TY - GEN A1 - Breuß, Michael A1 - Kleefeld, Andreas T1 - Implicit Monotone Difference Methods for Scalar Conservation Laws with Source Terms T2 - Acta Mathematica Vietnamica N2 - In this article, a concept of implicit methods for scalar conservation laws in one or more spatial dimensions allowing also for source terms of various types is presented. This material is a significant extension of previous work of the first author (Breuß SIAM J. Numer. Anal. 43(3), 970–986 2005). Implicit notions are developed that are centered around a monotonicity criterion. We demonstrate a connection between a numerical scheme and a discrete entropy inequality, which is based on a classical approach by Crandall and Majda. Additionally, three implicit methods are investigated using the developed notions. Next, we conduct a convergence proof which is not based on a classical compactness argument. Finally, the theoretical results are confirmed by various numerical tests. KW - Conservation laws KW - Finite difference methods KW - Implicit methods KW - Monotone methods KW - Source term KW - Entropy solution Y1 - 2020 U6 - https://doi.org/10.1007/s40306-019-00354-1 SN - 2315-4144 SN - 0251-4184 VL - 45 IS - 3 SP - 709 EP - 738 ER - TY - GEN A1 - Breuß, Michael A1 - Buhl, Johannes A1 - Mansouri Yarahmadi, Ashkan A1 - Bambach, Markus A1 - Peter, Pascal T1 - A Simple Approach to Stiffness Enhancement of a Printable Shape by Hamilton-Jacobi Skeletonization T2 - Procedia Manufacturing N2 - The 3D-Printing technology is ready to produce parts with specific properties like individual stiffness. Based on a predefined outer shape, the inner structure of a printed part defines mainly the mechanical features. By Hamilton-Jacobi skeletonization, a stiffness enhancement of a printable shape can be achieved in a way, that a novel AM-corner includes linear axis function. Originating in the field of shape analysis in computer vision and graphics, the so-called medial axis transform (MAT) is designed for the computation of a structure that resembles the bone structure of biological shapes. The input for MAT computation is typically a shape’s boundary. The arising topological skeletons have proven to provide a useful concept for many applications; however, their computation is generally intricate and also known to rely on many parameters, diminishing the accessibility of skeletonization methods. In this work, the classical Hamilton-Jacobi skeletonization approach is adopted to compute a stability enhancing shape structure. As the basic method has not been designed for the context of additive manufacturing, a set of suitable modifications are introduced to design an algorithm that suits our intended purpose. Unlike the traditional skeletonization schemes, the resulting method appears to be robust and in practice almost completely automated as we can identify useful generic parameter settings. By a finite element method (FEM) study, the elastic stress properties of the AM-corner with linear axis function is validated and printed with in metal (1.4404) with the 3D Selected Laser Melting (SLM) system AconityMIDI. The AM-knot with skeletonization guides approximately 20 times better than a standard knot. While the first obtained results are shown as 2.5 dimensional shapes, it is emphasized that the proposed algorithm offers many possibilities for extensions to three dimensions and variations in context of additive manufacturing. KW - Skeletonization KW - Hamilton-Jacobi skeletonization KW - stiffness enhancement KW - 3D printing KW - additive manufacturing Y1 - 2020 U6 - https://doi.org/10.1016/j.promfg.2020.04.147 SN - 2351-9789 N1 - 23rd International Conference on Material Forming (ESAFORM 2020) VL - Vol. 47 SP - 1190 EP - 1196 ER - TY - GEN A1 - Schneidereit, Toni A1 - Breuß, Michael T1 - Solving Ordinary Differential Equations using Artificial Neural Networks - A study on the solution variance T2 - Proceedings of the Conference Algoritmy 2020 N2 - Solving differential equations can be realised with simple artificial neural network architectures. Several methods make use of trial solutions with different construction approaches and can provide reliable results. However, many parameters, different optimisation methods and random weight initialisation result in a non constant variance to the exact solution. To our knowledge, this variance has not been studied yet. We investigate several parameters and constant versus random weight initialisation for two solution methods to determine their reliability with the use of backpropagation and ADAM optimisation. Y1 - 2020 UR - http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/1547/811 SP - 21 EP - 30 PB - Open Journal Systems , Public Knowledge Project ER - TY - GEN A1 - Bähr, Martin A1 - Buhl, Johannes A1 - Radow, Georg A1 - Schmidt, Johannes A1 - Bambach, Markus A1 - Breuß, Michael A1 - Fügenschuh, Armin T1 - Stable honeycomb structures and temperature based trajectory optimization for wire-arc additive manufacturing T2 - Optimization and Engineering N2 - We consider two mathematical problems that are connected and occur in the layer-wise production process of a workpiece using wire-arc additive manufacturing. As the first task, we consider the automatic construction of a honeycomb structure, given the boundary of a shape of interest. In doing this, we employ Lloyd’s algorithm in two different realizations. For computing the incorporated Voronoi tesselation we consider the use of a Delaunay triangulation or alternatively, the eikonal equation. We compare and modify these approaches with the aim of combining their respective advantages. Then in the second task, to find an optimal tool path guaranteeing minimal production time and high quality of the workpiece, a mixed-integer linear programming problem is derived. The model takes thermal conduction and radiation during the process into account and aims to minimize temperature gradients inside the material. Its solvability for standard mixed-integer solvers is demonstrated on several test-instances. The results are compared with manufactured workpieces. KW - Mixed-integer linear programming KW - Geometric optimization KW - Eikonal equation KW - Heat transmission KW - Additive manufacturing KW - Centroidal Voronoi tesselation Y1 - 2021 U6 - https://doi.org/10.1007/s11081-020-09552-5 SN - 1573-2924 VL - 22 IS - 2 SP - 913 EP - 974 ER - TY - CHAP A1 - Breuß, Michael A1 - Mansouri Yarahmadi, Ashkan ED - Durou, Jean-Denis ED - Falcone, Maurizio ED - Quéau, Yvain ED - Tozza, Silvia T1 - Perspective Shape from Shading : An Exposition on Recent Works with New Experiments T2 - Advances in Photometric 3D-Reconstruction N2 - Shape from Shading (SFS) is a fundamental task in computer vision. By given information about the reflectance of an object’s surface and the position of the light source, the SFS problem is to reconstruct the 3D depth of the object from a single grayscale 2D input image. A modern class of SFS models relies on the property that the camera performs a perspective projection. The corresponding perspective SFS methods have been the subject of many investigations within the last years. The goal of this chapter is to give an overview of these developments. In our discussion, we focus on important model aspects, and we investigate some prominent algorithms appearing in the literature in more detail than it was done in previous works. KW - Shape from Shading Perspective projection Hamilton Jacobi equations Numerical methods Fast marching method Y1 - 2020 SN - 978-3-030-51865-3 SN - 978-3-030-51866-0 U6 - https://doi.org/https://doi.org/10.1007/978-3-030-51866-0_2 SN - 2191-6586 SN - 2191-6594 SP - 31 EP - 72 PB - Springer CY - Cham ER - TY - GEN A1 - Bonhage, Alexander A1 - Eltaher, Mahmoud A1 - Raab, Thomas A1 - Breuß, Michael A1 - Raab, Alexandra A1 - Schneider, Anna T1 - A modified Mask region‐based convolutional neural network approach for the automated detection of archaeological sites on high‐resolution light detection and ranging‐derived digital elevation models in the North German Lowland T2 - Archaeological Prospection Y1 - 2021 U6 - https://doi.org/10.1002/arp.1806 SN - 1099-0763 VL - 28 IS - 2 SP - 177 EP - 186 ER - TY - GEN A1 - Köhler, Alexander A1 - Breuß, Michael ED - Elmoataz, Abderrahim ED - Fadili, Jalal ED - Quéau, Yvain ED - Rabin, Julien ED - Simon, Loïc T1 - Towards Efficient Time Stepping for Numerical Shape Correspondence T2 - Scale Space and Variational Methods in Computer Vision : 8th International Conference, SSVM 2021, Virtual Event, May 16–20, 2021, Proceedings N2 - The computation of correspondences between shapes is a principal task in shape analysis. To this end, methods based on partial differential equations (PDEs) have been established, encompassing e.g. the classic heat kernel signature as well as numerical solution schemes for geometric PDEs. In this work we focus on the latter approach. We consider here several time stepping schemes. The goal of this investigation is to assess, if one may identify a useful property of methods for time integration for the shape analysis context. Thereby we investigate the dependence on time step size, since the class of implicit schemes that are useful candidates in this context should ideally yield an invariant behaviour with respect to this parameter. To this end we study integration of heat and wave equation on a manifold. In order to facilitate this study, we propose an efficient, unified model order reduction framework for these models. We show that specific l0 stable schemes are favourable for numerical shape analysis. We give an experimental evaluation of the methods at hand of classical TOSCA data sets. Y1 - 2021 SN - 978-3-030-75548-5 SN - 978-3-030-75549-2 U6 - https://doi.org/10.1007/978-3-030-75549-2_14 SN - 1611-3349 SN - 0302-9743 SP - 165 EP - 176 PB - Springer International Publishing ER - TY - GEN A1 - Sridhar, Vivek A1 - Breuß, Michael A1 - Kahra, Marvin ED - Bebis, George T1 - Fast Approximation of Color Morphology T2 - Advances in Visual Computing : 16th International Symposium, ISVC 2021, Virtual Event, October 4-6, 2021, Proceedings, Part II N2 - The basic filters in mathematical morphology are dilation and erosion. They are defined by a flat or non-flat structuring element that is usually shifted pixel-wise over an image and a comparison process that takes place within the corresponding mask. The algorithmic complexity of fast algorithms that realise dilation and erosion for color images usually depends on size and shape of the structuring element. In this paper we propose and investigate an easy and fast way to make use of the fast Fourier transform for an approximate computation of dilation and erosion for color images. Similarly in construction as many other fast algorithms, the method extends a recent scheme proposed for single-channel filtering. It is by design highly flexible, as it can be used with flat and non-flat structuring elements of any size and shape. Moreover, its complexity only depends on the number of pixels in the filtered images. We analyse here some important aspects of the approximation, and we show experimentally that we obtain results of very reasonable quality while the method has very attractive computational properties. KW - Mathematical morphology Fourier transform Fast algorithms Y1 - 2022 SN - 978-3-030-90435-7 U6 - https://doi.org/10.1007/978-3-030-90436-4_39 SP - 488 EP - 499 PB - Springer CY - Cham ER - TY - BOOK A1 - Bähr, Martin A1 - Buhl, Johannes A1 - Radow, Georg A1 - Schmidt, Johannes A1 - Bambach, Markus A1 - Breuß, Michael A1 - Fügenschuh, Armin T1 - Stable honeycomb structures and temperature based trajectory optimization for wire-arc additive manufacturing Y1 - 2019 U6 - https://doi.org/10.26127/btuopen-5079 CY - Cottbus ER - TY - GEN A1 - Schneidereit, Toni A1 - Breuß, Michael T1 - Polynomial Neural Forms Using Feedforward Neural Networks for Solving Differential Equations T2 - Artificial Intelligence and Soft Computing N2 - Several neural network approaches for solving differential equations employ trial solutions with a feedforward neural network. There are different means to incorporate the trial solution in the construction, for instance one may include them directly in the cost function. Used within the corresponding neural network, the trial solutions define the so-called neural form. Such neural forms represent general, flexible tools by which one may solve various differential equations. In this article we consider time-dependent initial value problems, which requires to set up the trial solution framework adequately. The neural forms presented up to now in the literature for such a setting can be considered as first order polynomials. In this work we propose to extend the polynomial order of the neural forms. The novel construction includes several feedforward neural networks, one for each order. The feedforward neural networks are optimised using a stochastic gradient descent method (ADAM). As a baseline model problem we consider a simple yet stiff ordinary differential equation. In experiments we illuminate some interesting properties of the proposed approach. KW - Feedforward neural networks / Initial value problem / Trial solution /Differential equations Y1 - 2021 SN - 978-3-030-87985-3 U6 - https://doi.org/10.1007/978-3-030-87986-0_21 SN - 978-3-030-87986-0 SP - 236 EP - 245 PB - Springer CY - Cham ER - TY - GEN A1 - Sridhar, Vivek A1 - Breuß, Michael T1 - Sampling of Non-flat Morphology for Grey Value Images T2 - Computer Analysis of Images and Patterns N2 - Sampling is a basic operation in image processing. In previous literature, a morphological sampling theorem has been established showing how sampling interacts with image reconstruction by morphological operations. However, while many aspects of morphological sampling have been investigated for binary images in classic works, only some of them have been extended to grey scale imagery. Especially, previous attempts to study the relation between sampling and grey scale morphology are restricted by construction to flat morphological filters. In order to establish a sampling theory for non-flat morphology, we establish an alternative definition for grey scale opening and closing relying on the umbra notion. Making use of this, we prove a sampling theorem about the interaction of sampling with fundamental morphological operations for non-flat morphology. This allows to make precise corresponding relations between sampling and image reconstruction, extending classic results for flat morphology of grey value images. KW - Non-flat morphology/ Sampling theorem/ Mathematical morphology / Opening / Closing Y1 - 2021 SN - 978-3-030-89130-5 U6 - https://doi.org/10.1007/978-3-030-89131-2_8 SN - 978-3-030-89131-2 VL - 2021 SP - 88 EP - 97 PB - Springer CY - Cham ER - TY - GEN A1 - Mansouri Yarahmadi, Ashkan A1 - Breuß, Michael T1 - Automatic Watermeter Reading in Presence of Highly Deformed Digits T2 - Computer Analysis of Images and Patterns N2 - The task we face in this paper is to automate the reading of watermeters as can be found in large apartment houses. Typically water passes through such watermeters, so that one faces a wide range of challenges caused by water as the medium where the digits are positioned. One of the main obstacles is given by the frequently produced bubbles inside the watermeter that deform the digits. To overcome this problem, we propose the construction of a novel data set that resembles the watermeter digits with a focus on their deformations by bubbles. We report on promising experimental recognition results, based on a deep and recurrent network architecture performed on our data set. KW - Underwater digit recognition / Sequence models / Connectionist Temporal Classification Y1 - 2021 SN - 978-3-030-89130-5 U6 - https://doi.org/10.1007/978-3-030-89131-2_14 SN - 978-3-030-89131-2 SP - 153 EP - 163 PB - Springer CY - Cham ER - TY - GEN A1 - Kahra, Marvin A1 - Sridhar, Vivek A1 - Breuß, Michael T1 - Fast Morphological Dilation and Erosion for Grey Scale Images Using the Fourier Transform T2 - Scale Space and Variational Methods in Computer Vision N2 - The basic filters in mathematical morphology are dilation and erosion. They are defined by a flat or non-flat structuring element that is usually shifted pixel-wise over an image and a comparison process that takes place within the corresponding mask. Existing fast algorithms that realise dilation and erosion for grey value images are often limited with respect to size or shape of the structuring element. Usually their algorithmic complexity depends on these aspects. Many fast methods only address flat morphology. In this paper we propose a novel way to make use of the fast Fourier transform for the computation of dilation and erosion. Our method is by design highly flexible, as it can be used with flat and non-flat structuring elements of any size and shape. Moreover, its complexity does not depend on size or shape of the structuring element, but only on the number of pixels in the filtered images. We show experimentally that we obtain results of very reasonable quality with the proposed method. KW - Mathematical morphology / Fast Fourier Transform / Dilation / Erosion / Efficient algorithms Y1 - 2021 SN - 978-3-030-75548-5 U6 - https://doi.org/10.1007/978-3-030-75549-2_6 SN - 978-3-030-75549-2 SP - 65 EP - 77 PB - Springer CY - Cham ER - TY - GEN A1 - Schneidereit, Toni A1 - Breuß, Michael T1 - Collocation polynomial neural forms and domain fragmentation for solving initial value problems T2 - Neural Computing and Applications N2 - Several neural network approaches for solving differential equations employ trial solutions with a feedforward neural network. There are different means to incorporate the trial solution in the construction, for instance, one may include them directly in the cost function. Used within the corresponding neural network, the trial solutions define the so-called neural form. Such neural forms represent general, flexible tools by which one may solve various differential equations. In this article, we consider time-dependent initial value problems, which require to set up the neural form framework adequately. The neural forms presented up to now in the literature for such a setting can be considered as first-order polynomials. In this work, we propose to extend the polynomial order of the neural forms. The novel collocation-type construction includes several feedforward neural networks, one for each order. Additionally, we propose the fragmentation of the computational domain into subdomains. The neural forms are solved on each subdomain, whereas the interfacing grid points overlap in order to provide initial values over the whole fragmentation. We illustrate in experiments that the combination of collocation neural forms of higher order and the domain fragmentation allows to solve initial value problems over large domains with high accuracy and reliability. KW - Collocation neural forms KW - Polynomial neural forms KW - Trial solution KW - Initial value problems KW - Domain fragmentation Y1 - 2022 U6 - https://doi.org/10.1007/s00521-021-06860-4 SN - 1433-3058 SN - 0941-0643 VL - 34 IS - 9 SP - 7141 EP - 7156 ER -