Dr. rer. nat. Christoph von Tycowicz

Head of Research Group:

Geometric Data Analysis and Processing

Zuse Institute Berlin (ZIB)

Visual and Data-centric Computing

Takustr. 7

14195 Berlin

Mail: vontycowicz@zib.de

Phone: +49 30 84 185 350

Room 4013

Martin Hanik, Hans-Christian Hege, Anja Hennemuth, Christoph von Tycowicz:**Nonlinear Regression on Manifolds for Shape Analysis using Intrinsic Bézier Splines.**

Proc. Medical Image Computing and Computer Assisted Intervention (MICCAI), 2020.

We present a framework for nonlinear regression on manifolds by considering Riemannian splines, whose segments are Bézier curves, as trajectories. Unlike variational formulations that require time-discretization, we take a constructive approach that provides efficient and exact evaluation by virtue of the generalized de Casteljau algorithm. We validate our method in experiments on the reconstruction of periodic motion of the mitral valve as well as the analysis of femoral shape changes during the course of osteoarthritis, endorsing Bézier spline regression as an effective and flexible tool for manifold-valued regression.

Esfandiar Nava-Yazdani, Hans-Christian Hege, T. J. Sullivan, Christoph von Tycowicz:**Geodesic Analysis in Kendall’s Shape Space with Epidemiological Applications**.

Journal of Mathematical Imaging and Vision, 2020.

We analytically determine Jacobi fields and parallel transports and compute geodesic regression in Kendall’s shape space. Using the derived expressions, we can fully leverage the geometry via Riemannian optimization and thereby reduce the computational expense by several orders of magnitude over common, nonlinear constrained approaches. The methodology is demonstrated by performing a longitudinal statistical analysis of epidemiological shape data. As an example application we have chosen 3D shapes of knee bones, reconstructed from image data of the Osteoarthritis Initiative (OAI). Comparing subject groups with incident and developing osteoarthritis versus normal controls, we find clear differences in the temporal development of femur shapes. This paves the way for early prediction of incident knee osteoarthritis, using geometry data alone.

Christoph von Tycowicz:**Towards Shape-based Knee Osteoarthritis Classification using Graph Convolutional Networks**.

2020 IEEE 17th International Symposium on Biomedical Imaging (ISBI 2020), 2020.

We present a transductive learning approach for morphometric osteophyte grading based on geometric deep learning. We formulate the grading task as semi-supervised node classification problem on a graph embedded in shape space. To account for the high-dimensionality and non-Euclidean structure of shape space we employ a combination of an intrinsic dimension reduction together with a graph convolutional neural network. We demonstrate the performance of our derived classifier in comparisons to an alternative extrinsic approach.

Felix Ambellan, Stefan Zachow, Christoph von Tycowicz:**An as-invariant-as-possible GL+(3)-based Statistical Shape Model**.

Proc. 7th MICCAI workshop on Mathematical Foundations of Computational Anatomy (MFCA), pp. 219-228, Vol.11846, Lecture Notes in Computer Science, 2019.

We describe a novel nonlinear statistical shape model basedon differential coordinates viewed as elements of GL+(3). We adopt an as-invariant-as possible framework comprising a bi-invariant Lie group mean and a tangent principal component analysis based on a unique GL+(3)-left-invariant, O(3)-right-invariant metric. Contrary to earlier work that equips the coordinates with a specifically constructed group structure, our method employs the inherent geometric structure of the group-valued data and therefore features an improved statistical power in identifying shape differences. We demonstrate this in experiments on two anatomical datasets including comparison to the standard Euclidean as well as recent state-of-the-art nonlinear approaches to statistical shape modeling.

Felix Ambellan, Hans Lamecker, Christoph von Tycowicz, Stefan Zachow:**Statistical Shape Models – Understanding and Mastering Variation in Anatomy**.

Biomedical Visualisation, Paul M. Rea (Ed.), Springer Nature Switzerland AG, 1, pp. 67-84, 2019.

In our chapter we are describing how to reconstruct three-dimensional anatomy from medical image data and how to build Statistical 3D Shape Models out of many such reconstructions yielding a new kind of anatomy that not only allows quantitative analysis of anatomical variation but also a visual exploration and educational visualization. Future digital anatomy atlases will not only show a static (average) anatomy but also its normal or pathological variation in three or even four dimensions, hence, illustrating growth and/or disease progression.

Felix Ambellan, Stefan Zachow, and Christoph von Tycowicz: **A Surface-Theoretic Approach for Statistical Shape Modeling**.

Proc. Medical Image Computing and Computer Assisted Intervention (MICCAI), Lecture Notes in Computer Science, 2019.

We present a novel approach for nonlinear statistical shape modeling that is invariant under Euclidean motion and thus alignment-free. By analyzing metric distortion and curvature of shapes as elements of Lie groups in a consistent Riemannian setting, we construct a framework that reliably handles large deformations. Due to the explicit character of Lie group operations, our non-Euclidean method is very efficient allowing for fast and numerically robust processing. This facilitates Riemannian analysis of large shape populations accessible through longitudinal and multi-site imaging studies providing increased statistical power. We evaluate the performance of our model w.r.t. shape-based classification of pathological malformations of the human knee and show that it outperforms the standard Euclidean as well as a recent nonlinear approach especially in presence of sparse training data. To provide insight into the model’s ability of capturing natural biological shape variability, we carry out an analysis of specificity and generalization ability.

Esfandiar Nava-Yazdani, Hans-Christian Hege, and Christoph von Tycowicz: **A Geodesic Mixed Effects Model in Kendall’s Shape Space**.

Proc. 7th MICCAI workshop on Mathematical Foundations of Computational Anatomy (MFCA), Lecture Notes in Computer Science, 2019.

In many applications, geodesic hierarchical models are adequate for the study of temporal observations. We employ such a model derived for manifold-valued data to Kendall’s shape space. In particular, instead of the Sasaki metric, we adapt a functional-based metric, which increases the computational efficiency and does not require the implementation of the curvature tensor. We propose the corresponding variational time discretization of geodesics and apply the approach for the estimation of group trends and statistical testing of 3D shapes derived from an open access longitudinal imaging study on osteoarthritis.

Christoph von Tycowicz, Felix Ambellan, Anirban Mukhopadhyay, and Stefan Zachow: **An Efficient Riemannian Statistical Shape Model using Differential Coordinates**.

Medical Image Analysis, Volume 43, January 2018.

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.

Christopher Brandt, Christoph von Tycowicz, and Klaus Hildebrandt: **Geometric Flows of Curves in Shape Space for Processing Motion of Deformable Objects**, video.

Computer Graphics Forum 35(2) (Eurographics 2016).

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.

Christian Schulz, Christoph von Tycowicz, Hans-Peter Seidel, and Klaus Hildebrandt: **Animating Articulated Characters using Wiggly Splines**, video.

ACM SIGGRAPH/Eurographics Symposium on Computer Animation 2015, pages 101-109.

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.

Christoph von Tycowicz, Christian Schulz, Hans-Peter Seidel, and Klaus Hildebrandt:

**Real-Time Nonlinear Shape Interpolation**, video.

ACM Transactions on Graphics, Volume 34, Issue 3, May 2015, pages 34:1-34:10. (Presented at SIGGRAPH 2015)

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.

Christian Schulz, Christoph von Tycowicz, Hans-Peter Seidel, and Klaus Hildebrandt: **Animating Deformable Objects using Sparse Spacetime Constraints**, video (51 MB), supplementary material.

ACM Transactions on Graphics (SIGGRAPH 2014), Volume 33, Issue 4, July 2014, pages 109:1-109:10.

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.

Christoph von Tycowicz, Christian Schulz, Hans-Peter Seidel, and Klaus Hildebrandt:

**An Efficient Construction of Reduced Deformable Objects** (11 MB), video (64 MB).

ACM Transactions on Graphics (SIGGRAPH Asia 2013), Volume 32, Issue 6, November 2013, pages 213:1-213:10.

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.

Klaus Hildebrandt, Christian Schulz, Christoph von Tycowicz, and Konrad Polthier: **Interactive spacetime control of deformable objects** (4.6 MB), video (76 MB).

ACM Transactions on Graphics (SIGGRAPH 2012), Volume 31, Issue 4, July 2012, pages 71:1-71:8.

Creating motions of objects or characters that are physically plausible and follow an animator’s intent is a key task in computer animation. The spacetime constraints paradigm is a valuable approach to this problem, but it suffers from high computational costs. Based on spacetime constraints, we propose a technique that controls the motion of deformable objects and offers an interactive response. This is achieved by a model reduction of the underlying variational problem, which combines dimension reduction, multipoint linearization, and decoupling of ODEs. After a preprocess, the cost for creating or editing a motion is reduced to solving a number of one-dimensional spacetime problems, whose solutions are the wiggly splines introduced by Kass and Anderson [2008]. We achieve interactive response using a new fast and robust numerical scheme for solving a set of one-dimensional problems based on an explicit representation of the wiggly splines.

Klaus Hildebrandt, Christian Schulz, Christoph von Tycowicz, and Konrad Polthier: **Interactive Surface Modeling using Modal Analysis** (9.5 MB), video (57 MB).

ACM Transactions on Graphics, Volume 30, Issue 5, October 2011, pages 119:1-119:11. (Presented at SIGGRAPH 2012)

We propose a framework for deformation-based surface modeling that is interactive, robust and intuitive to use. The deformations are described by a non-linear optimization problem that models static states of elastic shapes under external forces which implement the user input. Interactive response is achieved by a combination of model reduction, a robust energy approximation, and an efficient quasi-Newton solver. Motivated by the observation that a typical modeling session requires only a fraction of the full shape space of the underlying model, we use second and third derivatives of a deformation energy to construct a low-dimensional shape space that forms the feasible set for the optimization. Based on mesh coarsening, we propose an energy approximation scheme with adjustable approximation quality. The quasi-Newton solver guarantees superlinear convergence without the need of costly Hessian evaluations during modeling. We demonstrate the effectiveness of the approach on different examples including the test suite introduced in [Botsch and Sorkine 2008].

Klaus Hildebrandt, Christian Schulz, Christoph von Tycowicz, and Konrad Polthier: **Modal Shape Analysis beyond Laplacian** (1.1 MB), video (7 MB).

Computer Aided Geometric Design, Volume 29, Issue 5, June 2012, Pages 204-218.

In recent years, substantial progress in shape analysis has been achieved through methods that use the spectra and eigenfunctions of discrete Laplace operators. In this work, we study spectra and eigenfunctions of discrete differential operators that can serve as an alternative to the discrete Laplacians for applications in shape analysis. We construct such operators as the Hessians of surface energies, which operate on a function space on the surface, or of deformation energies, which operate on a shape space. In particular, we design a quadratic energy such that, on the one hand, its Hessian equals the Laplace operator if the surface is a part of the Euclidean plane, and, on the other hand, the Hessian eigenfunctions are sensitive to the extrinsic curvature (e.g. sharp bends) on curved surfaces. Furthermore, we consider eigenvibrations induced by deformation energies, and we derive a closed form representation for the Hessian (at the rest state of the energy) for a general class of deformation energies. Based on these spectra and eigenmodes, we derive two shape signatures. One that measures the similarity of points on a surface, and another that can be used to identify features of surfaces.

Christoph von Tycowicz, Felix Kälberer, and Konrad Polthier: **Context-Based Coding of Adaptive Multiresolution Meshes** (1.4 MB).

Computer Graphics Forum, Volume 30, Issue 8, pages 2231-2245, December 2011.

Multiresolution meshes provide an efficient and structured representation of geometric objects. To increase the mesh resolution only at vital parts of the object, adaptive refinement is widely used. We propose a lossless compression scheme for these adaptive structures that exploits the parent-child relationships inherent to the mesh hierarchy. We use the rules that correspond to the adaptive refinement scheme and store bits only where some freedom of choice is left, leading to compact codes that are free of redundancy. Moreover, we extend the coder to sequences of meshes with varying refinement. The connectivity compression ratio of our method exceeds that of state-of-the-art coders by a factor of 2 to 7. For efficient compression of vertex positions we adapt popular wavelet-based coding schemes to the adaptive triangular and quadrangular cases to demonstrate the compatibility with our method. Akin to state-of-the-art coders, we use a zerotree to encode the resulting coefficients. Using improved context modeling we enhanced the zerotree compression, cutting the overall geometry data rate by 7% below those of the successful Progressive Geometry Compression. More importantly, by exploiting the existing refinement structure we achieve compression factors that are 4 times greater than those of coders which can handle irregular meshes.

Klaus Hildebrandt, Christian Schulz, Christoph von Tycowicz, and Konrad Polthier:** Eigenmodes of Surface Energies for Shape Analysis**

Advances in Geometric Modeling and Processing (Proceedings of Geometric Modeling and Processing 2010), Lecture Notes in Computer Science 6130, Springer, pages 296-314.

In this work, we study the spectra and eigenmodes of the Hessian of various discrete surface energies and discuss applications to shape analysis. In particular, we consider a physical model that describes the vibration modes and frequencies of a surface through the eigenfunctions and eigenvalues of the Hessian of a deformation energy, and we derive a closed form representation for the Hessian (at the rest state of the energy) for a general class of deformation energies. Furthermore, we design a quadratic energy, such that the eigenmodes of the Hessian of this energy are sensitive to the extrinsic curvature of the surface.

Based on these spectra and eigenmodes, we derive two shape signatures. One that measures the similarity of points on a surface, and another that can be used to identify features of the surface. In addition, we discuss a spectral quadrangulation scheme for surfaces.

Felix Kälberer, Christoph von Tycowicz, and Konrad Polthier:

Lossless Compression of Adaptive Multiresolution Meshes

Sibgrapi 2009 Technical Paper

We present a novel coder for lossless compression of adaptive multiresolution meshes that exploits their special hierarchical structure. The heart of our method is a new progressive connectivity coder that can be combined with leading geometry encoding techniques. The compressor uses the parent/child relationships inherent to the hierarchical mesh. We use the rules that accord to the refinement scheme and store bits only where it leaves freedom of choice, leading to compact codes that are free of redundancy. To illustrate our scheme we chose the widespread red-green refinement, but the underlying concepts can be directly transferred to other adaptive refinement schemes as well. The compression ratio of our method exceeds that of state-of-the-art coders by a factor of 2 to 3 on most of our benchmark models.

Christoph von Tycowicz, and Jörn Loviscach: **Measuring and Enhancing the Legibility of GPU-rendered Text**Eurographics 2008 Short Papers

The limited spatial resolution of standard computer displays impedes the legibility of text. To improve the rendering, we have developed a number of strategies that employ GPU-based shader programs and appropriate preprocessing steps. We devised and implemented a set of methods that may be used alone or in combination ranging from simple measures such as gamma-correct MIPmapping to complex ones such as RGB subpixel rendering combined with supersampling. We have researched into their performance in terms of both legibility and speed. We difinded a psychovisual test that quantifies legibility quickly enough to compare a selection of promising techniques. The results allow to choose the optimum blend of rendering methods for a given set of requirements.

C. Greenhalgh, S. Benford, A. Drozd, M. Flintham, A. Hampshire, L. Oppermann, K. Smith, and Christoph von Tycowicz: **Addressing Mobile Phone Diversity in Ubicomp Experience Development**UbiComp 2007 Paper

Mobile phones are a widely-available class of device with supporting communications infrastructure which can be appropriated and exploited to support ubicomp experiences. We explore how a single dimension of phone application type embodies the critical trade-off between capability and availability.

Felix Ambellan, Hans Lamecker, Christoph von Tycowicz, Stefan Zachow: **Statistical Shape Models – Understanding and Mastering Variation in Anatomy**Chapter in

In our chapter we are describing how to reconstruct three-dimensional anatomy from medical image data and how to build Statistical 3D Shape Models out of many such reconstructions yielding a new kind of anatomy that not only allows quantitative analysis of anatomical variation but also a visual exploration and educational visualization. Future digital anatomy atlases will not only show a static (average) anatomy but also its normal or pathological variation in three or even four dimensions, hence, illustrating growth and/or disease progression. Statistical Shape Models (SSMs) are geometric models that describe a collection of semantically similar objects in a very compact way. SSMs represent an average shape of many three-dimensional objects as well as their variation in shape. The creation of SSMs requires a correspondence mapping, which can be achieved e.g. by parameterization with a respective sampling. If a corresponding parameterization over all shapes can be established, variation between individual shape characteristics can be mathematically investigated. We will explain what Statistical Shape Models are and how they are constructed. Extensions of Statistical Shape Models will be motivated for articulated coupled structures. In addition to shape also the appearance of objects will be integrated into the concept. Appearance is a visual feature independent of shape that depends on observers or imaging techniques. Typical appearances are for instance the color and intensity of a visual surface of an object under particular lighting conditions, or measurements of material properties with computed tomography (CT) or magnetic resonance imaging (MRI). A combination of (articulated) statistical shape models with statistical models of appearance lead to articulated Statistical Shape and Appearance Models (a-SSAMs).After giving various examples of SSMs for human organs, skeletal structures, faces, and bodies, we will shortly describe clinical applications where such models have been successfully employed. Statistical Shape Models are the foundation for the analysis of anatomical cohort data, where characteristic shapes are correlated to demographic or epidemiologic data. SSMs consisting of several thousands of objects offer, in combination with statistical methods ormachine learning techniques, the possibility to identify characteristic clusters, thus being the foundation for advanced diagnostic disease scoring.

S. Götschel, Christoph von Tycowicz, K. Polthier, M. Weiser: **Reducing Memory Requirements in Scientific Computing and Optimal Control**Chapter in

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.

K. Polthier, A. Bobenko, K. Hildebrandt, R. Kornhuber, Christoph von Tycowicz, H. Yserentant, G. M. Ziegler: **Geometry processing**Chapter in

The book presents in seven chapters the highlights of the research work carried out in the MATHEON application areas: Life Sciences, Networks, Production, Electronic and Photonic Devices, Finance, Visualization, and Education. The chapters summarize many of the contributions, put them in the context of current mathematical research activities and outline their impact in various key technologies. To make some of the results more easily accessible to the general public, a large number of “showcases” are presented that illustrate a few success stories.

L. Oppermann, R. Jacobs, M. Watkins, R. Shackford, Christoph von Tycowicz, M. Wright, M. Capra, C. Greenhalgh, and S. Benford: **Love City: A Text-Driven, Location-Based Mobile Phone Game Played Between 3 Cities**Chapter in

Pervasive multi-player games have been location-based for the last couple of years. With the technology advancing at a rapid pace, the question is no longer how a location-based game could be built, but how it could be best delivered to the end-users – preferably on their own hardware and interwoven with their daily lives.

C. Greenhalgh, S. Benford, A. Drozd, M. Flintham, A. Hampshire, L. Oppermann, K. Smith, and Christoph von Tycowicz: **EQUIP2: A Platform for Mobile Phone-based Game Development**Chapter in

Mobile phones are a widely-available class of device with supporting communications infrastructure which can be appropriated and exploited to support ubicomp experiences. However mobile phones vary hugely in their capabilities, especially with regard to communication, context sensing and application hosting. We explore how a single dimension of ‘phone application type’ embodies the critical trade-off between capability and availability, i.e. between what can be done and the fraction of potential participants’ phones that can do this. We describe four different mobile phone ubicomp experiences that illustrate different points along this continuum (SMS, WAP/Web, and J2ME, Python and native applications). We introduce EQUIP2, a common software platform that has been co-developed to support these experiences. From this, and supported by a survey of the core experience developers, we identify four strategies for addressing mobile phone diversity: prioritise support for server development (including web integration), migrate functionality between server(s) and handset(s), support flexible communication options, and use a loosely coupled (data-driven and component-based) software approach.

Concepts and Algorithms for the Deformation, Analysis, and Compression of Digital Shapes

PhD Thesis, Freie Universität Berlin, 2014.

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.

Christoph von Tycowicz, Christian König, Stefan Zachow, Rainald M. Ehrig, Hans-Christian Hege, Georg N. Duda, Jürgen R. Reichenbach:**Measuring 3D knee dynamics using center out radial ultra-short echo time trajectories with a low cost experimental setup**SMRM (International Society for Magnetic Resonance in Medicine), 23rd Annual Meeting 2015, Toronto, Canada, 2015 (accepted for publication)

Christian Schulz, Christoph von Tycowicz, Hans-Peter Seidel, and Klaus Hildebrandt:**Proofs of two Theorems concerning Sparse Spacetime Constraints**Report

In the SIGGRAPH 2014 paper [SvTSH14] an approach for animating deformable objects using sparse spacetime constraints is introduced. This report contains the proofs of two theorems presented in the paper.

Christoph von Tycowicz, and Jörn Loviscach:**A Malleable Drum**SIGGRAPH 2008 Poster

Development of a virtual drum with a large range of interaction modes: The user can change the shape of the drum while playing it through a controller that offers hitting the virtual drum skin at 16 different points as well as simultaneously damping it and pushing it in. The real-time, low-latency simulation leverages the computing power of a GPU, which allows the resolution of the mesh to be increased greatly. Aiming at a minimum complexity for the GPU-based implementation, a finite-difference method is used, as known from the simulation of shallow water waves.

Christoph von Tycowicz, and Jörn Loviscach:**Designing No-Surprise Teapots**SIGGRAPH 2006 Poster

Many teapots behave rather jerky and don’t pour their content at a constant rate. I implemented design software that interactively displays pouring behavior characteristics of a 3D teapot model, allowing the user to apply changes to the model and immediately see the effect they have. This is enabled through a real- time analysis of a three-dimensional teapot model. The software determines the Tea Flow Rate, a physical quantity we introduced. The graph of the Tea Flow Rate of a teapot design will be displayed by the software. This allows the user to identify bad designs at first glance.

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