Winding and Euler numbers for 2D and 3D digital images |
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| Affiliation: | 1. Department of Mechanical Engineering and Science, Kyoto University, Nishikyo-ku, Kyoto 615-8540, Japan;2. Department of Engineering Mechanics & Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province, School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027, China;1. Beijing Key Lab of Precision/Ultra-precision Manufacturing Equipments and Control, Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China;2. State Key Laboratory of Tribology, Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China;3. Division of Advanced Manufacturing, Tsinghua Shenzhen International Graduate School, Tsinghua University, Shenzhen 518055, China |
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| Abstract: | New algorithms for computing the Euler number of a 3D digital image S are given, based on smoothing the image to a differentiable object and applying theorems of differential geometry and algebraic topology. They run in O(n) time, where n is the number of object elements of S with neighbors not in S. The basic idea is general and easily extended to images defined by other means, such as a hierarchical data structure or a union of isothetic (hyper) rectangles. |
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