Surface Parametrization and Curvature Measurement of Arbitrary 3-D Objects: Five Practical Methods
IEEE Transactions on Pattern Analysis and Machine Intelligence
On surface normal and Gaussian curvature approximations given data sampled from a smooth surface
Computer Aided Geometric Design
Optimizing 3D triangulations using discrete curvature analysis
Mathematical Methods for Curves and Surfaces
Intrinsic Surface Properties from Surface Triangulation
ECCV '92 Proceedings of the Second European Conference on Computer Vision
Hierarchical isosurface segmentation based on discrete curvature
VISSYM '03 Proceedings of the symposium on Data visualisation 2003
Surface Parameterization in Volumetric Images for Feature Classification
BIBE '00 Proceedings of the 1st IEEE International Symposium on Bioinformatics and Biomedical Engineering
Estimating the tensor of curvature of a surface from a polyhedral approximation
ICCV '95 Proceedings of the Fifth International Conference on Computer Vision
On the angular defect of triangulations and the pointwise approximation of curvatures
Computer Aided Geometric Design
Discrete Laplace-Beltrami operators and their convergence
Computer Aided Geometric Design - Special issue: Geometric modeling and processing 2004
Convergence analysis of a discretization scheme for Gaussian curvature over triangular surfaces
Computer Aided Geometric Design
Curvature estimation scheme for triangle meshes using biquadratic Bézier patches
Computer-Aided Design
Geometric accuracy analysis for discrete surface approximation
GMP'06 Proceedings of the 4th international conference on Geometric Modeling and Processing
Surface parameterization in volumetric images for curvature-based feature classification
IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics
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We propose a discrete approximation of Gaussian curvature over quadrilateral meshes using a linear combination of two angle deficits. Let g"i"j and b"i"j be the coefficients of the first and second fundamental forms of a smooth parametric surface F. Suppose F is sampled so that a surface mesh is obtained. Theoretically we show that for vertices of valence four, the considered two angle deficits are asymptotically equivalent to rational functions in g"i"j and b"i"j under some special conditions called the parallelogram criterion. Specifically, the numerators of the rational functions are homogenous polynomials of degree two in b"i"j with closed form coefficients, and the denominators are g"1"1g"2"2-g"1"2^2. Our discrete approximation of the Gaussian curvature derived from the combination of the angle deficits has quadratic convergence rate under the parallelogram criterion. Numerical results which justify the theoretical analysis are also presented.