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Fuzzy Vector Median-Based Surface Smoothing
May/June 2004 (vol. 10 no. 3)
pp. 252-265

Abstract—This paper proposes a novel approach for smoothing surfaces represented by triangular meshes. The proposed method is a two-step procedure: surface normal smoothing through fuzzy vector median (FVM) filtering followed by integration of surface normals for vertex position update based on the least square error (LSE) criteria. Median and Order Statistic-based filters are extensively used in signal processing, especially image processing, due to their ability to reject outliers and preserve features such as edges and monotonic regions. More recently, fuzzy ordering theory has been introduced to allow averaging among similarly valued samples. Fuzzy ordering theory leads naturally to the fuzzy median, which yields improved noise smoothing over traditional crisp median filters. This paper extends the fuzzy ordering concept to vector-based data and introduces the fuzzy vector median filter. The application of FVM filters to surface normal smoothing yields improved results over previously introduced normal smoothing algorithms. The improved filtering results, coupled with LSE vertex position update, produces surface smoothing that minimizes the effects of noise while simultaneously preserving detail features. The proposed method is simple to implement and relatively fast. Simulation results are presented showing the performance of the proposed method and its advantages over commonly used surface smoothing algorithms. Additionally, optimization procedures for FVM filters are derived and evaluated.

[1] M. Levoy, K. Pulli, B. Curless, S. Rusinkiewicz, D. Koller, L. Pereira, M. Ginzton, S. Anderson, J. Davis, J. Ginsberg, J. Shade, and D. Fulk, The Digital Michelangelo Project: 3D Scanning of Large Statues SIGGRAPH '00 Conf. Proc., pp. 131-144, July 2000.
[2] F. Bernardini, J. Mittleman, H. Rushmeler, and G. Taubin, Scanning Michelangelo's Florentine Pietà: Making the Results Usable Proc. Eurographics '99 Short Papers and Demos, Sept. 1999.
[3] F. Bernardini et al., "Building a Digital Model of Michelangelo'sFlorentine Pietà" IEEE Computer Graphics and Applications, Jan./Feb. 2002, pp. 59-67.
[4] W.E. Lorensen and H.E. Cline, Marching Cubes: A High Resolution 3D Surface Construction Algorithm SIGGRAPH '87 Conf. Proc., pp. 163-169, 1987.
[5] G. Taubin, A Signal Processing Approach for Fair Surface Design SIGGRAPH '95 Conf. Proc., pp. 351-358, Aug. 1995.
[6] L. Kobbelt, Discrete Fairing Proc. Seventh IMA Conf. Math. of Surfaces '97, pp. 101-130, 1997.
[7] L. Kobbelt, S. Campagna, J. Vorsatz, and H.-P. Seidel, Interactive Multi-Resolution Modeling on Arbitrary Meshes SIGGRAPH '98 Conf. Proc., pp. 105-114, 1998.
[8] M. Desbrun, M. Meyer, P. Schröder, and A.H. Barr, Implicit Fairing of Irregular Meshes Using Diffusion and Curvature Flow SIGGRAPH '99 Conf. Proc., pp. 317-324, May 1999.
[9] I. Guskov, W. Sweldens, and P. Schröder, Multiresolution Signal Processing for Meshes SIGGRAPH '99 Conf. Proc., pp. 325-334, Aug. 1999.
[10] U. Clarenz, U. Diewald, and M. Rumpf, Anisotropic Geometric Diffusion in Surface Processing Proc. IEEE Visualization, pp. 397-405, 2000.
[11] Y. Ohtake, A.G. Belyaev, and I.A. Bogaevski, Mesh Regularization and Adaptive Smoothing Computer-Aided Design, pp. 789-800, Sept. 2001.
[12] G. Taubin, IBM Research Report: Linear Anisotropic Mesh Filtering Technical Report RC22213, IBM Research Division T.J. Watson Research Center, Yorktown Heights, N.Y., Oct. 2001.
[13] T. Preusser and M. Rumpf, A Level Set Method for Anisotropic Geometric Diffusion in 3D Image Processing SIAM J. Applied Math., vol. 62, no. 5, pp. 1772-1793, 2002.
[14] T.R. Jones, F. Durand, and M. Desbrun, Non-Iterative, Feature-Preserving Mesh Smoothing ACM Trans. Graphics, vol. 22, pp. 943-949, July 2003.
[15] S. Fleishman, I. Drori, and D. Cohen-Or, Bilateral Mesh Denoising ACM Trans. Graphics, vol. 22, pp. 950-953, July 2003.
[16] G. Taubin, A. Guéziec, W. Horn, and F. Lazarus, Progressive Forest Split Compression SIGGRAPH '98 Conf. Proc., pp. 123-132, 1998.
[17] H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and W. Stuetzle, Mesh Optimization SIGGRAPH '93 Conf. Proc., pp. 19-26, Aug. 1993.
[18] M. Desbrun, M. Meyer, P. Schröder, and A.H. Barr, Anisotropic Feature-Preserving Denoising of Height Fields and Bivariate Data Graphics Interface, pp. 145-152, May 2000.
[19] P. Saint-Marc and G. Medioni, Adaptive Smoothing for Feature Extraction Proc. DARPA '88, pp. 1100-1113, 1988.
[20] P. Perona and J. Malik, "Scale-Space and Edge Detection Using Anisotropic Diffusion," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 12, no. 7, pp. 629639, July 1990.
[21] C. Tomasi and R. Manduchi,"Bilateral Filtering for Gray and Color Images," Proc. IEEE Int'l. Conf. Computer Vision, IEEE CS Press, 1998, pp. 836-846.
[22] T. Tasdizen, R. Whitaker, P. Burchard, and S. Osher, Geometric Surface Processing via Normal Maps ACM Trans. Graphics, vol. 22, no. 4, pp. 1012-1033, 2003.
[23] Nonlinear Image Processing, S.K. Mitra and G.L. Sicuranza, eds. San Diego, Calif.: Academic Press, 2001.
[24] A. Flaig, K.E. Barner, and G.R. Arce, Fuzzy Ranking: Theory and Applications Signal Processing, special issue on fuzzy processing, vol. 80, pp. 2849-2852, 2000.
[25] K.E. Barner and R. Hardie, Spatial-Rank Order Selection Filters Nonlinear Image Processing, pp. 69-110, San Diego, Calif.: Academic Press, 2001.
[26] K.E. Barner, Y. Nie, and W. An, Fuzzy Ordering Theory and Its Use in Filter Generalization EURASIP J. Applied Signal Processing, vol. 2001, pp. 206-218, Dec. 2001.
[27] J.W. Tukey, Nonlinear (Nonsuperimposable) Methods for Smoothing Data Proc. Conf. Rec. Eascon, p. 673, 1974.
[28] J. Astola, P. Haavisto, and Y. Neuvo, Vector Median Filters Proc. IEEE, vol. 78, no. 4, pp. 678-689, 1990.
[29] M. Barni, F. Buti, F. Bartolini, and V. Cappellini, A Quasi-Euclidean Norm to Speed Up Vector Median Filtering IEEE Trans. Image Processing, vol. 9, pp. 1704-1709, Oct. 2000.
[30] P. Trahanias and A. Venetsanopoulos, “Vector Directional Filters—A New Class of Multichannel Image Processing Filters,” IEEE Trans. Image Processing, vol. 2, pp. 528-534, Oct. 1993.
[31] K.E. Barner, Fuzzy Theory, Methods, and Applications in Nonlinear Signal Processing Signal, Image, and Speech Processing, vol. III of Intelligent Systems: Technology and Applications, pp. 49-101, Boca Raton, Fla.: CRC Press, 2003.
[32] E.D. Bloch, A First Course in Geometric Topology and Differential Geometry. Boston: Birkhaäuser, 1997.
[33] A. Nehorai and M. Hawkes, Performance Bounds for Estimating Vector Systems IEEE Trans. Image Processing, pp. 1737-1749, June 2000.
[34] P. Cignoni, C. Rocchini, and R. Scopigno, Metro: Measuring Error on Simplified Surfaces Computer Graphics Forum, vol. 17, no. 2, pp. 167-174, 1998.

Index Terms:
Surface, mesh, smoothing, fuzzy vector median, multivariate.
Citation:
Yuzhong Shen, Kenneth E. Barner, "Fuzzy Vector Median-Based Surface Smoothing," IEEE Transactions on Visualization and Computer Graphics, vol. 10, no. 3, pp. 252-265, May-June 2004, doi:10.1109/TVCG.2004.1272725
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