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A Closed-Form Solution to Tensor Voting: Theory and Applications
Aug. 2012 (vol. 34 no. 8)
pp. 1482-1495
Jiaya Jia, Dept. of Comput. Sci. & Eng., Chinese Univ. of Hong Kong, Shatin, China
Sai-Kit Yeung, Pillar of Inf. Syst. Technol. & Design, Singapore Univ. of Technol. & Design, Singapore, Singapore
Tai-Pang Wu, Enterprise & Consumer Electron., Hong Kong Appl. Sci. & Technol. Res. Inst., Shatin, China
Chi-Keung Tang, Dept. of Comput. Sci. & Eng., Hong Kong Univ. of Sci. & Technol., Hong Kong, China
G. Medioni, Univ. of Southern California, Los Angeles, CA, USA
We prove a closed-form solution to tensor voting (CFTV): Given a point set in any dimensions, our closed-form solution provides an exact, continuous, and efficient algorithm for computing a structure-aware tensor that simultaneously achieves salient structure detection and outlier attenuation. Using CFTV, we prove the convergence of tensor voting on a Markov random field (MRF), thus termed as MRFTV, where the structure-aware tensor at each input site reaches a stationary state upon convergence in structure propagation. We then embed structure-aware tensor into expectation maximization (EM) for optimizing a single linear structure to achieve efficient and robust parameter estimation. Specifically, our EMTV algorithm optimizes both the tensor and fitting parameters and does not require random sampling consensus typically used in existing robust statistical techniques. We performed quantitative evaluation on its accuracy and robustness, showing that EMTV performs better than the original TV and other state-of-the-art techniques in fundamental matrix estimation for multiview stereo matching. The extensions of CFTV and EMTV for extracting multiple and nonlinear structures are underway.

[1] S. Arya and D.M. Mount, "Approximate Nearest Neighbor Searching," Proc. ACM-SIAM Symp. Discrete Algorithms, pp. 271-280, 2003.
[2] J. Bilmes, "A Gentle Tutorial on the EM Algorithm and Its Application to Parameter Estimation for Gaussian Mixture and Hidden Markov Models," Technical Report ICSI-TR-97-021, ICSI, 1997.
[3] H. Chen and P. Meer, "Robust Regression with Projection Based m-Estimators," Proc. Ninth IEEE Int'l Conf. Computer Vision, vol. 2, pp. 878-885, 2003.
[4] O. Chum and J. Matas, "Optimal Randomized Ransac," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 30, no. 8, pp. 1472-1482, Aug. 2008.
[5] D. Comaniciu and P. Meer, "Mean Shift: A Robust Approach toward Feature Space Analysis," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 24, no. 5, pp. 603-619, May 2002.
[6] R. Dahyot, "Statistical Hough Transform," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 31, no. 8, pp. 1502-1509, Aug. 2009.
[7] M.A. Fischler and R.C. Bolles, "Random Sample Consensus: A Paradigm for Model Fitting with Applications to Image Analysis and Automated Cartography," Comm. ACM, vol. 24, pp. 381-395, 1981.
[8] D. Forsyth and J. Ponce, Computer Vision: A Modern Approach. Prentice Hall, 2003.
[9] E. Franken, M. van Almsick, P. Rongen, L. Florack, and B. ter Haar Romeny, "An Efficient Method for Tensor Voting Using Steerable Filters," Proc. Ninth European Conf. Computer Vision, vol. 4, pp. 228-240, 2006.
[10] Y. Furukawa and J. Ponce, "Accurate, Dense, and Robust Multiview Stereopsis," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 32, no. 8, pp. 1362-1376, Aug. 2010.
[11] B. Georgescu, I. Shimshoni, and P. Meer, "Mean Shift Based Clustering in High Dimensions: A Texture Classification Example," Proc. Ninth IEEE Int'l Conf. Computer Vision, pp. 456-463, 2003.
[12] R. Hartley, "In Defense of the Eight-Point Algorithm," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 19, no. 6, pp. 580-593, June 1997.
[13] P. Hough, "Machine Analysis of Bubble Chamber Pictures," Proc. Int'l Conf. High Energy Accelerators and Instrumentation, 1959.
[14] P.J. Huber, Robust Statistics. John Wiley & Sons, 1981.
[15] K. Lee, P. Meer, and R. Park, "Robust Adaptive Segmentation of Range Images," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 20, no. 2, pp. 200-205, Feb. 1998.
[16] M. Lourakis and A. Argyros, "The Design and Implementation of a Generic Sparse Bundle Adjustment Software Package Based on the Levenberg-Marquardt Algorithm," Technical Report 340, ICSFORTH, 2004.
[17] D. Lowe, "Distinctive Image Features from Scale-Invariant Keypoints," Int'l J. Computer Vision, vol. 60, no. 2, pp. 91-110, Nov. 2004.
[18] G.J. McLachlan and T. Krishnan, EM Algorithms and Extension. Elsevier, 1997.
[19] G. Medioni, M.S. Lee, and C.K. Tang, A Computational Framework for Segmentation and Grouping. Elsevier, 2000.
[20] P. Meer, "Robust Techniques for Computer Vision," Emerging Topics in Computer Vision, chapter 4, Prentice Hall, 2004.
[21] P. Mordohai and G. Medioni, "Dimensionality Estimation, Manifold Learning and Function Approximation Using Tensor Voting," J. Machine Learning Research, vol. 11, pp. 411-450, Jan. 2010.
[22] P. Rousseeuw, "Least Median of Squares Regression," J. Am. Statistics Assoc., vol. 79, pp. 871-880, 1984.
[23] P. Rousseeuw, Robust Regression and Outlier Detection. Wiley, 1987.
[24] K. Sim and R. Hartley, "Removing Outliers Using the Linfty Norm," Proc. IEEE Conf. Computer Vision and Pattern Recognition, vol. 1, pp. 485-494, 2006.
[25] N. Snavely, S. Seitz, and R. Szeliski, "Modeling the World from Internet Photo Collections," Int'l J. Computer Vision, vol. 80, no. 2, pp. 189-210, Nov. 2008.
[26] R. Subarao and P. Meer, "Beyond Ransac: User Independent Robust Regression," Proc. Workshop 25 Years of Random Sample Consensus, June 2006.
[27] W.S. Tong, C.K. Tang, and G. Medioni, "Simultaneous Two-View Epipolar Geometry Estimation and Motion Segmentation by 4D Tensor Voting," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 26, no. 9, pp. 1167-1184, Sept. 2004.
[28] H. Wang and D. Suter, "Robust Adaptive-Scale Parametric Model Estimation for Computer Vision," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 26, no. 11, pp. 1459-1474, Nov. 2004.
[29] T.P. Wu, S.K. Yeung, J. Jia, and C.K. Tang, "Quasi-Dense 3D Reconstruction Using Tensor-Based Multiview Stereo," Proc. IEEE Conf. Computer Vision and Pattern Recognition, 2010.

Index Terms:
tensors,computer vision,expectation-maximisation algorithm,image matching,Markov processes,parameter estimation,sampling methods,stereo image processing,multiview stereo matching,closed-form solution,CFTV,structure-aware tensor,salient structure detection,outlier attenuation,tensor voting convergence,Markov random field,MRF,MRFTV,expectation maximization algorithm,EM,single linear structure,robust parameter estimation,EMTV algorithm,fitting parameters,tensor parameters,random sampling,robust statistical techniques,fundamental matrix estimation,Tensile stress,Robustness,Closed-form solutions,Three dimensional displays,Vectors,Estimation,Convergence,multiview stereo.,Tensor voting,closed-form solution,structure inference,parameter estimation
Citation:
Jiaya Jia, Sai-Kit Yeung, Tai-Pang Wu, Chi-Keung Tang, G. Medioni, "A Closed-Form Solution to Tensor Voting: Theory and Applications," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 34, no. 8, pp. 1482-1495, Aug. 2012, doi:10.1109/TPAMI.2011.250
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