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Maximilian Fiedler

Publications and source records attributed to Maximilian Fiedler.

6 recordsLinked to original sources
arXiv2022

Turning grain maps into diagrams

The present paper studies mathematical models for representing, imaging, and analyzing polycrystalline materials. We introduce various techniques for converting grain maps into diagram or tessellation representations that rely on constrained clustering. In particular, we show how to significantly accelerate the generalized balanced power diagram method from [1] and how to extend it to allow for optimization over all relevant parameters. A comparison of the accuracies of the proposed approaches is given based on a 3D real-world data set of $339\times 339 \times 599$ voxels.

math.NA↗
arXiv2022

Dynamic grain models via fast heuristics for diagram representations

The present paper introduces a mathematical model for studying dynamic grain growth. In particular, we show how characteristic measurements, grain volumes, centroids, and central second-order moments at discrete moments in time can be turned quickly into a continuous description of the grain growth process in terms of geometric diagrams (which largely generalize the well-known Voronoi and Laguerre tessellations). We evaluate the computational behavior of our algorithm on real-world data.

physics.comp-ph↗
arXiv2022

On resolution coresets for constrained clustering

Specific data compression techniques, formalized by the concept of coresets, proved to be powerful for many optimization problems. In fact, while tightly controlling the approximation error, coresets may lead to significant speed up of the computations and hence allow to extend algorithms to much larger problem sizes. The present paper deals with a weight-balanced clustering problem from imaging in materials science. Here, the class of desired coresets is naturally confined to those which can be viewed as lowering the resolution of the data. Hence one would expect that such resolution coresets are inferior to unrestricted coreset. We show, however, that the restrictions are more than compensated by utilizing the underlying structure of the data. In particular, we prove bounds for resolution coresets which improve known bounds in the relevant dimensions and also lead to significantly faster algorithms practice.

math.OC↗
arXiv2022

Coresets for Weight-Constrained Anisotropic Assignment and Clustering

The present paper constructs coresets for weight-constrained anisotropic assignment and clustering. In contrast to the well-studied unconstrained least-squares clustering problem, approximating the centroids of the clusters no longer suffices in the weight-constrained anisotropic case, as even the assignment of the points to best sites is involved. This assignment step is often the limiting factor in materials science, a problem that partially motivates our work. We build on a paper by Har-Peled and Kushal, who constructed coresets of size $\mathcal{O}\bigl(\frac{k^3}{ε^{d+1}}\bigr)$ for unconstrained least-squares clustering. We generalize and improve on their results in various ways, leading to even smaller coresets with a size of only $\mathcal{O}\bigl(\frac{k^2}{ε^{d+1}}\bigr)$ for weight-constrained anisotropic clustering. Moreover, we answer an open question on coreset designs in the negative, by showing that the total sensitivity can become as large as the cardinality of the original data set in the constrained case. Consequently, many techniques based on importance sampling do not apply to weight-constrained clustering.

cs.DS↗
arXiv2021

Power-SLIC: Fast Superpixel Segmentations by Diagrams

Superpixel algorithms grouping pixels with similar color and other low-level properties are increasingly used for pre-processing in image segmentation. In recent years, a focus has been placed on developing geometric superpixel methods that facilitate the extraction and analysis of geometric image features. Diagram-based superpixel methods are important among the geometric methods as they generate compact and sparsely representable superpixels. Introducing generalized balanced power diagrams to the field of superpixels, we propose a diagram method called Power-SLIC. Power-SLIC is the first geometric superpixel method to generate piecewise quadratic boundaries. Its speed, competitive with fast state-of-the-art methods, is unprecedented for diagram approaches. Extensive computational experiments show that Power-SLIC outperforms existing diagram approaches in boundary recall, under segmentation error, achievable segmentation accuracy, and compression quality. Moreover, Power-SLIC is robust to Gaussian noise.

cs.CV↗