Abstract
In many applications, one is interested in the shape of an object, like the contour of a bone or the trajectory of joints of a tennis player, irrespective of the way these shapes are parameterized. However for analysis of these shape spaces, it is sometimes useful to have a parameterization at hand, in particular if one is interested in deforming shapes. The purpose of the paper is to examine three different methods that one can follow to endow shape spaces with a Riemannian metric that is measuring deformations in a parameterization independent way. The first is via Riemannian submersion on a quotient; the second is via isometric immersion on a particular slice; and the third is an alternative method that allows for an arbitrarily chosen complement to the vertical space and a metric degenerate along the fibers, which we call the gauge-invariant metric. This allows some additional flexibility in applications, as we describe.
Supported by FWF grant I 5015-N, Institut CNRS Pauli and University of Lille.
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References
Bauer, M., Charon, N., Klassen, E., Kurtek, S., Needham, T., Pierron, T.: Elastic metrics on spaces of Euclidean curves: theory and algorithms (2022). arXiv:2209.09862v1
Ciuclea, I., Tumpach, A.B., Vizman, C.: Shape spaces of non-linear flags. To appear in GSI23
Drira, H., Tumpach, A.B., Daoudi, M.: Gauge invariant framework for trajectories analysis. In: DIFFCV Workshop 2015 (2015). https://doi.org/10.5244/C.29.DIFFCV.6. GIFcurve.pdf
Mio, W., Srivastava, A., Joshi, S.H.: On shape of plane elastic curves. Int. J. Comput. Vision 73, 307–324 (2007)
Lang, S.: Fundamentals of Differential Geometry. Springer, New York (1999). https://doi.org/10.1007/978-1-4612-0541-8
Preston, S.C.: The geometry of whips. Ann. Global Anal. Geometry 41(3) (2012)
Tumpach, A.B.: Gauge invariance of degenerate Riemannian metrics. Not. Amer. Math. Soc. (2016). http://www.math.univ-lille1.fr/~tumpach/Site/researchfiles/Noticesfull.pdf
Tumpach, A.B., Drira, H., Daoudi, M., Srivastava, A.: Gauge invariant framework for shape analysis of surfaces. IEEE Trans. Pattern Anal. Mach. Intell. 38(1) (2016)
Tumpach, A.B.: On canonical parameterizations of \(2D\)-shapes To appear in GSI23
Tumpach, A.B., Preston, S.C.: Quotient elastic metrics on the manifold of arc-length parameterized plane curves. J. Geom. Mech. 9(2), 227–256 (2017)
Tumpach, A.B., Preston, S.C.: Riemannian metrics on shape spaces: comparison of different constructions. (in preparation)
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Tumpach, A.B., Preston, S.C. (2023). Three Methods to Put a Riemannian Metric on Shape Space. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2023. Lecture Notes in Computer Science, vol 14071. Springer, Cham. https://doi.org/10.1007/978-3-031-38271-0_1
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DOI: https://doi.org/10.1007/978-3-031-38271-0_1
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