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Issue No.11 - November (2010 vol.32)
pp: 2085-2099
Heung-Sun Ng , Hong Kong University of Science and Technology, Hong Kong
Tai-Pang Wu , Hong Kong University of Science and Technology, Hong Kong
Chi-Keung Tang , Hong Kong University of Science and Technology, Hong Kong
ABSTRACT
Representative surface reconstruction algorithms taking a gradient field as input enforce the integrability constraint in a discrete manner. While enforcing integrability allows the subsequent integration to produce surface heights, existing algorithms have one or more of the following disadvantages: They can only handle dense per-pixel gradient fields, smooth out sharp features in a partially integrable field, or produce severe surface distortion in the results. In this paper, we present a method which does not enforce discrete integrability and reconstructs a 3D continuous surface from a gradient or a height field, or a combination of both, which can be dense or sparse. The key to our approach is the use of kernel basis functions, which transfer the continuous surface reconstruction problem into high-dimensional space, where a closed-form solution exists. By using the Gaussian kernel, we can derive a straightforward implementation which is able to produce results better than traditional techniques. In general, an important advantage of our kernel-based method is that the method does not suffer discretization and finite approximation, both of which lead to surface distortion, which is typical of Fourier or wavelet bases widely adopted by previous representative approaches. We perform comparisons with classical and recent methods on benchmark as well as challenging data sets to demonstrate that our method produces accurate surface reconstruction that preserves salient and sharp features. The source code and executable of the system are available for downloading.
INDEX TERMS
Surface from gradients, integrability, kernel methods, basis functions.
CITATION
Heung-Sun Ng, Tai-Pang Wu, Chi-Keung Tang, "Surface-from-Gradients without Discrete Integrability Enforcement: A Gaussian Kernel Approach", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.32, no. 11, pp. 2085-2099, November 2010, doi:10.1109/TPAMI.2009.183
REFERENCES
[1] A. Agrawal, R. Chellappa, and R. Raskar, "An Algebraic Approach to Surface Reconstruction from Gradient Fields," Proc. IEEE Int'l Conf. Computer Vision, vol. 1, pp. 174-181, 2005.
[2] A. Agrawal, R. Raskar, and R. Chellappa, "What Is the Range of Surface Reconstructions from a Gradient Field?" Proc. European Conf. Computer Vision, pp. 578-591, 2006.
[3] N. Aronszajn, "Theory of Reproducing Kernels," Trans. Am. Math. Soc., vol. 68, no. 3, pp. 337-404, 1950.
[4] A. Blake and A. Zisserman, Visual Reconstruction. MIT Press, 1987.
[5] F. Cucker and S. Smale, "On the Mathematical Foundations of Learning," Bull. Am. Math. Soc., vol. 39, no. 1, pp. 1-49, 2001.
[6] H.Q. Dinh, G. Turk, and G. Slabaugh, "Reconstructing Surfaces by Volumetric Regularization Using Radial Basis Functions," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 24, no. 10, pp. 1358-1371, Oct. 2002.
[7] L. Fang and D.C. Gossard, "Multidimensional Curve Fitting to Unorganized Data Points by Nonlinear Minimization," Computer-Aided Design, vol. 27, no. 1, pp. 48-58, Jan. 1995.
[8] R.T. Frankot and R. Chellappa, "A Method for Enforcing Integrability in Shape from Shading Algorithms," Proc. IEEE Int'l Conf. Computer Vision, pp. 118-128, 1987.
[9] F. Girosi, M. Jones, and T. Poggio, "Priors Stabilizers and Basis Functions: From Regularization to Radial, Tensor and Additive Splines," technical report, 1993.
[10] D.B. Goldman, B. Curless, A. Hertzmann, and S.M. Seitz, "Shape, Spatially-Varying BRDFs from Photometric Stereo," Proc. IEEE Int'l Conf. Computer Vision, pp. 341-348, 2005.
[11] R.C. Gonzalez and R.E. Woods, Digital Image Processing, third ed. Prentice Hall, 2008.
[12] M.J. Harker and P.L. O'Leary, "Least Squares Surface Reconstruction from Measured Gradient Fields," Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 1-7, 2008.
[13] B. Karaçali and W.E. Snyder, "Partial Integrability in Surface Reconstruction from a Given Gradient Field," Proc. Int'l Conf. Image Processing, pp. 525-528, 2002.
[14] B. Karaçali and W.E. Snyder, "Noise Reduction in Surface Reconstruction from a Given Gradient Field," Int'l J. Computer Vision, vol. 60, no. 1, pp. 25-44, Oct. 2004.
[15] J.J. Koenderink, "Pictorial Relief," Royal, vol. 356, no. 1740, pp. 1071-1086, 1998.
[16] P. Kovesi, "Shapelets Correlated with Surface Normals Produce Surfaces," Proc. IEEE Int'l Conf. Computer Vision, pp. 994-1001, 2005.
[17] J. Mercer, "Functions of Positive and Negative Type and Their Connection with the Theory of Integral Equation," Philosophical Trans. Royal Soc., 1909.
[18] C.A. Micchelli, "Interpolation of Scattered Data: Distance Matrices and Conditionally Positive Definite Functions," Constructive Approximation, vol. 2, no. 1, pp. 11-22, Dec. 1986.
[19] E.N. Mortensen and W.A. Barrett, "Intelligent Scissors for Image Composition," Proc. ACM SIGGRAPH '95, pp. 191-198, 1995.
[20] D. Nehab, S. Rusinkiewicz, J. Davis, and R. Ramamoorthi, "Efficiently Combining Positions and Normals for Precise 3D Geometry," Proc. ACM SIGGRAPH '05, vol. 24, no. 3, Aug. 2005.
[21] H.S. Ng, T.P. Wu, and C.K. Tang, "Surface-from-Gradients with Incomplete Data for Single View Modeling," Proc. IEEE Int'l Conf. Computer Vision, 2007.
[22] Y. Ohtake, A. Belyaev, M. Alexa, G. Turk, and H.-P. Seidel, "Multi-Level Partition of Unity Implicits," ACM Trans. Graphics, vol. 22, no. 3, pp. 463-470, 2003.
[23] Y. Ohtake, A. Belyaev, and H.-P. Seidel, "Ridge-Valley Lines on Meshes via Implicit Surface Fitting," Proc. ACM SIGGRAPH '04, vol. 23, no. 3, pp. 609-612, 2004.
[24] N. Petrovic, I. Cohen, B.J. Frey, R. Koetter, and T.S. Huang, "Enforcing Integrability for Surface Reconstruction Algorithms Using Belief Propagation in Graphical Models," Proc. IEEE Conf. Computer Vision and Pattern Recognition, vol. 1, no. 743, 2001.
[25] M. Prasad, A. Zisserman, and A.W. Fitzgibbon, "Single View Reconstruction of Curved Surfaces," Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 1345-1354, 2006.
[26] V.V. Savchenko, E.A. Pasko, O.G. Okunev, and T.L. Kunii, "Function Representation of Solids Reconstructed from Scattered Surface Points and Contours," Computer Graphics Forum, vol. 14, pp. 181-188, 1995.
[27] B. Schoelkopf and A. Smola, Learning with Kernels. MIT Press, 2002.
[28] T. Simchony, R. Chellappa, and M. Shao, "Direct Analytical Methods for Solving Poisson Equations in Computer Vision Problems," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 12, no. 5, pp. 435-446, May 1990.
[29] R. Szeliski, "Fast Surface Interpolation Using Hierarchical Basis Functions," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 12, no. 6, pp. 513-528, June 1990.
[30] D. Terzopoulos, "The Computation of Visible-Surface Representations," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 10, no. 4 pp. 417-438, July 1988.
[31] D. Terzopoulos, A. Witkin, and M. Kass, "Symmetry-Seeking Models for 3D Object Recognition," Int'l J. Computer Vision, vol. 1, no. 3, pp. 211-221, Oct. 1987.
[32] A.N. Tikhonov and V.Y. Arsenin, Solutions of Ill-Posed Problems. Vh Winston, 1977.
[33] G. Wahba, Spline Models for Observational Data. Cambridge Univ. Press, 1990.
[34] T.P. Wu and C.K. Tang, "Dense Photometric Stereo by Expectation Maximization," Proc. European Conf. Computer Vision, pp. 159-172, 2006.
[35] T.P. Wu and C.K. Tang, "Visible Surface Reconstruction from Normals with Discontinuity Consideration," Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 1793-1800, 2006.
[36] T.P. Wu, C.K. Tang, M.S. Brown, and H.Y. Shum, "Shapepalettes: Interactive Normal Transfer via Sketching," Proc. ACM SIGGRAPH '07, Aug. 2007.
[37] L. Zhang, G. Dugas-Phocion, J.S. Samson, and S.M. Seitz, "Single View Modeling of Free-Form Scenes," Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 990-997, 2001.
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