TY - GEN A1 - Radow, Georg A1 - Hoeltgen, Laurent A1 - Quéau, Yvain A1 - Breuß, Michael T1 - Optimisation of Classic Photometric Stereo by Non-convex Variational Minimisation T2 - Journal of Mathematical Imaging and Vision N2 - Estimating shape and appearance of a three-dimensional object from a given set of images is a classic research topic that is still actively pursued. Among the various techniques available, photometric stereo is distinguished by the assumption that the underlying input images are taken from the same point of view but under different lighting conditions. The most common techniques are conceptually close to the classic photometric stereo problem, meaning that the modelling encompasses a linearisation step and that the shape information is computed in terms of surface normals. In this work, instead of linearising we aim to stick to the original formulation of the photometric stereo problem, and we propose to minimise a much more natural objective function, namely the reprojection error in terms of depth. Minimising the resulting non-trivial variational model for photometric stereo allows to recover the depth of the photographed scene directly. As a solving strategy, we follow an approach based on a recently published optimisation scheme for non-convex and non-smooth cost functions. The main contributions of our paper are of theoretical nature. A technical novelty in our framework is the usage of matrix differential calculus. We supplement our approach by a detailed convergence analysis of the resulting optimisation algorithm and discuss possibilities to ease the computational complexity. At hand of an experimental evaluation we discuss important properties of the method. Overall, our strategy achieves more accurate results than other approaches that rely on the classic photometric stereo assumptions. The experiments also highlight some practical aspects of the underlying optimisation algorithm that may be of interest in a more general context. KW - Non-convex minimisation KW - Computer vision KW - Photometric stereo Y1 - 2019 U6 - https://doi.org/10.1007/s10851-018-0828-7 SN - 1573-7683 VL - 61 IS - 1 SP - 84 EP - 105 ER - TY - GEN A1 - Dachsel, Robert A1 - Jöster, Annika A1 - Breuß, Michael ED - Lee, Chilwoo ED - Su, Zhixun ED - Sugimoto, Akihiro T1 - Real-Time Retinal Vessel Segmentation on High-Resolution Fundus Images Using Laplacian Pyramids T2 - Image and Video Technology, 9th Pacific-Rim Symposium, PSIVT 2019, Sydney, NSW, Australia, November 18–22, 2019, Proceedings N2 - In ophthalmology, fundus images are commonly used to examine the human eye. The image data shows among others the capillary system of the retina. Recognising alternations in the retinal blood vessels is pivotal to diagnosing certain diseases. The visual inspection of those fundus images is a time-consuming process and a challenging task which has to be done by medical experts. Furthermore, rapid advances in medical imaging allow for generating fundus images of increased quality and resolution. Therefore, the support by computers for the analysis and evaluation of complex fundus image information is growing in importance and there is a corresponding need for fast and efficient algorithms. In this paper, we present a well-engineered, robust real-time segmentation algorithm which is adapted to the recent and upcoming challenges of high resolution fundus images. Thereby we make use of the multiscale representation of the Laplacian pyramid which is fast to compute and useful for detecting coarse as well as finely branched blood vessels. It is possible to process images of size 3504×2336 pixels in 0.8 s on a standard desktop computer and 0.3 on a Nvidia Titan XP GPU. By a detailed evaluation at hand of an accessible high-resolution data set we demonstrate that our approach is competitive in quality to state of the art methods for segmenting blood vessels but much faster. KW - Laplacian pyramids Vessel segmentation High-resolution fundus images Real-time retinal imaging Y1 - 2019 SN - 978-3-030-34878-6 SN - 978-3-030-34879-3 U6 - https://doi.org/10.1007/978-3-030-34879-3_26 SN - 0302-9743 SN - 1611-3349 SP - 337 EP - 350 PB - Springer CY - Cham ER - TY - GEN A1 - Hoeltgen, Laurent A1 - Kleefeld, Andreas A1 - Harries, Isaac A1 - Breuß, Michael T1 - Theoretical foundation of the weighted laplace inpainting problem T2 - Journal of Applications of Mathematics N2 - Laplace interpolation is a popular approach in image inpainting using partial differential equations. The classic approach considers the Laplace equation with mixed boundary conditions. Recently a more general formulation has been proposed, where the differential operator consists of a point-wise convex combination of the Laplacian and the known image data. We provide the first detailed analysis on existence and uniqueness of solutions for the arising mixed boundary value problem. Our approach considers the corresponding weak formulation and aims at using the Theorem of Lax-Milgram to assert the existence of a solution. To this end we have to resort to weighted Sobolev spaces. Our analysis shows that solutions do not exist unconditionally. The weights need some regularity and must fulfil certain growth conditions. The results from this work complement findings which were previously only available for a discrete setup. KW - image inpainting KW - image reconstruction KW - Laplace equation KW - Laplace interpolation KW - mixed boundary condition KW - partial differential equation KW - weighted Sobolev space Y1 - 2019 U6 - https://doi.org/10.21136/AM.2019.0206-18 SN - 1572-9109 SN - 0862-7940 VL - 64 IS - 3 SP - 281 EP - 300 ER - TY - GEN A1 - Welk, Martin A1 - Breuß, Michael A1 - Sridhar, Vivek T1 - Matrix Morphology with Extremum Principle T2 - Mathematical Morphology and Its Applications to Signal and Image Processing : 14th International Symposium, ISMM 2019, Saarbrücken, Germany, July 8-10, 2019, Proceedings N2 - The fundamental operations of mathematical morphology are dilation and erosion. In previous works, these operations have been generalised in a discrete setting to work with fields of symmetric matrices, and also corresponding methods based on partial differential equations have been constructed. However, the existing methods for dilation and erosion in the matrix-valued setting are not overall satisfying. By construction they may violate a discrete extremum principle, which means that results may leave the convex hull of the matrices that participate in the computation. This may not be desirable from the theoretical point of view, as the corresponding property is fundamental for discrete and continuous-scale formulations of dilation and erosion in the scalar setting. Moreover, if such a principle could be established in the matrix-valued framework, this would help to make computed solutions more interpretable. In our paper we address this issue. We show how to construct a method for matrix-valued morphological dilation and erosion that satisfies a discrete extremum principle. We validate the construction by showing experimental results on synthetic data as well as colour images, as the latter can be cast as fields of symmetric matrices. KW - Dilation Erosion Matrix valued images Extremum principle Y1 - 2019 SN - 978-3-030-20866-0 SN - 978-3-030-20867-7 U6 - https://doi.org/10.1007/978-3-030-20867-7_14 SP - 177 EP - 188 PB - Springer CY - Cham ER - TY - GEN A1 - Welk, Martin A1 - Breuß, Michael T1 - The convex-hull-stripping median approximates affine curvature motion T2 - Scale Space and Variational Methods in Computer Vision : 7th International Conference, SSVM 2019, Hofgeismar, Germany, June 30 – July 4, 2019, Proceedings N2 - The median filter is one of the fundamental filters in image processing. Its standard realisation relies on a rank ordering of given data which is easy to perform if the given data are scalar values. However, the generalisation of the median filter to multivariate data is a delicate issue. One of the methods of potential interest for computing a multivariate median is the convex-hull-stripping median from the statistics literature. Its definition is of purely algorithmical nature, and it offers the advantageous property of affine equivariance. While it is a classic result that the standard median filter approximates mean curvature motion, no corresponding assertion has been established up to now for the convex-hull-stripping median. The aim of our paper is to close this gap in the literature. In order to provide a theoretical foundation for the convex-hull-stripping median of multivariate images, we investigate its continuous-scale limit. It turns out that the resulting evolution is described by the well-known partial differential equation of affine curvature motion. Thus we have established in this paper a relation between two important models from image processing and statistics. We also present some experiments that support our theoretical findings. KW - Median filter Convex hull stripping Partial differential equations Curve evolution Y1 - 2019 SN - 978-3-030-22367-0 U6 - https://doi.org/10.1007/978-3-030-22368-7_16 SP - 199 EP - 210 PB - Springer CY - Cham ER - TY - GEN A1 - Eltaher, Mahmoud A1 - Bonhage, Alexander A1 - Raab, Thomas A1 - Breuß, Michael T1 - A modified Mask R-CNN approach for automated detection of archaeological sites on high resolution LiDAR-derived DEMs T2 - Anthropogenetische Geomorphologie - Geomorophologie im Anthropozän : Virtuelle Jahrestagung des Arbeitskreises für Geomorphologie 2020, 28./29. September 2020, BTU Cottbus - Senftenberg KW - Anthropogenetische Geomorphologie; Anthropozän; BTU; Geopedologie Anthropocene; Anthropogenic geomorphology; Geopedology Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:co1-opus4-53633 SN - 2196-4122 SP - 50 PB - Cottbus ER - TY - GEN A1 - Breuß, Michael A1 - Hoeltgen, Laurent A1 - Radow, Georg T1 - Towards PDE-Based Video Compression with Optimal Masks Prolongated by Optic Flow T2 - Journal of Mathematical Imaging and Vision N2 - Lossy image compression methods based on partial differential equations have received much attention in recent years. They may yield high-quality results but rely on the computationally expensive task of finding an optimal selection of data. For the possible extension to video compression, this data selection is a crucial issue. In this context, one could either analyse the video sequence as a whole or perform a frame-by-frame optimisation strategy. Both approaches are prohibitive in terms of memory and run time. In this work, we propose to restrict the expensive computation of optimal data to a single frame and to approximate the optimal reconstruction data for the remaining frames by prolongating it by means of an optic flow field. In this way, we achieve a notable decrease in the computational complexity. As a proof-of-concept, we evaluate the proposed approach for multiple sequences with different characteristics. In doing this, we discuss in detail the influence of possible computational setups. We show that the approach preserves a reasonable quality in the reconstruction and is very robust against errors in the flow field. KW - Partial differential equations KW - Inpainting KW - Laplace interpolation KW - Optic flow KW - Video reconstruction Y1 - 2021 U6 - https://doi.org/10.1007/s10851-020-00973-6 SN - 1573-7683 SN - 0924-9907 VL - 63 IS - 2 SP - 144 EP - 156 ER - 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 -