@article{TycowiczSchulzSeideletal., author = {Tycowicz, Christoph von and Schulz, Christian and Seidel, Hans-Peter and Hildebrandt, Klaus}, title = {Real-time Nonlinear Shape Interpolation}, series = {ACM Transactions on Graphics}, volume = {34}, journal = {ACM Transactions on Graphics}, number = {3}, doi = {10.1145/2729972}, pages = {34:1 -- 34:10}, abstract = {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.}, language = {en} } @article{TycowiczSchulzSeideletal., author = {Tycowicz, Christoph von and Schulz, Christian and Seidel, Hans-Peter and Hildebrandt, Klaus}, title = {An Efficient Construction of Reduced Deformable Objects}, series = {ACM Transactions on Graphics}, volume = {32}, journal = {ACM Transactions on Graphics}, number = {6}, publisher = {ACM}, doi = {10.1145/2508363.2508392}, pages = {213:1 -- 213:10}, abstract = {Many efficient computational methods for physical simulation are based on model reduction. We propose new model reduction techniques for the approximation of reduced forces and for the construction of reduced shape spaces of deformable objects that accelerate the construction of a reduced dynamical system, increase the accuracy of the approximation, and simplify the implementation of model reduction. Based on the techniques, we introduce schemes for real-time simulation of deformable objects and interactive deformation-based editing of triangle or tet meshes. We demonstrate the effectiveness of the new techniques in different experiments with elastic solids and shells and compare them to alternative approaches.}, language = {en} } @inproceedings{SchulzTycowiczSeideletal., author = {Schulz, Christian and Tycowicz, Christoph von and Seidel, Hans-Peter and Hildebrandt, Klaus}, title = {Animating articulated characters using wiggly splines}, series = {ACM SIGGRAPH / Eurographics Symposium on Computer Animation}, booktitle = {ACM SIGGRAPH / Eurographics Symposium on Computer Animation}, doi = {10.1145/2786784.2786799}, pages = {101 -- 109}, abstract = {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.}, language = {en} } @article{SchulzTycowiczSeideletal., author = {Schulz, Christian and Tycowicz, Christoph von and Seidel, Hans-Peter and Hildebrandt, Klaus}, title = {Animating Deformable Objects Using Sparse Spacetime Constraints}, series = {ACM Transactions on Graphics}, volume = {33}, journal = {ACM Transactions on Graphics}, number = {4}, doi = {10.1145/2601097.2601156}, pages = {109:1 -- 109:10}, abstract = {We propose a scheme for animating deformable objects based on spacetime optimization. The main feature is that it robustly and quickly (within a few seconds) generates interesting motion from a sparse set of spacetime constraints. Providing only partial (as opposed to full) keyframes for positions and velocities is sufficient. The computed motion satisfies the constraints and the remaining degrees of freedom are determined by physical principles using elasticity and the spacetime constraints paradigm. Our modeling of the spacetime optimization problem combines dimensional reduction, modal coordinates, wiggly splines, and rotation strain warping. Controlling the warped motion requires the derivative of the warp map. We derive a representation of the derivative that can be efficiently and robustly evaluated. Our solver is based on a theorem that characterizes the solutions of the optimization problem and allows us to restrict the optimization to very low-dimensional search spaces. This treatment of the optimization problem avoids a time discretization and the resulting method can robustly deal with sparse input and wiggly motion.}, language = {en} }