Abstract
The computation of multidimensional persistent Betti numbers for a sublevel filtration on a suitable topological space equipped with a ℝn-valued continuous filtering function can be reduced to the problem of computing persistent Betti numbers for a parameterized family of one-dimensional filtering functions. A notion of continuity for points in persistence diagrams exists over this parameter space excluding a discrete number of so-called singular parameter values. We have identified instances of nontrivial monodromy over loops in nonsingular parameter space. In other words, following cornerpoints of the persistence diagrams along nontrivial loops can result in them switching places. This has an important incidence, e.g., in computer-assisted shape recognition, as we believe that new, improved distances between shape signatures can be defined by considering continuous families of matchings between cornerpoints along paths in nonsingular parameter space. Considering that nonhomotopic paths may yield different matchings will therefore be necessary. In this contribution we will discuss theoretical properties of the monodromy in question and give an example of a filtration in which it can be shown to be nontrivial.
Chapter PDF
Similar content being viewed by others
References
Bendich, P., Edelsbrunner, H., Kerber, M.: Computing robustness and persistence for images. IEEE Transactions on Visualization and Computer Graphics 16(6), 1251–1260 (2010)
Biasotti, S., Cerri, A., Frosini, P., Giorgi, D., Landi, C.: Multidimensional size functions for shape comparison. J. Math. Imaging Vision 32, 161–179 (2008)
Biasotti, S., De Floriani, L., Falcidieno, B., Frosini, P., Giorgi, D., Landi, C., Papaleo, L., Spagnuolo, M.: Describing shapes by geometrical-topological properties of real functions. ACM Comput. Surv. 40(4), 1–87 (2008)
Bronstein, A., Bronstein, M., Kimmel, R.: Numerical Geometry of Non-Rigid Shapes, 1st edn. Springer Publishing Company, Incorporated (2008)
Cagliari, F., Di Fabio, B., Ferri, M.: One-dimensional reduction of multidimensional persistent homology. Proc. Amer. Math. Soc. 138, 3003–3017 (2010)
Carlsson, G.: Topology and data. Bulletin of the American Mathematical Society 46(2), 255–308 (2009)
Carlsson, G., Zomorodian, A.: The theory of multidimensional persistence. Discr. Comput. Geom. 42(1), 71–93 (2009)
Cerri, A., Di Fabio, B., Ferri, M., Frosini, P., Landi, C.: Betti numbers in multidimensional persistent homology are stable functions. Math. Method. Appl. Sci. (in press), doi:10.1002/mma.2704
Cerri, A., Ferri, M., Giorgi, D.: Retrieval of trademark images by means of size functions. Graph. Models 68(5), 451–471 (2006)
Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discr. Comput. Geom. 37(1), 103–120 (2007)
Di Fabio, B., Landi, C.: Persistent homology and partial similarity of shapes. Pattern Recognition Letters 33, 1445–1450 (2012)
Edelsbrunner, H., Harer, J.: Computational Topology: An Introduction. American Mathematical Society (2009)
Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discrete Comput. Geom. 28(4), 511–533 (2002)
Edelsbrunner, H., Symonova, O.: The adaptive topology of a digital image. In: 2012 Ninth International Symposium on Voronoi Diagrams in Science and Engineering (ISVD), pp. 41–48 (2012)
Frosini, P., Landi, C.: Size functions and formal series. Appl. Algebra Engrg. Comm. Comput. 12(4), 327–349 (2001)
Paris, S., Durand, F.: A topological approach to hierarchical segmentation using mean shift. In: IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2007, pp. 1–8 (2007)
Robins, V., Wood, P.J., Sheppard, A.P.: Theory and algorithms for constructing discrete Morse complexes from grayscale digital images. IEEE Trans. Pattern Anal. Mach. Intell. 33(8), 1646–1658 (2011)
Zheng, Y., Gu, S., Edelsbrunner, H., Tomasi, C., Benfey, P.: Detailed reconstruction of 3D plant root shape. In: Metaxas, D.N., Quan, L., Sanfeliu, A., Gool, L.J.V. (eds.) ICCV. pp. 2026–2033. IEEE (2011)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cerri, A., Ethier, M., Frosini, P. (2013). A Study of Monodromy in the Computation of Multidimensional Persistence. In: Gonzalez-Diaz, R., Jimenez, MJ., Medrano, B. (eds) Discrete Geometry for Computer Imagery. DGCI 2013. Lecture Notes in Computer Science, vol 7749. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37067-0_17
Download citation
DOI: https://doi.org/10.1007/978-3-642-37067-0_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-37066-3
Online ISBN: 978-3-642-37067-0
eBook Packages: Computer ScienceComputer Science (R0)