Abstract
In the elastic shape analysis approach to shape matching and object classification, plane curves are represented as points in an infinite-dimensional Riemannian manifold, wherein shape dissimilarity is measured by geodesic distance. A remarkable result of Younes, Michor, Shah and Mumford says that the space of closed planar shapes, endowed with a natural metric, is isometric to an infinite-dimensional Grassmann manifold via the so-called square root transform. This result facilitates efficient shape comparison by virtue of explicit descriptions of Grassmannian geodesics. In this paper, we extend this shape analysis framework to treat shapes of framed space curves. By considering framed curves, we are able to generalize the square root transform by using quaternionic arithmetic and properties of the Hopf fibration. Under our coordinate transformation, the space of closed framed curves corresponds to an infinite-dimensional complex Grassmannian. This allows us to describe geodesics in framed curve space explicitly. We are also able to produce explicit geodesics between closed, unframed space curves by studying the action of the loop group of the circle on the Grassmann manifold. We apply our results to compute means for collections of space curves and to perform statistical analysis of circular DNA molecule shapes.
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Notes
In particular, we used the MATLAB implementation available on the FSU Statistical Shape Analysis and Modeling Group website http://ssamg.stat.fsu.edu
References
Bauer, M., Bruveris, M., Harms, P., Michor, P.W.: Vanishing geodesic distance for the riemannian metric with geodesic equation the kdv-equation. Ann. Glob. Anal. Geom. 41(4), 461–472 (2012)
Bauer, M., Bruveris, M., Marsland, S., Michor, P.W.: Constructing reparameterization invariant metrics on spaces of plane curves. Differ. Geom. Appl. 34, 139–165 (2014)
Bauer, M., Bruveris, M., Michor, P.W.: Why use sobolev metrics on the space of curves. In: Turaga, P.K., Srivastava, A. (eds.) Riemannian Computing in Computer Vision, pp. 233–255. Springer, Berlin (2016)
Bauer, M., Eslitzbichler, M., Grasmair, M.: Landmark-guided elastic shape analysis of human character motions. Inverse Probl. Imaging 11(4), 601–621 (2017)
Bergou, M., Audoly, B., Vouga, E., Wardetzky, M., Grinspun, E.: Discrete viscous threads. ACM Trans. Graph. (TOG) 29(4), 116 (2010)
Bertails, F., Audoly, B., Cani, M.P., Querleux, B., Leroy, F., Lévêque, J.L.: Super-helices for predicting the dynamics of natural hair. In: ACM Transactions on Graphics (TOG), vol. 25, pp. 1180–1187. ACM (2006)
Bishop, R.L.: There is more than one way to frame a curve. Am. Math. Mon. 82(3), 246–251 (1975)
Boumal, N., Mishra, B., Absil, P.A., Sepulchre, R.: Manopt, a matlab toolbox for optimization on manifolds. J. Mach. Learn. Res. 15(1), 1455–1459 (2014)
Bruveris, M.: Optimal reparametrizations in the square root velocity framework. SIAM J. Math. Anal. 48(6), 4335–4354 (2016)
Celledoni, E., Eslitzbichler, M., Schmeding, A.: Shape analysis on lie groups with applications in computer animation. J. Geom. Mech. 8(3), 273–304 (2016)
Dichmann, D.J., Li, Y., Maddocks, J.H.: Hamiltonian formulations and symmetries in rod mechanics. In: Mesirov, J.P., Schulten, K., Sumners, D.W. (eds.) Mathematical Approaches to Biomolecular Structure and Dynamics, pp. 71–113. Springer, Berlin (1996)
Dryden, I.L., Mardia, K.V.: Statistical Shape Analysis: with Applications in R, 2nd edn. Wiley, New York (2016)
Edelman, A., Arias, T.A., Smith, S.T.: The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20(2), 303–353 (1998)
Gelfand, I.M., Minlos, R.A., Shapiro, Z.Y.: Representations of the Rotation and Lorentz Groups and their Applications. Courier Dover Publications, New York (2018)
Goemans, O., Overmars, M.: Automatic generation of camera motion to track a moving guide. In: Erdmann, M., Heu, D., Overmars, M., van der Stappen, A.F. (eds.) Algorithmic Foundations of Robotics VI, pp. 187–202. Springer, Berlin (2004)
Hamilton, R.S.: The inverse function theorem of nash and moser. Bull. Am. Math. Soc. 7(1), 65–222 (1982)
Hanson, A.J.: Visualizing quaternions. In: ACM SIGGRAPH 2005 Courses, p. 1. ACM (2005)
Harms, P., Mennucci, A.C.: Geodesics in infinite dimensional stiefel and grassmann manifolds. Comptes Rendus Mathematique 350(15–16), 773–776 (2012)
Hu, S., Lundgren, M., Niemi, A.J.: Discrete frenet frame, inflection point solitons, and curve visualization with applications to folded proteins. Phys. Rev. E 83(6), 061908 (2011)
Irobalieva, R.N., Fogg, J.M., Catanese Jr., D.J., Sutthibutpong, T., Chen, M., Barker, A.K., Ludtke, S.J., Harris, S.A., Schmid, M.F., Chiu, W., et al.: Structural diversity of supercoiled dna. Nat. Commun. 6, 8440 (2015)
Jermyn, I.H., Kurtek, S., Klassen, E., Srivastava, A.: Elastic shape matching of parameterized surfaces using square root normal fields. In: European Conference on Computer Vision, pp. 804–817. Springer (2012)
Joshi, S.H., Klassen, E., Srivastava, A., Jermyn, I.: A novel representation for riemannian analysis of elastic curves in rn. In: Computer Vision and Pattern Recognition, 2007. CVPR’07. IEEE Conference on, pp. 1–7. IEEE (2007)
Kehrbaum, S., Maddocks, J.: Elastic rods, rigid bodies, quaternions and the last quadrature. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 355(1732), 2117–2136 (1997)
Kendall, D.G.: Shape manifolds, procrustean metrics, and complex projective spaces. Bull. Lond. Math. Soc. 16(2), 81–121 (1984)
Kovar, L., Gleicher, M.: Flexible automatic motion blending with registration curves. In: Proceedings of the 2003 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, pp. 214–224. Eurographics Association (2003)
Kurtek, S., Needham, T.: Simplifying transforms for general elastic metrics on the space of plane curves. arXiv preprint arXiv:1803.10894 (2018)
Kurtek, S., Srivastava, A., Klassen, E., Laga, H.: Landmark-guided elastic shape analysis of spherically-parameterized surfaces. In: Computer Graphics Forum, vol. 32, pp. 429–438. Wiley Online Library (2013)
Lahiri, S., Robinson, D., Klassen, E.: Precise matching of pl curves in \({\mathbb{R}}^n\) in the square root velocity framework. Geom. Imaging Comput. 2(3), 133–186 (2015)
Le Brigant, A.: A discrete framework to find the optimal matching between manifold-valued curves. J. Math. Imaging Vis. 61(1), 40–70 (2019)
Mio, W., Srivastava, A., Joshi, S.: On shape of plane elastic curves. Int. J. Comput. Vis. 73(3), 307–324 (2007)
Needham, T.: Knot types of generalized kirchhoff rods. To appear, J. Knot Theory Ramif., arXiv preprint arXiv:1708.09124 (2017)
Needham, T.: Kähler structures on spaces of framed curves. Ann. Glob. Anal. Geom. 54(1), 123–153 (2018)
Needham, T.R.: Grassmannian geometry of framed curve spaces. Ph.D. thesis, University of Georgia (2016)
Neretin, Y.A.: On jordan angles and the triangle inequality in grassmann manifolds. Geometriae Dedicata 86(1–3), 81–91 (2001)
Pennec, X.: Intrinsic statistics on riemannian manifolds: basic tools for geometric measurements. J. Math. Imaging Vis. 25(1), 127 (2006)
Srivastava, A., Klassen, E., Joshi, S.H., Jermyn, I.H.: Shape analysis of elastic curves in euclidean spaces. IEEE Trans. Pattern Anal. Mach. Intell. 33(7), 1415–1428 (2011)
Srivastava, A., Klassen, E.P.: Functional and Shape Data Analysis. Springer, Berlin (2016)
Srivastava, A., Turaga, P., Kurtek, S.: On advances in differential-geometric approaches for 2d and 3d shape analyses and activity recognition. Image Vis. Comput. 30(6–7), 398–416 (2012)
Su, Z., Klassen, E., Bauer, M.: The square root velocity framework for curves in a homogeneous space. In: 2017 IEEE Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), pp. 680–689. IEEE (2017)
Wang, W., Jüttler, B., Zheng, D., Liu, Y.: Computation of rotation minimizing frames. ACM Trans. Graph. (TOG) 27(1), 2 (2008)
Younes, L., Michor, P.W., Shah, J.M., Mumford, D.B.: A metric on shape space with explicit geodesics. Rendiconti Lincei-Matematica e Applicazioni 19(1), 25–57 (2008)
Acknowledgements
I would like to thank Muyuan Chen, Steven Ludtke and Lynn Zechiedrich for graciously providing me with the very interesting DNA minicircles data. Next I would like to thank Michael Tychonievich for his help in developing a GUI for the framed curves matching program used to produce the numerical experiments. Many thanks are also due to various colleagues with whom I have had conversations about elastic shape analysis and framed curves over the years, including Jason Cantarella, Sebastian Kurtek, Erik Schreyer and Clayton Shonkwiler. Finally, I thank the anonymous reviewers for several helpful comments and suggestions on the first version of the paper.
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Appendix
Appendix
1.1 Proof of Theorem 2
Let \(g^{\mathrm {SO}(3)}\) denote the standard bi-invariant metric on \(\mathrm {SO}(3)\) induced by the Euclidean metric on \({{\mathfrak {s}}}{{\mathfrak {o}}}(3) \approx {\mathbb {R}}^3\), let \(g^{{\mathbb {R}}^+}\) denote the bi-invariant metric on \({\mathbb {R}}^+\) induced by
on \(T_1 {\mathbb {R}}^+ = {\mathbb {R}}\) and let \(g^{\mathrm {SO}(3)} \otimes g^{{\mathbb {R}}^+}\) denote the product metric on \(\mathrm {SO}(3) \times {\mathbb {R}}^+\). A natural \(L^2\)-type metric on \({\mathcal {P}}(\mathrm {SO}(3) \times {\mathbb {R}}^+)\) is given by
where \((A,r) \in {\mathcal {P}}(\mathrm {SO}(3) \times {\mathbb {R}}^+)\), and the arguments of \(g_{(A,r)}\) are elements of
and \(\mathrm {d}s = r(t)\mathrm {d}t\). It is straightforward to show that the pullback of g to \(\widehat{{\mathcal {S}}}_o\) via (1) is exactly the metric \(g^{\mathcal {S}}\).
Now let h denote the classical Hopf map (3) and let \(\mathrm {sq}\) denote the squaring map \(r \mapsto r^2\) for \(r \in {\mathbb {R}}^+\). It is a classical fact that h satisfies \(h^*g^{\mathrm {SO}(3)} = 4 g^{\mathrm {SU}(2)}\), where \(g^{\mathrm {SU}(2)}\) is the standard metric on \(\mathrm {SU}(2)\), which is isometric to the round metric \(g^{S^3}\) on \(S^3 \approx \mathrm {SU}(2)\). It follows that
where \(g^{S^3} \otimes g^{{\mathbb {R}}^+}\) is the product metric on \(S^3 \times {\mathbb {R}}^+\). Let \(f:{\mathbb {H}}\setminus \{\mathbf {0}\} \rightarrow S^3 \times {\mathbb {R}}^+\) denote the polar coordinate map \(q \mapsto (q/\Vert q\Vert _{{\mathbb {H}}},\Vert q\Vert _{{\mathbb {H}}})\). An elementary computation shows
Note that the map \(\mathrm {H}\) is obtained by applying \((h \times \mathrm {sq}) \circ f\) pointwise and then composing the result with the inverse of (1). Consider the following calculation, in which \(q \in {\mathcal {P}}{\mathbb {H}}^*\) satisfies \(\mathrm {H}(q)=(\gamma ,V)\) and \((\gamma ,V) \mapsto (A,r)\) under (1):
Using the fact that
(as operators on metrics), integrating over I against \(\mathrm {d}t\) and using \(\mathrm {d}s=r(t)\mathrm {d}t\) then yields
and this concludes the proof.
1.2 Proof of Proposition 1
Since \(\mathrm {SU}(2)\) acts by \(L^2\) isometries, we seek the minimizer \({\widehat{A}}\) of \(\arccos \langle q_0,q_1 \cdot A\rangle _{L^2}\), which is equivalent to finding the maximizer of \(\langle q_0,q_1 \cdot A\rangle _{L^2}\). The latter quantity is equal to
where the second equality follows by cyclic permutation invariance of the real part of quaternionic arithmetic. The quantity is therefore maximized by \({\widehat{A}} \in S^3 \approx \mathrm {SU}(2)\) with conjugate in the same direction as \(\int _I \overline{q_1} \cdot q_0 \; \mathrm {d}t\), and this completes the proof.
1.3 Proof of Theorem 4
Let \(q \in S_{\sqrt{2}}\). The horizontal tangent space to q is the subset of tangent vectors in
which are \(L^2\)-orthogonal to the \({\mathcal {P}}S^1\)-orbit directions at q. These orbit directions are of the form \(i \xi \cdot q\), where \(\xi :{\mathbb {R}}\rightarrow {\mathbb {R}}\) is a smooth function. A tangent vector p is therefore horizontal if and only if \(\langle p,i\xi q\rangle _{L^2} = 0\) for all \(\xi \). Switching to complex coordinates \(q=(z,w)\) and \(p=(u,v)\), this condition becomes
for all smooth \(\xi \). By the standard argument from the calculus of variations, we conclude that \(p=(u,v)\) is horizontal if and only if \(\mathrm {Im} \langle (u,v),(z,w)\rangle _{{\mathbb {C}}^2}\) is identically zero.
Consider elements \(q_0=(z_0,w_0)\) and \(q_1=(z_1,w_1)\) of \(S_{\sqrt{2}}^*\) which do not lie in the same \({\mathcal {P}}S^1\)-orbit and with
for all t. We seek \({\widehat{q}}_1 = e^{i\psi } \cdot q_1\) in the \({\mathcal {P}}S^1\)-orbit of \(q_1\) such that the geodesic \(q_u\) joining \(q_0\) and \({\widehat{q}}_1\) in \(S_{{\sqrt{2}}}\) is horizontal for all u. Since \({\mathcal {P}}S^1\) acts by isometries, if the geodesic starts horizontal then it will stay horizontal—that is, it suffices to find \({\widehat{q}}_1\) so that \(\left. \frac{d}{du}\right| _{u=0} q_u\) is \({\mathcal {P}}S^1\)–horizontal at \(q_0\).
The geodesic joining \(q_0\) and \({\widehat{q}}_1\) is given by (10). The derivative at \(u=0\) of this geodesic is given by
Writing \(q_0=(z_0,w_0)\), \(q_1=(z_1,w_1)\) and recalling that \({\widehat{q}}_1=e^{i\psi } \cdot q_1\) for some \(\psi :{\mathbb {R}}\rightarrow {\mathbb {R}}\), the desired horizontality condition reduces to
and this condition is achieved by taking
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Needham, T. Shape Analysis of Framed Space Curves. J Math Imaging Vis 61, 1154–1172 (2019). https://doi.org/10.1007/s10851-019-00895-y
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DOI: https://doi.org/10.1007/s10851-019-00895-y