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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.

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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
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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