practical Formula and Affine Covariant Metrics

In this paper we study some nonlocal variational models for different image inpainting tasks. Nonlocal methods for denoising and inpainting have gained considerable attention due to their good performance on textured images, a known weakness of classical local methods which are performant in recovering the geometric structure of the image. We first review a general variational framework for the problem of nonlocal inpainting that exploits the self-similarity of natural images to copy information in a consistent way from the known parts of the image. We single out two particular methods depending on the information we copy: either the gray level (or color) information or its gradient. We review the main properties of the corresponding energies and their minima. Then we discuss three other applications: we consider the problem of stereo inpainting, some simple cases of video inpainting, and the problem of interpolation of incomplete depth maps knowing a reference image. Incomplete depth maps can be obtained as a result of stereo algorithms, or given for instance by Time-of-Flight cameras (in that case the interpolated result can be used to generate the images of the stereo pair). We discuss the basic algorithms to minimize the energies and we display some numerical experiments illustrating the main properties of the proposed models.

On stereo-rectification of pushbroom images

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.