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15
On The Theory Of Planar Shape
 SIAM MULTISCALE MODELING AND SIMULATION
, 2003
"... 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. Thi ..."
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Cited by 17 (2 self)
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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.
Some Qualitative Properties for the Total Variational Flow
"... 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 s ..."
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Cited by 13 (0 self)
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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 pLaplacian 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 ...
On the Level Lines and Geometry of VectorValued Images
, 2000
"... In this letter, we extend the concept of level lines of scalar images to vectorvalued data. Consistent with the scalar case, we define the levellines of vectorvalued images as the integral curves of the directions of minimal vectorial change. This direction, and the magnitude of the change, are ..."
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Cited by 8 (1 self)
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In this letter, we extend the concept of level lines of scalar images to vectorvalued data. Consistent with the scalar case, we define the levellines of vectorvalued 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 vectorvalued images with a scalar image.
Grain Filters
, 2001
"... 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 t ..."
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Cited by 4 (1 self)
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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.
The Mcomponents of level sets of continuous functions in WBV
 in wbv. Publicacions Matemàtiques
, 2001
"... 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 Mconnected components of its level sets, coincides ..."
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Cited by 2 (1 self)
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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 Mconnected 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 Mcomponents of [u ]. Thus the notion of Mcomponent 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...
The Γlimit of the twodimensional OhtaKawasaki energy. II. Droplet arrangement via the renormalized energy
"... This is the first in a series of two papers in which we derive a Γexpansion for a twodimensional nonlocal GinzburgLandau energy with Coulomb repulsion, also known as the OhtaKawasaki model in connection with diblock copolymer systems. In that model, two phases appear, which interact via a nonlo ..."
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This is the first in a series of two papers in which we derive a Γexpansion for a twodimensional nonlocal GinzburgLandau energy with Coulomb repulsion, also known as the OhtaKawasaki model in connection with diblock copolymer systems. In that model, two phases appear, which interact via a nonlocal Coulomb type energy. We focus on the regime where one of the phases has very small volume fraction, thus creating small “droplets ” of the minority phase in a “sea ” of the majority phase. In this paper we show that an appropriate setting for Γconvergence in the considered parameter regime is via weak convergence of the suitably normalized charge density in the sense of measures. We prove that, after a suitable rescaling, the OhtaKawasaki energy functional Γconverges to a quadratic energy functional of the limit charge density generated by the screened Coulomb kernel. A consequence of our results is that minimizers (or almost minimizers) of the energy have droplets which are almost all asymptotically round, have the same radius and are uniformly distributed in the domain. The proof relies mainly on the analysis of the sharp interface version of the energy, with the connection to the original diffuse interface model obtained via matching upper and lower bounds for the energy. We thus also obtain a characterization of the limit charge density for the energy minimizers in the diffuse interface model. 1
Topographic Maps of Color Images
, 2000
"... We address the problem of extending topographic maps to color images. A topographic map gives a morphological and a geometrical representation of the information contained in natural images. Two approaches are presented and discussed. The first one is new and consists in defining a total order in IR ..."
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We address the problem of extending topographic maps to color images. A topographic map gives a morphological and a geometrical representation of the information contained in natural images. Two approaches are presented and discussed. The first one is new and consists in defining a total order in IR in accordance with the human visual perception of shapes. This allows to define color topographic maps in the same way that what it has been done for graylevel topographic maps. It has the advantage of leading all properties known in the graylevel case to remain true in the color case. But the map contains a so huge quantity of data that it has to be drastically simplified. The second approach, based on a so far unpublished result [4], allows to build a simplified representation by using the geometry given by the luminance component only. We present experiments which illustrate the advantages and the drawbacks of each method.
On the shape of liquid drops and crystals in the small mass regime. Preprint at http://cvgmt.sns.it/papers/figmag10/fmsmall4.pdf
"... Abstract. We consider liquid drops or crystals lying in equilibrium under the action of a potential energy. For small masses, the proximity of the resulting minimizers from the Wulff shape associated to the surface tension is quantitatively controlled in terms of the smallness of the mass and with r ..."
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Abstract. We consider liquid drops or crystals lying in equilibrium under the action of a potential energy. For small masses, the proximity of the resulting minimizers from the Wulff shape associated to the surface tension is quantitatively controlled in terms of the smallness of the mass and with respect to the natural notions of distance induced by the regularity of the Wulff shape. Stronger results are proved in the twodimensional case. For instance, it is shown that a planar crystal undergoing the action of a small exterior force field remains convex, and admits only small translations parallel to its faces. Contents
VARIATIONAL APPROACH TO IMAGE SEGMENTATION
"... Abstract. This paper focuses on a second order functional depending on free discontinuity and free gradientdiscontinuity, 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 und ..."
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Abstract. This paper focuses on a second order functional depending on free discontinuity and free gradientdiscontinuity, 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 halfspace. 1.