Computer Vision, Graphics, and Image Processing
Triangulating topological spaces
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
A-shapes of a finite point set
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Surface reconstruction with anisotropic density-scaled alpha shapes
Proceedings of the conference on Visualization '98
A simple provable algorithm for curve reconstruction
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
Weighted alpha shapes
Provably good surface sampling and approximation
Proceedings of the 2003 Eurographics/ACM SIGGRAPH symposium on Geometry processing
Contour interpolation by straight skeletons
Graphical Models
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Stability of persistence diagrams
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Provably good sampling and meshing of Lipschitz surfaces
Proceedings of the twenty-second annual symposium on Computational geometry
Shape reconstruction from unorganized cross-sections
SGP '07 Proceedings of the fifth Eurographics symposium on Geometry processing
Provable surface reconstruction from noisy samples
Computational Geometry: Theory and Applications
Isotopic reconstruction of surfaces with boundaries
SGP '09 Proceedings of the Symposium on Geometry Processing
The power crust, unions of balls, and the medial axis transform
Computational Geometry: Theory and Applications
Automatic recognition of 2D shapes from a set of points
ICIAR'11 Proceedings of the 8th international conference on Image analysis and recognition - Volume Part I
Shape reconstruction from an unorganized point cloud with outliers
ICIAR'12 Proceedings of the 9th international conference on Image Analysis and Recognition - Volume Part I
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This paper deals with the problem of reconstructing shapes from an unorganized set of sample points (called S). First, we give an intuitive notion for gathering sample points in order to reconstruct a shape. Then, we introduce a variant of α-shape [1] which takes into account that the density of the sample points varies from place to place, depending on the required amount of details. The Locally-Density-Adaptive-α-hull (LDA-α-hull) is formally defined and some nice properties are proven. It generates a monotone family of hulls for α ranging from 0 to 1. Afterwards, from LDA-α-hull, we formally define the LDA-α-shape, describing the boundaries of the reconstructed shape, and the LDA-α-complex, describing the shape and its interior. Both describe a monotone family of subgraphs of the Delaunay triangulation of S (called Del(S)). That is, for α varying from 0 to 1, LDA-α-shape (resp. LDA-α-complex) goes from the convex hull of S (resp. Del(S)) to S. These definitions lead to a very simple and efficient algorithm to compute LDA-α-shape and LDA-α-complex in O(n log(n)). Finally, a few meaningful examples show how a shape is reconstructed and underline the stability of the reconstruction in a wide range of α even if the density of the sample points varies from place to place.