Awesome list for polyhedra recovery: A curated list of awesome papers about polyhedra recovery.
Recovery based on support function measurements Support function estimation means the reconstruction of convex body from the set of its support function measurements.
Least squares estimator for support function values
Type
Title
Comment
Prince J.L., Willsky A.S. Reconstructing convex sets from support line measurements // IEEE Transactions on Pattern Analysis and Machine Intelligence. Apr. 1990. 12(4). P. 377-389 2D reconstruction problem for uniformly distributed angles
Prince J.L. Geometric model-based estimation from projections. Doctoral dissertation, Massachusetts Institute of Technology, 1988. Prince's PhD thesis with included theory from above [Prince & Willsky 1990 ].
Lele A.S., Kulkarni S.R., Willsky A.S. Convex-polygon estimation from support-line measurements and applications to target reconstruction from laser-radar data // JOSA A. 1992 Oct 1;9(10):1693-714. The generalization of [Prince & Willsky 1990 ] to non-uniform distribution of angles
Karl W.C., Kulkarni S.R., Verghese G.C., Willsky A.S. Local tests for consistency of support hyperplane data // Journal of Mathematical Imaging and Vision. 1996 Jun 1;6(2-3):249-67. 3D problem, constraints consistency criterion. No testing on real data.
Gregor J., Rannou F.R. Three‐dimensional support function estimation and application for projection magnetic resonance imaging // International journal of imaging systems and technology. 2002 Jan 1;12(1):43-50. 3D real data testing of [Karl et al. 1996 ].
Gregor J., Rannou F. Least-squares framework for projection MRI reconstruction // In Medical Imaging 2001: Image Processing 2001 Jul 3 (Vol. 4322, pp. 888-899). International Society for Optics and Photonics. Details of above paper.
Other consistent estimators Statistical works: LSE for tangient points, tractable and adaptive estimators
Type
Title
Comment
Gardner R.J., Kiderlen M., Milanfar P. Convergence of algorithms for reconstructing convex bodies and directional measures // The Annals of Statistics. 2006 Jun 1:1331-74. First paper about speed of convergence of the estimation algorithm.
Gardner R.J., Kiderlen M. A new algorithm for 3D reconstruction from support functions // IEEE transactions on pattern analysis and machine intelligence. 2009 Mar;31(3):556-62. Another, more efficient way to solve consistency problem in 3D
Guntuboyina A. Optimal rates of convergence for convex set estimation from support functions // The Annals of Statistics. 2012;40(1):385-411. Optimal rates, related to [Gardner et al. 2006 ].
Cai T., Guntuboyina A., Wei Y. Adaptive estimation of planar convex sets // arXiv preprint arXiv:1508.03744. 2015 Aug. Older, draft preprint version of the below paper.
Cai T.T., Guntuboyina A., Wei Y. Adaptive estimation of planar convex sets // The Annals of Statistics. 2018;46(3):1018-49. Estimate that has optimal rate in point-wise and body-wise sense.
Soh Y.S., Chandrasekaran V . Fitting tractable convex sets to support function evaluations // arXiv preprint arXiv:1903.04194. 2019 Mar 11. Fitting affine images of fixed bodies (e.g. simplicies)
yssoh/cvxreg Code for the above paper
Soh Y.S . Fitting Convex Sets to Data: Algorithms and Applications. Doctoral dissertation, California Institute of Technology, 2019 PhD thesis containing above paper's results and some more
Ghosh A. et al . Max-Affine Regression: Provable, Tractable, and Near-Optimal Statistical Estimation // arXiv preprint arXiv:1906.09255. 2019 Jun 21. General algorithm for max-affine regression, which is a generalization of SFE
Gregor et al (2002) writes about the usage of KKVW's algorithm of support function estimation for the problem of focus of attention estimation in MRI. However, in further research these authors propose other techinuques, that are less precise, but are more efficient. They are:
Some related works that research probing of convex bodies. Probing is equal to support function measurement in the dual space. See Greschak's thesis for details.
Convex support estimation TODO: more papers about this area
Reconstruction from self shadows and reflections
Type
Title
Comment
Hatzitheodorou M., Kender J.R . An optimal algorithm for the derivation of shape from shadows // Proceedings CVPR'88: The Computer Society Conference on Computer Vision and Pattern Recognition 1988 Jun 5 (pp. 486-491) ?
Savarese S . Shape reconstruction from shadows and reflections. PhD dissertation. California Institute of Technology, 2005 Concave objects reconstruction from self-shadows, and specular shape reconstruction.
Savarese S., Chen M., Perona P . Second order local analysis for 3d reconstruction of specular surfaces // Proceedings of First International Symposium on 3D Data Processing Visualization and Transmission 2002 Jun 19 (pp. 356-361) part of above thesis
Savarese S., Chen M., Perona P . Recovering local shape of a mirror surface from reflection of a regular grid // European Conference on Computer Vision 2004 May 11 (pp. 468-481) part of above thesis
Savarese S., Li F.F., Perona P . Can we see the shape of a mirror? // Journal of Vision. 2003 Oct 1;3(9):74-. part of above thesis
Savarese S., Perona P . Local analysis for 3d reconstruction of specular surfaces // Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR 2001 2001 Dec 8 (Vol. 2, pp. II-II) part of above thesis
Savarese S., Perona P . Local analysis for 3d reconstruction of specular surfaces — part ii // European Conference on Computer Vision 2002 May 28 (pp. 759-774). part of above thesis
Savarese S., Rushmeier H., Bernardini F., Perona P . Shadow carving // Proceedings Eighth IEEE International Conference on Computer Vision. ICCV 2001 2001 Jul 7 (Vol. 1, pp. 190-197) part of above thesis
Savarese S., Rushmeier H., Bernardini F., Perona P . Implementation of a shadow carving system for shape capture // Proceedings of First International Symposium on 3D Data Processing Visualization and Transmission 2002 Jun 19 (pp. 12-23) part of above thesis
Savarese S. et al . 3d reconstruction by shadow carving: Theory and practical evaluation // International journal of computer vision. 2007 Mar 1;71(3):305-36. part of above thesis
Reconstruction from silhouettes Scholar query: polyhedron reconstruction from silhouette
Bayesian inference for inverse problems in tomography Scholar query: bayesian inference for polyhedra reconstruction
