Three-dimensional alpha shapes
ACM Transactions on Graphics (TOG)
Preserving Topology by a Digitization Process
Journal of Mathematical Imaging and Vision
The crust and the &Bgr;-Skeleton: combinatorial curve reconstruction
Graphical Models and Image Processing
A simple algorithm for homeomorphic surface reconstruction
Proceedings of the sixteenth annual symposium on Computational geometry
Proceedings of the sixth ACM symposium on Solid modeling and applications
Weighted alpha shapes
Approximations of Shape and Configuration Space
Approximations of Shape and Configuration Space
Continuous Shape Transformation and Metrics on Regions
Fundamenta Informaticae - Qualitative Spatial Reasoning
Finding the Homology of Submanifolds with High Confidence from Random Samples
Discrete & Computational Geometry
Smooth manifold reconstruction from noisy and non-uniform approximation with guarantees
Computational Geometry: Theory and Applications
Learning from Positive and Unlabeled Examples: A Survey
ISIP '08 Proceedings of the 2008 International Symposiums on Information Processing
A Sampling Theory for Compact Sets in Euclidean Space
Discrete & Computational Geometry
Provably correct reconstruction of surfaces from sparse noisy samples
Pattern Recognition
Towards a general sampling theory for shape preservation
Image and Vision Computing
Image Digitization and its Influence on Shape Properties in Finite Dimensions
Image Digitization and its Influence on Shape Properties in Finite Dimensions
Reconstructing shapes with guarantees by unions of convex sets
Proceedings of the twenty-sixth annual symposium on Computational geometry
Vietoris-rips complexes also provide topologically correct reconstructions of sampled shapes
Proceedings of the twenty-seventh annual symposium on Computational geometry
On the shape of a set of points in the plane
IEEE Transactions on Information Theory
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The problem of reconstructing a region from a set of sample points is common in many geometric applications, including computer vision. It is very helpful to be able to guarantee that the reconstructed region ''approximates'' the true region, in some sense of approximation. In this paper, we study a general category of reconstruction methods, called ''locally-based reconstruction functions of radius @a,'' and we consider two specific functions, J"@a(S) and F"@a(S), within this category. We consider a sample S, either finite or infinite, that is specified to be within a given Hausdorff distance @d of the true region R, and we prove a number of theorems which give conditions on R, @d that are sufficient to guarantee that the reconstructed region is an approximation of the true region. Specifically, we prove:1.For any R, if F is any locally-based reconstruction method of radius @a where @a is small enough, and if the Hausdorff distance from S to R is small enough, then the dual-Hausdorff distance from F(S) to R, the Hausdorff distance between their boundaries, and the measure of their symmetric difference are guaranteed to be small. 2.If R is r-regular, then for any @e,@f0, if @a is small enough, and the Hausdorff distance from S to R is small enough, then each of the regions J"@a(S) and F"@a(S) is @e-similar to R and is an (@e,@f)-approximation in tangent of R.