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
Implicit fairing of irregular meshes using diffusion and curvature flow
Proceedings of the 26th annual conference on Computer graphics and interactive techniques
Laplacian Eigenmaps for dimensionality reduction and data representation
Neural Computation
Fair morse functions for extracting the topological structure of a surface mesh
ACM SIGGRAPH 2004 Papers
Local and Global Comparison of Continuous Functions
VIS '04 Proceedings of the conference on Visualization '04
Geometry-Aware Bases for Shape Approximation
IEEE Transactions on Visualization and Computer Graphics
Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing
Persistence-sensitive simplification functions on 2-manifolds
Proceedings of the twenty-second annual symposium on Computational geometry
Spectral surface quadrangulation
ACM SIGGRAPH 2006 Papers
Computer Aided Geometric Design - Special issue: Applications of geometric modeling in the life sciences
Salient critical points for meshes
Proceedings of the 2007 ACM symposium on Solid and physical modeling
Describing shapes by geometrical-topological properties of real functions
ACM Computing Surveys (CSUR)
Spectral quadrangulation with orientation and alignment control
ACM SIGGRAPH Asia 2008 papers
Laplace-Beltrami spectra as 'Shape-DNA' of surfaces and solids
Computer-Aided Design
A topological hierarchy for functions on triangulated surfaces
IEEE Transactions on Visualization and Computer Graphics
Topology- and error-driven extension of scalar functions from surfaces to volumes
ACM Transactions on Graphics (TOG)
Technical Section: Shape approximation by differential properties of scalar functions
Computers and Graphics
SMI 2012: Full Local approximation of scalar functions on 3D shapes and volumetric data
Computers and Graphics
Empirical mode decomposition on surfaces
Graphical Models
wFEM heat kernel: Discretization and applications to shape analysis and retrieval
Computer Aided Geometric Design
SMI 2013: Steepest descent paths on simplicial meshes of arbitrary dimensions
Computers and Graphics
Surface- and volume-based techniques for shape modeling and analysis
SIGGRAPH Asia 2013 Courses
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In engineering, geographical applications, scientific visualization, and bio-informatics, a variety of phenomena is described by a large set of data modeled as the values of a scalar function f:M-R defined on a surface M. A low quality of the discrete representations of the input data, unstable computations, numerical approximations, and noise might produce functions with a high number of critical points. In this context, we propose an algorithmic framework for smoothing an arbitrary scalar function, while simplifying its redundant critical points and preserving those that are mandatory for its description. From our perspective, the critical points of f are a natural choice to guide the approximation scheme; in fact, they usually represent relevant information about the behavior of f or the shape itself. To address the aforementioned aims, we compute a smooth approximation f@?:M-R of f whose set of critical points contains those that have been preserved by the simplification process. The idea behind the proposed approach is to combine smoothing techniques, critical points, and spectral properties of the Laplacian matrix. Inserting constraints in the smoothing of f allows us to overcome the traditional error-driven approximation of f, which does not provide constraints on the preserved topological features. Finally, the computational cost of the proposed approach is O(nlogn), where n is the number of vertices of M.