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A geometric model for active contours in image processing   总被引:50,自引:0,他引:50  
Summary We propose a new model for active contours based on a geometric partial differential equation. Our model is intrinsec, stable (satisfies the maximum principle) and permits a rigorous mathematical analysis. It enables us to extract smooth shapes (we cannot retrieve angles) and it can be adapted to find several contours simultaneously. Moreover, as a consequence of the stability, we can design robust algorithms which can be engineed with no parameters in applications. Numerical experiments are presented.  相似文献
2.
Minimal surfaces: a geometric three dimensional segmentation approach   总被引:2,自引:0,他引:2  
Summary. A novel geometric approach for three dimensional object segmentation is presented. The scheme is based on geometric deformable surfaces moving towards the objects to be detected. We show that this model is related to the computation of surfaces of minimal area (local minimal surfaces). The space where these surfaces are computed is induced from the three dimensional image in which the objects are to be detected. The general approach also shows the relation between classical deformable surfaces obtained via energy minimization and geometric ones derived from curvature flows in the surface evolution framework. The scheme is stable, robust, and automatically handles changes in the surface topology during the deformation. Results related to existence, uniqueness, stability, and correctness of the solution to this geometric deformable model are presented as well. Based on an efficient numerical algorithm for surface evolution, we present a number of examples of object detection in real and synthetic images. Received January 4, 1996 / Revised version received August 2, 1996  相似文献
3.
This paper contains a systematic analysis of a natural measure theoretic notion of connectedness for sets of finite perimeter in ℝ 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(Ω) of functions of weakly bounded variation in Ω, and show that these filters are also well behaved in the classical Sobolev and BV spaces. Received July 27, 1999 / final version received June 8, 2000?Published online November 8, 2000  相似文献
4.
We prove that ifE is a Banach lattice andS, T ∈ ℒ (E) are such that 0≦sT,r(s)=r(T) andr(T) is a Riesz point ofσ(T) thenr(S) is a Riesz point ofσ(S). We prove also some results on compact positive perturbations of positive irreducible operators and lattice homomorphisms.  相似文献
5.
The aim of this paper is to study the isoperimetric problem with fixed volume inside convex sets and other related geometric variational problems in the Gauss space, in both the finite and infinite dimensional case. We first study the finite dimensional case, proving the existence of a maximal Cheeger set which is convex inside any bounded convex set. We also prove the uniqueness and convexity of solutions of the isoperimetric problem with fixed volume inside any convex set. Then we extend these results in the context of the abstract Wiener space, and for that we study the total variation denoising problem in this context.  相似文献
6.
We prove existence and uniqueness of entropy solutions for the nonhomogeneous Dirichlet problem associated to the relativistic heat equation.  相似文献
7.
We prove that if C⊂RNCRN is an open bounded convex set, then there is only one Cheeger set inside CC and it is convex. A Cheeger set of CC is a set which minimizes the ratio perimeter over volume among all subsets of CC.  相似文献
8.
The tree of shapes of an image is an ordered structure which permits an efficient manipulation of the level sets of an image, modeled as a real continuous function defined on a rectangle of , N ≥ 2. In this paper we construct the tree of shapes of an image by fusing both trees of connected components of upper and lower level sets. We analyze the branch structure of both trees and we construct the tree of shapes by joining their branches in a suitable way. This was the algorithmic approach for 2D images introduced by F. Guichard and P. Monasse in their initial paper, though other efficient approaches were later developed in this case. In this paper, we prove the well-foundedness of this approach for the general case of multidimensional images. This approach can be effectively implemented in the case of 3D images and can be applied for segmentation, but this is not the object of this paper. Devoted to the memory of Professor H.H. Schaefer  相似文献
9.
The transportation problem can be formalized as the problem of finding the optimal paths to transport a measure μ + onto a measure μ with the same mass. In contrast with the Monge–Kantorovich formalization, recent approaches model the branched structure of such supply networks by an energy functional whose essential feature is to favor wide roads. Given a flow s in a road or a tube or a wire, the transportation cost per unit length is supposed to be proportional to s α with 0 < α < 1. For the Monge–Kantorovich energy α = 1 so that it is equivalent to have two roads with flow 1/2 or a larger one with flow 1. If instead 0 < α < 1, a road with flow is preferable to two individual roads s 1 and s 2 because . Thus, this very simple model intuitively leads to branched transportation structures. Such a branched structure is observable in ground transportation networks, in draining and irrigation systems, in electric power supply systems and in natural objects like the blood vessels or the trees. When such structures can irrigate a whole bounded open set of . The aim of this paper is to give a mathematical proof of several structure and regularity properties empirically observed in transportation networks. It is first proven that optimal transportation networks have a tree structure and can be monotonically approximated by finite graphs. An interior regularity result is then proven according to which an optimal network is a finite graph away from the irrigated measure. It is also proven that the branching number of optimal networks has everywhere a universal explicit bound. These results answer questions raised in two recent papers by Xia.  相似文献
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