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Object-oriented persistent homology. (English) Zbl 1349.55004

Summary: Persistent homology provides a new approach for the topological simplification of big data via measuring the life time of intrinsic topological features in a filtration process and has found its success in scientific and engineering applications. However, such a success is essentially limited to qualitative data classification and analysis. Indeed, persistent homology has rarely been employed for quantitative modeling and prediction. Additionally, the present persistent homology is a passive tool, rather than a proactive technique, for classification and analysis. In this work, we outline a general protocol to construct object-oriented persistent homology methods. By means of differential geometry theory of surfaces, we construct an objective functional, namely, a surface free energy defined on the data of interest. The minimization of the objective functional leads to a Laplace-Beltrami operator which generates a multiscale representation of the initial data and offers an objective oriented filtration process. The resulting differential geometry based object-oriented persistent homology is able to preserve desirable geometric features in the evolutionary filtration and enhances the corresponding topological persistence. The cubical complex based homology algorithm is employed in the present work to be compatible with the Cartesian representation of the Laplace-Beltrami flow. The proposed Laplace-Beltrami flow based persistent homology method is extensively validated. The consistence between Laplace-Beltrami flow based filtration and Euclidean distance based filtration is confirmed on the Vietoris-Rips complex for a large amount of numerical tests. The convergence and reliability of the present Laplace-Beltrami flow based cubical complex filtration approach are analyzed over various spatial and temporal mesh sizes. The Laplace-Beltrami flow based persistent homology approach is utilized to study the intrinsic topology of proteins and fullerene molecules. Based on a quantitative model which correlates the topological persistence of fullerene central cavity with the total curvature energy of the fullerene structure, the proposed method is used for the prediction of fullerene isomer stability. The efficiency and robustness of the present method are verified by more than 500 fullerene molecules. It is shown that the proposed persistent homology based quantitative model offers good predictions of total curvature energies for ten types of fullerene isomers. The present work offers the first example to design object-oriented persistent homology to enhance or preserve desirable features in the original data during the filtration process and then automatically detect or extract the corresponding topological traits from the data.

MSC:

55N35 Other homology theories in algebraic topology
49Q05 Minimal surfaces and optimization
53B50 Applications of local differential geometry to the sciences
58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
92E10 Molecular structure (graph-theoretic methods, methods of differential topology, etc.)

Software:

javaPlex; Perseus
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Full Text: DOI arXiv Link

References:

[1] Bates, P. W.; Chen, Z.; Sun, Y. H.; Wei, G. W.; Zhao, S., Geometric and potential driving formation and evolution of biomolecular surfaces, J. Math. Biol., 59, 193-231 (2009) · Zbl 1311.92212
[2] Bates, P. W.; Wei, G. W.; Zhao, S., The minimal molecular surface (2006)
[3] Bates, P. W.; Wei, G. W.; Zhao, S., The minimal molecular surface (2006), Midwest Quantitative Biology Conference, Mission Point Resort, Mackinac Island, MI, September 29-October 1
[4] Bates, P. W.; Wei, G. W.; Zhao, Shan, Minimal molecular surfaces and their applications, J. Comput. Chem., 29, 3, 380-391 (2008)
[5] Bendich, Paul; Edelsbrunner, Herbert; Kerber, Michael, Computing robustness and persistence for images, IEEE Trans. Vis. Comput. Graph., 16, 1251-1260 (2010)
[6] Blomgren, P.; Chan, T. F., Color TV: total variation methods for restoration of vector-valued images, IEEE Trans. Image Process., 7, 3, 304-309 (1998)
[7] Bubenik, Peter; Kim, Peter T., A statistical approach to persistent homology, Homol. Homotopy Appl., 19, 337-362 (2007) · Zbl 1136.55004
[8] Carlsson, G., Topology and data, Am. Math. Soc., 46, 2, 255-308 (2009) · Zbl 1172.62002
[9] Carlsson, G.; Ishkhanov, T.; Silva, V.; Zomorodian, A., On the local behavior of spaces of natural images, Int. J. Comput. Vis., 76, 1, 1-12 (2008) · Zbl 1477.68463
[10] Carstensen, V.; Kimmel, R.; Sapiro, G., Geodesic active contours, Int. J. Comput. Vis., 22, 61-79 (1997) · Zbl 0894.68131
[11] Cecil, Thomas, A numerical method for computing minimal surfaces in arbitrary dimension, J. Comput. Phys., 206, 2, 650-660 (2005) · Zbl 1070.65048
[12] Chang, H. W.; Bacallado, S.; Pande, V. S.; Carlsson, G. E., Persistent topology and metastable state in conformational dynamics, PLoS ONE, 8, 4, Article e58699 pp. (2013)
[13] Chen, Duan; Chen, Zhan; Wei, G. W., Quantum dynamics in continuum for proton transport II: variational solvent-solute interface, Int. J. Numer. Methods Biomed. Eng., 28, 25-51 (2012) · Zbl 1319.92064
[14] Chen, Duan; Wei, G. W., Quantum dynamics in continuum for proton transport—generalized correlation, J. Chem. Phys., 136, Article 134109 pp. (2012)
[15] Chen, Z.; Baker, N. A.; Wei, G. W., Differential geometry based solvation models I: Eulerian formulation, J. Comput. Phys., 229, 8231-8258 (2010) · Zbl 1229.92030
[16] Chen, Z.; Baker, N. A.; Wei, G. W., Differential geometry based solvation models II: Lagrangian formulation, J. Math. Biol., 63, 1139-1200 (2011) · Zbl 1284.92025
[17] Chen, Z.; Wei, G. W., Differential geometry based solvation models III: quantum formulation, J. Chem. Phys., 135, Article 194108 pp. (2011)
[18] Chen, Z.; Zhao, Shan; Chun, J.; Thomas, D. G.; Baker, N. A.; Bates, P. B.; Wei, G. W., Variational approach for nonpolar solvation analysis, J. Chem. Phys., 137, Article 084101 pp. (2012)
[19] Cheng, L. T.; Dzubiella, Joachim; McCammon, Andrew J.; Li, B., Application of the level-set method to the implicit solvation of nonpolar molecules, J. Chem. Phys., 127, 8 (2007)
[20] Chopp, David L., Computing minimal surfaces via level set curvature flow, J. Comput. Phys., 106, 1, 77-91 (1993) · Zbl 0786.65015
[21] Dabaghian, Y.; Memoli, F.; Frank, L.; Carlsson, G., A topological paradigm for hippocampal spatial map formation using persistent homology, PLoS Comput. Biol., 8, 8, Article e1002581 pp. (08 2012)
[22] Dey, T. K.; Li, K. Y.; Sun, J.; David, C. S., Computing geometry aware handle and tunnel loops in 3d models, ACM Trans. Graph., 27 (2008)
[23] Dey, Tamal K.; Wang, Y. S., Reeb graphs: approximation and persistence, Discrete Comput. Geom., 49, 1, 46-73 (2013) · Zbl 1261.68095
[24] Di Fabio, Barbara; Landi, Claudia, A Mayer-Vietoris formula for persistent homology with an application to shape recognition in the presence of occlusions, Found. Comput. Math., 11, 499-527 (2011) · Zbl 1231.55004
[25] Edelsbrunner, H.; Letscher, D.; Zomorodian, A., Topological persistence and simplification, Discrete Comput. Geom., 28, 511-533 (2002) · Zbl 1011.68152
[26] Edelsbrunner, Herbert; Harer, John, Computational Topology: An Introduction (2010), American Mathematical Soc. · Zbl 1193.55001
[27] Feng, X.; Prohl, A., Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits, Math. Comput., 73, 541-567 (2004) · Zbl 1115.76049
[28] Feng, X.; Xia, K. L.; Tong, Y. Y.; Wei, G. W., Multiscale geometric modeling of macromolecules II: Lagrangian representation, J. Comput. Chem., 34, 2100-2120 (2013)
[29] Frosini, Patrizio; Landi, Claudia, Size theory as a topological tool for computer vision, Pattern Recognit. Image Anal., 9, 4, 596-603 (1999) · Zbl 0995.68092
[30] Frosini, Patrizio; Landi, Claudia, Persistent Betti numbers for a noise tolerant shape-based approach to image retrieval, Pattern Recognit. Lett., 34, 863-872 (2013)
[31] Gameiro, M.; Hiraoka, Y.; Izumi, S.; Kramar, M.; Mischaikow, K.; Nanda, V., Topological measurement of protein compressibility via persistence diagrams, Jpn. J. Ind. Appl. Math., 32, 1-17 (2014)
[32] Ghrist, R., Barcodes: the persistent topology of data, Bull. Am. Math. Soc., 45, 61-75 (2008) · Zbl 1391.55005
[33] Gomes, J.; Faugeras, O. D., Using the vector distance functions to evolve manifolds of arbitrary codimension, (Lecture Notes in Computer Science, vol. 2106 (2001)), 1-13 · Zbl 0991.68099
[34] Greer, J. B.; Bertozzi, A. L., H-1 solutions of a class of fourth order nonlinear equations for image processing, Discrete Contin. Dyn. Syst., 10, 349-366 (2004) · Zbl 1159.68619
[35] Greer, J. B.; Bertozzi, A. L., Traveling wave solutions of fourth order PDEs for image processing, SIAM J. Math. Anal., 36, 38-68 (2004) · Zbl 1082.35080
[37] Harker, Shaun; Mischaikow, Konstantin; Mrozek, Marian; N, Vidit; Wagner, Hubert; Juda, Mateusz, The efficiency of a homology algorithm based on discrete Morse theory and coreductions, (Proceeding of the 3rd International Workshop on Computational Topology in Image Context, Image A (2010)), 41-47
[38] Harker, Shaun; Mischaikow, Konstantin; Mrozek, Marian; Nanda, Vidit, Discrete Morse theoretic algorithms for computing homology of complexes and maps, Found. Comput. Math. (2013) · Zbl 1387.55010
[39] Hatcher, Allen, Algebraic Topology (2001), Cambridge University Press · Zbl 1044.55001
[40] Horak, D.; Maletic, S.; Rajkovic, M., Persistent homology of complex networks, J. Stat. Mech. Theory Exp., 2009, 03, Article P03034 pp. (2009) · Zbl 1459.55004
[41] Jin, Z. M.; Yang, X. P., Strong solutions for the generalized Perona-Malik equation for image restoration, Nonlinear Anal., Theory Methods Appl., 73, 1077-1084 (2010) · Zbl 1194.35503
[42] Kaczynski, T.; Mischaikow, K.; Mrozek, M., Computational Homology (2004), Springer-Verlag · Zbl 1039.55001
[43] Kasson, P. M.; Zomorodian, A.; Park, S.; Singhal, N.; Guibas, L. J.; Pande, V. S., Persistent voids a new structural metric for membrane fusion, Bioinformatics, 23, 1753-1759 (2007)
[44] Krishnamoorthy, Bala; Provan, Scott; Tropsha, Alexander, A topological characterization of protein structure, (Data Mining in Biomedicine, Springer Optimization and Its Applications (2007)), 431-455
[45] Lee, H.; Kang, H.; Chung, M. K.; Kim, B.; Lee, D. S., Persistent brain network homology from the perspective of dendrogram, IEEE Trans. Med. Imaging, 31, 12, 2267-2277 (Dec 2012)
[46] Li, Y.; Santosa, F., A computational algorithm for minimizing total variation in image restoration, IEEE Trans. Image Process., 5, 6, 987-995 (1996)
[47] Liu, Xu; Xie, Zheng; Yi, Dongyun, A fast algorithm for constructing topological structure in large data, Homol. Homotopy Appl., 14, 221-238 (2012) · Zbl 1243.68162
[48] Mikula, Karol; Sevcovic, Daniel, A direct method for solving an anisotropic mean curvature flow of plane curves with an external force, Math. Methods Appl. Sci., 27, 13, 1545-1565 (2004) · Zbl 1049.35019
[49] Mischaikow, K.; Mrozek, M.; Reiss, J.; Szymczak, A., Construction of symbolic dynamics from experimental time series, Phys. Rev. Lett., 82, 1144-1147 (1999)
[50] Mischaikow, K.; Nanda, V., Morse theory for filtrations and efficient computation of persistent homology, Discrete Comput. Geom., 50, 2, 330-353 (2013) · Zbl 1278.57030
[51] Mumford, David; Shah, Jayant, Optimal approximations by piecewise smooth functions and associated variational problems, Commun. Pure Appl. Math., 42, 5, 577-685 (1989) · Zbl 0691.49036
[52] Nanda, Vidit, Perseus: the persistent homology software, Software available at: · Zbl 1278.57030
[53] Niyogi, P.; Smale, S.; Weinberger, S., A topological view of unsupervised learning from noisy data, SIAM J. Comput., 40, 646-663 (2011) · Zbl 1230.62085
[54] Opron, K.; Xia, K. L.; Wei, G. W., Fast and anisotropic flexibility-rigidity index for protein flexibility and fluctuation analysis, J. Chem. Phys., 140, 234105 (2014)
[55] Osher, S.; Sethian, J. A., Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79, 1, 12-49 (1988) · Zbl 0659.65132
[56] Osher, Stanley; Fedkiw, Ronald P., Level set methods: an overview and some recent results, J. Comput. Phys., 169, 2, 463-502 (2001) · Zbl 0988.65093
[57] Osher, Stanley; Rudin, Leonid I., Feature-oriented image enhancement using shock filters, SIAM J. Numer. Anal., 27, 4, 919-940 (1990) · Zbl 0714.65096
[58] Pachauri, D.; Hinrichs, C.; Chung, M. K.; Johnson, S. C.; Singh, V., Topology-based kernels with application to inference problems in Alzheimer’s disease, IEEE Trans. Med. Imaging, 30, 10, 1760-1770 (Oct 2011)
[59] Park, J. K.; Wei, G. W., A molecular level prototype for mechanoelectrical transducers in mammalian hair cells, J. Comput. Neurosci., 35, 231-241 (2014)
[60] Rieck, Bastian; Mara, Hubert; Leitte, Heike, Multivariate data analysis using persistence-based filtering and topological signatures, IEEE Trans. Vis. Comput. Graph., 18, 2382-2391 (2012)
[61] Robins, Vanessa, Towards computing homology from finite approximations, (Topology Proceedings, vol. 24 (1999)), 503-532 · Zbl 1026.55009
[62] Rudin, Leonid I.; Osher, Stanley; Fatemi, Emad, Nonlinear total variation based noise removal algorithms, (Proceedings of the Eleventh Annual International Conference of the Center for Nonlinear Studies on Experimental Mathematics: Computational Issues in Nonlinear Science (1992), Elsevier North-Holland, Inc.: Elsevier North-Holland, Inc. Amsterdam, The Netherlands), 259-268 · Zbl 0780.49028
[63] Sapiro, G.; Ringach, D. L., Anisotropic diffusion of multivalued images with applications to color filtering, IEEE Trans. Image Process., 5, 11, 1582-1586 (1996)
[64] Sarti, A.; Malladi, R.; Sethian, J. A., Subjective surfaces: a geometric model for boundary completion, Int. J. Comput. Vis., 46, 3, 201-221 (2002) · Zbl 1012.68727
[65] Sethian, J. A., Evolution, implementation, and application of level set and fast marching methods for advancing fronts, J. Comput. Phys., 169, 2, 503-555 (2001) · Zbl 0988.65095
[66] Silva, V. D.; Ghrist, R., Blind swarms for coverage in 2-d, (Proceedings of Robotics: Science and Systems (2005)), 01
[67] Singh, G.; Memoli, F.; Ishkhanov, T.; Sapiro, G.; Carlsson, G.; Ringach, D. L., Topological analysis of population activity in visual cortex, J. Vis., 8, 8 (2008)
[68] Smereka, Peter, Semi-implicit level set methods for curvature and surface diffusion motion, J. Sci. Comput., 19, 1, 439-456 (2003) · Zbl 1035.65098
[69] Sochen, N.; Kimmel, R.; Malladi, R., A general framework for low level vision, IEEE Trans. Image Process., 7, 3, 310-318 (1998) · Zbl 0973.94502
[70] Strombom, D., Persistent homology in the cubical setting (2007), Lulea University of Technology, Master’s thesis
[71] Tausz, Andrew; Vejdemo-Johansson, Mikael; Adams, Henry, Javaplex: a research software package for persistent (co)homology (2011), Software available at: · Zbl 1402.65186
[72] Wagner, C.; Chen, C.; Vucini, E., Efficient computation of persistent homology for cubical data, (Topological Methods in Data Analysis and Visualization II (2012), Springer: Springer Heidelberg, Dordrecht, London, New York) · Zbl 1246.68245
[73] Wang, Bei; Summa, Brian; Pascucci, Valerio; Vejdemo-Johansson, M., Branching and circular features in high dimensional data, IEEE Trans. Vis. Comput. Graph., 17, 1902-1911 (2011)
[74] Wang, Y.; Wei, G. W.; Yang, Si-Yang, Partial differential equation transform - variational formulation and Fourier analysis, Int. J. Numer. Methods Biomed. Eng., 27, 1996-2020 (2011) · Zbl 1254.94012
[75] Wang, Y.; Wei, G. W.; Yang, Si-Yang, Mode decomposition evolution equations, J. Sci. Comput., 50, 495-518 (2012) · Zbl 1457.65152
[76] Wang, Y.; Wei, G. W.; Yang, Si-Yang, Selective extraction of entangled textures via adaptive PDE transform, Int. J. Biomed. Imaging, Article 958142 pp. (2012)
[77] Wei, G. W., Generalized Perona-Malik equation for image restoration, IEEE Signal Process. Lett., 6, 7, 165-167 (1999)
[78] Wei, G. W., Differential geometry based multiscale models, Bull. Math. Biol., 72, 1562-1622 (2010) · Zbl 1198.92001
[79] Wei, G. W.; Jia, Y. Q., Synchronization-based image edge detection, Europhys. Lett., 59, 6, 814-819 (2002)
[80] Wei, G. W.; Sun, Y. H.; Zhou, Y. C.; Feig, M., Molecular multiresolution surfaces (2005), pp. 1-11
[81] Wei, Guo-Wei, Multiscale, multiphysics and multidomain models I: basic theory, J. Theor. Comput. Chem., 12, 8, 1341006 (2013)
[82] Wei, Guo-Wei; Zheng, Qiong; Chen, Zhan; Xia, Kelin, Variational multiscale models for charge transport, SIAM Rev., 54, 4, 699-754 (2012) · Zbl 1306.92021
[83] Willmore, T. J., Riemannian Geometry (1997), Oxford University Press: Oxford University Press USA · Zbl 0797.53002
[84] Xia, K. L.; Feng, X.; Tong, Y. Y.; Wei, G. W., Multiscale geometric modeling of macromolecules I: Cartesian representation, J. Comput. Phys., 275, 912-936 (2014) · Zbl 1349.92016
[85] Xia, K. L.; Feng, X.; Tong, Y. Y.; Wei, G. W., Persistent homology for the quantitative prediction of fullerene stability, J. Comput. Chem., 36, 408-422 (2015)
[86] Xia, K. L.; Wei, G. W., Persistent homology analysis of protein structure, flexibility and folding, Int. J. Numer. Methods Biomed. Eng., 30, 814-844 (2014)
[87] Xia, K. L.; Wei, G. W., Multidimensional persistence in biomolecular data, J. Comput. Chem., 36, 1502-1520 (2015)
[88] Xia, K. L.; Wei, G. W., Persistent topology for cryo-EM data analysis, Int. J. Numer. Methods Biomed. Eng., 31, Article e02719 pp. (2015)
[89] Xia, K. L.; Zhao, Z. X.; Wei, G. W., Multiresolution topological simplification, J. Comput. Biol., 22, 1-5 (2015)
[90] Xu, M.; Zhou, S. L., Existence and uniqueness of weak solutions for a fourth-order nonlinear parabolic equation, J. Math. Anal. Appl., 325, 1, 636-654 (2007) · Zbl 1107.35038
[91] Yao, Y.; Sun, J.; Huang, X. H.; Bowman, G. R.; Singh, G.; Lesnick, M.; Guibas, L. J.; Pande, V. S.; Carlsson, G., Topological methods for exploring low-density states in biomolecular folding pathways, J. Chem. Phys., 130, Article 144115 pp. (2009)
[92] Yu, Z. Y.; Bajaj, C., Computational approaches for automatic structural analysis of large biomolecular complexes, IEEE/ACM Trans. Comput. Biol. Bioinform., 5, 568-582 (2008)
[93] Zhang, Y.; Xu, G.; Bajaj, C., Quality meshing of implicit solvation models of biomolecular structures, Comput. Aided Geom. Des., 23, 6, 510-530 (2006) · Zbl 1098.92034
[94] Zhao, Shan, Pseudo-time-coupled nonlinear models for biomolecular surface representation and solvation analysis, Int. J. Numer. Methods Biomed. Eng., 27, 1964-1981 (2011) · Zbl 1241.92007
[95] Zhao, Shan, Operator splitting ADI schemes for pseudo-time coupled nonlinear solvation simulations, J. Comput. Phys., 257, 1000-1021 (2014) · Zbl 1349.78105
[96] Zheng, Q.; Yang, S. Y.; Wei, G. W., Molecular surface generation using PDE transform, Int. J. Numer. Methods Biomed. Eng., 28, 291-316 (2012)
[97] Zheng, Qiong; Chen, Duan; Wei, G. W., Second-order Poisson-Nernst-Planck solver for ion transport, J. Comput. Phys., 230, 5239-5262 (2011) · Zbl 1222.82073
[98] Zheng, Qiong; Wei, G. W., Poisson-Boltzmann-Nernst-Planck model, J. Chem. Phys., 134, Article 194101 pp. (2011)
[99] Zomorodian, A.; Carlsson, G., Computing persistent homology, Discrete Comput. Geom., 33, 249-274 (2005) · Zbl 1069.55003
[100] Zomorodian, Afra; Carlsson, Gunnar, Localized homology, Comput. Geom. Theory Appl., 41, 3, 126-148 (2008) · Zbl 1155.65021
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