TY - CHAP A1 - Kleefeld, Andreas A1 - Ju, Yong Chul A1 - Tozza, Silvia A1 - Breuß, Michael A1 - Bruhn, Andrés ED - Burghardt, Tilo ED - Damen, Dima ED - Mayol-Cuevas, Walterio ED - Mirmehdi, Majid T1 - Generalised Perspective Shape from Shading with Oren-Nayar Reflectance T2 - Proceedings 24th British Machine Vision Conference, BMVC 2013, Bristol, UK, September 2013 N2 - In spite of significant advances in Shape from Shading (SfS) over the last years, it is still a challenging task to design SfS approaches that are flexible enough to handle a wide range of input scenes. In this paper, we address this lack of flexibility by proposing a novel model that extends the range of possible applications. To this end, we consider the class of modern perspective SfS models formulated via partial differential equations (PDEs). By combining a recent spherical surface parametrisation with the advanced non-Lambertian Oren-Nayar reflectance model, we obtain a robust approach that allows to deal with an arbitrary position of the light source while being able to handle rough surfaces and thus more realistic objects at the same time. To our knowledge, the resulting model is currently the most advanced and most flexible approach in the literature on PDE-based perspective SfS. Apart from deriving our model, we also show how the corresponding set of sophisticated Hamilton-Jacobi equations can be efficiently solved by a specifically tailored fast marching scheme. Experiments with medical real-world data demonstrate that our model works in practice and that is offers the desired flexibility. Y1 - 2013 UR - http://www.bmva.org/bmvc/2013/Papers/paper0042/index.html U6 - https://doi.org/10.5244/C.27.42 SP - 42.1 EP - 42.11 PB - BMVA Press ER - TY - GEN A1 - Breuß, Michael A1 - Dietrich, Dominik T1 - On the Optimization of Flux Limiter Schemes for Hyperbolic Conservation Laws T2 - Numerical Methods for Partial Differential Equations Y1 - 2013 SN - 1098-2426 VL - Vol. 29 IS - 3 SP - 884 EP - 896 ER - TY - GEN A1 - Breuß, Michael A1 - Cristiani, Emiliano A1 - Durou, Jean-Denis A1 - Falcone, Maurizio A1 - Vogel, Oliver T1 - Perspective Shape from Shading: Ambiguity Analysis and Numerical Approximations T2 - SIAM Journal on Imaging Sciences Y1 - 2012 SN - 1936-4954 VL - 5 IS - 1 SP - 311 EP - 342 ER - TY - CHAP A1 - Ju, Yong Chul A1 - Breuß, Michael A1 - Bruhn, Andrés A1 - Galiani, Silvano ED - Browden, R. ED - Collomosse, J. ED - Mikolajczyk, K. T1 - Shape from Shading for Rough Surfaces: Analysis of the Oren-Nayar Model T2 - Proceedings 23rd British Machine Vision Conference, BMVC 2012, Surrey, UK, September 3-7, 2012 Y1 - 2012 SN - 1-901725-46-4 U6 - https://doi.org/10.5244/C.26.104 SP - 104.1 EP - 104.11 PB - BMVA Press ER - TY - CHAP A1 - Breuß, Michael A1 - Burgeth, Bernhard A1 - Pizarro, Luis ED - Ansorge, Rainer T1 - Numerical Analysis, Tensor-Valued Extension and Application in Image Processing T2 - Recent Developments in the Numerics of Hyperbolic Conservation Laws Y1 - 2013 SN - 978-3-642-33220-3 SP - 73 EP - 87 PB - Springer CY - Berlin [u.a.] ER - TY - CHAP A1 - Galliani, S. A1 - Breuß, Michael A1 - Ju, Yong Chul ED - Bowden, R. ED - Collomosse, J. ED - Mikolajczyk, K. T1 - Fast and Robust Surface Normal Integration by a Discrete Eikonal Equation T2 - Proceedings, 23rd British Machine Vision Conference, BMVC 2012, Surrey, UK, September 3-7, 2012 Y1 - 2012 SN - 1-901725-46-4 U6 - https://doi.org/10.5244/C.26.106 SP - 106.1 EP - 106.11 PB - BMVA Press ER - TY - CHAP A1 - Galliani, S. A1 - Ju, Yong Chul A1 - Breuß, Michael A1 - Bruhn, Andrés ED - Kuijper, A. ED - Pock, T. ED - Bredies, K. ED - Bischof, H. T1 - Generalised Perspective Shape from Shading in Spherical Coordinates T2 - Proceedings Fourth International Conference on Scale Space and Variational Methods in Computer Vision, SSVM 2013, Graz, Austria, June 2013 Y1 - 2013 SN - 978-3-642-38266-6 SP - 222 EP - 233 PB - Springer CY - Berlin ER - TY - CHAP A1 - Welk, Martin A1 - Breuß, Michael ED - Hawkes, Peter W. T1 - Morphological Amoebas and Partial Differential Equations T2 - Advances in Imaging and Electron Physics Y1 - 2014 SN - 978-0-12-800144-8 SP - 139 EP - 212 PB - Elsevier CY - Amsterdam ER - TY - CHAP A1 - Rhodin, Helge A1 - Breuß, Michael ED - Kuijper, A. ED - Pock, T. ED - Bredies, K. ED - Bischof, H. T1 - A Mathematically Justified Algorithm for Shape from Texture T2 - Proceedings Fourth International Conference on Scale Space and Variational Methods in Computer Vision, SSVM 2013, Graz, Austria, June 2013 Y1 - 2013 SN - 978-3-642-38266-6 SN - 978-3-642-38267-3 SP - 294 EP - 305 PB - Springer CY - Berlin ER - TY - CHAP A1 - Weickert, Joachim A1 - Hagenburg, Kai Uwe A1 - Breuß, Michael A1 - Vogel, Oliver ED - Heyden, A. ED - Kahl, F. ED - Olsson, C. ED - Oskarsson, M. ED - Tai, X.-C. T1 - Linear osmosis models for visual computing T2 - Energy Minimization Methods in Computer Vision and Pattern Recognition, EMMCVPR 2013, Lund, Sweden, August 2013 Y1 - 2013 SN - 978-3-642-40394-1 SN - 978-3-642-40395-8 SP - 26 EP - 39 PB - Springer CY - Berlin ER - TY - CHAP A1 - Peter, Pascal A1 - Breuß, Michael ED - Breuß, Michael ED - Bruckstein, Alfred ED - Maragos, Petros T1 - Refined Homotopic Thinning Algorithms and Quality Measures for Skeletonisation Methods T2 - Innovations for Shape Analysis : Models and Algorithms Y1 - 2013 SN - 978-3-642-34140-3 SP - 77 EP - 92 PB - Springer CY - Berlin ER - TY - CHAP A1 - Kleefeld, Andreas A1 - Breuß, Michael A1 - Welk, Martin A1 - Burgeth, Bernhard ED - Trémeau, Alain ED - Schettini, Raimondo ED - Tominaga, Shoji T1 - Adaptive Filters for Color Images: Median Filtering and Its Extensions T2 - Computational Color Imaging N2 - In this paper we are concerned with robust structure-preserving denoising filters for color images. We build on a recently proposed transformation from the RGB color space to the space of symmetric 2×2 matrices that has already been used to transfer morphological dilation and erosion concepts from matrix-valued data to color images. We investigate the applicability of this framework to the construction of color-valued median filters. Additionally, we introduce spatial adaptivity into our approach by morphological amoebas that offer excellent capabilities for structure-preserving filtering. Furthermore, we define color-valued amoeba M-smoothers as a generalization of the median-based concepts. Our experiments confirm that all these methods work well with color images. They demonstrate the potential of our approach to define color processing tools based on matrix field techniques. KW - Matrix field KW - Color image KW - Median filter KW - M-smoother KW - Amoeba filter Y1 - 2015 UR - http://link.springer.com/chapter/10.1007/978-3-319-15979-9_15# SN - 978-3-319-15978-2 U6 - https://doi.org/10.1007/978-3-319-15979-9_15 SP - 149 EP - 158 PB - Springer CY - Berlin ER - TY - CHAP A1 - Sharifi Boroujerdi, Ali A1 - Breuß, Michael A1 - Burgeth, Bernhard A1 - Kleefeld, Andreas ED - Aujol, Jean-François ED - Nikolova, Mila ED - Papadakis, Nicolas T1 - PDE-Based Color Morphology Using Matrix Fields T2 - Scale Space and Variational Methods in Computer Vision, 5th International Conference, SSVM 2015, Lège-Cap Ferret, France, May 31 - June 4, 2015, Proceedings N2 - In this work, we propose a novel way for performing operations of mathematical morphology on color images. To this end, we convert pixelwise the rgb-values into symmetric 2x2 matrices. The new color space can be interpreted geometrically as a biconal color space structure. Motivated by the formulation of the fundamental morphological operations dilation and erosion in terms of partial differential equations (PDEs), we show how to define finite difference schemes making use of the matrix field formulation. The computation of a pseudo supremum and a pseudo infimum of three color matrices is a crucial step for setting up advanced PDE-based methods. We show that this can be achieved for our goal by an algebraic technique. We investigate our approach by dedicated experiments and confirm useful properties of the new PDE-based color morphology operations. KW - PDE-based morphology KW - Matrix fields KW - Color morphology KW - Finite difference schemes KW - FCT scheme KW - Pseudo supremum KW - Pseudo infimum Y1 - 2015 SN - 978-3-319-18460-9 U6 - https://doi.org/10.1007/978-3-319-18461-6_37 SP - 461 EP - 473 PB - Springer CY - Berlin ER - TY - CHAP A1 - Welk, Martin A1 - Kleefeld, Andreas A1 - Breuß, Michael ED - Benediktsson, Jón Atli ED - Chanussot, Jocelyn ED - Najman, Laurent ED - Talbot, Hugues T1 - Non-adaptive and Amoeba Quantile Filters for Colour Images T2 - Mathematical Morphology and Its Applications to Signal and Image Processing N2 - Quantile filters, or rank-order filters, are local image filters which assign quantiles of intensities of the input image within neighbourhoods as output image values. Combining a multivariate quantile definition developed in matrix-valued morphology with a recently introduced mapping between the RGB colour space and the space of symmetric 2×2 matrices, we state a class of colour image quantile filters, along with a class of morphological gradient filters derived from these. Using amoeba structuring elements, we devise image-adaptive versions of both filter classes. Experiments demonstrate the favourable properties of the filters. KW - Quantile KW - Rank-order filter KW - Color image KW - Matrix field KW - Amoebas Y1 - 2015 SN - 978-3-319-18719-8 SN - 978-3-319-18720-4 U6 - https://doi.org/10.1007/978-3-319-18720-4_34 SP - 398 EP - 409 PB - Springer International Publishing CY - Berlin ER - TY - CHAP A1 - Breuß, Michael A1 - Quèau, Yvain A1 - Bähr, Martin A1 - Durou, Jean-Denis T1 - Highly Efficient Surface Normal Integration T2 - Proceedings of the Conference Algoritmy 2016, 20th Conference on Scientific Computing, Vysoké Tatry - Podbanské, Slovakia, March 14 - 18 KW - surface normal integration KW - Poisson integration KW - Krylov subspace methods KW - fast marching methods Y1 - 2016 UR - http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/409 SN - 978-80-227-4544-4 SP - 204 EP - 213 PB - Publishing House of Slovak University of Technology in Bratislava CY - Bratislava ER - TY - CHAP A1 - Hoeltgen, Laurent A1 - Breuß, Michael ED - Handlovičová, A. ED - Ševčovič, D. T1 - Efficient Co-Domain Quantisation for PDE-Based Image Compression T2 - Proceedings of the Conference Algoritmy 2016, 20th Conference on Scientific Computing, Vysoké Tatry - Podbanské, Slovakia, March 14 - 18 KW - Laplace interpolation KW - inpainting KW - compression KW - quantisation KW - clustering KW - partial differential equations Y1 - 2016 UR - http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/408 SN - 978-80-227-4544-4 SP - 194 EP - 203 PB - Publishing House of Slovak University of Technology in Bratislava CY - Bratislava ER - TY - CHAP A1 - Bähr, Martin A1 - Breuß, Michael ED - Gall, Juergen ED - Gehler, Peter ED - Leibe, Bastian T1 - An Improved Eikonal Method for Surface Normal Integration T2 - Pattern Recognition, 37th German Conference, GCPR 2015, Aachen, Germany, October 7-10, 2015, Proceedings Y1 - 2015 UR - http://link.springer.com/chapter/10.1007%2F978-3-319-24947-6_22 SN - 978-3-319-24946-9 U6 - https://doi.org/10.1007/978-3-319-24947-6_22 SP - 274 EP - 284 PB - Springer International Publishing CY - Berlin ER - TY - CHAP A1 - Chul Ju, Yong A1 - Bruhn, Andrés A1 - Breuß, Michael ED - Aujol, Jean-François ED - Nikolova, Mila ED - Papadakis, Nicolas T1 - Variational Perspective Shape from Shading T2 - Scale Space and Variational Methods in Computer Vision, 5th International Conference, SSVM 2015, Lège-Cap Ferret, France, May 31 - June 4, 2015, Proceedings KW - Shape from shading KW - Variational methods KW - Perspective camera model Y1 - 2015 UR - http://link.springer.com/chapter/10.1007%2F978-3-319-18461-6_43 SN - 978-3-319-18460-9 U6 - https://doi.org/10.1007/978-3-319-18461-6_43 SP - 538 EP - 550 PB - Springer International Publishing CY - Berlin ER - TY - CHAP A1 - Khanian, Maryam A1 - Sharifi Boroujerdi, Ali A1 - Breuß, Michael ED - Meriaudeau, Fabrice ED - Aubreton, Olivier T1 - Perspective photometric stereo beyond Lambert T2 - Twelfth International Conference on Quality Control by Artificial Vision, 3 - 5 June 2015, Le Creusot, France KW - Cameras KW - Reflection KW - Reflectivity KW - Specular reflections Y1 - 2015 UR - http://proceedings.spiedigitallibrary.org/proceeding.aspx?articleid=2290117 SN - 978-80-227-4544-4 U6 - https://doi.org/10.1117/12.2182839 PB - SPIE CY - Bellingham, Wash. ER - TY - CHAP A1 - Khanian, Maryam A1 - Breuß, Michael A1 - Davari, Ali ED - Simos, Theodore E. ED - Tsitouras, Charalambos T1 - An improved stopping criterion for anisotropic diffusion T2 - Proceedings of the International Conference on Numerical Analysis and Applied Mathematics 2014 (ICNAAM-2014), Rhodes, Greece, 22-28 September 2014 KW - Amoeba filter Matrix field KW - Color image KW - M-smoother KW - Matrix field KW - Median filter Y1 - 2015 UR - http://scitation.aip.org/content/aip/proceeding/aipcp/10.1063/1.4919928 SN - 978-0-7354-1287-3 U6 - https://doi.org/10.1063/1.4919928 PB - AIP Publishing CY - Melville, New York ER - TY - CHAP A1 - Ju, Yong Chul A1 - Maurer, Daniel A1 - Breuß, Michael A1 - Bruhn, Andrés ED - Breuß, Michael ED - Bruckstein, Alfred ED - Maragos, Petros ED - Wuhrer, Stefanie T1 - Direct variational perspective shape from shading with Cartesian depth parametrisation T2 - Perspectives in Shape Analysis, Mathematics and Visualization Y1 - 2016 SN - 978-3-319-24724-3 U6 - https://doi.org/10.1007/978-3-319-24726-7 SP - 43 EP - 72 PB - Springer International Publishing CY - Cham ER - TY - GEN A1 - Hoeltgen, Laurent A1 - Queau, Yvain A1 - Breuß, Michael A1 - Radow, Georg T1 - Optimised photometric stereo via non-convex variational minimisation Y1 - 2016 UR - http://www.bmva.org/bmvc/2016/papers/paper036/ N1 - Poster für die BMVC in York, September 2016 PB - BMVA Press CY - York, UK ER - TY - GEN A1 - Maurer, Daniel A1 - Ju, Yong Chul A1 - Breuß, Michael A1 - Bruhn, Andrés T1 - Combining Shape from Shading and Stereo: A Variational Approach for the Joint Estimation of Depth, Illumination and Albedo N2 - Shape from shading (SfS) and stereo are two fundamentally different strategies for image-based 3-D reconstruction. While approaches for SfS infer the depth solely from pixel intensities, methods for stereo are based on a matching process that establishes correspondences across images. In this paper we propose a joint variational method that combines the advantages of both strategies. By integrating recent stereo and SfS models into a single minimisation framework, we obtain an approach that exploits shading information to improve upon the reconstruction quality of robust stereo methods. To this end, we fuse a Lambertian SfS approach with a robust stereo model and supplement the resulting energy functional with a detail-preserving anisotropic second-order smoothness term. Moreover, we extend the novel model in such a way that it jointly estimates depth, albedo and illumination. This in turn makes it applicable to objects with non-uniform albedo as well as to scenes with unknown illumination. Experiments for synthetic and real-world images show the advantages of our combined approach: While the stereo part overcomes the albedo-depth ambiguity inherent to all SfS methods, the SfS part improves the degree of details of the reconstruction compared to pure stereo methods. Y1 - 2016 N1 - Das Poster wurde auf der BMVC in York im September 2016 präsentiert PB - BMVA Press CY - York, UK ER - TY - GEN A1 - Welk, Martin A1 - Kleefeld, Andreas A1 - Breuß, Michael T1 - Quantile Filtering of Colour Images via Symmetric Matrices T2 - Mathematical Morphology - Theory and Applications N2 - Quantile filters, or rank-order filters, are local image filters which assign quantiles of intensities of the input image within neighbourhoods as output image values. Combining a multivariate quantile definition developed in matrix-valued morphology with a recently introduced mapping between the RGB colour space and the space of symmetric 2 × 2 matrices, we state a class of colour image quantile filters, along with a class of morphological gradient filters derived from these.We consider variants of these filters based on three matrix norms – the nuclear, Frobenius, and spectral norm – and study their differences. We investigate the properties of the quantile and gradient filters and their links to dilation and erosion operators. Using amoeba structuring elements,we devise image-adaptive versions of our quantile and gradient filters. Experiments are presented to demonstrate the favourable properties of the filters, and compare them to existing approaches in colour morphology. KW - quantile KW - rank-order filter KW - colour image KW - matrix field KW - amoebas Y1 - 2016 U6 - https://doi.org/10.1515/mathm-2016-0008 SN - 2353-3390 VL - 1 IS - 1 SP - 136 EP - 174 ER - TY - CHAP A1 - Radow, Georg A1 - Breuß, Michael A1 - Hoeltgen, Laurent A1 - Fischer, Thomas ED - Sharma, Puneet ED - Filippo, Maria Bianchi T1 - Optimised Anisotropic Poisson Denoising T2 - Image Analysis, 20th Scandinavian Conference, SCIA 2017, Tromsø, Norway, June 12–14, 2017, Proceedings, Part I Y1 - 2017 SN - 978-3-319-59126-1 SN - 978-3-319-59125-4 U6 - https://doi.org/10.1007/978-3-319-59126-1_42 SP - 502 EP - 514 PB - Springer CY - Cham ER - TY - CHAP A1 - Dachsel, Robert A1 - Breuß, Michael A1 - Hoeltgen, Laurent ED - Felsberg, Michael ED - Heyden, Andreas ED - Krüger, Norbert T1 - The Classic Wave Equation Can Do Shape Correspondence T2 - Computer Analysis of Images and Patterns, CAIP International Conference on Computer Analysis of Images and Patterns, Ystad,Sweden, 2017 N2 - A major task in non-rigid shape analysis is to retrieve correspondences between two almost isometric 3D objects. An important tool for this task are geometric feature descriptors. Ideally, a feature descriptor should be invariant under isometric transformations and robust to small elastic deformations. A successful class of feature descriptors employs the spectral decomposition of the Laplace-Beltrami operator. Important examples are the heat kernel signature using the heat equation and the more recent wave kernel signature applying the Schrödinger equation from quantum mechanics. In this work we propose a novel feature descriptor which is based on the classic wave equation that describes e.g. sound wave propagation. We explore this new model by discretizing the underlying partial differential equation. Thereby we consider two different time integration methods. By a detailed evaluation at hand of a standard shape data set we demonstrate that our approach may yield significant improvements over state of the art methods for finding correct shape correspondences. KW - Feature descriptor KW - Shape analysis KW - Wave equation Y1 - 2017 SN - 978-3-319-64688-6 U6 - https://doi.org/10.1007/978-3-319-64689-3_22 SP - 264 EP - 275 PB - Springer International Publishing CY - Cham ER - TY - CHAP A1 - Dachsel, Robert A1 - Breuß, Michael A1 - Hoeltgen, Laurent T1 - Shape Matching by Time Integration of Partial Differential Equations T2 - Scale Space and Variational Methods in Computer Vision, 6th International Conference on Scale Space and Variational Methods in Computer Vision (SSVM 2017, Kolding, Denmark, June 2017), proceedings N2 - The main task in three dimensional shape matching is to retrieve correspondences between two similar three dimensional objects. To this end, a suitable point descriptor which is invariant under isometric transformations is required. A commonly used descriptor class relies on the spectral decomposition of the Laplace-Beltrami operator. Important examples are the heat kernel signature and the more recent wave kernel signature. In previous works, the evaluation of the descriptor is based on eigenfunction expansions. Thereby a significant practical aspect is that computing a complete expansion is very time and memory consuming. Thus additional strategies are usually introduced that enable to employ only part of the full expansion. In this paper we explore an alternative solution strategy. We discretise the underlying partial differential equations (PDEs) not only in space as in the mentioned approaches, but we also tackle temporal parts by using time integration methods. Thus we do not perform eigenfunction expansions and avoid the use of additional strategies and corresponding parameters. We study here the PDEs behind the heat and wave kernel signature, respectively. Our shape matching experiments show that our approach may lead to quality improvements for finding correct correspondences in comparison to the eigenfunction expansion methods. KW - Shape matching Point descriptor Shape analysis Heat equation Schrödinger equation Y1 - 2017 SN - 978-3-319-58770-7 U6 - https://doi.org/10.1007/978-3-319-58771-4_53 SP - 669 EP - 680 PB - Springer International Publishing CY - Cham ER - TY - CHAP A1 - Hoeltgen, Laurent A1 - Harris, I. A1 - Breuß, Michael A1 - Kleefeld, Andreas T1 - Analytic Existence and Uniqueness Results for PDE-Based Image Reconstruction with the Laplacian T2 - Scale Space and Variational Methods in Computer Vision, 6th International Conference on Scale Space and Variational Methods in Computer Vision (SSVM 2017, Kolding, Denmark, June 2017), proceedings N2 - Partial differential equations are well suited for dealing with image reconstruction tasks such as inpainting. One of the most successful mathematical frameworks for image reconstruction relies on variations of the Laplace equation with different boundary conditions. In this work we analyse these formulations and discuss the existence and uniqueness of solutions of corresponding boundary value problems, as well as their regularity from an analytic point of view. Our work not only sheds light on useful aspects of the well posedness of several standard problem formulations in image reconstruction but also aggregates them in a common framework. In addition, the performed analysis guides us to specify two new formulations of the classic image reconstruction problem that may give rise to new developments in image reconstruction. KW - Partial differential equations Laplace equation Mixed boundary conditions Image reconstruction Image inpainting Y1 - 2017 SN - 978-3-319-58770-7 U6 - https://doi.org/10.1007/978-3-319-58771-4_6 SP - 66 EP - 79 PB - Springer International Publishing CY - Cham ER - TY - CHAP A1 - Breuß, Michael A1 - Hoeltgen, Laurent A1 - Kleefeld, Andreas T1 - Matrix-Valued Levelings for Colour Images T2 - Mathematical morphology and its applications to signal and image processing, 13th international symposium, ISMM 2017, Fontainebleau, France, May 15-17, 2017, proceedings N2 - Morphological levelings represent a useful tool for the decomposition of an image into cartoon and texture components. Moreover, they can be used to construct a morphological scale space. However, the classic construction of levelings is limited to the use of grey scale images, since an ordering of pixel values is required. In this paper we propose an extension of morphological levelings to colour images. To this end, we consider the formulation of colour images as matrix fields and explore techniques based on the Loewner order for formulating morphological levelings in this setting. Using the matrix-valued colours we study realisations of levelings relying on both the completely discrete construction and the formulation using a partial differential equation. Experimental results confirm the potential of our matrix-based approaches for analysing texture in colour images and for extending the range of applications of levelings in a convenient way to colour image processing. KW - Mathematical morphology Dilation Erosion Loewner order Leveling Y1 - 2017 SN - 978-3-319-57239-0 U6 - https://doi.org/10.1007/978-3-319-57240-6_24 SP - 296 EP - 308 PB - Springer International Publishing CY - Cham ER - TY - GEN A1 - Bähr, Martin A1 - Breuß, Michael A1 - Quèau, Yvain A1 - Sharifi Boroujerdi, Ali A1 - Durou, Jean-Denis T1 - Fast and accurate surface normal integration on non-rectangular domains T2 - Computational Visual Media N2 - The integration of surface normals for the purpose of computing the shape of a surface in 3D space is a classic problem in computer vision. However, even nowadays it is still a challenging task to devise a method that is flexible enough to work on non-trivial computational domains with high accuracy, robustness, and computational efficiency. By uniting a classic approach for surface normal integration with modern computational techniques, we construct a solver that fulfils these requirements. Building upon the Poisson integration model, we use an iterative Krylov subspace solver as a core step in tackling the task. While such a method can be very efficient, it may only show its full potential when combined with suitable numerical preconditioning and problem-specific initialisation. We perform a thorough numerical study in order to identify an appropriate preconditioner for this purpose. To provide suitable initialisation, we compute this initial state using a recently developed fast marching integrator. Detailed numerical experiments illustrate the benefits of this novel combination. In addition, we show on real-world photometric stereo datasets that the developed numerical framework is flexible enough to tackle modern computer vision applications. Y1 - 2017 U6 - https://doi.org/10.1007/s41095-016-0075-z SN - 2096-0433 VL - 3 IS - 2 SP - 107 EP - 129 ER - TY - GEN A1 - Hoeltgen, Laurent A1 - Peter, Pascal A1 - Breuß, Michael T1 - Clustering-based quantisation for PDE-based image compression T2 - Signal, Image and Video Processing N2 - Optimal known pixel data for inpainting in compression codecs based on partial differential equations is real-valued and thereby expensive to store. Thus, quantisation is required for efficient encoding. In this paper, we interpret the quantisation step as a clustering problem. Due to the global impact of each known pixel and correlations between spatial and tonal data, we investigate the central question, which kind of feature vectors should be used for clustering with popular strategies such as k-means. Our findings show that the number of colours can be reduced significantly without impacting the reconstruction quality. Surprisingly, these benefits are negated by an increased coding cost in compression applications. KW - Laplace interpolation KW - Inpainting KW - Compression KW - Quantisation KW - Clustering KW - Partial differential equations Y1 - 2018 U6 - https://doi.org/10.1007/s11760-017-1173-9 SN - 1863-1703 SN - 1863-1711 VL - 12 IS - 3 SP - 411 EP - 419 ER - TY - GEN A1 - Hoeltgen, Laurent A1 - Breuß, Michael A1 - Herold, Gert A1 - Sarradj, Ennes T1 - Sparse ℓ1 regularisation of matrix valued models for acoustic source characterisation T2 - Optimization and Engineering N2 - We present a strategy for the recovery of a sparse solution of a common problem in acoustic engineering, which is the reconstruction of sound source levels and locations applying microphone array measurements. The considered task bears similarities to the basis pursuit formalism but also relies on additional model assumptions that are challenging from a mathematical point of view. Our approach reformulates the original task as a convex optimisation model. The sought solution shall be a matrix with a certain desired structure. We enforce this structure through additional constraints. By combining popular splitting algorithms and matrix differential theory in a novel framework we obtain a numerically efficient strategy. Besides a thorough theoretical consideration we also provide an experimental setup that certifies the usability of our strategy. Finally, we also address practical issues, such as the handling of inaccuracies in the measurement and corruption of the given data. We provide a post processing step that is capable of yielding an almost perfect solution in such circumstances. KW - Convex optimisation KW - Sparse recovery KW - Split Bregman KW - Microphone array KW - Matrix differentiation KW - Acoustic source characterisation Y1 - 2018 U6 - https://doi.org/10.1007/s11081-017-9357-2 SN - 1389-4420 SN - 1573-2924 VL - 19 IS - 1 SP - 39 EP - 70 ER - TY - CHAP A1 - Bähr, Martin A1 - Dachsel, Robert A1 - Breuß, Michael ED - Welk, Martin ED - Urschler, Martin ED - Roth, Peter M. T1 - Fast Solvers for Solving Shape Matching by Time Integration T2 - Proceedings of the OAGM Workshop 2018 Medical Image Analysis, May 15 - 16, 2018, Hall/Tyrol, Austria N2 - The main task in three-dimensional non-rigid shape correspondence is to retrieve similarities between two or more similar three-dimensional objects. An important building block of many methods constructed to achieve this goal is a simplified shape representation called feature descriptor, which is invariant under almost isometric transformations. A recent feature descriptor relies on the full numerical integration of the geometric heat equation. This approach involves to solve a system of linear equations with multiple right-hand sides. To this end, it is necessary to find a fast and accurate numerical scheme in conjunction with the solution of a sparse linear system and many different right sides. In this paper we evaluate direct, iterative and model order reduction (MOR) methods and their influence to shape correspondence applications which will be validated on standard shape data sets with different resolutions. Y1 - 2018 SN - 978-3-85125-603-1 U6 - https://doi.org/10.3217/978-3-85125-603-1-14 SP - 65 EP - 72 PB - Verlag der TU Graz CY - Graz ER - TY - CHAP A1 - Dachsel, Robert A1 - Breuß, Michael A1 - Hoeltgen, Laurent ED - Welk, Martin ED - Urschler, Martin ED - Roth, Peter M. T1 - A Study of Spectral Expansion for Shape Correspondence T2 - Proceedings of the OAGM Workshop 2018 Medical Image Analysis, May 15 - 16, 2018, Hall/Tyrol, Austria N2 - The main task in three dimensional non-rigid shape correspondence is to retrieve similarities between two or more similar three dimensional objects. A useful way to tackle this problem is to construct a simplified shape representation, called feature descriptor, which is invariant under deformable transformations. A successful class of such feature descriptors is based on physical phenomena, concretely by the heat equation for the heat kernel signature and the Schrödinger equation for the wave kernel signature. Both approaches employ the spectral decomposition of the Laplace-Beltrami operator, meaning that solutions of the corresponding equations are expressed by a series expansion in terms of eigenfunctions. The feature descriptor is then computed at hand of those solutions. In this paper we explore the influence of the amount of used eigenfunctions on shape correspondence applications, as this is a crucial point with respect to accuracy and overall computational efficiency of the method. Our experimental study will be performed at hand of a standard shape data set. Y1 - 2018 SN - 978-3-85125-603-1 U6 - https://doi.org/10.3217/978-3-85125-603-1-15 SP - 73 EP - 79 PB - Verlag der TU Graz CY - Graz ER - TY - CHAP A1 - Sharifi Boroujerdi, Ali A1 - Khanian, Maryam A1 - Breuß, Michael T1 - Deep Interactive Region Segmentation and Captioning T2 - 2017 13th International Conference on Signal-Image Technology & Internet-Based Systems (SITIS), Jaipur, India N2 - Based on recent developments in dense image captioning, it is now possible to describe every object of a photographed scene with a caption while objects are determined by bounding boxes. However, the user interpretation of such an output is not trivial due to the existence of many overlapping bounding boxes. Furthermore, in current captioning frameworks, the user is not able to involve personal preferences to exclude areas that are out of interest. In this paper, we propose a novel hybrid deep learning architecture for interactive region segmentation and captioning whereby the user is able to specify an arbitrary region of the image that should be highlighted and described. To this end, we trained three different highly deep architectures on our special training data to identify the User Intention Region (UIR). In parallel, a dense image captioning model is utilized to locate all the objects of the scene by drawing bounding boxes and produce their linguistic descriptions. During our fusion approach, the detected UIR will be explained with the caption of the best match bounding box. To the best of our knowledge, this is the first work that provides such a comprehensive output. Our experiments show the superiority of the proposed approach over state-of-the-art interactive segmentation methods on several well-known segmentation benchmarks. In addition, replacement of the bounding boxes with the result of the interactive segmentation leads to a better understanding of the dense image captioning output as well as an enhancement in object localization accuracy. KW - Machine learning KW - deep learning Y1 - 2018 SN - 978-1-5386-4283-2 U6 - https://doi.org/10.1109/SITIS.2017.27 SP - 103 EP - 110 PB - IEEE CY - Piscataway, NJ ER - TY - CHAP A1 - Scheffler, Robert A1 - Mansouri Yarahmadi, Ashkan A1 - Breuß, Michael A1 - Köhler, Ekkehard ED - Pelillo, Marcello ED - Hancock, Edwin T1 - A Graph Theoretic Approach for Shape from Shading T2 - Energy minimization methods in computer vision and pattern recognition , 11th International Conference, EMMCVPR 2017, Venice, Italy, October 30 – November 1, 2017 N2 - Resolving ambiguities is a fundamental problem in shape from shading (SFS). The classic SFS approach allows to reconstruct the surface locally around singular points up to an ambiguity of convex, concave or saddle point type. In this paper we follow a recent approach that seeks to resolve the local ambiguities in a global graph-based setting so that the complete surface reconstruction is consistent. To this end, we introduce a novel graph theoretic formulation for the underlying problem that allows to prove for the first time in the literature that the underlying surface orientation problem is NP-complete. Moreover, we show that our novel framework allows to define an algorithmic framework that solves the disambiguation problem. It makes use of cycle bases for dealing with the graph construction and enables an easy embedding into an optimization method that amounts here to a linear program. KW - Shape from shading KW - Ambiguity KW - Configuration graph KW - Cycle basis Y1 - 2018 SN - 978-3-319-78198-3 U6 - https://doi.org/10.1007/978-3-319-78199-0_22 SP - 328 EP - 341 PB - Springer CY - Cham ER - TY - GEN A1 - Khanian, Maryam A1 - Sharifi Boroujerdi, Ali A1 - Breuß, Michael T1 - Photometric stereo for strong specular highlights T2 - Computational Visual Media N2 - Photometric stereo is a fundamental technique in computer vision known to produce 3D shape with high accuracy. It uses several input images of a static scene taken from one and the same camera position but under varying illumination. The vast majority of studies in this 3D reconstruction method assume orthographic projection for the camera model. In addition, they mainly use the Lambertian reflectance model as the way that light scatters at surfaces. Thus, providing reliable photometric stereo results from real world objects still remains a challenging task. We address 3D reconstruction by use of a more realistic set of assumptions, combining for the first time the complete Blinn–Phong reflectance model and perspective projection. Furthermore, we compare two different methods of incorporating the perspective projection into our model. Experiments are performed on both synthetic and real world images; the latter do not benefit from laboratory conditions. The results show the high potential of our method even for complex real world applications such as medical endoscopy images which may include many specular highlights. KW - photometric stereo (PS) KW - complete Blinn–Phong model KW - perspective projection KW - diffuse reflection KW - specular reflection Y1 - 2018 U6 - https://doi.org/10.1007/s41095-017-0101-9 SN - 2096-0433 SN - 2096-0662 VL - 4 IS - 1 SP - 83 EP - 102 ER - TY - GEN A1 - Maurer, Daniel A1 - Ju, Yong Chul A1 - Breuß, Michael A1 - Bruhn, Andrés T1 - Combining Shape from Shading and Stereo: A Joint Variational Method for Estimating Depth, Illumination and Albedo T2 - International Journal of Computer Vision N2 - Shape from shading (SfS) and stereo are two fundamentally different strategies for image-based 3-D reconstruction. While approaches for SfS infer the depth solely from pixel intensities, methods for stereo are based on a matching process that establishes correspondences across images. This difference in approaching the reconstruction problem yields complementary advantages that are worthwhile being combined. So far, however, most “joint” approaches are based on an initial stereo mesh that is subsequently refined using shading information. In this paper we follow a completely different approach. We propose a joint variational method that combines both cues within a single minimisation framework. To this end, we fuse a Lambertian SfS approach with a robust stereo model and supplement the resulting energy functional with a detail-preserving anisotropic second-order smoothness term. Moreover, we extend the resulting model in such a way that it jointly estimates depth, albedo and illumination. This in turn makes the approach applicable to objects with non-uniform albedo as well as to scenes with unknown illumination. Experiments for synthetic and real-world images demonstrate the benefits of our combined approach: They not only show that our method is capable of generating very detailed reconstructions, but also that joint approaches are feasible in practice. KW - Stereo reconstruction KW - Shape from Shading KW - Variational methods KW - Illumination estimation KW - Albedo estimation KW - Joint reasoning Y1 - 2018 U6 - https://doi.org/10.1007/s11263-018-1079-1 SN - 0920-5691 SN - 1573-1405 VL - 126 IS - 12 SP - 1342 EP - 1366 ER - TY - GEN A1 - Breuß, Michael A1 - Kemm, Friedemann A1 - Vogel, Oliver T1 - A numerical study of Newton interpolation with extremely high degrees T2 - Kybernetika : international journal published by Institute of Information Theory and Automation N2 - In current textbooks the use of Chebyshev nodes with Newton interpolation is advocated as the most efficient numerical interpolation method in terms of approximation accuracy and computational effort. However, we show numerically that the approximation quality obtained by Newton interpolation with Fast Leja (FL) points is competitive to the use of Chebyshev nodes, even for extremely high degree interpolation. This is an experimental account of the analytic result that the limit distribution of FL points and Chebyshev nodes is the same when letting the number of points go to infinity. Since the FL construction is easy to perform and allows to add interpolation nodes on the fly in contrast to the use of Chebyshev nodes, our study suggests that Newton interpolation with FL points is currently the most efficient numerical technique for polynomial interpolation. Moreover, we give numerical evidence that any reasonable function can be approximated up to machine accuracy by Newton interpolation with FL points if desired, which shows the potential of this method. KW - polynomial interpolation KW - Newton interpolation KW - interpolation nodes KW - Chebyshev nodes KW - Leja ordering KW - fast Leja points Y1 - 2018 U6 - https://doi.org/10.14736/kyb-2018-2-0279 SN - 0023-5954 VL - 54 IS - 2 SP - 279 EP - 288 ER - TY - GEN A1 - Hoeltgen, Laurent A1 - Breuß, Michael A1 - Radow, Georg ED - Lellmann, Jan ED - Burger, Martin ED - Modersitzki, Jan T1 - Towards PDE-Based Video Compression with Optimal Masks and Optic Flow T2 - Scale Space and Variational Methods in Computer Vision : 7th International Conference, SSVM 2019, Hofgeismar, Germany, June 30 – July 4, 2019, Proceedings 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 optimal data. For the possible extension to video compression, the 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. 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. We show that the method preserves a reasonable quality in the reconstruction, and is very robust against errors in the flow field. KW - Partial differential equations Inpainting Laplace interpolation Optic flow Video reconstruction Y1 - 2019 SN - 978-3-030-22367-0 SN - 978-3-030-22368-7 U6 - https://doi.org/10.1007/978-3-030-22368-7_7 SP - 79 EP - 91 PB - Springer CY - Cham ER - TY - GEN A1 - Radow, Georg A1 - Breuß, Michael ED - Vento, Mario ED - Percannella, Gennaro T1 - Variational Optical Flow: Warping and Interpolation Revisited T2 - Computer Analysis of Images and Patterns 18th International Conference, CAIP 2019, Salerno, Italy, September 3–5, 2019, Proceedings, Part I N2 - One of the fundamental problems in computer vision is to attain the apparent motion in image sequences, the optical flow. As evaluations at hand of recent benchmarks show, this field is highly competitive. High ranking variational methods often consist of a combination of techniques, where frequently the presentation in the literature focuses on novel contributions in modelling. In this paper we investigate the warping technique and related algorithmic design choices that are fundamental for practical implementation. At hand of a detailed yet straightforward derivation we discuss different warping variations. These are evaluated in numerical experiments, and furthermore we investigate the impact of a variety of interpolation methods that can be used. KW - Optical flow Variational methods Warping Interpolation B-splines Y1 - 2019 SN - 978-3-030-29887-6 U6 - https://doi.org//10.1007/978-3-030-29888-3_33 SN - 978-3-030-29888-3 SP - 409 EP - 420 PB - Springer CY - Cham ER - TY - GEN A1 - Bähr, Martin A1 - Radow, Georg A1 - Breuß, Michael A1 - Fügenschuh, Armin ED - Fortz, Bernard ED - Labbé, Martine T1 - Computation of Stable Honeycomb Structures for Additive Manufacturing T2 - Operations Research Proceedings 2018, Selected Papers of the Annual International Conference of the German Operations Research Society (GOR), Brussels, Belgium, September 12-14, 2018 N2 - In certain additive manufacturing processes of industrial interest, the task arises to build up structures layer-wise in a purely vertical manner. The question arises how to construct such a structure in a convenient way so that it is structurally as stable as possible. In this paper, 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 realisations. For computing the incorporated Voronoi tessellation we consider the use of a Delaunay triangulation or the Eikonal equation. As a main point of our paper, we give a comparison of these two methods. We show that one can make use of the arising graph of the honeycomb structure as input for a specific routing scheme that enhances printability when the printing material stays soft for some time during the printing process. KW - Additive manufacturing Centroidal, Voronoi tessellation, Energy minimisation, Geometric optimisation, Eikonal equation, Fast marching Y1 - 2019 SN - 978-3-030-18499-5 U6 - https://doi.org/10.1007/978-3-030-18500-8_45 SN - 978-3-030-18500-8 SP - 363 EP - 369 PB - Springer CY - Cham ER - 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 -