A Heuristic Method for Region Reconstruction from Noisy Samples

We consider the problem of reconstructing a region in the plane from a noisy sample of points in it. Figure 1 shows the setting: Λ is a region of R2 and points are sampled in or near Λ. Note three things about the sampling: the points are well distributed in the interior of Λ; there are sample points outside Λ (these are the effect of noise in the sampling); and the boundary of Λ is not sampled at all, except by accident. The classical geometrical solutions for shape reconstruction from points, such as α-shapes4 and β -skeletons,8 work well in the absence of noise but are too sensitive to the presence of noise, because they use all sample points in the reconstruction graph. We seek a method that can automatically identify points that are definitely in the interior of the region (these are trustworthy) and points that are near the boundary (these are less trustworthy because of noise). To handle noise and to quantify the trustworthiness of each point, we approach the region reconstruction problem as a function reconstruction problem: