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In high accuracy numerical simulations and optimal control of time-dependent processes, often both many time steps and fine spatial discretizations are needed. Adjoint gradient computation, or post-processing of simulation results, requires the storage of the solution trajectories over the whole time, if necessary together with the adaptively refined spatial grids. In this paper we discuss various techniques to reduce the memory requirements, focusing first on the storage of the solution data, which typically are double precision floating point values. We highlight advantages and disadvantages of the different approaches. Moreover, we present an algorithm for the efficient storage of adaptively refined, hierarchic grids, and the integration with the compressed storage of solution data.

We propose a novel Riemannian framework for statistical analysis of shapes that is able to account for the nonlinearity in shape variation. By adopting a physical perspective, we introduce a differential representation that puts the local geometric variability into focus. We model these differential coordinates as elements of a Lie group thereby endowing our shape space with a non-Euclidean structure. A key advantage of our framework is that statistics in a manifold shape space becomes numerically tractable improving performance by several orders of magnitude over state-of-the-art. We show that our Riemannian model is well suited for the identification of intra-population variability as well as inter-population differences. In particular, we demonstrate the superiority of the proposed model in experiments on specificity and generalization ability. We further derive a statistical shape descriptor that outperforms the standard Euclidean approach in terms of shape-based classification of morphological disorders.

We introduce techniques for the processing of motion and animations of non-rigid shapes. The idea is to regard animations of deformable objects as curves in shape space. Then, we use the geometric structure on shape space to transfer concepts from curve processing in R^n to the processing of motion of non-rigid shapes. Following this principle, we introduce a discrete geometric flow for curves in shape space. The flow iteratively replaces every shape with a weighted average shape of a local neighborhood and thereby globally decreases an energy whose minimizers are discrete geodesics in shape space. Based on the flow, we devise a novel smoothing filter for motions and animations of deformable shapes. By shortening the length in shape space of an animation, it systematically regularizes the deformations between consecutive frames of the animation. The scheme can be used for smoothing and noise removal, e.g., for reducing jittering artifacts in motion capture data. We introduce a reduced-order method for the computation of the flow. In addition to being efficient for the smoothing of curves, it is a novel scheme for computing geodesics in shape space. We use the scheme to construct non-linear “Bézier curves” by executing de Casteljau’s algorithm in shape space.

We introduce techniques for the processing of motion and animations of non-rigid shapes. The idea is to regard animations of deformable objects as curves in shape space. Then, we use the geometric structure on shape space to transfer concepts from curve processing in Rn to the processing of motion of non-rigid shapes. Following this principle, we introduce a discrete geometric flow for curves in shape space. The flow iteratively replaces every shape with a weighted average shape of a local neighborhood and thereby globally decreases an energy whose minimizers are discrete geodesics in shape space. Based on the flow, we devise a novel smoothing filter for motions and animations of deformable shapes. By shortening the length in shape space of an animation, it systematically regularizes the deformations between consecutive frames of the animation. The scheme can be used for smoothing and noise removal, e.g., for reducing jittering artifacts in motion capture data. We introduce a reduced-order method for the computation of the flow. In addition to being efficient for the smoothing of curves, it is a novel scheme for computing geodesics in shape space. We use the scheme to construct non-linear Bézier curves by executing de Casteljau's algorithm in shape space.

We introduce a scheme for real-time nonlinear interpolation of a set of shapes. The scheme exploits the structure of the shape interpolation problem, in particular, the fact that the set of all possible interpolated shapes is a low-dimensional object in a high-dimensional shape space. The interpolated shapes are defined as the minimizers of a nonlinear objective functional on the shape space. Our approach is to construct a reduced optimization problem that approximates its unreduced counterpart and can be solved in milliseconds. To achieve this, we restrict the optimization to a low-dimensional subspace that is specifically designed for the shape interpolation problem. The construction of the subspace is based on two components: a formula for the calculation of derivatives of the interpolated shapes and a Krylov-type sequence that combines the derivatives and the Hessian of the objective functional. To make the computational cost for solving the reduced optimization problem independent of the resolution of the example shapes, we combine the dimensional reduction with schemes for the efficient approximation of the reduced nonlinear objective functional and its gradient. In our experiments, we obtain rates of 20-100 interpolated shapes per second even for the largest examples which have 500k vertices per example shape.

We propose a new framework for spacetime optimization that can generate artistic motion with a long planning horizon for complex virtual characters. The scheme can be used for generating general types of motion and neither requires motion capture data nor an initial motion that satisfies the constraints. Our modeling of the spacetime optimization combines linearized dynamics and a novel warping scheme for articulated characters. We show that the optimal motions can be described using a combination of vibration modes, wiggly splines, and our warping scheme. This enables us to restrict the optimization to low-dimensional spaces of explicitly parametrized motions. Thereby the computation of an optimal motion is reduced to a low-dimensional non-linear least squares problem, which can be solved with standard solvers. We show examples of motions created by specifying only a few constraints for positions and velocities.

This thesis concerns model reduction techniques for the efficient numerical treatment
of physical systems governing the deformation behavior of geometrically complex shapes.
We present new strategies for the construction of simplified,
low-dimensional models that capture the main features of the original complex system
and are suitable for use in interactive computer graphics applications.
To demonstrate the effectiveness of the new techniques we propose frameworks
for real-time simulation and interactive deformation-based modeling
of elastic solids and shells and compare them to alternative approaches.
In addition, we investigate differential operators that are derived from
the physical models and hence can serve as alternatives to the Laplace-Beltrami
operator for applications in modal shape analysis.
Furthermore, this thesis addresses the compression of digital shapes.
In particular, we present a lossless compression scheme that is adapted to
the special characteristics of adaptively refined, hierarchical meshes.

Geometry processing
(2014)