Abstract

In the shape analysis approach to computer vision problems, one treats shapes as points in an infinite-dimensional Riemannian manifold, thereby facilitating algorithms for statistical calculations such as geodesic distance between shapes and averaging of a collection of shapes. The performance of these algorithms depends heavily on the choice of the Riemannian metric. In the setting of plane curve shapes, attention has largely been focused on a two-parameter family of first order Sobolev metrics, referred to as elastic metrics. They are particularly useful due to the existence of simplifying coordinate transformations for particular parameter values, such as the well-known square-root velocity transform. In this paper, we extend the transformations appearing in the existing literature to a family of isometries, which take any elastic metric to the flat $L^2$ metric. We also extend the transforms to treat piecewise linear curves and demonstrate the existence of optimal matchings over the diffeomorphism group in this setting. We conclude the paper with multiple examples of shape geodesics for open and closed curves. We also show the benefits of our approach in a simple classification experiment.

Keywords

  1. elastic shape analysis
  2. statistical shape analysis
  3. infinite-dimensional geometry
  4. Sobolev metrics
  5. curve matching

MSC codes

  1. Primary
  2. 58B20
  3. 58E50; Secondary
  4. 68U05

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
E. V. Anoshkina, A. G. Belyaev, and H.-P. Seidel, Asymptotic analysis of three-point approximations of vertex normals and curvatures, in Proceedings of Vision, Modeling, and Visualization, Aka GmbH, 2002, pp. 211--216.
2.
B. Ayers, E. Luders, N. Cherbuin, and S. H. Joshi, Corpus callosum thickness estimation using elastic shape matching, in International Symposium on Biomedical Imaging, 2015, pp. 1518--1521.
3.
M. Bauer, M. Bruveris, N. Charon, and J. Møller-Andersen, A relaxed approach for curve matching with elastic metrics, ESAIM Control Optim. Calc. Var., 25 (2019), 72.
4.
M. Bauer, M. Bruveris, P. Harms, and J. Møller-Andersen, A numerical framework for Sobolev metrics on the space of curves, SIAM J. Imaging Sci., 10 (2017), pp. 47--73, https://doi.org/10.1137/16M1066282.
5.
M. Bauer, M. Bruveris, S. Marsland, and P. W. Michor, Constructing reparameterization invariant metrics on spaces of plane curves, Differential Geom. Appl., 34 (2014), pp. 139--165.
6.
M. Bauer, M. Bruveris, and P. W. Michor, Why use Sobolev metrics on the space of curves, in Riemannian Computing in Computer Vision, Springer, 2016, pp. 233--255.
7.
M. Bauer, P. Harms, and P. W. Michor, Almost local metrics on shape space of hypersurfaces in $n$-space, SIAM J. Imaging Sci., 5 (2012), pp. 244--310, https://doi.org/10.1137/100807983.
8.
M. Bauer, P. Harms, and P. W. Michor, Curvature weighted metrics on shape space of hypersurfaces in $n$-space, Differential Geom. Appl., 30 (2012), pp. 33--41.
9.
D. M. Boyer, Y. Lipman, E. S. Clair, J. Puente, B. A. Patel, T. Funkhouser, J. Jernvall, and I. Daubechies, Algorithms to automatically quantify the geometric similarity of anatomical surfaces, Proc. Natl. Acad. Sci. USA, 108 (2011), pp. 18221--18226.
10.
A. M. Bronstein, M. M. Bronstein, R. Kimmel, M. Mahmoudi, and G. Sapiro, A Gromov-Hausdorff framework with diffusion geometry for topologically-robust non-rigid shape matching, Int. J. Comput. Vis., 89 (2010), pp. 266--286.
11.
M. Bruveris, Optimal reparametrizations in the square root velocity framework, SIAM J. Math. Anal., 48 (2016), pp. 4335--4354, https://doi.org/10.1137/15M1014693.
12.
M. Bruveris, P. W. Michor, and D. Mumford, Geodesic completeness for Sobolev metrics on the space of immersed plane curves, in Forum of Mathematics, Sigma, Vol. 2, Cambridge University Press, 2014, e19.
13.
E. Celledoni, M. Eslitzbichler, and A. Schmeding, Shape analysis on Lie groups with applications in computer animation, J. Geom. Mech., 8 (2016), pp. 273--304.
14.
N. Charon and A. Trouvé, The varifold representation of nonoriented shapes for diffeomorphic registration, SIAM J. Imaging Sci., 6 (2013), pp. 2547--2580, https://doi.org/10.1137/130918885.
15.
G. Charpiat, O. Faugeras, and R. Keriven, Shape statistics for image segmentation with prior, in IEEE Computer Vision and Pattern Recognition, 2007, pp. 1--6.
16.
F. Chazal, D. Cohen-Steiner, L. J. Guibas, F. Mémoli, and S. Y. Oudot, Gromov-Hausdorff stable signatures for shapes using persistence, Computer Graphics Forum, 28 (2009), pp. 1393--1403.
17.
I. L. Dryden and K. V. Mardia, Statistical Shape Analysis: With Applications in R, 2nd ed., Wiley, New York, 2016.
18.
A. Duncan, E. Klassen, and A. Srivastava, Statistical shape analysis of simplified neuronal trees, Ann. Appl. Stat., 12 (2018), pp. 1385--1421.
19.
J. Glaunès, A. Qiu, M. I. Miller, and L. Younes, Large deformation diffeomorphic metric curve mapping, Int. J. Comput. Vis., 80 (2008), pp. 317--336.
20.
F. Gronwald, On non-Riemannian parallel transport in Regge calculus, Classical Quantum Gravity, 12 (1995), pp. 1181--1189.
21.
S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, Optimal mass transport for registration and warping, Int. J. Comput. Vis., 60 (2004), pp. 225--240.
22.
R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc., 7 (1982), pp. 65--222.
23.
I. H. Jermyn, S. Kurtek, E. Klassen, and A. Srivastava, Elastic Shape Matching of Parameterized Surfaces Using Square Root Normal Fields, in European Conference on Computer Vision, Springer, 2012, pp. 804--817.
24.
S. H. Joshi, E. Klassen, A. Srivastava, and I. H. Jermyn, A novel representation for Riemannian analysis of elastic curves in $\mathbb{R}^n$, in IEEE Computer Vision and Pattern Recognition, 2007, pp. 1--7.
25.
D. G. Kendall, Shape manifolds, Procrustean metrics, and complex projective shapes, Bull. London Math. Soc., 16 (1984), pp. 81--121.
26.
E. Klassen, A. Srivastava, W. Mio, and S. H. Joshi, Analysis of planar shapes using geodesic paths on shape spaces, IEEE Trans. Pattern Anal. Mach. Intell., 26 (2004), pp. 372--383.
27.
A. Kriegl and P. W. Michor, The Convenient Setting of Global Analysis, Math. Surveys Monogr. 53, AMS, 1997.
28.
S. Kurtek, A geometric approach to pairwise Bayesian alignment of functional data using importance sampling, Electron. J. Stat., 11 (2017), pp. 502--531.
29.
S. Kurtek, E. Klassen, J. C. Gore, Z. Ding, and A. Srivastava, Elastic geodesic paths in shape space of parametrized surfaces, IEEE Trans. Pattern Anal. Mach. Intell., 34 (2012), pp. 1717--1730.
30.
H. Laga, S. Kurtek, A. Srivastava, M. Golzarian, and S. J. Miklavcic, A Riemannian elastic metric for shape-based plant leaf classification, in 2012 International Conference on Digital Image Computing Techniques and Applications (DICTA), 2012, pp. 1--7.
31.
S. Lahiri, D. Robinson, and E. Klassen, Precise matching of PL curves in $\mathbb{R}^n$ in the square root velocity framework, Geom. Imaging Comput., 2 (2015), pp. 133--186.
32.
Y. Lu, R. Herbei, and S. Kurtek, Bayesian registration of functions with a Gaussian process prior, J. Comput. Graph. Statist., 26 (2017), pp. 894--904.
33.
F. Mémoli, Gromov--Wasserstein distances and the metric approach to object matching, Found. Comput. Math., 11 (2011), pp. 417--487.
34.
P. W. Michor and D. Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms, Doc. Math., 10 (2005), pp. 217--245.
35.
P. W. Michor and D. Mumford, An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach, Appl. Comput. Harmon. Anal., 23 (2007), pp. 74--113.
36.
M. I. Miller and L. Younes, Group actions, homeomorphisms, and matching: A general framework, Int. J. Comput. Vis., 41 (2001), pp. 61--84.
37.
W. Mio, A. Srivastava, and S. Joshi, On shape of plane elastic curves, Int. J. Comput. Vis., 73 (2007), pp. 307--324.
38.
T. Needham, Kähler structures on spaces of framed curves, Ann. Global Anal. Geom., 54 (2018), pp. 123--153.
39.
T. Regge, General relativity without coordinates, Nuovo Cimento (10), 19 (1961), pp. 558--571.
40.
D. T. Robinson, Functional Data Analysis and Partial Shape Matching in the Square Root Velocity Framework, Ph.D. thesis, Florida State University, 2012.
41.
H. L. Royden and P. Fitzpatrick, Real Analysis, Vol. 2, Macmillan, New York, 1968.
42.
L. Schumaker, Spline Functions: Basic Theory, Cambridge University Press, 2007.
43.
C. G. Small, The Statistical Theory of Shape, Springer, 1996.
44.
A. Srivastava, E. Klassen, S. H. Joshi, and I. H. Jermyn, Shape analysis of elastic curves in Euclidean spaces, IEEE Trans. Pattern Anal. Mach. Intell., 33 (2011), pp. 1415--1428.
45.
S. Srivastava, S. B. Lal, D. Mishra, U. Angadi, K. Chaturvedi, S. N. Rai, and A. Rai, An efficient algorithm for protein structure comparison using elastic shape analysis, Algorithms Molecular Biol., 11 (2016), 27.
46.
G. Sundaramoorthi, A. Yezzi, and A. C. Mennucci, Sobolev active contours, Int. J. Comput. Vis., 73 (2007), pp. 345--366.
47.
A. Trouvé and L. Younes, Diffeomorphic matching problems in one dimension: Designing and minimizing matching functionals, in European Conference on Computer Vision, 2000, pp. 573--587.
48.
A. Trouvé and L. Younes, On a class of diffeomorphic matching problems in one dimension, SIAM J. Control Optim., 39 (2000), pp. 1112--1135, https://doi.org/10.1137/S036301299934864X.
49.
A. B. Tumpach and S. C. Preston, Quotient elastic metrics on the manifold of arc-length parameterized plane curves, J. Geom. Mech., 9 (2017), pp. 227--256.
50.
H. Whitney, On regular closed curves in the plane, Compositio Math., 4 (1937), pp. 276--284.
51.
D. Yeung, H. Chang, Y. Xiong, S. George, R. Kashi, T. Matsumoto, and G. Rigoll, SVC$2004$: First International Signature Verification Competition, in Biometric Authentication, 2004, pp. 16--22.
52.
Y. You, W. Huang, K. A. Gallivan, and P.-A. Absil, A Riemannian approach for computing geodesies in elastic shape analysis, in Signal and Information Processing, 2015, pp. 727--731.
53.
L. Younes, Computable elastic distances between shapes, SIAM J. Appl. Math., 58 (1998), pp. 565--586, https://doi.org/10.1137/S0036139995287685.
54.
L. Younes, Elastic Distance between Curves under the Metamorphosis Viewpoint, preprint, https://arxiv.org/abs/1804.10155, 2018.
55.
L. Younes, P. W. Michor, J. M. Shah, and D. B. Mumford, A metric on shape space with explicit geodesics, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 19 (2008), pp. 25--57.
56.
C. T. Zahn and R. Z. Roskies, Fourier descriptors for plane closed curves, IEEE Trans. Comput., 21 (1972), pp. 269--281.

Information & Authors

Information

Published In

cover image SIAM Journal on Imaging Sciences
SIAM Journal on Imaging Sciences
Pages: 445 - 473
ISSN (online): 1936-4954

History

Submitted: 30 May 2019
Accepted: 13 January 2020
Published online: 12 March 2020

Keywords

  1. elastic shape analysis
  2. statistical shape analysis
  3. infinite-dimensional geometry
  4. Sobolev metrics
  5. curve matching

MSC codes

  1. Primary
  2. 58B20
  3. 58E50; Secondary
  4. 68U05

Authors

Affiliations

Funding Information

National Institutes of Health https://doi.org/10.13039/100000002
National Science Foundation https://doi.org/10.13039/100000001 : DMS-1613054, CCF-1740761, CCF-1839252

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.