A metric on shape space with explicit geodesics

Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni

Volumn 19, Issue 1, 2008, Pages 25-57

  Younes, Laurent ^a   Michor, Peter W ^b,c   Shah, Jayant ^d   Mumford, David ^e  

^a JOHNS HOPKINS UNIVERSITY   (United States)

^b UNIVERSITY OF VIENNA   (Austria)

^c Erwin Schrodinger International Institute for Mathematical Physics   (Austria)

^d NORTHEASTERN UNIVERSITY   (United States)

^e BROWN UNIVERSITY   (United States)

Author keywords

Diffeomorphism group; Riemannian metric; Shape space; Stiefel manifold

Indexed keywords

EID: 84887244054     PISSN: 11206330     EISSN: 17200768     Source Type: Journal    
DOI: 10.4171/RLM/506     Document Type: Article

References (19)

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.
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.
6. S. HELGASON, Differential Geometry, Lie Groups and Symmetric Spaces. Grad. Stud. Math. 34, Amer. Math. Soc., 1978 (revised version, 2001).
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.
9. A. KRIEGL - P. W. MICHOR, The Convenient Setting of Global Analysis. Math. Surveys Monogr. 53, Amer. Math. Soc., 1997.
10. H. KRIM - A. YEZZI (eds.), Statistics and Analysis of Shapes. Birkhäuser, 2006.
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.
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. (Pubitemid 33634791)
14. B. O'NEILL, The fundamental equations of a submersion. Michigan Math. J. (1966), 459-469.
15. E. SHARON - D. MUMFORD, 2d-shape analysis using conformal mapping. Int. J. Comput. Vision 70 (2006), 55-75. (Pubitemid 43855867)
16. A. TROUVÉ - 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. TROUVÉ - L. YOUNES, On a class of diffeomorphic matching problems in one dimension. SIAM J. Control Optim. 39 (2002), 1112-1135.
18. L. YOUNES, Computable elastic distances between shapes. SIAM J. Appl. Math. 58 (1998), 565-586. (Pubitemid 128496905)
19. L. YOUNES, Optimal matching between shapes via elastic deformations. Image and Vision Computing 17 (1999), 381-389.