Konstantin Poelke**Hodge-Type Decompositions for Piecewise Constant Vector Fields on Simplicial Surfaces and Solids with Boundary**. PhD Thesis, Freie Universität Berlin, 2017

For surfaces with boundary that come from a closed surface of genus g=0, one obtains a complete, L2-orthogonal five-term decomposition where both the spaces H_N and H_D appear as L2-orthogonal subspaces. On the other hand, if g>0, these two spaces are no longer orthogonal to each other, and there are now two orthogonal decompositions - one involving H_N, the other one involving H_D.

A deep result in the smooth world states that the spaces of Neumann and Dirichlet forms always have a trivial intersection, but the corresponding result for the discrete spaces does not hold in general.

Surprisingly, at this stage the combinatorics of the triangulating grid comes into play, and it becomes a matter of how the triangulation connects topologically rich regions with the boundary components to decide whether the statement holds true or not. Based on results by Lovász and Benjamini on weighted networks, a criterion is derived that guarantees the validity of the trivial-intersection result. Next, convergence of the derived decomposition statements is proved under the assumption that the simplicial geometry converges metrically against a smooth geometry. To this end, a seminal convergence result by Dodziuk on the approximation of smooth Hodge decomposition components by Whitney forms is generalized to include the refined discrete decompositions. One obtains convergence with respect to the L2-norm which is of linear order in the mesh size.

The dissertation concludes with two central applications of this discrete Hodge theory: the computation of harmonic cohomology representatives and the computational decomposition of a given PCVF.

For both applications, algorithms are presented and evaluated on various test models and the numerical aspects of the involved solving steps are discussed.

Anna Wawrzinek**On Isoparametric Catmull-Clark Finite Elements for Mean Curvature Flow**. PhD Thesis, Freie Universität Berlin, 2016

Faniry H. Razafindrazaka**Quad Layout Generation and Symmetric Tilings of Closed Surfaces**. PhD Thesis, Freie Universität Berlin, 2016

Maria Ella Kadas**Methods to extract and quantify retinal Blood Vessels and Optic Nerve Head from optical Coherence Tomography Data in Neurological Disorders**. PhD Thesis, Freie Universität Berlin, 2016

However, current OCT technology being mainly applied in the analysis and quantification of ophthalmological diseases lacks tailored image analysis methods for many changes caused by neurological disorders. The focus of this thesis lies on the development of segmentation and analysis methods to quantify two major components of the retina in confocal scanning laser ophthalmoscopy (cSLO data - 2D image) and in OCT data (3D OCT volume data), the retinal blood vessels, and the optic nerve head (ONH). The difficulty in developing robust and accurate methods for detecting these structures consists in the heterogeneous aspect of the data, coming from the natural anatomical diversity of the subjects, artifacts during data acquisition, especially in patients rather than in data from healthy control, and most importantly from certain structural changes that occur in the data during the disease course.

We present four approaches for extracting features from the retinal vasculature and for the ONH in multiple sclerosis (with its subtypes), neuromyelitis optica spectrum disorder and idiopathic intracranial hypertension. The first two approaches focus on the detection of the vasculature in SLO images. We propose a new 2D model of the vessel profile that accounts for the central reflex seen in this particular image type in order to quantify the vessel inner and outer boundary. Furthermore, we developed new filter response measures for vessel enhancement based on Morlet wavelet, the Hessian tensor, and an optimal oriented flux approach, and tested their capability of correctly detecting the vessel inner and outer boundary, curvature especially in junction regions. In the case of the ONH, we present a robust approach to detect a reference surface for the volume computation in atrophic and swelled ONH. Moreover, we present a novel algorithm for the detection of the ONH center directly in the 3D OCT volume. The basic idea of this method is to use the information from the computed reference surface to reduce the computation to a sub-volume (a reduced volume) in the ONH region. Furthermore, we address several challenges present in our data: motion artifacts due to eye/head movements by using a modified thin plate spline fitting that is able to model the natural curvature of the retina, artifacts arising from the shadows created by the presence of blood vessel by incorporating contextual textural features in a 3D grow-cut setting.

We evaluate our methods in various clinical settings. To demonstrate the effectiveness of our novel methods, we applied them to various patient and healthy control datasets.

Christoph von Tycowicz**Concepts and Algorithms for the Deformation, Analysis, and Compression of Digital Shapes**. PhD Thesis, Freie Universität Berlin, 2014

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.

Stefan W. von Deylen**Numerical Approximation in Riemannian Manifolds by Karcher Means**. PhD Thesis, Freie Universität Berlin, 2013

Felix Kälberer**Low Distortion Surface Parameterization**. PhD Thesis, Freie Universität Berlin, 2013

Parameterization maps usually have to meet a number of quality criteria, important examples are small angle and length distortion. In addition, it is often demanded that the gradient of the parameterization function are aligned with the direction of surface features such as sharp bends and edges.

Our QuadCover method, which forms the basis of this thesis, generates a parameterization automatically from a tensor field of feature directions. The method builds on the fact that such multi-dimensional direction fields can be interpreted as one-dimensional vector fields on a branched covering of the surface. In this way, known results about vector fields, such as Hodge decomposition, can be applied. On this basis, QuadCover finds a parameterization that aligns as close as possible to a given direction field.

Parameterizations of highest quality additionally require that length and angle distortion are minimized. For this, the number and location of branch points of the direction field is critical. In this work, we are pursuing several approaches: First, we show with a new method that the movement and especially the creation of branch points can drastically reduce distortion. Second, the distortion that is caused by the existence of branch points is reduced significantly by using a sophisticated rounding method. The third approach opposes the different types of distortion of the two former steps, and infers the optimal number of branch points out of them. The combination of the approaches makes it possible surpass even recent algorithms in terms of distortion.

Christian Schulz**Interactive Spacetime Control of Deformable Objects and Modal Shape Analysis beyond Laplacian**. PhD Thesis, Freie Universität Berlin, 2013

Klaus Hildebrandt**Discretization and Approximation of the Shape Operator, the Laplace--Beltrami Operator, and the Willmore Energy of Surfaces**. PhD Thesis, Freie Universität Berlin, 2012

Matthias Nieser**Parameterization and Tiling of Polyhedral Surfaces**. PhD Thesis, Freie Universität Berlin, 2012

Frame fields are multi-valued functions. In this thesis the equivalence of a frame field to a (single-valued) vector field on a Riemannian surface is shown. This observation lays the theoretical foundation for the QuadCover algorithm.

QuadCover automatically generates a global parameterization of an arbitrary two-dimensional polyhedral manifold. The resulting parameter lines form a mesh which is globally continuous and allows to remesh the surface into a mesh of high quality. The faces of the mesh consist of either quadrilaterals, triangles or hexagons.

Another application of QuadCover is texturing. The parameterization is used to cover the surface with an arbitrary tileable pattern. Thereby, the parameterization must be compatible with the rotational symmetry of the texture pattern. This thesis presents a method for tiling a surface with regular quadrilateral-, hexagonal-, triangular-, or stripe-patterns.

Often, a user has special demands on a parameterization which makes it necessary for the user to interact with the parameterization software. QuadCover allows to move singularities on the surface. Furthermore, the user can specify curves on the surface as geometric constraint, which means that a parameter line will exactly follow the given curve. There are also combinatorial constraints which allow to manipulate the topology of the generated mesh.

Max Wardetzky**Discrete Differential Operators on Polyhedral Surfaces - Convergence and Approximation**. PhD Thesis, Freie Universität Berlin, 2006