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 .