Abstract not found

One of the aims of computer vision in the past 30 years has been to recognize shapes by numerical algorithms. Now, what are the geometric features on which shape recognition can be based? In this paper, we review the mathematical arguments leading to a unique definition of planar shape elements. This definition is derived from the invariance requirement to not less than five classes of perturbations, namely noise, a#ne distortion, contrast changes, occlusion, and background. This leads to a single possibility: shape elements as the normalized, affine smoothed pieces of level lines of the image. As a main possible application, we show the existence of a generic image comparison technique able to find all shape elements common to two images.

We prove the existence of a nite extinction time for the solutions of the Dirichlet problem for the total variational ow. For the Neumann problem, we prove that the solutions reach the average of its initial datum in nite time. The asymptotic prole of the solutions of the Dirichlet problem is also studied. It is shown that the proles are non zero solutions of an eigenvalue type problem which seems to be unexplored in the previous literature. The propagation of the support is analyzed in the radial case showing a behaviour enterely dierent to the case of the problem associated to the p-Laplacian operator. Finally, the study of the radially symmetric case allows us to point out other qualitative properties which are peculiar of this special class of quasilinear equations. Key words: Total variation ow, nonlinear parabolic equations, asymptotic behaviour, eigenvalue type problem, propagation of the support. AMS (MOS) subject classication: 35K65, 35K55. 1 Introduction Let be a ...

In this letter, we extend the concept of level lines of scalar images to vector-valued data. Consistent with the scalar case, we define the level-lines of vector-valued images as the integral curves of the directions of minimal vectorial change. This direction, and the magnitude of the change, are computed using classical Riemannian geometry. As an example of the use of this new concept, we show how to visualize the basic geometry of vector-valued images with a scalar image.

Motivated by operators simplifying the topographic map of a function, we study the theoretical properties of two kinds of "grain" filters. The first category, discovered by L. Vincent, de nes grains as connected components of level sets and removes those of small area. This category is composed of two filters, the maxima filter and the minima filter. However, they do not commute. The second kind of filter, introduced by Masnou, works on "shapes", which are based on connected components of level sets. This filter has the additional property that it acts in the same manner on upper and lower level sets, that is, it commutes with an inversion of contrast. We illustrate this study with examples comparing the latter and the former ones.

We prove that the topographic map structure of upper semicontinuous functions, defined in terms of classical connected components of its level sets, and of functions of bounded variation (or a generalization, the WBV functions), defined in terms of M-connected components of its level sets, coincides when the function is a continuous function in WBV . Both function spaces are frequently used as models for images. Thus, if the domain of the image is Jordan domain, a rectangle, for instance, and the image u 2 C( \WBV( (beingconstantnear@ 1 we prove that for almost all levels of u, the classical connected components of positive measure of [u ] coincide with the M-components of [u ]. Thus the notion of M-component can be seen as a relaxation of the classical notion of connected component when going from C( to WBV( 4 1 Introduction An image can be realistically modelled as a real function u(x) where x represents an arbitrary point of IR N (N = 2 for usual snapshots, 3 for me...

Abstract. This paper focuses on a second order functional depending on free discontinuity and free gradient-discontinuity, whose minimizers provide a variational solution to contour detection problem in image segmentation. We briefly resume the state of the art about Blake & Zisserman functional under different types of boundary condition which are related to contour enhancement in image segmentation. We prove a new Caccioppoli inequality suitable to study regularity of minimizers of related boundary value problems in any dimension n ≥ 1 and deduce that there are no nontrivial local minimizers in half-space. 1.

In this work, we compare two models that were proposed to improve the well-known model of Rudin, Osher, and Fatemi [Phys. D, 60 (1992), pp. 259–268]. The first one, proposed by Meyer [Oscillating Patterns in Image Processing and in Some Nonlinear Evolution Equations. The ... RI, 2001], is the BV − G model, and the second one is the BV − L 1 model. We state the similitudes between both models. In particular, we prove a characterization theorem for optimal decompositions for the BV − L 1 model. We then compare these models in the particular case of radial functions.

Abstract. The aim of this paper is to study the minimal perimeter problem for sets containing a fixed set E in R 2 in a very general setting, and to give the explicit solution.