The medial axis is an important shape representation that finds a wide range of applications in shape analysis. For large-scale shapes of high resolution, a progressive medial axis representation that starts with the lowest resolution and gradually adds more details is desired. In this paper, we propose a fast and robust geometric algorithm that computes progressive medial axes of a large-scale planar shape. The key ingredient of our method is a novel structural analysis of merging medial axes of two planar shapes along a shared boundary. Our method is robust by separating the analysis of topological structure from numerical computation. Our method is also fast and we show that the time complexity of merging two medial axes is O(n log nv), where n is the number of total boundary generators, nv is strictly smaller than n and behaves as a small constant in all our experiments. Experiments on large-scale polygonal data and comparison with state-of-the-art methods show the efficiency and effectiveness of the proposed method.

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Let X be a compact connected strongly pseudoconvex CR manifold of real dimension 2n . 1 in CN. It has been an interesting question to find an intrinsic smoothness criteria for the complex Plateau problem. For n  3 and N = n+1, Yau found a necessary and sufficient condition for the interior regularity of the HarveyLawson solution to the complex Plateau problem by means of Kohn–Rossi cohomology groups on X in 1981. For n = 2 and N  n + 1, the problem has been open for over 30 years. In this paper we introduce a new CR invariant g(1,1)(X) of X. The vanishing of this invariant will give the interior regularity of the Harvey–Lawson solution up to normalization. In the case n = 2 and N = 3, the vanishing of this invariant is enough to give the interior regularity.

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No abstract uploaded!