A metric on shape space with explicit geodesics. (English) Zbl 1142.58013
Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 19, No. 1, 25-57 (2008).
Two of the authors studied several classes of Riemannian metrics on spaces of shapes, see [P. Michor and D. Mumford, Appl. Comput. Harmon. Anal. 23, No. 1, 74–113 (2007; Zbl 1116.58007)]. In this paper, Sobolev 1-type metric is studied as a specific metric of spaces of shapes, and explict descriptions of geodesics and computaitons of sectional curvatures are given. As a consequence, the space of closed curves modulo rotation and change of variables is shown to have positive curvature.
Throughout the paper, curves mean curves in the complex plane. Identify the tangent space of the space of immersions of curves to the complex plane modulo translation with the set of vector field \(h:S^1\to C\) along \(c\) modulo constant vector fields. The specific metric \(G_c(h,h)\) is \(\frac{1}{\ell(c)}\int_{S^1}|D_sh|^2\,ds\). The geodesic equation on this metric is computed in the Appendix. The key fact on this metric is that it is induced from the usual \(L^2\)-metric by the map \(\Phi:(e,f)\to\frac{1}{2}\int_0^\theta (e(x);f(x))^2\,dx\) (Theorem 2.7.). \(\Phi(e,f)\) is an immersion if \(Z(e,f)=\{\theta:e(\theta)=f(\theta)=0\}=\emptyset\), and periodic if \((e,f)\) is an orthonormal pair. So the property of \(G_c\) is related to the property of classical manifolds such as sphere, complex projective space, Stiefel manifolds and Grassmann manifolds. On the other hand, owing to the exclusion of \(Z(e,f)\), \(G_c\) is not a complete metric. To overcome this difficulty, authors introduce a Fréchet equivalence based on a monotone relation (3.5). As a price of this extension of equivalence, geodesics may have curves with singularities, which are shown in examples.
Applying these general discussions, the construction of geodesics and the calculation of sectional curvatures are presented in Sections 4 and 5. The existence problem of geodesics reduces to a variational problem for a nonlinear functional. As for this problem, the authors refer to A. Trouvé and L. Younes [SIAM J. Control Optim. 39, No. 4, 1112–1135 (2000; Zbl 0983.49009)]. The calculation of the sectional curvature uses the formula of B. O’Neill [Mich. Math. J. 13, 459–469 (1966; Zbl 0145.18602)], together with explicit calculation of O’Neill’s correction term ((20) and (21)). Geodesic distance defined in §4 is computed in a short time by dynamic programming. This is explained in §6 [cf. A. Trouvé and L. Younes, Lect. Notes Comput. Sci. 1842. 573–587 (2000; Zbl 1045.68945)].
Throughout the paper, curves mean curves in the complex plane. Identify the tangent space of the space of immersions of curves to the complex plane modulo translation with the set of vector field \(h:S^1\to C\) along \(c\) modulo constant vector fields. The specific metric \(G_c(h,h)\) is \(\frac{1}{\ell(c)}\int_{S^1}|D_sh|^2\,ds\). The geodesic equation on this metric is computed in the Appendix. The key fact on this metric is that it is induced from the usual \(L^2\)-metric by the map \(\Phi:(e,f)\to\frac{1}{2}\int_0^\theta (e(x);f(x))^2\,dx\) (Theorem 2.7.). \(\Phi(e,f)\) is an immersion if \(Z(e,f)=\{\theta:e(\theta)=f(\theta)=0\}=\emptyset\), and periodic if \((e,f)\) is an orthonormal pair. So the property of \(G_c\) is related to the property of classical manifolds such as sphere, complex projective space, Stiefel manifolds and Grassmann manifolds. On the other hand, owing to the exclusion of \(Z(e,f)\), \(G_c\) is not a complete metric. To overcome this difficulty, authors introduce a Fréchet equivalence based on a monotone relation (3.5). As a price of this extension of equivalence, geodesics may have curves with singularities, which are shown in examples.
Applying these general discussions, the construction of geodesics and the calculation of sectional curvatures are presented in Sections 4 and 5. The existence problem of geodesics reduces to a variational problem for a nonlinear functional. As for this problem, the authors refer to A. Trouvé and L. Younes [SIAM J. Control Optim. 39, No. 4, 1112–1135 (2000; Zbl 0983.49009)]. The calculation of the sectional curvature uses the formula of B. O’Neill [Mich. Math. J. 13, 459–469 (1966; Zbl 0145.18602)], together with explicit calculation of O’Neill’s correction term ((20) and (21)). Geodesic distance defined in §4 is computed in a short time by dynamic programming. This is explained in §6 [cf. A. Trouvé and L. Younes, Lect. Notes Comput. Sci. 1842. 573–587 (2000; Zbl 1045.68945)].
Reviewer: Akira Asada (Takarazuka)
MSC:
| 58E10 | Variational problems in applications to the theory of geodesics (problems in one independent variable) |
| 58B20 | Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds |
| 58D15 | Manifolds of mappings |
| 58E40 | Variational aspects of group actions in infinite-dimensional spaces |
References:
| [1] | P. ALEXANDROFF - H. HOPF, Topologie. I . Springer, Berlin, 1974. |
| [2] | R. BASRI - L. COSTA - D. GEIGER - D. JACOBS, Determining the similarity of deformable shapes . In: IEEE Workshop on Physics based Modeling in Computer Vision, 1995, 135-143. |
| [3] | R. D. BENGURIA - M. LOSS, Connection between the Lieb-Thirring conjecture for Schrö- dinger operators and an isoperimetric problem for ovals in the plane . In: Partial Differential Equations and Inverse Problems, Contemp. Math. 362, Amer. Math. Soc., 2004, 53-61. · Zbl 1087.81042 |
| [4] | M. P. DO CARMO, Riemannian Geometry . Birkhäuser, 1992. |
| [5] | U. GRENANDER - D. M. KEENAN, On the shape of plane images . SIAM J. Appl. Math. 53 (1991), 1072-1094. · Zbl 0780.60007 · doi:10.1137/0153054 |
| [6] | S. HELGASON, Differential Geometry, Lie Groups and Symmetric Spaces . Grad. Stud. Math. 34, Amer. Math. Soc., 1978 (revised version, 2001). · Zbl 0993.53002 |
| [7] | E. KLASSEN - A. SRIVASTAVA - W. MIO - S. JOSHI, Analysis of planar shapes using geodesic paths on shape spaces . IEEE Trans. Pattern. Anal. Machine Intell. 26 (2002), no. 3. |
| [8] | A. KRIEGL - M. LOSIK - P. W. MICHOR, Choosing roots of polynomials smoothly, II . Israel J. Math. 139 (2004), 183-188. · Zbl 1071.26009 · doi:10.1007/BF02787548 |
| [9] | A. KRIEGL - P. W. MICHOR, The Convenient Setting of Global Analysis . Math. Surveys Monogr. 53, Amer. Math. Soc., 1997. · Zbl 0889.58001 |
| [10] | H. KRIM - A. YEZZI (eds.), Statistics and Analysis of Shapes . Birkhäuser, 2006. 57 · Zbl 1091.00003 · doi:10.1007/0-8176-4481-4 |
| [11] | P. MICHOR - D. MUMFORD, An overview of the riemannian metrics on spaces of curves using the hamiltonian approach . Appl. Comput. Harmonic Anal. 23 (2007), 64-113. · Zbl 1116.58007 · doi:10.1016/j.acha.2006.07.004 |
| [12] | W. MIO - A. SRIVASTAVA - S. JOSHI, On the shape of plane elastic curves . Tech. report, Florida State Univ., 1995. |
| [13] | Y. A. NERETIN, On Jordan angles and the triangle inequality in Grassmann manifolds . Geom. Dedicata 86 (2001), 81-92. · Zbl 1001.53037 · doi:10.1023/A:1011974705094 |
| [14] | B. O’NEILL, The fundamental equations of a submersion . Michigan Math. J. (1966), 459-469. · Zbl 0145.18602 · doi:10.1307/mmj/1028999604 |
| [15] | E. SHARON - D. MUMFORD, 2d-shape analysis using conformal mapping . Int. J. Comput. Vision 70 (2006), 55-75. |
| [16] | A. TROUV É - L. YOUNES, Diffeomorphic matching in 1d: designing and minimizing matching functionals . In: ECCV 2000, Proc. 6th European Conf. on Computer Vision (Dublin, 2000), D. Vernon (ed.), Lecture Notes in Comput. Sci. 1842, Springer, 2000, 573-587. |
| [17] | A. TROUV É - L. YOUNES, On a class of diffeomorphic matching problems in one dimension . SIAM J. Control Optim. 39 (2002), 1112-1135. · Zbl 0983.49009 · doi:10.1137/S036301299934864X |
| [18] | L. YOUNES, Computable elastic distances between shapes . SIAM J. Appl. Math. 58 (1998), 565-586. · Zbl 0907.68158 · doi:10.1137/S0036139995287685 |
| [19] | L. YOUNES, Optimal matching between shapes via elastic deformations . Image and Vision Computing 17 (1999), 381-389. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
