Marching cubes: A high resolution 3D surface construction algorithm
SIGGRAPH '87 Proceedings of the 14th annual conference on Computer graphics and interactive techniques
The use of the L-curve in the regularization of discrete ill-posed problems
SIAM Journal on Scientific Computing
A signal processing approach to fair surface design
SIGGRAPH '95 Proceedings of the 22nd annual conference on Computer graphics and interactive techniques
Machine Learning
Guaranteeing the topology of an implicit surface polygonization for interactive modeling
Proceedings of the 24th annual conference on Computer graphics and interactive techniques
An equivalence between sparse approximation and support vector machines
Neural Computation
Topology preserving and controlled topology simplifying multiresolution isosurface extraction
Proceedings of the conference on Visualization '00
Reconstruction and representation of 3D objects with radial basis functions
Proceedings of the 28th annual conference on Computer graphics and interactive techniques
Introduction to Implicit Surfaces
Introduction to Implicit Surfaces
Modelling with implicit surfaces that interpolate
ACM Transactions on Graphics (TOG)
Exploring scalar fields using critical isovalues
Proceedings of the conference on Visualization '02
Volume Data Mining Using 3D Field Topology Analysis
IEEE Computer Graphics and Applications
Estimating surface normals in noisy point cloud data
Proceedings of the nineteenth annual symposium on Computational geometry
Laplacian Eigenmaps for dimensionality reduction and data representation
Neural Computation
Multi-level partition of unity implicits
ACM SIGGRAPH 2003 Papers
SMI '01 Proceedings of the International Conference on Shape Modeling & Applications
Fair morse functions for extracting the topological structure of a surface mesh
ACM SIGGRAPH 2004 Papers
Interpolating and approximating implicit surfaces from polygon soup
ACM SIGGRAPH 2004 Papers
Surface Reconstruction of Noisy and Defective Data Sets
VIS '04 Proceedings of the conference on Visualization '04
Local and Global Comparison of Continuous Functions
VIS '04 Proceedings of the conference on Visualization '04
3D scattered data interpolation and approximation with multilevel compactly supported RBFs
Graphical Models - Special issue on SMI 2003
Hierarchical Clustering for Unstructured Volumetric Scalar Fields
Proceedings of the 14th IEEE Visualization 2003 (VIS'03)
IEEE Computer Graphics and Applications
SIMS: a Multi-Level Approach to Surface Reconstruction with Sparse Implicits
SMI '06 Proceedings of the IEEE International Conference on Shape Modeling and Applications 2006
Persistence-sensitive simplification functions on 2-manifolds
Proceedings of the twenty-second annual symposium on Computational geometry
Spectral surface quadrangulation
ACM SIGGRAPH 2006 Papers
Harmonic volumetric mapping for solid modeling applications
Proceedings of the 2007 ACM symposium on Solid and physical modeling
Salient critical points for meshes
Proceedings of the 2007 ACM symposium on Solid and physical modeling
An adaptive MLS surface for reconstruction with guarantees
SGP '05 Proceedings of the third Eurographics symposium on Geometry processing
SGP '05 Proceedings of the third Eurographics symposium on Geometry processing
Hierarchical error-driven approximation of implicit surfaces from polygonal meshes
SGP '06 Proceedings of the fourth Eurographics symposium on Geometry processing
IEEE Transactions on Visualization and Computer Graphics
Topological Landscapes: A Terrain Metaphor for Scientific Data
IEEE Transactions on Visualization and Computer Graphics
Efficient Computation of Morse-Smale Complexes for Three-dimensional Scalar Functions
IEEE Transactions on Visualization and Computer Graphics
Volumetric parameterization and trivariate b-spline fitting using harmonic functions
Proceedings of the 2008 ACM symposium on Solid and physical modeling
Describing shapes by geometrical-topological properties of real functions
ACM Computing Surveys (CSUR)
Technical Section: Computing smooth approximations of scalar functions with constraints
Computers and Graphics
Fast orthogonal least squares algorithm for efficient subset modelselection
IEEE Transactions on Signal Processing
A topological hierarchy for functions on triangulated surfaces
IEEE Transactions on Visualization and Computer Graphics
Interactively visualizing procedurally encoded scalar fields
VISSYM'04 Proceedings of the Sixth Joint Eurographics - IEEE TCVG conference on Visualization
Defining, contouring, and visualizing scalar functions on point-sampled surfaces
Computer-Aided Design
SMI 2012: Full Local approximation of scalar functions on 3D shapes and volumetric data
Computers and Graphics
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The behavior of a variety of phenomena measurable on the boundary of 3D shapes is studied by modeling the set of known measurements as a scalar function f :P → R, defined on a surface P. Furthermore, the large amount of scientific data calls for efficient techniques to correlate, describe, and analyze this data. In this context, we focus on the problem of extending the measures captured by a scalar function f, defined on the boundary surface P of a 3D shape, to its surrounding volume. This goal is achieved by computing a sequence of volumetric functions that approximate f up to a specified accuracy and preserve its critical points. More precisely, we compute a smooth map g : R3 → R such that the piecewise linear function h :=gP : P → R, which interpolates the values of g at the vertices of the triangulated surface P, is an approximation of f with the same critical points. In this way, we overcome the limitation of traditional approaches to function approximation, which are mainly based on a numerical error estimation and do not provide measurements of the topological and geometric features of f. The proposed approximation scheme builds on the properties of f related to its global structure, that is, its critical points, and ignores the local details of f, which can be successively introduced according to the target approximation accuracy.