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by Hong-kai Zhao, Stanley Oshery, Barry Merrimany, Myungjoo Kangy
Computer Vision and Image Understanding
ftp://ftp.math.ucla.edu/pub/camreport/cam98-7.ps.gz
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Abstract:
In this paper we consider a fundamental visualization problem which arises in computer vision, computer graphics and numerical simulation. The problem is to find a curve in two dimensions, or a surface in three dimensions which can be regarded as the shape represented by a set of unorganized points, and/or curves, and/or surface patches. We do not assume any knowledge of the ordering, connectivity or topology of the data sets or of the true shape. Only the location of each point or general Hausdorff distance to the data set is known. The key idea in our approach is to find an implicit nonparametric representation of the curve or surface on a fixed rectangular grid. With this representation of surfaces we can easily (a) find the closest point and distance from any point to the surface (useful in illumination and many other applications), (b) find the intersection curve of two surfaces which is guaranteed to lie on both surfaces in our representation, and (c) perform any Boolean operation (see Figure 17). We use a version of the variational level set method [ZCMO, ZMOW] to interpret the desired shape as one or more elastic membranes attached to the data set. We may initialize our problem with a single large membrane enclosing this set. We then apply a gradient descent algorithm to the energy of this membrane, which results in an interesting and simple motion of the shape involving curvature times
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