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<p>In this paper, we consider the geometrical analysis of oriented digital surfaces, which form the boundary of connected voxel sets. We present a fast and exact method for the detection and reconstruction of its convex and concave parts. The main ingredient is the notion of visibility, which joins some pairs of points whenever the straight segment between them stays sufficiently close to the input surface without crossing it. It entails a definition of local convexity/concavity. Then, outer and inner candidate segments are assembled separately into visible facets, thus building polyhedral approximations of the local convex and concave zones. Furthermore we propose two different optimizations: the first one is a probing-based method to accelerate the detection of locally convex and concave vertices and the other is a top-down hierarchical filtering approach to reduce the number of visibility tests. Combined together, they induce considerable speed-up and enable the processing of large 3D images.</p>

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Fast and Exact Convex and Concave Decomposition of Digitized Shapes

  • 期刊论文

  • 发表日期：

    2026年7月12日

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  • Jacques-Olivier Lachaud,
  • Tristan Roussillon

摘要

In this paper, we consider the geometrical analysis of oriented digital surfaces, which form the boundary of connected voxel sets. We present a fast and exact method for the detection and reconstruction of its convex and concave parts. The main ingredient is the notion of visibility, which joins some pairs of points whenever the straight segment between them stays sufficiently close to the input surface without crossing it. It entails a definition of local convexity/concavity. Then, outer and inner candidate segments are assembled separately into visible facets, thus building polyhedral approximations of the local convex and concave zones. Furthermore we propose two different optimizations: the first one is a probing-based method to accelerate the detection of locally convex and concave vertices and the other is a top-down hierarchical filtering approach to reduce the number of visibility tests. Combined together, they induce considerable speed-up and enable the processing of large 3D images.