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Elastic Shape Analysis of Functions, Curves and Trajectories

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Riemannian Computing in Computer Vision

Abstract

We present a Riemannian framework for geometric shape analysis of curves, functions, and trajectories on nonlinear manifolds. Since scalar functions and trajectories can also have important geometric features, we use shape as an all-encompassing term for the descriptors of curves, scalar functions and trajectories. Our framework relies on functional representation and analysis of curves and scalar functions, by square-root velocity fields (SRVF) under the Fisher–Rao metric, and of trajectories by transported square-root vector fields (TSRVF). SRVFs are general functional representations that jointly capture both the shape (geometry) and the reparameterization (sampling speed) of curves, whereas TSRVFs also capture temporal reparameterizations of time-indexed shapes. The space of SRVFs for shapes of curves becomes a subset of a spherical Riemannian manifold under certain special constraints. A fundamental tool in shape analysis is the construction and implementation of geodesic paths between shapes. This is used to accomplish a variety of tasks, including the definition of a metric to compare shapes, the computation of intrinsic statistics for a set of shapes, and the definition of probability models on shape spaces. We demonstrate our approach using several applications from computer vision and medical imaging including the analysis of (1) curves, (2) human growth, (3) bird migration patterns, and (4) human actions from video surveillance images and skeletons from depth images.

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References

  1. An X, Liu Z, Shi Y, Li N, Wang Y, Joshi SH (2012) Modeling dynamic cellular morphology in images. In: Medical image computing and computer-assisted intervention–MICCAI 2012. Springer, Berlin, pp 340–347

    Chapter  Google Scholar 

  2. Amor BB, Su J, Srivastava A (2014) Action recognition using rate-invariant analysis of skeletal shape trajectories. IEEE Trans Pattern Anal Mach Intell 1(99):1, 10.1109/TPAMI.2015.2439257

    Google Scholar 

  3. Dryden IL, Mardia K (1998) Statistical shape analysis. Wiley series in probability and statistics: probability and statistics. Wiley, Chichester

    Google Scholar 

  4. Grenander U, Miller MI (1998) Computational anatomy: an emerging discipline. Q Appl Math 56(4): 617–694

    MATH  MathSciNet  Google Scholar 

  5. Jones HE, Bayley N (1941) The Berkeley growth study. Child Dev 12(2):167–173

    Google Scholar 

  6. Joshi SH, Srivastava A (2009) Intrinsic Bayesian active contours for extraction of object boundaries in images. Int J Comput Vis 81(3):331–355

    Article  Google Scholar 

  7. Joshi SH, Klassen E, Srivastava A, Jermyn I (2007) A novel representation for Riemannian analysis of elastic curves in R n. In: IEEE conference on computer vision and pattern recognition (CVPR). IEEE, New York, pp 1–7

    Google Scholar 

  8. Joshi SH, Klassen E, Srivastava A, Jermyn I (2007) Removing shape-preserving transformations in square-root elastic (SRE) framework for shape analysis of curves. In: Energy minimization methods in computer vision and pattern recognition (EMMCVPR), pp 387–398

    Google Scholar 

  9. Joshi SH, Cabeen RP, Joshi AA, Sun B, Dinov I, Narr KL, Toga AW, Woods RP (2012) Diffeomorphic sulcal shape analysis on the cortex. IEEE Trans Med Imaging 31(6):1195–1212

    Article  Google Scholar 

  10. Joshi SH, Narr KL, Philips OR, Nuechterlein KH, Asarnow RF, Toga AW, Woods RP (2013) Statistical shape analysis of the corpus callosum in schizophrenia. Neuroimage 64:547–559

    Article  Google Scholar 

  11. Karcher H (1977) Riemannian center of mass and mollifier smoothing. Commun Pure Appl Math 30:509–541

    Article  MATH  MathSciNet  Google Scholar 

  12. Kendall DG (1984) Shape manifolds, procrustean metrics, and complex projective spaces. Bull Lond Math Soc 16(2):81–121

    Article  MATH  MathSciNet  Google Scholar 

  13. Kenobi K, Dryden IL, Le H (2010) Shape curves and geodesic modeling. Biometrika 97(3):567–584

    Article  MATH  MathSciNet  Google Scholar 

  14. Kneip A, Ramsay JO Combining registration and fitting for functional models. J Am Stat Assoc 103(483):1155–1165 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Le H (2003) Unrolling shape curves. J Lond Math Soc 68(2):511–526

    Article  MATH  Google Scholar 

  16. Li W, Zhang Z, Liu Z (2010) Action recognition based on a bag of 3D points. In: IEEE international workshop on CVPR for human communicative behavior analysis, pp 9–14

    Google Scholar 

  17. Liu W, Srivastava A, Zhang J (2011) A mathematical framework for protein structure comparison. PLoS Comput Biol 7(2):e1001,075

    Article  MathSciNet  Google Scholar 

  18. Michor PW, Mumford D (2006) Riemannian geometries on spaces of plane curves. J Eur Math Soc 8:1–48

    Article  MATH  MathSciNet  Google Scholar 

  19. Mio W, Srivastava A, Joshi SH (2007) On shape of plane elastic curves. Int J Comput Vis 73(3):307–324

    Article  Google Scholar 

  20. Ramsay JO, Li X (1998) Curve registration. J R Stat Soc Ser B (Stat Methodol) 60(2):351–363

    Article  MATH  MathSciNet  Google Scholar 

  21. Ramsay JO, Silverman BW (2005) Functional data analysis. Springer series in statistics, 2nd edn. Springer, New York

    Google Scholar 

  22. Rao RC (1945) Information and the accuracy attainable in the estimation of statistical parameters. Bull Calcutta Math Soc 37:81–91

    MATH  MathSciNet  Google Scholar 

  23. Rossi F, Villa N (2006) Support vector machine for functional data classification. Neurocomputing 69(7–9):730–742; New issues in Neurocomputing: 13th European symposium on artificial neural networks 13th European symposium on artificial neural networks, 2005

    Google Scholar 

  24. Shotton J, Sharp T, Kipman A, Fitzgibbon A, Finocchio M, Blake A, Cook M, Moore R (2013) Real-time human pose recognition in parts from single depth images. Commun ACM 56(1):116–124

    Article  Google Scholar 

  25. Srivastava A, Klassen E, Joshi SH, Jermyn IH (2011) Shape analysis of elastic curves in Euclidean spaces. IEEE Trans Pattern Anal Mach Intell 33:1415–1428

    Article  Google Scholar 

  26. Srivastava A, Wu W, Kurtek S, Klassen E, Marron JS (2011) Registration of functional data using Fisher-Rao metric. arXiv:1103.3817v2

    Google Scholar 

  27. Su J, Kurtek S, Klassen E, Srivastava A (2014) Statistical analysis of trajectories on Riemannian manifolds: bird migration, hurricane tracking, and video surveillance. Ann Appl Stat 8(1):530–552

    Article  MATH  MathSciNet  Google Scholar 

  28. Su J, Srivastava A, de Souza FD, Sarkar S (2014) Rate-invariant analysis of trajectories on Riemannian manifolds with application in visual speech recognition. In: IEEE conference on computer vision and pattern recognition (CVPR). IEEE, New York, pp 620–627

    Google Scholar 

  29. Sundaramoorthi G, Mennucci A, Soatto S, Yezzi A (2011) A new geometric metric in the space of curves, and applications to tracking deforming objects by prediction and filtering. SIAM J Imaging Sci 4(1):109–145

    Article  MATH  MathSciNet  Google Scholar 

  30. Tang R, Muller HG (2008) Pairwise curve synchronization for functional data. Biometrika 95(4):875–889

    Article  MATH  MathSciNet  Google Scholar 

  31. Thompson DW (1943) On growth and form. Cambridge University Press, Cambridge

    Google Scholar 

  32. Trouve A, Younes L (2000) On a class of diffeomorphic matching problems in one dimension. SIAM J Control Optim 39(4):1112–1135

    Article  MATH  MathSciNet  Google Scholar 

  33. Tucker JD, Wu W, Srivastava A (2013) Generative models for functional data using phase and amplitude separation. Comput Stat Data Anal 61:50–66

    Article  MathSciNet  Google Scholar 

  34. Veeraraghavan A, Srivastava A, Roy-Chowdhury AK, Chellappa R (2009) Rate-invariant recognition of humans and their activities. IEEE Trans Image Process 8(6):1326–1339

    Article  MathSciNet  Google Scholar 

  35. Younes L (1998) Computable elastic distance between shapes. SIAM J Appl Math 58(2):565–586

    Article  MATH  MathSciNet  Google Scholar 

  36. Zhang Z (2012) Microsoft kinect sensor and its effect. IEEE Multimedia 19(2):4–10

    Article  Google Scholar 

  37. Zhang Z, Klassen E, Srivastava A, Turaga PK, Chellappa R (2011) Blurring-invariant Riemannian metrics for comparing signals and images. In: Proceedings of ICCV, pp 1770–1775

    Google Scholar 

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Author information

Authors and Affiliations

  1. University of California, Los Angeles, CA, USA

    Shantanu H. Joshi

  2. Texas Tech University, Lubbock, TX, USA

    Jingyong Su

  3. Florida State University, Tallahassee, FL, USA

    Zhengwu Zhang

  4. Institut Mines-Télécom/Télécom Lille, CRIStAL (UMR CNRS 9189), Lille, France

    Boulbaba Ben Amor

Authors
  1. Shantanu H. Joshi
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  2. Jingyong Su
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  3. Zhengwu Zhang
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  4. Boulbaba Ben Amor
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Corresponding author

Correspondence to Shantanu H. Joshi .

Editor information

Editors and Affiliations

  1. Arizona State University, Tempe, Arizona, USA

    Pavan K. Turaga

  2. Florida State University, Tallahassee, Florida, USA

    Anuj Srivastava

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Joshi, S.H., Su, J., Zhang, Z., Ben Amor, B. (2016). Elastic Shape Analysis of Functions, Curves and Trajectories. In: Turaga, P., Srivastava, A. (eds) Riemannian Computing in Computer Vision. Springer, Cham. https://doi.org/10.1007/978-3-319-22957-7_10

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  • DOI: https://doi.org/10.1007/978-3-319-22957-7_10

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-22956-0

  • Online ISBN: 978-3-319-22957-7

  • eBook Packages: EngineeringEngineering (R0)

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