# Posters

### Velimir Khlebnikov's Laws of Time

#### Henriette Lipschütz, Ulrich Reitebuch, and Martin Skrodzki

##### At: Illustrating Dynamics and Probability Workshop, 11 - 15 November 2019, Providence, RI, USA

The artist Kristina Paustian honored the Russian futurist Velimir Khlebnikov with her solo exhibition \emph{Laws of Time. The future calculations by Velimir Khlebnikov} in Berlin, Germany. Amongst other aspects, she focused on his poetry and the mathematics used in his works. On this poster, we investigate his “Laws of Time” and motivate a subsequent question in probability.

### Processing of Point Set Surfaces

#### Martin Skrodzki, Eric Zimmermann, Ulrich Reitebuch, Sunil Yadav, and Konrad Polthier

##### At: Illustrating Geometry and Topology Workshop, 16 - 20 September 2019, Providence, RI, USA

Point Sets are acquired as representations of surfaces in R3 e.g. via laser-scanning or LiDaR. At first, they are unstructured and neighborhood relations have to be established. Then, noise added during generation has to be removed robustly also on non-uniform input while retaining features. The obtained cleaned point set can then be used for e.g. surface reconstruction and subsequently for 3D printing.

### Variational Shape Approximation of Point Set Surfaces

#### Martin Skrodzki, Eric Zimmermann, and Konrad Polthier

##### At: IGS 2019 International Geometry Summit, 17 - 21 June 2019, Vancouver, Canada

##### Awarded the “Best Poster Award”

This work proposes an algorithm for point set segmentation based on the concept of Variational Shape Approximation (VSA), which uses the k-means approach. It iteratively selects seeds, grows flat planar proxy regions according to normal similarity, and updates the proxies. It is known that this algorithm does not converge in general. We provide a concrete example showing that the utilized error measure can indeed grow during the run of the algorithm. To reach convergence, we propose a modification of the original VSA. Further, we provide two new operations applied to the proxy regions, namely split and merge, which enqueue in the pipeline and act according to a user-given parameter. The advantages over regular VSA are independence of both a prescribed number of proxies and a (manual) selection of seeds. Especially the latter is a common drawback of region-growing approaches in segmentation.

### Velimir Khlebnikov's Laws of Time

#### Henriette Lipschütz, Ulrich Reitebuch, and Martin Skrodzki

##### At: Art exhibition “Laws of Time. The future calculations by Velimir Khlebnikov”, 16 - 23 March 2019, Feldfünf, Berlin, Germany

The (video) artist Kristina Paustian honored the Russian futurist Velimir Khlebnikov with her solo exhibition “Laws of Time. The future calculations by Velimir Khlebnikov” in Berlin, Germany. Among other aspects, she focused on his poetry and the mathematics used in his works. In this paper, we present the connections of Khlebnikov's writings to mathematics and his thoughts about foretelling the future by analyzing several events which took place in the past. On this poster, his ideas are visualized and evaluated in comparison to randomly generated data.

### Analysis of NVT-based Point Set Denoising in Parameter Space

#### Eric Zimmermann, Sunil Yadav, Martin Skrodzki, Ulrich Reitebuch, and Konrad Polthier

##### At: SMI 2018 Conference, 6 - 8 June 2018, Lisbon, Portugal

Our point set denoising algorithm is an iterative, three-phase algorithm for noisy point sets. Its parameters offer a variety of tuning opportunities. Used models are the gargoyle (real, noisy, irregular), the Chinese ball (real, noisy, many features) and rabbit (real, noisy, many features), the fan disk (synthetic, noisy, sharp features, near-flat areas), the sphere (synthetic, noisy), and the cube (synthetic, noisy, sharp features). We compare the effect of different parameter values on the obtained result as well as on the number of iterations taken by the algorithm.

### Asymptotical & Combinatorial Results on the Neighborhood Grid Data Structure

#### Martin Skrodzki, Ulrich Reitebuch, and Konrad Polthier

##### At: Einstein Workshop Discrete Geometry and Topology, 13 - 16 March 2018, Berlin, Germany and Workshop Mathematics and Image Analysis 2018, 15 - 17 January 2018, Berlin, Germany

The Neighborhood Grid approximates neighborhood information. A (quadratic) matrix contains the coordinates of the points such that in each row the x-values are increasing while in each column the y-values are increasing.

For the algorithm, the order of the points suffices, the exact coordinates are irrelevant. If the above ordering is given, we call it a “stable state”.

We provide both an upper and a lower bound on the building time for such a stable state. Furthermore, we present some combinatorial results on open research questions concerning the data structure.

### Computational and Structural Aspects of Point Set Surfaces

#### Martin Skrodzki, Ulrich Reitebuch, and Konrad Polthier

##### At: BMS Days 2018, 19 - 20 February 2018, Berlin, Germany

The Neighborhood Grid approximates neighborhood information. A (quadratic) matrix contains the coordinates of the points such that in each row the x-values are increasing while in each column the y-values are increasing.

For the algorithm, the order of the points suffices, the exact coordinates are irrelevant. If the above ordering is given, we call it a “stable state”.

We provide results on the convergence of a parallel building algorithm for stable states. Furthermore, we investigate the uniqueness of stable states.

### Computational and Structural Aspects of Point Set Surfaces

#### Martin Skrodzki and Ulrich Reitebuch

##### At: SIAM Conference on Industrial and Applied Geometry, 8 - 13 Juli 2017, Pittsburgh, USA

Based on previous, well-established, and successfully used discretization schemes of differential geometric structures and operators on triangulated meshes, we take the next step and transfer this calculus to point set surface data. These arise naturally in 3D acquisition processes. We aim at a theoretical framework, which mimics a many features as possible of smooth manifolds in the setting of point sets.

The Neighborhood Grid approximates neighborhood information. A (quadratic) matrix contains the coordinates of the points such that in each row the x-values are increasing while in each column the y-values are increasing.

For the algorithm, the order of the points suffices, the exact coordinates are irrelevant. If the above ordering is given, we call it a “stable state”.

We provide results on the convergence of a parallel building algorithm for stable states. Furthermore, we investigate the uniqueness of stable states.

### Computational and Structural Aspects of Point Set Surfaces

#### Martin Skrodzki

##### At: BMS Days 2017, 20 - 21 February 2017, Berlin, Germany

Based on previous, well-established, and successfully used discretization schemes of differential geometric structures and operators on triangulated meshes, we take the next step and transfer this calculus to point set surface data. These arise naturally in 3D acquisition processes. We aim at a theoretical framework, which mimics a many features as possible of smooth manifolds in the setting of point sets.