Sunil Kumar Yadav, Ulrich Reitebuch and Konrad Polthier
Mesh Denoising based on Normal Voting Tensor and Binary Optimization
To appear in: IEEE Transactions on Visualization and Computer Graphics (Volume: PP, Issue: 99)
This paper presents a two-stage mesh denoising algorithm. Unlike other traditional averaging approaches, our approach uses an element-based normal voting tensor to compute smooth surfaces. By introducing a binary optimization on the proposed tensor together with a local binary neighborhood concept, our algorithm better retains sharp features and produces smoother umbilical regions than previous approaches. On top of that, we provide a stochastic analysis on the different kinds of noise based on the average edge length. The quantitative results demonstrate that the performance of our method is better compared to state-of-the-art smoothing approaches.
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Sunil Kumar Yadav, Seyedamirhosein Motamedi, Timm Oberwahrenbrock, Frederike Cosima Oertel, Konrad Polthier, Friedemann Paul, Ella Maria Kadas, and Alexander U. Brandt
CuBe: parametric modeling of 3D foveal shape using cubic Bézier
To appear in: Biomedical Optics Express, Vol. 8, Issue 9, pp. 4181-4199, 2017
Optical coherence tomography (OCT) allows three-dimensional (3D) imaging of the retina, and is commonly used for assessing pathological changes of fovea and macula in many diseases. Many neuroinflammatory conditions are known to cause modifications to the fovea shape. In this paper, we propose a method for parametric modeling of the foveal shape. Our method exploits invariant features of the macula from OCT data and applies a cubic Bézier polynomial along with a least square optimization to produce a best fit parametric model of the fovea. Additionally, we provide several parameters of the foveal shape based on the proposed 3D parametric modeling. Our quantitative and visual results show that the proposed model is not only able to reconstruct important features from the foveal shape, but also produces less error compared to the state-of-the-art methods. Finally, we apply the model in a comparison of healthy control eyes and eyes from patients with neuroinflammatory central nervous system disorders and optic neuritis, and show that several derived model parameters show significant differences between the two groups.
Martin Skrodzki and Konrad Polthier,
Turing-Like Patterns Revisited: A Peek Into The Third Dimension
To appear in: Proceedings of Bridges, 2017
Beginning with their introduction in 1952 by Alan Turing, Turing-like patterns have inspired research in several different fields. One of these is the field of cellular automata, which have been utilized to create Turing-like patterns by David A. Young and others. In this paper we provide a generalization of these patterns to the third dimension. Several visualizations are given to illustrate the created models.
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Proceedings of the Forum "Math-for-Industry" 2015.
Editors: Bob Anderssen, Philip Broadbridge, Yasuhide Fukumoto. Naoyuki Kamiyama, Yoshihiro Mizoguchi, Konrad Polthier, Osamu Saeki
Georg Glaeser and Konrad Polthier
2. Aufl. 2010. Nachdruck 2014, XII, 340 S.
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Proceedings of the Forum of Mathematics for Industry 2013. Series: Mathematics for Industry, Vol. 1. Wakayama, M., Anderssen, R.S., Cheng, J., Fukumoto, Y., McKibbin, R., Polthier, K., Takagi, T., Toh, K.-C. (Eds.). 2014, XV, 369 p. 113 illus., 84 illus. in color.
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Hodge-Type Decompositions for Piecewise Constant Vector Fields on Simplicial Surfaces and Solids with Boundary. PhD Thesis, Freie Universität Berlin, 2017
This dissertation develops a theory for Hodge-type decompositions of piecewise constant vector fields (PCVF) on simplicial surfaces and solids with boundary in R3 which is structurally consistent with the corresponding results in the smooth world. First, the necessary differential-geometric and topological foundations of Hodge decomposition statements on smooth manifolds with boundary are reviewed, with a particular focus on a recent result by Shonkwiler which classifies certain harmonic fields as cohomology representatives for either the cohomology induced by the boundary components, or the "inner topology" of the manifold. Next, based on linear Lagrange, Crouzeix-Raviart and Nédélec elements, a discrete Hodge-Morrey-Friedrichs decomposition for PCVFs on simplicial surfaces and solids with boundary is derived. Of particular importance are the spaces H_N and H_D of discrete harmonic Neumann and Dirichlet fields, respectively, as they represent certain cohomology classes of the manifold and are therefore deeply linked to the topology of the simplicial manifold.
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.
On Isoparametric Catmull-Clark Finite Elements for Mean Curvature Flow. PhD Thesis, Freie Universität Berlin, 2016
In this thesis, we deal with the study of subdivision finite element methods for the solution of differential equations on curved surfaces based on the Catmull-Clark subdivisions. The focus is on quadrangular control grids and the characteristic parameterization of the limit surfaces. These are descried using the generalized B-spline basis functions of the Catmull-Clark type. In particular, we present a new finite element approach compatible with the classical definition of the subdivision surfaces. Compared to the previously used natural finite elements, the form of the grid and, consequently, the stability of the limit surface remains resistant. This is achieved as the characteristic finite elements inherit the continuity properties of the subdivision surfaces classically generated by grid refinements.
Faniry H. Razafindrazaka
Quad Layout Generation and Symmetric Tilings of Closed Surfaces. PhD Thesis, Freie Universität Berlin, 2016
This thesis concerns two fundamental concepts in surface topology. The first part proposes a solution of the problem of generating an all quadrilateral patch layout on a given surface. We approach the problem from a combinatorial graph optimization point of view. Mainly, finding a nice quad layout of a given surface is equivalent to solving a minimum weight perfect matching problem with additional quad guarantee constraints. The results are of high quality in terms of coarseness and alignment to important features of the geometry which can be used for wide range of applications such as hierarchical subdivision or high order surface fitting. The second part suggests an algorithm to symmetrically generate high genus surfaces suitable for space models of regular maps. It is based on a novel identification in hyperbolic space to derive directly the tubular neighborhood of the edge of a tiling directly the the hyperbolic representation followed by a spring relaxation procedure with intersection-free guarantee. We succeed to produce new embeddings of regular maps ranging from genus 5 to 85.
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