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Issue No.06 - November/December (2010 vol.16)
pp: 1251-1260
ABSTRACT
We are interested in 3-dimensional images given as arrays of voxels with intensity values. Extending these values to acontinuous function, we study the robustness of homology classes in its level and interlevel sets, that is, the amount of perturbationneeded to destroy these classes. The structure of the homology classes and their robustness, over all level and interlevel sets, can bevisualized by a triangular diagram of dots obtained by computing the extended persistence of the function. We give a fast hierarchicalalgorithm using the dual complexes of oct-tree approximations of the function. In addition, we show that for balanced oct-trees, thedual complexes are geometrically realized in $R^3$ and can thus be used to construct level and interlevel sets. We apply these tools tostudy 3-dimensional images of plant root systems.
INDEX TERMS
voxel arrays, oct-trees, persistent homology, persistence diagrams, level sets, robustness, approximations, plant roots
CITATION
Paul Bendich, Herbert Edelsbrunner, Michael Kerber, "Computing Robustness and Persistence for Images", IEEE Transactions on Visualization & Computer Graphics, vol.16, no. 6, pp. 1251-1260, November/December 2010, doi:10.1109/TVCG.2010.139
REFERENCES
[1] C. L. Bajaj, A. Gillette, and S. Goswami, Topology based selection and curation of level sets. In Topology-Based Methods in Visualization II, H.-C. Hege, K. Polthier, G. Scheuermann (eds.), Springer-Verlag, 45–58, 2009.
[2] C. L. Bajaj, V. Pascucci, and D. R. Schikore, The contour spec-trum. In "Proc. 8th IEEE Conf. Visualization, 1997", 167–173.
[3] P. Bendich, H. Edelsbrunner, D. Morozov, and A. Patel, Robustness of level and interlevel sets. Manuscript, IST Austria, Klosterneuburg, Austria, 2009.
[4] M. Bern, D. Eppstein, and J. Gilbert, Provably good mesh generation. J. Comput Sys. Sci. 48 (1994), 384–409.
[5] G. Carlsson, T. Ishkhanov, V. de Silva, and A. Zomorodian, On the local behavior of spaces of local images. Internat. J. Comput Vision 76 (2008), 1–12.
[6] A. Cerri, M. Ferri, and D. Giorgi, Retrieval of trademark images by means of size functions. Graphical Models 68 (2006), 451–471.
[7] M. K. Chung, P. Bubenik, and P. T. Kim, Persistence diagrams of cortical surface data. In Information Processing in Medical Imaging, Springer-Verlag, LNCS 5636, 2009, 386–397.
[8] V. deSilvaand, R. Ghrist, Coverage in sensor networks via persistent homology. Alg. Geom. Topology 7 (2007), 339–358.
[9] M.-L. Dequèant, S. Ahnert, H. Edelsbrunner, T. M. A. Fink, E. F. Glynn, G. Hattem, A. Kudlicki, Y. Mileyko, J. Morton, A. R. Mushegian, L. Pachter, M. Rowicka, A. Shiu, B. Sturm-fels AND O. Pourqui, É. Comparison of pattern detection methods in microarray time series of the segmentation clock. PLoS ONE 3 (2008), e2856, doi:10.1371/journal.pone.000 2856.
[10] H. Edelsbrunner, and J. L. Harer, Computational Topology. An Introduction. Amer. Math. Soc., Providence, Rhode Island, 2009.
[11] H. Edelsbrunner, D. Morozov, and A. Patel Quantifying transversality by measuring the robustness of intersections. Manuscript, Dept. Comput. Sci., Duke Univ., Durham, North Carolina, 2009.
[12] A. Gyulassy, V. Natarajan, V. Pascucci, P.-T. Bremer, and B. Ham, ANN. A, topological approach to simplification of three-dimensional scalar functions. IEEE Trans. Vis. Comput. Graph. 12 (2006), 474–484.
[13] D. Morozov, Persistence algorithm takes cubic time in worst case. In BioGeometry News, Dept. Comput. Sci., Duke Univ., Durham, North Carolina, 2005.
[14] J. R. Munkres, Elements of Algebraic Topology. Perseus, Cambridge, Massachusetts, 1984.
[15] T. S. Newman and H. Yi, A survey of the marching cube algorithm. Computers and Graphics 30 (2006), 854–879.
[16] V. Pascucci, G. Scorzelli, P.-T. Bremer, and A. Mascaren-has, Robust on-line computation of Reeb graphs: simplicity and speed. ACM Trans. Graphics 26 (2007), 58.
[17] H. Samet, The Design and Analysis of Spatial Data Structures. Addison-Wesley, Reading, Massachusetts, 1990.
[18] M. Sonka, V. Hlavac, and R. Boyle, Image Processing, Analysis and Machine Vision. Second edition, PWS Publishing, Pacific Grove, California, 1999.
[19] M. van Krefeld, R. van Oostrum, C. L. Bajaj, V. Pascucci, and D. R. Schikore, Contour trees and small seed sets for isosurface traversal. In "Proc. 13th Ann. Sympos. Comput. Geom., 1997", 212–220.
[20] Y. Wang, P. K. Agarwal, P. Brown, H. Edelsbrunner, and J. Rudolph, Coarse and reliable geometric alignment for protein docking. In "Proc. Pacific Sympos. Biocomput., 2005", 65–75.
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