摘要 :
Medial axis transform (MAT) is very sensitive to noise, in the sense that, even if a shape is perturbed only slightly, the Hausdorff distance between the MATs of the original shape and the perturbed one may be large. But it turns ...
展开
Medial axis transform (MAT) is very sensitive to noise, in the sense that, even if a shape is perturbed only slightly, the Hausdorff distance between the MATs of the original shape and the perturbed one may be large. But it turns out that MAT is stable, if we view this phenomenon with the one-sided Hausdorff distance, rather than with the two-sided Hausdorff distance. In this paper, we show that, if the original domain is weakly injective, which means that the MAT of the domain has no end point which is the center of an inscribed circle osculating the boundary at only one point, the one-sided Hausdorff distance of the original domain's MAT with respect to that of the perturbed one is bounded linearly with the Hausdorff distance of the perturbation. We also show by example that the linearity of this bound cannot be achieved for the domains which are not weakly injective. In particular, these results apply to the domains with sharp corners, which were excluded in the past. One consequence of these results is that we can clarify theoretically the notion of extracting "the essential part of the MAT", which is the heart of the existing pruning methods.
收起
摘要 :
Despite its usefulness in many applications, the medial axis transform (MAT) is very sensitive to the change of the boundary in the sense that, even if a shape is perturbed only slightly, the Hausdorff distance between the MATs of...
展开
Despite its usefulness in many applications, the medial axis transform (MAT) is very sensitive to the change of the boundary in the sense that, even if a shape is perturbed only slightly, the Hausdorff distance between the MATs of the original shape and the perturbed one may be large. However, it is known that MATs of 2D domains are stable if we view this phenomenon with the one-sided Hausdorff distance. This result depends on the fact that MATs are stable if the differences between them are measured with the recently introduced hyperbolic Hausdorff distance. In this paper, we extend the result for the one-sided stability of the MAT to a class of 3D domains called weakly injective, which contains many important 3D shapes typically appearing in solid modeling. Especially, the weakly injective 3D domains can have sharp features like corners or edges. In fact, by using the stability of the MAT under the hyperbolic Hausdorff distance, we obtain an explicit bound for the one-sided Hausdorff distance of the MAT of a weakly injective 3D domain with respect to that of a perturbed domain, which is linear with respect to the domain perturbation. We discuss some consequences of this result concerning the computation and the approximation of the MAT of 3D objects.
收起
摘要 :
We study the hyperbolization problem of two hyperspaces over a metric space and prove that they can be hyperbolized by asymptotically PT-1 metrics. (c) 2016 Elsevier Inc. All rights reserved.
摘要 :
We introduce an asymmetric distance function, which we call the left Hausdorff distance function, on the space of geodesic laminations on a closed hyperbolic surface of genus at least 2. This distance is an asymmetric version of t...
展开
We introduce an asymmetric distance function, which we call the left Hausdorff distance function, on the space of geodesic laminations on a closed hyperbolic surface of genus at least 2. This distance is an asymmetric version of the Hausdorff distance between compact subsets of a metric space. We prove a rigidity result for the action of the extended mapping class group of the surface on the space of geodesic laminations equipped with the topology induced from this distance. More specifically, we prove that there is a natural homomorphism from the extended mapping class group into the group of bijections of the space of geodesic laminations that preserve left Hausdorff convergence and that this homomorphism is an isomorphism.
收起
摘要 :
The purpose of this paper is to present some new (hybrid) fixed point theorems involving multivalued operators which satisfy weakly generalized contractive conditions in an ordered complete metric space. An example of the existenc...
展开
The purpose of this paper is to present some new (hybrid) fixed point theorems involving multivalued operators which satisfy weakly generalized contractive conditions in an ordered complete metric space. An example of the existence of solutions for a perturbed impulsive hyperbolic differential inclusion with variable times is given to illustrate the usability of our results.
收起
摘要 :
A well known result in the analysis of finite metric spaces due to Gromov says that given any metric space (X, d(x)) there exists a tree metric t(x) on X such that vertical bar d(x) - t(x)vertical bar(infinity) is bounded above by...
展开
A well known result in the analysis of finite metric spaces due to Gromov says that given any metric space (X, d(x)) there exists a tree metric t(x) on X such that vertical bar d(x) - t(x)vertical bar(infinity) is bounded above by twice hyp(X) . log(2 vertical bar X vertical bar). Here hyp(X) is the hyperbolicity of X, a quantity that measures the treeness of 4-tuples of points in X. This bound is known to be asymptotically tight.
收起
摘要 :
We study a uniform attractor A for a dissipative wave equation in a bounded domain Ω □□n under the assumption that the external force singularly oscillates in time; more precisely, it is of the form g 0(x, t)+ ε?α g 1 (x, t/ε...
展开
We study a uniform attractor A for a dissipative wave equation in a bounded domain Ω □□n under the assumption that the external force singularly oscillates in time; more precisely, it is of the form g 0(x, t)+ ε?α g 1 (x, t/ε), x ∈ Ω, t ∈ □, where α > 0, 0 < ε ≤ 1. In E = H 0 1 × L 2, this equation has an absorbing set B ε estimated as ‖B ε‖ E ≤C 1+C 2ε?α and, therefore, can increase without bound in the norm of E as ε → 0+. Under certain additional constraints on the function g 1(x, z), x ∈ Ω, z ∈ □, we prove that, for 0 < α ≤ α 0, the global attractors of such an equation are bounded in E, i.e.,A~ε||E≤C_3 , 0 < ε ≤ 1. Along with the original equation, we consider a “limiting” wave equation with external force g 0(x, t) that also has a global attractor . For the case in which g 0(x, t) = g 0(x) and the global attractor of the limiting equation is exponential, it is established that, for 0 < α ≤ α 0, the Hausdorff distance satisfies the estimate dist _E(A~ε,A~o)≤Cεn(a), where η(α) > 0. For η(α) and α 0, explicit formulas are given. We also study the nonautonomous case in which g 0 = g 0(x, t). It is assumed that sufficient conditions are satisfied for which the “limiting” nonautonomous equation has an exponential global attractor. In this case, we obtain upper bounds for the Hausdorff distance of the attractors A_ε from A~o , similar to those given above.
收起
原文获取