Algebraic topology for computer vision

Computer Vision

Volumn , Issue , 2011, Pages 239-268

  Freedman, Daniel ^a   Chen, Chao ^b  

^a HEWLETT PACKARD LABORATORIES   (United States)

^b VIENNA UNIVERSITY OF TECHNOLOGY   (Austria)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84891997271     PISSN: None     EISSN: None     Source Type: Book    
DOI: None     Document Type: Chapter

References (49)

1. H. Edelsbrunner, D. Letscher, and A. Zomorodian. Topological persistence and simplif cation. Discrete and Computational Geometry, 28(4):511-533, 2002.
2. H. Edelsbrunner. Biological applications of computational topology. Handbook of Discrete and Computational Geometry, pages 1395-1412, 2004.
3. D. Cohen-Steiner, H. Edelsbrunner, and D. Morozov. Vines and vineyards by updating persistence in linear time. In Proceedings of the Twenty-Second Annual Symposium on Computational Geometry, pages 119-126, 2006.
4. R. Ghrist and A. Muhammad. Coverage and hole-detection in sensor networks via homology. In Proceedings of the 4th International Symposium on Information Processing in Sensor Networks, 2005.
5. V. de Silva, R. Ghrist, and A. Muhammad. Blind swarms for coverage in 2-D. In Robotics: Science and Systems, 2005.
6. T.K. Dey, K. Li, J. Sun, and D. Cohen-Steiner. Computing geometry-aware handle and tunnel loops in 3D models. ACM Transactions on Graphics (TOG), 27(3):45:1-45:9, 2008.
7. H. Edelsbrunner. Surface tiling with differential topology. In Proceedings of the Third Eurographics Symposium on Geometry Processing, pages 9-11, 2005.
8. J. R. Munkres. Elements of Algebraic Topology. Addison-Wesley, Redwood City, California, 1984.
9. A. Hatcher. Algebraic Topology. Cambridge University Press, 2002.
10. H. Edelsbrunner and J. Harer. Persistent homology - a survey. In Surveys on Discrete and Computational Geometry. Twenty Years Later. Contemporary Mathematics, volume 453, pages 257-282. American Mathematical Society, 2008.
11. R. Ghrist. Barcodes: The persistent topology of data. Bulletin of the American Mathematical Society, 45(1):61-75, 2008.
12. G. Vegter. Computational topology. Handbook of Discrete and Computational Geometry, pages 719-742, 2004.
13. A. Zomorodian. Computational topology, volume 2 of Algorithms and Theory of Computation Handbook, chapter 3. Chapman & Hall/CRC, 2010.
14. G. Carlsson. Topology and data. Bulletin of the American Mathematical Society, 46(2):255-308, 2009.
15. R. Kannan and A. Bachem. Polynomial algorithms for computing the Smith and Hermite normal forms of an integer matrix. SIAM Journal on Computing, 8:499-507, 1979.
16. T. Kaczyński, K.M.Mischaikow, and M.Mrozek. Computational Homology. Springer Verlag, 2004.
17. Afra Zomorodian and Gunnar Carlsson. Computing persistent homology. Discrete and Computational Geometry, 33(2):249-274, 2005.
18. Afra Zomorodian. Topology for Computing. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, 2005.
19. D. Cohen-Steiner, H. Edelsbrunner, and J. Harer. Stability of persistence diagrams. Discrete and Computational Geometry, 37(1):103-120, 2007.
20. F. Chazal, D. Cohen-Steiner, M. Glisse, L.J. Guibas, and S.Y. Oudot. Proximity of persistence modules and their diagrams. In Proceedings of the 25th Annual Symposium on Computational Geometry, pages 237-246, 2009.
21. A. Zomorodian and G. Carlsson. Localized homology. Computational Geometry, 41(3):126-148, 2008.
22. D. Zhang and G. Lu. Review of shape representation and description techniques. Pattern Recognition, 37(1):1-19, 2004.
23. J.W.H. Tangelder and R.C. Veltkamp. A survey of content based 3D shape retrieval methods. Multimedia Tools and Applications, 39(3):441-471, 2008.
24. G. Carlsson, A. Zomorodian, A. Collins, and L. Guibas. Persistence barcodes for shapes. International Journal of Shape Modeling, 11(2):149-187, 2005.
25. A. Collins, A. Zomorodian, G. Carlsson, and L.J. Guibas. A barcode shape descriptor for curve point cloud data. Computers & Graphics, 28(6):881-894, 2004.
26. J. Leray. Sur la forme des espaces topologiques et sur les points f xes des repŕesentations. J. Math. Pures Appl., IX. Śer., 24:95-167, 1945.
27. F. Chazal, D. Cohen-Steiner, L.J. Guibas, F. Ḿemoli, and S.Y. Oudot. Gromov-Hausdorff stable signatures for shapes using persistence. In Computer Graphics Forum, volume 28, pages 1393-1403, 2009.
28. D.J. Field. Relations between the statistics of natural images and the response properties of cortical cells. Journal of the Optical Society of America A, 4(12):2379-2394, 1987.
29. B. Olshausen and D. Field. Natural image statistics and eff cient coding. Network: Computation in Neural Systems, 7(2):333-339, 1996.
30. A. Van der Schaaf and JH Van Hateren. Modelling the power spectra of natural images: statistics and information. Vision Research, 36(17):2759-2770, 1996.
31. J. Huang and D. Mumford. Statistics of natural images and models. In Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, volume 1, pages 541-547, 1999.
32. A.B. Lee, K.S. Pedersen, and D. Mumford. The nonlinear statistics of high-contrast patches in natural images. International Journal of Computer Vision, 54(1):83-103, 2003.
33. Y. Weiss and WT Freeman. What makes a good model of natural images? In IEEE Conference on Computer Vision and Pattern Recognition, 2007. CVPR'07, pages 1-8, 2007.
34. T.S. Cho, M. Butman, S. Avidan, and W.T. Freeman. The patch transform and its applications to image editing. In IEEE Conference on Computer Vision and Pattern Recognition, 2008. CVPR 2008, pages 1-8, 2008.
35. G. Carlsson, T. Ishkhanov, V. De Silva, and A. Zomorodian. On the local behavior of spaces of natural images. International Journal of Computer Vision, 76(1):1-12, 2008.
36. G. Singh, F. Memoli, T. Ishkhanov, G. Sapiro, G. Carlsson, and D.L. Ringach. Topological analysis of population activity in visual cortex. Journal of Vision, 8(8):1-18, 2008.
37. D. Attali, M. Glisse, S. Hornus, F. Lazarus, and D. Morozov. Persistence-sensitive simplif cation of functions on surfaces in linear time. In Proceedings of the TopoInVis Workshop, 2009.
38. J.W. Milnor. Morse Theory. Princeton University Press, 1963.
39. Y. Matsumoto. An Introduction to Morse Theory. American Mathematical Society, 2002.
40. H. Edelsbrunner, D. Morozov, and V. Pascucci. Persistence-sensitive simplif cation functions on 2-manifolds. In Proceedings of the Twenty-Second Annual Symposium on Computational Geometry, pages 127-134, 2006.
41. K. Fukunaga and L. Hostetler. The estimation of the gradient of a density function, with applications in pattern recognition. IEEE Transactions on Information Theory, 21(1):32-40, 1975.
42. Y. Cheng. Mean shift, mode seeking, and clustering. IEEE Transactions on Pattern Analysis and Machine Intelligence, 17(8):790-799, 1995.
43. D. Comaniciu and P. Meer. Mean shift: a robust approach toward feature space analysis. IEEE Transactions on Pattern Analysis and Machine Intelligence, pages 603-619, 2002.
44. F. Chazal, L.J. Guibas, S.Y. Oudot, and P. Skraba. Analysis of scalar f elds over point cloud data. In Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1021-1030. Society for Industrial and Applied Mathematics Philadelphia, PA, USA, 2009.
45. S. Paris and F. Durand. A topological approach to hierarchical segmentation using mean shift. In IEEE Conference on Computer Vision and Pattern Recognition, 2007. CVPR'07, pages 1-8, 2007.
46. F. Chazal and S.Y. Oudot. Towards persistence-based reconstruction in Euclidean spaces. In Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry, pages 232-241, 2008.
47. H. Edelsbrunner, J. Harer, and A. Zomorodian. Hierarchical Morse-Smale complexes for piecewise linear 2-manifolds. Discrete and Computational Geometry, 30(1):87- 107, 2003.
48. P.T. Bremer, B. Hamann, H. Edelsbrunner, and V. Pascucci. A topological hierarchy for functions on triangulated surfaces. IEEE Transactions on Visualization and Computer Graphics, 10(4):385-396, 2004.
49. A. Gyulassy, V. Natarajan, V. Pascucci, P.T. Bremer, and B. Hamann. Topology-based simplif cation for feature extraction from 3d scalar f elds. In Proceedings of the IEEE Conference on Visualization, pages 535-542, 2005.