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.

Abstract: We prove existence and uniqueness of optimal maps on RCD∗(K,N) spaces under the assumption that the starting measure is absolutely continuous. We also discuss how this result naturally leads to the notion of exponentiation and to the local-to-global property of RCD∗(K,N) bounds.

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.

Abstract: The plastic component of the deformation gradient plays a central role in finite kinematic models of plasticity. However, its characterization has been the source of extended debates in the literature and many important issues still remain unresolved. Some examples are the micromechanical understanding of $F=F^eF^p$ with multiple active slip systems, the uniqueness of the decomposition, or the characterization of the plastic deformation without reference to the so-called intermediate configuration. In this paper, we shed some light to these issues via a two-dimensional kinematic analysis of the plastic deformation induced by discrete slip surfaces and the corresponding dislocation structures. In particular, we supply definitions for the elastic and plastic components of the deformation gradient as a function of the active slip systems without any a priori assumption on the decomposition of the total deformation gradient. These definitions are explicitly and uniquely given from the microstructure and do not make use of any unrealizable intermediate configuration. The analysis starts from a semi-continuous mathematical description of the deformation at the microscale, where the displacements are considered continuous everywhere in the domain except at the discrete slip surfaces, over which there is a displacement jump. At this scale, where the microstructure is resolved, the deformation is uniquely characterized from purely kinematic considerations and the elastic and plastic components of the deformation gradient can be defined based on physical arguments. These quantities are then passed to the continuous limit via homogenization, i.e., by increasing the number of slip surfaces to infinity and reducing the lattice parameter to zero. This continuum limit is computed for several illustrative examples, where the well-known multiplicative decomposition of the total deformation gradient is recovered. Additionally, by similar arguments, an expression of the dislocation density tensor is obtained as the limit of discrete dislocation densities which are well characterized within the semi-continuous model.

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.