@techreport{KahraBreuss2022, type = {Working Paper}, author = {Kahra, Marvin and Breuß, Michael}, title = {Properties of morphological dilation in max-plus and plus-prod algebra in connection with the Fourier transformation}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:co1-opus4-58483}, year = {2022}, abstract = {The basic filters in mathematical morphology are dilation and erosion. They are defined by a structuring element that is usually shifted pixel-wise over an image, together with a comparison process that takes place within the corresponding mask. This comparison is made in the grey value case by means of maximum or minimum formation. Hence, there is easy access to max-plus algebra and, by means of an algebra change, also to the theory of linear algebra. We show that an approximation of the maximum function forms a commutative semifield (with respect to multiplication) and corresponds to the maximum again in the limit case. In this way, we demonstrate a novel access to the logarithmic connection between the Fourier transform and the slope transformation. In addition, we prove that the dilation by means of a Fast Fourier Transform depends only on the size of the structuring element used. Moreover, we derive a bound above which the Fourier approximation yields results that are exact in terms of grey value quantization.}, subject = {Mathematical morphology; Fourier transform; Dilation; Max-plus algebra; Slope transform; Mathematische Morphologie; Fourier-Transformation; Dilatation; Max-Plus-Algebra; Steigungstransformation; Dilatation ; Fourier-Transformation; Mathematische Morphologie}, language = {en} } @article{EltaherBreuss2025, author = {Eltaher, Mahmoud and Breuß, Michael}, title = {Deep learning for unsupervised 3D shape representation with superquadrics}, series = {AI}, volume = {6}, journal = {AI}, number = {12}, publisher = {MDPI}, address = {Lausanne}, issn = {2673-2688}, doi = {10.3390/ai6120317}, year = {2025}, abstract = {The representation of 3D shapes from point clouds remains a fundamental challenge in computer vision. A common approach decomposes 3D objects into interpretable geometric primitives, enabling compact, structured, and efficient representations. Building upon prior frameworks, this study introduces an enhanced unsupervised deep learning approach for 3D shape representation using superquadrics. The proposed framework fits a set of superquadric primitives to 3D objects through a fully integrated, differentiable pipeline that enables efficient optimization and parameter learning, directly extracting geometric structure from 3D point clouds without requiring ground-truth segmentation labels. This work introduces three key advancements that substantially improve representation quality, interpretability, and evaluation rigor: (1) A uniform sampling strategy that enhances training stability compared with random sampling used in earlier models; (2) An overlapping loss that penalizes intersections between primitives, reducing redundancy and improving reconstruction coherence; and (3) A novel evaluation framework comprising Primitive Accuracy, Structural Accuracy, and Overlapping Percentage metrics. This new metric design transitions from point-based to structure-aware assessment, enabling fairer and more interpretable comparison across primitive-based models. Comprehensive evaluations on benchmark 3D shape datasets demonstrate that the proposed modifications yield coherent, compact, and semantically consistent shape representations, establishing a robust foundation for interpretable and quantitative evaluation in primitive-based 3D reconstruction.}, subject = {3D shape representation; Superquadrics; Deep learning; Unsupervised learning; Point clouds; Geometric modeling}, language = {en} } @article{SchneidereitBreuss2023, author = {Schneidereit, Toni and Breuß, Michael}, title = {Adaptive neural-domain refinement for solving time-dependent differential equations}, doi = {10.1186/s13662-023-03789-x}, year = {2023}, abstract = {A classic approach for solving differential equations with neural networks builds upon neural forms, which employ the differential equation with a discretisation of the solution domain. Making use of neural forms for time-dependent differential equations, one can apply the recently developed method of domain segmentation. That is, the domain may be split into several subdomains, on which the optimisation problem is solved. In classic adaptive numerical methods, the mesh as well as the domain may be refined or decomposed, in order to improve the accuracy. Also, the degree of approximation accuracy may be adapted. Therefore, it is desirable to transfer such important and successful strategies to the field of neural-network-based solutions. In the presented work, we propose a novel adaptive neural approach to meet this aim for solving time-dependent problems. To this end, each subdomain is reduced in size until the optimisation is resolved up to a predefined training accuracy. In addition, while the neural networks employed are by default small, we propose a means to adjust also the number of neurons in an adaptive way. We introduce conditions to automatically confirm the solution reliability and optimise computational parameters whenever it is necessary. Results are provided for several initial-value problems that illustrate important computational properties of the method.}, subject = {Neural forms; Differential equations; Physics-informed neural networks; Adaptive neural refinement; Domain decomposition; Neuronale Formen; Differentialgleichungen; Physikalisch informierte neuronale Netze; Adaptive neuronale Verfeinerung; Gebietszerlegung; Differentialgleichung; Neuronales Netz}, language = {en} }