Preserving geometric properties in reconstructing regions from internal and nearby points

Abstract

The problem of reconstructing a region from a set of sample points is common in many geometric applications, including computer vision. It is very helpful to be able to guarantee that the reconstructed region “approximates” the true region, in some sense of approximation. In this paper, we study a general category of reconstruction methods, called “locally-based reconstruction functions of radius α,” and we consider two specific functions, Jα(S) and Fα(S), within this category. We consider a sample S, either finite or infinite, that is specified to be within a given Hausdorff distance δ of the true region R, and we prove a number of theorems which give conditions on R, δ that are sufficient to guarantee that the reconstructed region is an approximation of the true region. Specifically, we prove:1.For any R, if F is any locally-based reconstruction method of radius α where α is small enough, and if the Hausdorff distance from S to R is small enough, then the dual-Hausdorff distance from F(S) to R, the Hausdorff distance between their boundaries, and the measure of their symmetric difference are guaranteed to be small.2.If R is r-regular, then for any ϵ,ϕ>0, if α is small enough, and the Hausdorff distance from S to R is small enough, then each of the regions Jα(S) and Fα(S) is ϵ-similar to R and is an (ϵ,ϕ)-approximation in tangent of R.