This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Higher-Order Nonlinear Priors for Surface Reconstruction
July 2004 (vol. 26 no. 7)
pp. 878-891

Abstract—For surface reconstruction problems with noisy and incomplete range data, a Bayesian estimation approach can improve the overall quality of the surfaces. The Bayesian approach to surface estimation relies on a likelihood term, which ties the surface estimate to the input data, and the prior, which ensures surface smoothness or continuity. This paper introduces a new high-order, nonlinear prior for surface reconstruction. The proposed prior can smooth complex, noisy surfaces, while preserving sharp, geometric features, and it is a natural generalization of edge-preserving methods in image processing, such as anisotropic diffusion. An exact solution would require solving a fourth-order partial differential equation (PDE), which can be difficult with conventional numerical techniques. Our approach is to solve a cascade system of two second-order PDEs, which resembles the original fourth-order system. This strategy is based on the observation that the generalization of image processing to surfaces entails filtering the surface normals. We solve one PDE for processing the normals and one for refitting the surface to the normals. Furthermore, we implement the associated surface deformations using level sets. Hence, the algorithm can accommodate very complex shapes with arbitrary and changing topologies. This paper gives the mathematical formulation and describes the numerical algorithms. We also show results using range and medical data.

[1] D. Mumford and J. Shah, Boundary Detection by Minimizing Functionals Proc. IEEE Conf. Computer Vision and Pattern Recognition, 1985.
[2] D. Mumford and J. Shah, Optimal Approximations by Piecewise Smooth Functions and Associated Variational Problems Comm. Pure and Applied Math., vol. 42, pp. 577-685, 1989.
[3] N. Nordstrom, Biased Anisotropic Diffusion A Unified Regularization and Diffusion Approach to Edge Detection Image and Vision Computing, vol. 8, no. 4, pp. 318-327, 1990.
[4] J. Shah, Segmentation by Nonlinear Diffusion Proc. Conf. Computer Vision and Pattern Recognition, pp. 202-207, 1991.
[5] M. Black, G. Sapiro, D. Marimont, and D. Heeger, “Robust Anisotropic Diffusion,” IEEE Trans. Image Processing, vol. 7, no. 3, pp. 421-432, 1998.
[6] T. Tasdizen, R. Whitaker, P. Burchard, and S. Osher, Geometric Surface Smoothing via Anisotropic Diffusion of Normals Proc. IEEE Visualization, pp. 125-132, Oct. 2002.
[7] R.M. Bolle and D.B. Cooper, On Optimally Combining Pieces of Information, with Application to Estimating 3D Complex-Object Position from Range Data IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 8, no. 5, pp. 619-638, Sept. 1986.
[8] R. Bajcsy and F. Solina, Three Dimensional Object Representation Revisited Proc. First Int'l Conf. Computer Vision, pp. 231-240, June 1987.
[9] A.P. Pentland, Recognition by Parts Proc. First Int'l Conf. Computer Vision, pp. 612-620, June 1987.
[10] G. Taubin,“Estimation of planar curves, surfaces, and nonplanar space curves defined by implicit equations with applications to edge and range image segmentation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 13, no. 11, pp. 1115-1137, Nov. 1991.
[11] D. DeCarlo and D. Metaxas, "Adaptive Shape Evolution Using Blending," IEEE Proc. Int'l Conf. Computer Vision, pp. 834-839, 1995.
[12] G. Turk and M. Levoy, Zippered Polygon Meshes from Range Images Proc. SIGGRAPH, pp. 311-318, July 1994.
[13] Y. Chen and G. Médioni, Fitting a Surface to 3D Points Using an Inflating Ballon Model Proc. Second CAD-Based Vision Workshop, A. Kak and K. Ikeuchi, eds., vol. 13, pp. 266-273, 1994.
[14] B. Curless and M. Levoy, A Volumetric Method for Building Complex Models from Range Images Proc. SIGGRAPH (Computer Graphics), July 1996.
[15] H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and W. Stuetzle, Surface Reconstruction from Unorganized Points Computer Graphics, vol. 26, no. 2, pp. 71-78, 1992.
[16] A. Hilton, A.J. Stoddart, J. Illingworth, and T. Windeatt, Reliable Surface Reconstruction from Multiple Range Images Proc. European Conf. Computer Vision, 1996.
[17] R.T. Whitaker, A Level-Set Approach to 3D Reconstruction from Range Data Int'l J. Computer Vision, vol. 29, no. 3, pp. 203-231, 1998.
[18] R. Whitaker and J. Gregor, A Maximum Likelihood Surface Estimator for Dense Range Data IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 24, Oct. 2002.
[19] J. Gregor and R.T. Whitaker, Indoor Scene Reconstruction from Sets of Noisy Range Images Graphical Models, vol. 63, pp. 304-332, 2002.
[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] D. Geiger and A. Yuille, A Common Framework for Image Segmentation Int'l J. Computer Vision, vol. 6, no. 3, pp. 227-243, 1991.
[22] W. Snyder, Y.-S. Han, G. Bilbro, R. Whitaker, and S. Pizer, Image Relaxation: Restoration and Feature Extraction IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 17, no. 6, pp. 620-624, June 1995.
[23] R.T. Whitaker, Volumetric Deformable Models: Active Blobs Visualization in Biomedical Computing, R.A. Robb, ed., SPIE, 1994.
[24] L. Lorigo, O. Faugeras, E. Grimson, R. Keriven, R. Kikinis, A. Nabavi, and C.-F. Westin, Co-Dimension 2 Geodesic Active Contours for the Segmentation of Tubular Strucures Proc. Computer Vision and Pattern Recognition, 2000.
[25] G. Taubin, A Signal Processing Approach to Fair Surface Design Proc. SIGGRAPH, pp. 351-358, 1995.
[26] U. Clarenz, U. Diewald, and M. Rumpf, Anisotropic Geometric Diffusion in Surface Processing Proc. IEEE Visualization, pp. 397-405, 2000.
[27] Y. Ohtake, A.G. Belyaev, and I.A. Bogaevski, Polyhedral Surface Smoothing with Simultaneous Mesh Regularization Geometric Modeling and Processing, 2000.
[28] G. Taubin, Linear Anisotropic Mesh Filtering Technical Report RC22213, IBM Research Division, Oct. 2001.
[29] D.L. Chopp and J.A. Sethian, Motion by Intrinsic Laplacian of Curvature Interfaces and Free Boundaries, vol. 1, pp. 1-18, 1999.
[30] W. Welch and A. Witkin, Free-Form Shape Design Using Triangulated Surfaces Proc. SIGGRAPH '94, pp. 247-256, 1994.
[31] R. Schneider and L. Kobbelt, Generating Fair Meshes with$g^1$Boundary Conditions Proc. Geometric Modeling and Processing, pp. 251-261, 2000.
[32] R. Whitaker and V. Elangovan, A Direct Approach to Estimating Surfaces in Tomographic Data J. Medical Image Analysis, vol. 6, no. 3, pp. 235-249, 2002.
[33] G. Sapiro, Geometric Partial Differential Equations and Image Analysis. Cambridge Univ. Press, 2001.
[34] D. Geiger and A. Yuille, A Common Framework for Image Segmentation Int'l J. Computer Vision, vol. 6, no. 3, pp. 227-243, 1991.
[35] A. Polden, Compact Surfaces of Least Total Curvature technical report, Univ. of Tubingen, Germany, 1997.
[36] L. Ambrosio and S. Masnou, A Direct Variational Approach to a Problem Arising in Image Reconstruction Interfaces and Free Boundaries, vol. 5, no. 1, pp. 63-81, Jan. 2003.
[37] C. Ballester, M. Bertalmio, V. Caselles, G. Sapiro, and J. Verdera, Filling-In by Joint Interpolation of Vector Fields and Gray Levels IEEE Trans. Image Processing, vol. 10, pp. 1200-1211, Aug. 2001.
[38] M.P. DoCarmo, Differential Geometry of Curves and Surfaces. Prentice Hall, 1976.
[39] M. Bertalmio, L.-T. Cheng, S. Osher, and G. Sapiro, Variational Methods and Partial Differential Equations on Implicit Surfaces J. Computational Physics, vol. 174, pp. 759-80, 2001.
[40] B. Tang, G. Sapiro, and V. Caselles, Diffusion of General Data on Non-Flat Manifolds via Harmonic Maps Theory: The Direction Diffusion Case Int'l J. Computer Vision, vol. 36, no. 2, pp. 149-161, 2000.
[41] A. Lefohn, J.M. Kniss, C. Hansen, and R. Whitaker, Interactive Deformation and Visualization of Level Set Surfaces Using Graphics Hardware Proc. IEEE Visualization, pp. 75-82, 2003.
[42] A. Lefohn, J. Cates, and R. Whitaker, Interactive, GPU-Based Level Sets for 3D Brain Tumor Segmentation Medical Image Computing and Computer Assisted Intervention, pp. 564-572, 2003.
[43] M. Rumpf and R. Strzodka, Level Set Segmentation in Graphics Hardware Proc. Int'l Conf. Image Processing, pp. 1103-1106, 2001.

Index Terms:
Surface reconstruction, robust estimation, anisotropic diffusion, level sets.
Citation:
Tolga Tasdizen, Ross Whitaker, "Higher-Order Nonlinear Priors for Surface Reconstruction," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 26, no. 7, pp. 878-891, July 2004, doi:10.1109/TPAMI.2004.31
Usage of this product signifies your acceptance of the Terms of Use.