Shape space – a paradigm for character animation in computer graphics

Nowadays 3D computer animation is increasingly realistic as the models used for the characters become more and more complex. These models are typically represented by meshes of hundreds of thousands or even millions of triangles. The mathematical notion of a shape space allows us to effectively model, manipulate, and animate such meshes. Once an appropriate notion of dissimilarity measure between different triangular meshes is defined, various useful tools in character modeling and animation turn out to coincide with basic geometric operations derived from this definition

Numerical Methods in Shape Spaces and Optimal Branching Patterns

The contribution of this thesis is twofold. The main part deals with numerical methods in the context of shape space analysis, where the shape space at hand is considered as a Riemannian manifold. In detail, we apply and extend the time-discrete geodesic calculus (established by Rumpf and Wirth [WBRS11, RW15]) to the space of discrete shells, i.e. triangular meshes with fixed connectivity. The essential building block is a variational time-discretization of geodesic curves, which is based on a local approximation of the squared Riemannian distance on the manifold. On physical shape spaces this approximation can be derived e.g. from a dissimilarity measure. The dissimilarity measure between two shell surfaces can naturally be defined as an elastic deformation energy capturing both membrane and bending distortions. Combined with a non-conforming discretization of a physically sound thin shell model the time-discrete geodesic calculus applied to the space of discrete shells is shown to be suitable to solve important problems in computer graphics and animation. To extend the existing calculus, we introduce a generalized spline functional based on the covariant derivative along a curve in shape space whose minimizers can be considered as Riemannian splines. We establish a corresponding time-discrete functional that fits perfectly into the framework of Rumpf and Wirth, and prove this discretization to be consistent. Several numerical simulations reveal that the optimization of the spline functional—subject to appropriate constraints—can be used to solve the multiple interpolation problem in shape space, e.g. to realize keyframe animation. Based on the spline functional, we further develop a simple regression model which generalizes linear regression to nonlinear shape spaces. Numerical examples based on real data from anatomy and botany show the capability of the model. Finally, we apply the statistical analysis of elastic shape spaces presented by Rumpf and Wirth [RW09, RW11] to the space of discrete shells. To this end, we compute a Fréchet mean within a class of shapes bearing highly nonlinear variations and perform a principal component analysis with respect to the metric induced by the Hessian of an elastic shell energy. The last part of this thesis deals with the optimization of microstructures arising e.g. at austenite-martensite interfaces in shape memory alloys. For a corresponding scalar problem, Kohn and Müller [KM92, KM94] proved existence of a minimizer and provided a lower and an upper bound for the optimal energy. To establish the upper bound, they studied a particular branching pattern generated by mixing two different martensite phases. We perform a finite element simulation based on subdivision surfaces that suggests a topologically different class of branching patterns to represent an optimal microstructure. Based on these observations we derive a novel, low dimensional family of patterns and show—numerically and analytically—that our new branching pattern results in a significantly better upper energy bound

NeuroMorph: Unsupervised Shape Interpolation and Correspondence in One Go

We present NeuroMorph, a new neural network architecture that takes as input two 3D shapes and produces in one go, i.e. in a single feed forward pass, a smooth interpolation and point-to-point correspondences between them. The interpolation, expressed as a deformation field, changes the pose of the source shape to resemble the target, but leaves the object identity unchanged. NeuroMorph uses an elegant architecture combining graph convolutions with global feature pooling to extract local features. During training, the model is incentivized to create realistic deformations by approximating geodesics on the underlying shape space manifold. This strong geometric prior allows to train our model end-to-end and in a fully unsupervised manner without requiring any manual correspondence annotations. NeuroMorph works well for a large variety of input shapes, including non-isometric pairs from different object categories. It obtains state-of-the-art results for both shape correspondence and interpolation tasks, matching or surpassing the performance of recent unsupervised and supervised methods on multiple benchmarks.Comment: Published at the IEEE/CVF Conference on Computer Vision and Pattern Recognition 202

Approximation of Splines in Wasserstein Spaces

This paper investigates a time discrete variational model for splines in Wasserstein spaces to interpolate probability measures. Cubic splines in Euclidean space are known to minimize the integrated squared acceleration subject to a set of interpolation constraints. As generalization on the space of probability measures the integral over the squared acceleration is considered as a spline energy and regularized by addition of the usual action functional. Both energies are then discretized in time using local Wasserstein-2 distances and the generalized Wasserstein barycenter. The existence of time discrete regularized splines for given interpolation conditions is established. On the subspace of Gaussian distributions, the spline interpolation problem is solved explicitly and consistency in the discrete to continuous limit is shown. The computation of time discrete splines is implemented numerically, based on entropy regularization and the Sinkhorn algorithm. A variant of the iPALM method is applied for the minimization of the fully discrete functional. A variety of numerical examples demonstrate the robustness of the approach and show striking characteristics of the method. As a particular application the spline interpolation for synthesized textures is presented.Comment: 25 pages, 9 figure