Spherical Diffusion for 3D Surface Smoothing

A diffusion-based approach to surface smoothing is presented. Surfaces are represented as scalar functions defined on the sphere. The approach is equivalent to Gaussian smoothing on the sphere and is computationally efficient since it does not require iterative smoothing. Furthermore, it does not suffer from the well-known shrinkage problem. Evolution of important shape features (parabolic curves) under diffusion is demonstrated. A nonlinear modification of the diffusion process is introduced in order to improve smoothing behavior of elongated and poorly centered objects.

[1]  Thomas Bülow.  Multiscale Image Processing on the Sphere , 2002, DAGM-Symposium.

[2]  Berthold K. P. Horn.  Extended Gaussian images , 1984, Proceedings of the IEEE.

[3]  Mutsuo Sano,et al.  3D object representation using spherical harmonic functions , 1993, Proceedings of 1993 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS '93).

[4]  Andrew P. Witkin,et al.  Scale-space filtering: A new approach to multi-scale description , 1984, ICASSP.

[5]  D. Healy,et al.  Computing Fourier Transforms and Convolutions on the 2-Sphere , 1994 .

[6]  Richard Courant,et al.  Methods of Mathematical Physics, 1 , 1955 .

[7]  Willi Freeden,et al.  Non-orthogonal expansions on the sphere , 1995 .

[8]  S. Rosenberg.  The Laplacian on a Riemannian Manifold: The Laplacian on a Riemannian Manifold , 1997 .

[9]  Jan Sijbers,et al.  Efficient algorithm fo the computation of 3D Fourier descriptors , 2002, Proceedings. First International Symposium on 3D Data Processing Visualization and Transmission.

[10]  M. Tinkham.  Group Theory and Quantum Mechanics , 1964 .

[11]  Sarp Ertürk,et al.  Efficient Representation of 3D Human Head Models , 1999, BMVC.

[12]  T. Bulow.  Spherical diffusion for 3D surface smoothing , 2002, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[13]  Gabriel Taubin,et al.  Geometric Signal Processing on Polygonal Meshes , 2000, Eurographics.

[14]  A. J. Barret,et al.  Methods of Mathematical Physics, Volume I . R. Courant and D. Hilbert. Interscience Publishers Inc., New York. 550 pp. Index. 75s. net. , 1954, The Journal of the Royal Aeronautical Society.

[15]  Atsushi Imiya,et al.  On the History of Gaussian Scale-Space Axiomatics , 1997, Gaussian Scale-Space Theory.

[16]  Guido Gerig,et al.  Parametrization of Closed Surfaces for 3-D Shape Description , 1995, Comput. Vis. Image Underst..

[17]  T. Lindeberg,et al.  Scale-Space Theory : A Basic Tool for Analysing Structures at Different Scales , 1994 .

[18]  J. Koenderink.  The structure of images , 2004, Biological Cybernetics.

[19]  Ross T. Whitaker,et al.  Geometric surface smoothing via anisotropic diffusion of normals , 2002, IEEE Visualization, 2002. VIS 2002..

[20]  Dmitry B. Goldgof,et al.  The Use of Three- and Four-Dimensional Surface Harmonics for Rigid and Nonrigid Shape Recovery and Representation , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[21]  Gabriel Taubin,et al.  A signal processing approach to fair surface design , 1995, SIGGRAPH.

[22]  Mark Meyer,et al.  Implicit fairing of irregular meshes using diffusion and curvature flow , 1999, SIGGRAPH.

[23]  Thomas Bülow,et al.  Surface Representations Using Spherical Harmonics and Gabor Wavelets on the Sphere , 2001 .

[24]  Andrew P. Witkin,et al.  Scale-Space Filtering , 1983, IJCAI.

[25]  Yutaka Ohtake,et al.  A comparison of mesh smoothing methods , 2003 .

[26]  Dana H. Ballard,et al.  Computer Vision , 1982 .

[27]  A. Makadia,et al.  Image processing in catadioptric planes: spatiotemporal derivatives and optical flow computation , 2002, Proceedings of the IEEE Workshop on Omnidirectional Vision 2002. Held in conjunction with ECCV'02.

[28]  Frédo Durand,et al.  Non-iterative, feature-preserving mesh smoothing , 2003, ACM Trans. Graph..

[29]  Katsushi Ikeuchi,et al.  The Complex EGI: A New Representation for 3-D Pose Determination , 1993, IEEE Trans. Pattern Anal. Mach. Intell..

[30]  G. Chirikjian,et al.  Engineering Applications of Noncommutative Harmonic Analysis: With Emphasis on Rotation and Motion Groups , 2000 .

[31]  Joachim Weickert,et al.  Anisotropic diffusion in image processing , 1996 .

[32]  Nasser Khalili,et al.  Curvature Computation on Free-Form 3-D Meshes at Multiple Scales , 2001, Comput. Vis. Image Underst..

[33]  J. J. Koenderink,et al.  Dynamic shape , 1986, Biological Cybernetics.

[34]  Makoto Sato,et al.  Scale space filtering on spherical pattern , 1992, Proceedings., 11th IAPR International Conference on Pattern Recognition. Vol. III. Conference C: Image, Speech and Signal Analysis,.