Matthias Erbar, Jan Maas and Melchior Wirth On the geometry of geodesics in discrete optimal transport Calc. Var. Partial Differential Equations, 58(1): Art. 19, 19 2019 10.1007/s00526-018-1456-1
2017
Behrend Heeren, Martin Rumpf and Benedikt Wirth Variational time discretization of Riemannian splines IMA J. Numer. Anal. 2017 https://arxiv.org/abs/1711.06069
2015
Alexander Effland, Martin Rumpf, Stefan Simon, Kirsten Stahn and Benedikt Wirth Bézier curves in the space of images In Proceedings Scale Space and Variational Methods in Computer Vision, Volume 9087 of Lecture Notes in Computer Science
page 372-384.
Publisher: Springer International
2015 http://dx.doi.org/10.1007/978-3-319-18461-6_30
Abstract: Bézier curves are a widespread tool for the design of curves in Euclidian space. This paper generalizes the notion of Bézier curves to the infinite-dimensional space of images. To this end the space of images is equipped with a Riemannian metric which measures the cost of image transport and intensity variation in the sense of the metamorphosis model by Miller and Younes. Bézier curves are then computed via the Riemannian version of de Casteljau's algorithm, which is based on a hierarchical scheme of convex combination along geodesic curves. Geodesics are approximated using a variational discretization of the Riemannian path energy. This leads to a generalized de Casteljau method to compute suitable discrete Bézier curves in image space. Selected test cases demonstrate qualitative properties of the approach. Furthermore, a Bézier approach for the modulation of face interpolation and shape animation via image sketches is presented.
Martin Rumpf and Benedikt Wirth Variational time discretization of geodesic calculus IMA J. Numer. Anal., 35(3): 1011-1046 2015 http://dx.doi.org/10.1093/imanum/dru027
Abstract: We analyze a variational time discretization of geodesic calculus on finite- and certain classes of infinite-dimensional Riemannian manifolds. We investigate the fundamental properties of discrete geodesics, the associated discrete logarithm, discrete exponential maps, and discrete parallel transport, and we prove convergence to their continuous counterparts. The presented analysis is based on the direct methods in the calculus of variation, on -convergence, and on weighted finite element error estimation. The convergence results of the discrete geodesic calculus are experimentally confirmed for a basic model on a two-dimensional Riemannian manifold. This provides a theoretical basis for the application to shape spaces in computer vision, for which we present one specific example.
2014
Martin Rumpf and Benedikt Wirth Discrete geodesic calculus in the space of viscous fluidic objects SIAM J. Imaging Sci., 6(4): 2581-2602 2014 http://www.arxiv.org/abs/1210.0822
Abstract: Based on a local approximation of the Riemannian distance on a manifold by a computationally cheap dissimilarity measure, a time discrete geodesic calculus is developed, and applications to shape space are explored. The dissimilarity measure is derived from a deformation energy whose Hessian reproduces the underlying Riemannian metric, and it is used to define length and energy of discrete paths in shape space. The notion of discrete geodesics defined as energy minimizing paths gives rise to a discrete logarithmic map, a variational definition of a discrete exponential map, and a time discrete parallel transport. This new concept is developed in the context of shape spaces with shapes that are described via deformations of a given reference shape, and it is applied to a particular shape space in which shapes are considered as boundary contours of physical objects consisting of viscous material. The flexibility and computational efficiency of the approach is demonstrated for topology preserving shape morphing, the representation of paths in shape space via local shape variations as path generators, shape extrapolation via discrete geodesic flow, and the transfer of geometric features.
2013
Benjamin Berkels, Tom Fletcher, Behrend Heeren, Martin Rumpf and Benedikt Wirth Discrete geodesic regression in shape space In Anders Heyden, Fredrik Kahl, Carl Olsson, Magnus Oskarsson, Xue-Cheng Tai, editor, Energy Minimization Methods in Computer Vision and Pattern Recognition, Volume 8081 of Lecture Notes in Computer Science
page 108-122.
Publisher: Springer International
2013 http://dx.doi.org/10.1007/978-3-642-40395-8_9
Abstract: A new approach for the effective computation of geodesic re- gression curves in shape spaces is presented. Here, one asks for a geodesic curve on the shape manifold that minimizes a sum of dissimilarity mea- sures between given two- or three-dimensional input shapes and corre- sponding shapes along the regression curve. The proposed method is based on a variational time discretization of geodesics. Curves in shape space are represented as deformations of suitable reference shapes, which renders the computation of a discrete geodesic as a PDE constrained optimization for a family of deformations. The PDE constraint is de- duced from the discretization of the covariant derivative of the velocity in the tangential direction along a geodesic. Finite elements are used for the spatial discretization, and a hierarchical minimization strategy together with a Lagrangian multiplier type gradient descent scheme is implemented. The method is applied to the analysis of root growth in botany and the morphological changes of brain structures due to aging.