TY - JOUR A1 - Kofler, Andreas A1 - Wald, Christian A1 - Kolbitsch, Christoph A1 - von Tycowicz, Christoph A1 - Ambellan, Felix T1 - Joint Reconstruction and Segmentation in Undersampled 3D Knee MRI combining Shape Knowledge and Deep Learning JF - Physics in Medicine and Biology N2 - Task-adapted image reconstruction methods using end-to-end trainable neural networks (NNs) have been proposed to optimize reconstruction for subsequent processing tasks, such as segmentation. However, their training typically requires considerable hardware resources and thus, only relatively simple building blocks, e.g. U-Nets, are typically used, which, albeit powerful, do not integrate model-specific knowledge. In this work, we extend an end-to-end trainable task-adapted image reconstruction method for a clinically realistic reconstruction and segmentation problem of bone and cartilage in 3D knee MRI by incorporating statistical shape models (SSMs). The SSMs model the prior information and help to regularize the segmentation maps as a final post-processing step. We compare the proposed method to a state-of-the-art (SOTA) simultaneous multitask learning approach for image reconstruction and segmentation (MTL) and to a complex SSMs-informed segmentation pipeline (SIS). Our experiments show that the combination of joint end-to-end training and SSMs to further regularize the segmentation maps obtained by MTL highly improves the results, especially in terms of mean and maximal surface errors. In particular, we achieve the segmentation quality of SIS and, at the same time, a substantial model reduction that yields a five-fold decimation in model parameters and a computational speedup of an order of magnitude. Remarkably, even for undersampling factors of up to R=8, the obtained segmentation maps are of comparable quality to those obtained by SIS from ground-truth images. Y1 - 2024 U6 - https://doi.org/10.1088/1361-6560/ad3797 ER - TY - GEN A1 - Götschel, Sebastian A1 - von Tycowicz, Christoph A1 - Polthier, Konrad A1 - Weiser, Martin T1 - Reducing Memory Requirements in Scientific Computing and Optimal Control N2 - 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. T3 - ZIB-Report - 13-64 KW - optimal control KW - trajectory storage KW - mesh compression KW - compression Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-42695 SN - 1438-0064 ER - TY - JOUR A1 - von Tycowicz, Christoph A1 - Ambellan, Felix A1 - Mukhopadhyay, Anirban A1 - Zachow, Stefan T1 - An Efficient Riemannian Statistical Shape Model using Differential Coordinates JF - Medical Image Analysis N2 - 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. Y1 - 2018 U6 - https://doi.org/10.1016/j.media.2017.09.004 VL - 43 IS - 1 SP - 1 EP - 9 ER - TY - GEN A1 - Brandt, Christopher A1 - Tycowicz, Christoph von A1 - Hildebrandt, Klaus T1 - Geometric Flows of Curves in Shape Space for Processing Motion of Deformable Objects N2 - 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. T3 - ZIB-Report - 16-29 Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-59504 SN - 1438-0064 ER - TY - JOUR A1 - Brandt, Christopher A1 - Tycowicz, Christoph von A1 - Hildebrandt, Klaus T1 - Geometric Flows of Curves in Shape Space for Processing Motion of Deformable Objects JF - Computer Graphics Forum N2 - 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. Y1 - 2016 U6 - https://doi.org/10.1111/cgf.12832 VL - 35 IS - 2 ER - TY - JOUR A1 - Tycowicz, Christoph von A1 - Schulz, Christian A1 - Seidel, Hans-Peter A1 - Hildebrandt, Klaus T1 - Real-time Nonlinear Shape Interpolation JF - ACM Transactions on Graphics N2 - 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. Y1 - 2015 U6 - https://doi.org/10.1145/2729972 VL - 34 IS - 3 SP - 34:1 EP - 34:10 ER - TY - CHAP A1 - Schulz, Christian A1 - Tycowicz, Christoph von A1 - Seidel, Hans-Peter A1 - Hildebrandt, Klaus T1 - Animating articulated characters using wiggly splines T2 - ACM SIGGRAPH / Eurographics Symposium on Computer Animation N2 - 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. Y1 - 2015 U6 - https://doi.org/10.1145/2786784.2786799 SP - 101 EP - 109 ER - TY - CHAP A1 - Götschel, Sebastian A1 - Tycowicz, Christoph von A1 - Polthier, Konrad A1 - Weiser, Martin ED - Carraro, T. ED - Geiger, M. ED - Koerkel, S. ED - Rannacher, R. T1 - Reducing Memory Requirements in Scientific Computing and Optimal Control T2 - Multiple Shooting and Time Domain Decomposition Methods Y1 - 2015 SP - 263 EP - 287 PB - Springer ER - TY - THES A1 - Tycowicz, Christoph von T1 - Concepts and Algorithms for the Deformation, Analysis, and Compression of Digital Shapes N2 - 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. Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:188-fudissthesis000000096721-7 UR - http://www.diss.fu-berlin.de/diss/receive/FUDISS_thesis_000000096721 ER - TY - CHAP A1 - Polthier, Konrad A1 - Bobenko, Alexander A1 - Hildebrandt, Klaus A1 - Kornhuber, Ralf A1 - Tycowicz, Christoph von A1 - Yserentant, Harry A1 - Ziegler, Günter M. T1 - Geometry processing T2 - MATHEON - Mathematics for Key Technologies Y1 - 2014 SN - 978-3-03719-137-8 U6 - https://doi.org/10.4171/137 VL - 1 SP - 341 EP - 355 PB - EMS Publishing House ER -