Object recognition, robotic vision, occluding noise removal or photograph design require the ability to perform disocclusion. W e call disocclusion the recovery of hidden parts of objects in a digital image by interpolation from the vicinity of the occluded area. It is shown in this paper how disocclusion can be performed by means of level lines structure, which offers a reliable, complete and contrast-invariant representation of image, in contrast to edges. Level lines based disocclusion yields a solution that may have strong discontinuities, which is not possible with PDE-based interpolation. Moreover, the proposed method is fully compatible with Kanizsa ' s theory of "amodal completion".

Abstract. This paper contains a systematic analysis of a natural measure theoretic notion of connectedness for sets of finite perimeter in IR N , introduced by H. Federer in the more general framework of the theory of currents. We provide a new and simpler proof of the existence and uniqueness of the decomposition into the so-called M-connected components. Moreover, we study carefully the structure of the essential boundary of these components and give in particular a reconstruction formula of a set of finite perimeter from the family of the boundaries of its components. In the two dimensional case we show that this notion of connectedness is comparable with the topological one, modulo the choice of a suitable representative in the equivalence class. Our strong motivation for this study is a mathematical justification of all those operations in image processing that involve connectedness and boundaries. As an application, we use this weak notion of connectedness to provide a rigorous mathematical basis to a large class of denoising filters acting on connected components of level sets. We introduce a natural domain for these filters, the space WBV¢ ¤ £ ¦ ¥ of functions of weakly bounded variation in £ , and show that these filters are also well behaved in the classical Sobolev and BV spaces.

Object recognition, robot vision, image and film restoration may require the ability to perform disocclusion. We call disocclusion the recovery of occluded areas in a digital image by interpolation from their vicinity. It is shown in this paper how disocclusion can be performed by means of the level-lines structure, which offers a reliable, complete and contrast-invariant representation of images. Level-lines based disocclusion yields a solution that may have strong discontinuities. The proposed method is compatible with Kanizsa's amodal completion theory.

Exemplar-based methods have proven their efficiency for the reconstruction of missing parts in a digital image. Texture as well as local geometry are often very well restored. Some applications, however, require the ability to reconstruct non local geometric features, e.g. long edges. We propose in this paper to endow a particular instance of exemplar-based method with a geometric guide. The guide is obtained by a prior interpolation of a simplified version of the image using straight lines or Euler spirals. We derive from it an additional geometric penalization for the metric associated with the exemplar-based algorithm. We discuss the details of the method and show several examples of reconstruction.

We consider a variational approach to the problem of recovering missing parts in a panchromatic digital image. Representing the image by a scalar function u, we propose a model based on the relaxation of the energywhich takes into account the perimeter of the level sets of u as well as the L p norm of the mean curvature along their boundaries. We investigate the properties of this variational model and the existence of minimizing functions in BV. We also address related issues for integral varifolds with generalized mean curvature in L p .

It has been recently conjectured that, in the context of the Heisenberg groupHn endowed with its Carnot–Carathéodory metric and Haar measure, the isoperimetricsets (i.e., minimizers of the H-perimeter among sets of constant Haar measure) couldcoincide with the solutions to a “restricted” isoperimetric problem within the class ofsets having finite perimeter, smooth boundary, and cylindrical symmetry. In this paper,we derive new properties of these restricted isoperimetric sets, which we call Heisenbergbubbles. In particular, we show that their boundary has constant mean H-curvature and, quitesurprisingly, that it is foliated by the family of minimal geodesics connecting two specialpoints. In view of a possible strategy for proving that Heisenberg bubbles are actuallyisoperimetric among the whole class of measurable subsets of Hn, we turn our attentionto the relationship between volume, perimeter, and -enlargements. In particular, we provea Brunn–Minkowski inequality with topological exponent as well as the fact that the Hperimeterof a bounded, open set F ⊂ Hn of class C2 can be computed via a generalizedMinkowski content, defined by means of any bounded set whose horizontal projection is the2n-dimensional unit disc. Some consequences of these properties are discussed

Among all methods for reconstructing missing regions in a digital image, the so-called exemplar-based algorithms are very efficient and often produce striking results. They are based on the simple idea-initially used for texture synthesis-that the unknown part of an image can be reconstructed by simply pasting samples extracted from the known part. Beyond heuristic considerations, there have been very few contributions in the literature to explain from a mathematical point of view the performances of these purely algorithmic and discrete methods. More precisely, a recent paper by Levina and Bickel [64] provides a theoretical explanation of their ability to recover very well the texture, but nothing equivalent has been done so far for the recovery of geometry. Our purpose in this paper is twofold: 1. To propose well-posed variational models in the continuous domain that can be naturally associated to exemplar-based algorithms; 2. To investigate their ability to reconstruct either local or long-range geometric features like edges. In particular, we propose several optimization models in R N , we discuss their relation with the original algorithms, and show the existence of minimizers in the framework of functions of bounded variation. Focusing on a simple 2D situation, we provide experimental evidences that basic exemplar-based algorithms are able to reconstruct a local geometric information whereas the minimization of the proposed variational models allows a global reconstruction of geometry and in particular of smooth edges. The derivation of globally minimizing algorithms associated to these models is still an open problem. Yet the results presented in this paper are a first step towards new inpainting algorithms with an improved quality of geometry reconstruction and no loss of quality for texture reconstruction.

We present the first method to handle curvature regularity in region-based image segmentation and inpainting that is independent of initialization.To this end we start from a new formulation of length-based optimization schemes, based on surface continuation constraints, and discuss the connections to existing schemes. The formulation is based on a cell complex and considers basic regions and boundary elements. The corresponding optimization problem is cast as an integer linear program.We then show how the method can be extended to include curvature regularity, again cast as an integer linear program. Here, we are considering pairs of boundary elements to reflect curvature. Moreover, a constraint set is derived to ensure that the boundary variables indeed reflect the boundary of the regions described by the region variables.We show that by solving the linear programming relaxation one gets quite close to the global optimum, and that curvature regularity is indeed much better suited in the presence of long and thin objects compared to standard length regularity.