In the Computer Vision community it is a common belief that higher order smoothness, such as curvature, should be modeled using higher order interactions. For example, 2nd order derivatives for deformable (active) contours are represented by triple cliques. Similarly, the 2nd order regularization methods in stereo predominantly use MRF models with scalar (1D) disparity labels and triple clique interactions. In this paper we give an overview of an energy minimization framework for tangential approximation of surfaces developed in [21, 22]. The framework uses higher dimensional labels to encode second order smoothness with pairwise interactions. Hence, many generic optimization algorithms (e.g. message passing, graph cut, etc.) can be used to optimize the proposed regularization functionals. The accuracy of our approach for representing curvature is demonstrated by theoretical and empirical results on real data sets from multi-view reconstruction and stereo.
@article{ACIRM_2013__3_1_51_0,
author = {Carl Olsson and Yuri Boykov},
title = {Tangential {Approximation} of {Surfaces}},
journal = {Actes des rencontres du CIRM},
pages = {51--60},
publisher = {CIRM},
volume = {3},
number = {1},
year = {2013},
doi = {10.5802/acirm.54},
zbl = {06938602},
language = {en},
url = {https://acirm.centre-mersenne.org/articles/10.5802/acirm.54/}
}
Carl Olsson; Yuri Boykov. Tangential Approximation of Surfaces. Actes des rencontres du CIRM, Volume 3 (2013) no. 1, pp. 51-60. doi : 10.5802/acirm.54. https://acirm.centre-mersenne.org/articles/10.5802/acirm.54/
[1] Computing and rendering point set surfaces, IEEE Transactions on Visualization and Computer Graphics, Volume 9 (2003), pp. 3-15 | DOI
[2] Multiway cut for stereo and motion with slanted surfaces, International Conference on Computer Vision (1999)
[3] Visual Reconstruction, MIT Press, Cambridge, USA, 1987 | DOI
[4] Pseudo-boolean optimization, Discrete applied mathematics, Volume 123 (2002) no. 1, pp. 155-225 | DOI | MR
[5] Fast Approximate Energy Minimization via Graph Cuts, IEEE Transations on Pattern Analysis and Machine Intelligence (2001) | DOI
[6] Convex Relaxation of a Class of Vertex Penalizing Functionals, Journal of Mathematical Imaging and Vision (2012), pp. 1-25 | Zbl
[7] Epi-convergence of discrete elastica, Applicable Analysis, Volume 79 (2001) no. 1-2, pp. 137-171 | DOI | MR | Zbl
[8] Fast Approximate Energy Minimization with Label Costs, International Journal of Computer Vision, Volume 96 (2012) no. 1, pp. 1-27 | DOI | MR | Zbl
[9] Fast Global Optimization of Curvature, Proc. of CVPR 2010 (2010)
[10] Efficient Belief Propagation for Early Vision, Int. J. Comput. Vision, Volume 70 (2006) no. 1, pp. 41-54 | DOI
[11] Multiple View Geometry in Computer Vision, Cambridge University Press, 2004 | DOI | Zbl
[12] Consolidation of Unorganized Point Clouds for Surface Reconstruction, ACM Trans. on Graphics (2009) | DOI
[13] Energy-Based Geometric Multi-model Fitting, International Journal of Computer Vision, Volume 97 (2012) no. 2, pp. 123-147 | DOI | Zbl
[14] Multi-camera Scene Reconstruction via Graph Cuts, European Conf. on Computer Vision, Volume III (2002), pp. 82-96 | Zbl
[15] Convergent Tree-Reweighted Message Passing for Energy Minimization, IEEE Trans. Pattern Anal. Mach. Intell., Volume 28 (2006), pp. 1568-1583 | DOI
[16] Anisotropic smoothing of point sets, Computer Aided Geometric Design, Volume 22 (2005), 2005 pages | MR | Zbl
[17] Fusion Moves for Markov Random Field Optimization, IEEE Trans. Pattern Anal. Mach. Intell., Volume 32 (2010) no. 8, pp. 1392-1405 | DOI
[18] Differential geometric inference in surface stereo, Pattern Analysis and Machine Intelligence, IEEE Transactions on, Volume 32 (2010) no. 1, pp. 72-86 | DOI
[19] Parameterization-free Projection for Geometry Reconstruction, ACM Trans. om Graphics. (2007) | DOI
[20] Over-Parameterized Variational Optical Flow, Int. J. Comput. Vision, Volume 76 (2008) no. 2, pp. 205-216
[21] Curvature-based regularization for surface approximation, IEEE Conference on Computer Vision and Pattern Recognition (2012)
[22] In Defense of 3D-Label Stereo, IEEE Conference on Computer Vision and Pattern Recognition (2013)
[23] Over-parameterized optical flow using a stereoscopic constraint, SSVM (2011)
[24] Optimizing Binary MRFs via Extended Roof Duality, IEEE conf. on Computer Vision and Pattern Recognition (2007)
[25] Curvature Regularity for Region-based Image Segmentation and Inpainting: A Linear Programming Relaxation, Int. Conf. on Computer Vision (2009)
[26] Curvature Regularization for Curves and Surfaces in a Global Optimization Framework, EMMCVPR (2011), pp. 205-218
[27] Modeling surfaces of arbitrary topology with dynamic particles, Computer Vision and Pattern Recognition, 1993. Proceedings CVPR ’93., 1993 IEEE Computer Society Conference on (1993)
[28] Non-Rigid Structure from Locally-Rigid Motion, IEEE Int. Conf. of Computer Vision and Pattern Recognition (2010)
[29] Multiview Stereo via Volumetric Graph-Cuts and Occlusion Robust Photo-Consistency, IEEE Trans. Pattern Anal. Mach. Intell., Volume 29 (2007) no. 12, pp. 2241-2246 | DOI
[30] Global Stereo Reconstruction under Second Order Smoothness Priors, IEEE Transactions on Pattern Analysis and Machine Intelligence (2009) | DOI
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