Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations
Journal of Computational Physics
A set of level 3 basic linear algebra subprograms
ACM Transactions on Mathematical Software (TOMS)
Using particles to sample and control implicit surfaces
SIGGRAPH '94 Proceedings of the 21st annual conference on Computer graphics and interactive techniques
Shape transformation using variational implicit functions
Proceedings of the 26th annual conference on Computer graphics and interactive techniques
A Generalization of Algebraic Surface Drawing
ACM Transactions on Graphics (TOG)
Multidimensional binary search trees used for associative searching
Communications of the ACM
Scale-Space Theory in Computer Vision
Scale-Space Theory in Computer Vision
Introduction to Implicit Surfaces
Introduction to Implicit Surfaces
Adaptive surface reconstruction based on implicit PHT-splines
Proceedings of the 14th ACM Symposium on Solid and Physical Modeling
Parallel and adaptive surface reconstruction based on implicit PHT-splines
Computer Aided Geometric Design
Fast and smooth interactive segmentation of medical images using variational interpolation
EG VCBM'10 Proceedings of the 2nd Eurographics conference on Visual Computing for Biology and Medicine
Accurate reconstruction of 3D cardiac geometry from coarsely-sliced MRI
Computer Methods and Programs in Biomedicine
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We describe algebraic methods for creating implicit surfaces using linear combinations of radial basis interpolants to form complex models from scattered surface points. Shapes with arbitrary topology are easily represented without the usual interpolation or aliasing errors arising from discrete sampling. These methods were first applied to implicit surfaces by Savchenko, et al. and later developed independently by Turk and O'Brien as a means of performing shape interpolation. Earlier approaches were limited as a modeling mechanism because of the order of the computational complexity involved. We explore and extend these implicit interpolating methods to make them suitable for systems of large numbers of scattered surface points by using compactly supported radial basis interpolants. The use of compactly supported elements generates a sparse solution space, reducing the computational complexity and making the technique practical for large models. The local nature of compactly supported radial basis functions permits the use of computational techniques and data structures such as k-d trees for spatial subdivision, promoting fast solvers and methods to divide and conquer many of the subproblems associated with these methods. Moreover, the representation of complex models permits the exploration of diverse surface geometry. This reduction in computational complexity enables the application of these methods to the study of shape properties of large complex shapes.