@inproceedings{PeterHoffmannNedwedetal., author = {Peter, Pascal and Hoffmann, Sebastian and Nedwed, Frank and Hoeltgen, Laurent and Weickert, Joachim}, title = {From Optimised Inpainting with Linear PDEs Towards Competitive Image Compression Codecs}, series = {Image and Video Technology, 7th Pacific-Rim Symposium, PSIVT 2015, Auckland, New Zealand, November 25-27, 2015}, booktitle = {Image and Video Technology, 7th Pacific-Rim Symposium, PSIVT 2015, Auckland, New Zealand, November 25-27, 2015}, publisher = {Springer}, address = {Cham}, isbn = {978-3-319-29450-6}, doi = {10.1007/978-3-319-29451-3_6}, pages = {63 -- 74}, abstract = {For inpainting with linear partial differential equations (PDEs) such as homogeneous or biharmonic diffusion, sophisticated data optimisation strategies have been found recently. These allow high-quality reconstructions from sparse known data. While they have been explicitly developed with compression in mind, they have not entered actual codecs so far: Storing these optimised data efficiently is a nontrivial task. Since this step is essential for any competetive codec, we propose two new compression frameworks for linear PDEs: Efficient storage of pixel locations obtained from an optimal control approach, and a stochastic strategy for a locally adaptive, tree-based grid. Suprisingly, our experiments show that homogeneous diffusion inpainting can surpass its often favoured biharmonic counterpart in compression. Last but not least, we demonstrate that our linear approach is able to beat both JPEG2000 and the nonlinear state-of-the-art in PDE-based image compression.}, language = {en} } @inproceedings{RadowBreussHoeltgenetal., author = {Radow, Georg and Breuß, Michael and Hoeltgen, Laurent and Fischer, Thomas}, title = {Optimised Anisotropic Poisson Denoising}, series = {Image Analysis, 20th Scandinavian Conference, SCIA 2017, Troms{\o}, Norway, June 12-14, 2017, Proceedings, Part I}, booktitle = {Image Analysis, 20th Scandinavian Conference, SCIA 2017, Troms{\o}, Norway, June 12-14, 2017, Proceedings, Part I}, editor = {Sharma, Puneet and Filippo, Maria Bianchi}, publisher = {Springer}, address = {Cham}, isbn = {978-3-319-59126-1}, doi = {10.1007/978-3-319-59126-1_42}, pages = {502 -- 514}, language = {en} } @inproceedings{DachselBreussHoeltgen, author = {Dachsel, Robert and Breuß, Michael and Hoeltgen, Laurent}, title = {The Classic Wave Equation Can Do Shape Correspondence}, series = {Computer Analysis of Images and Patterns, CAIP International Conference on Computer Analysis of Images and Patterns, Ystad,Sweden, 2017}, booktitle = {Computer Analysis of Images and Patterns, CAIP International Conference on Computer Analysis of Images and Patterns, Ystad,Sweden, 2017}, editor = {Felsberg, Michael and Heyden, Andreas and Kr{\"u}ger, Norbert}, publisher = {Springer International Publishing}, address = {Cham}, isbn = {978-3-319-64688-6}, doi = {10.1007/978-3-319-64689-3_22}, pages = {264 -- 275}, abstract = {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 Schr{\"o}dinger 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.}, language = {en} } @misc{Hoeltgen, author = {Hoeltgen, Laurent}, title = {Understanding image inpainting with the help of the Helmholtz equation}, series = {Mathematical Sciences}, volume = {11}, journal = {Mathematical Sciences}, number = {1}, issn = {2008-1359}, doi = {10.1007/s40096-017-0207-3}, pages = {73 -- 77}, abstract = {Partial differential equations have recently been used for image compression purposes. One of the most successful frameworks solves the Laplace equation using a weighting scheme to determine the importance of individual pixels. We provide a physical interpretation of this approach in terms of the Helmholtz equation which explains its superiority. For better reconstruction quality, we subsequently formulate an optimisation task for the corresponding finite difference discretisation to maximise the influence of the physical traits of the Helmholtz equation. Our findings show that sharper contrasts and lower errors in the reconstruction are possible.}, language = {en} } @misc{HoeltgenPeterBreuss, author = {Hoeltgen, Laurent and Peter, Pascal and Breuß, Michael}, title = {Clustering-based quantisation for PDE-based image compression}, series = {Signal, Image and Video Processing}, volume = {12}, journal = {Signal, Image and Video Processing}, number = {3}, issn = {1863-1703}, doi = {10.1007/s11760-017-1173-9}, pages = {411 -- 419}, abstract = {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.}, language = {en} } @misc{HoeltgenBreussHeroldetal., author = {Hoeltgen, Laurent and Breuß, Michael and Herold, Gert and Sarradj, Ennes}, title = {Sparse ℓ1 regularisation of matrix valued models for acoustic source characterisation}, series = {Optimization and Engineering}, volume = {19}, journal = {Optimization and Engineering}, number = {1}, issn = {1389-4420}, doi = {10.1007/s11081-017-9357-2}, pages = {39 -- 70}, abstract = {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.}, language = {en} } @inproceedings{DachselBreussHoeltgen, author = {Dachsel, Robert and Breuß, Michael and Hoeltgen, Laurent}, title = {A Study of Spectral Expansion for Shape Correspondence}, series = {Proceedings of the OAGM Workshop 2018 Medical Image Analysis, May 15 - 16, 2018, Hall/Tyrol, Austria}, booktitle = {Proceedings of the OAGM Workshop 2018 Medical Image Analysis, May 15 - 16, 2018, Hall/Tyrol, Austria}, editor = {Welk, Martin and Urschler, Martin and Roth, Peter M.}, publisher = {Verlag der TU Graz}, address = {Graz}, isbn = {978-3-85125-603-1}, doi = {10.3217/978-3-85125-603-1-15}, pages = {73 -- 79}, abstract = {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 Schr{\"o}dinger 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.}, language = {en} } @inproceedings{BreussHoeltgenKleefeld, author = {Breuß, Michael and Hoeltgen, Laurent and Kleefeld, Andreas}, title = {Matrix-Valued Levelings for Colour Images}, series = {Mathematical morphology and its applications to signal and image processing, 13th international symposium, ISMM 2017, Fontainebleau, France, May 15-17, 2017, proceedings}, booktitle = {Mathematical morphology and its applications to signal and image processing, 13th international symposium, ISMM 2017, Fontainebleau, France, May 15-17, 2017, proceedings}, publisher = {Springer International Publishing}, address = {Cham}, isbn = {978-3-319-57239-0}, doi = {10.1007/978-3-319-57240-6_24}, pages = {296 -- 308}, abstract = {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.}, language = {en} } @inproceedings{HoeltgenHarrisBreussetal., author = {Hoeltgen, Laurent and Harris, I. and Breuß, Michael and Kleefeld, Andreas}, title = {Analytic Existence and Uniqueness Results for PDE-Based Image Reconstruction with the Laplacian}, series = {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}, booktitle = {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}, publisher = {Springer International Publishing}, address = {Cham}, isbn = {978-3-319-58770-7}, doi = {10.1007/978-3-319-58771-4_6}, pages = {66 -- 79}, abstract = {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.}, language = {en} } @inproceedings{DachselBreussHoeltgen, author = {Dachsel, Robert and Breuß, Michael and Hoeltgen, Laurent}, title = {Shape Matching by Time Integration of Partial Differential Equations}, series = {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}, booktitle = {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}, publisher = {Springer International Publishing}, address = {Cham}, isbn = {978-3-319-58770-7}, doi = {10.1007/978-3-319-58771-4_53}, pages = {669 -- 680}, abstract = {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.}, language = {en} } @inproceedings{HoeltgenBreuss, author = {Hoeltgen, Laurent and Breuß, Michael}, title = {Efficient Co-Domain Quantisation for PDE-Based Image Compression}, series = {Proceedings of the Conference Algoritmy 2016, 20th Conference on Scientific Computing, Vysok{\´e} Tatry - Podbansk{\´e}, Slovakia, March 14 - 18}, booktitle = {Proceedings of the Conference Algoritmy 2016, 20th Conference on Scientific Computing, Vysok{\´e} Tatry - Podbansk{\´e}, Slovakia, March 14 - 18}, editor = {Handlovičov{\´a}, A. and Ševčovič, D.}, publisher = {Publishing House of Slovak University of Technology in Bratislava}, address = {Bratislava}, isbn = {978-80-227-4544-4}, pages = {194 -- 203}, language = {en} } @misc{BreussHoeltgenRadow, author = {Breuß, Michael and Hoeltgen, Laurent and Radow, Georg}, title = {Towards PDE-Based Video Compression with Optimal Masks Prolongated by Optic Flow}, series = {Journal of Mathematical Imaging and Vision}, volume = {63}, journal = {Journal of Mathematical Imaging and Vision}, number = {2}, issn = {1573-7683}, doi = {10.1007/s10851-020-00973-6}, pages = {144 -- 156}, abstract = {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.}, language = {en} } @misc{RadowHoeltgenQueauetal., author = {Radow, Georg and Hoeltgen, Laurent and Qu{\´e}au, Yvain and Breuß, Michael}, title = {Optimisation of Classic Photometric Stereo by Non-convex Variational Minimisation}, series = {Journal of Mathematical Imaging and Vision}, volume = {61}, journal = {Journal of Mathematical Imaging and Vision}, number = {1}, issn = {1573-7683}, doi = {10.1007/s10851-018-0828-7}, pages = {84 -- 105}, abstract = {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.}, language = {en} } @misc{HoeltgenKleefeldHarriesetal., author = {Hoeltgen, Laurent and Kleefeld, Andreas and Harries, Isaac and Breuß, Michael}, title = {Theoretical foundation of the weighted laplace inpainting problem}, series = {Journal of Applications of Mathematics}, volume = {64}, journal = {Journal of Applications of Mathematics}, number = {3}, issn = {1572-9109}, doi = {10.21136/AM.2019.0206-18}, pages = {281 -- 300}, abstract = {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.}, language = {en} } @misc{HoeltgenBreussRadow, author = {Hoeltgen, Laurent and Breuß, Michael and Radow, Georg}, title = {Towards PDE-Based Video Compression with Optimal Masks and Optic Flow}, series = {Scale Space and Variational Methods in Computer Vision : 7th International Conference, SSVM 2019, Hofgeismar, Germany, June 30 - July 4, 2019, Proceedings}, journal = {Scale Space and Variational Methods in Computer Vision : 7th International Conference, SSVM 2019, Hofgeismar, Germany, June 30 - July 4, 2019, Proceedings}, editor = {Lellmann, Jan and Burger, Martin and Modersitzki, Jan}, publisher = {Springer}, address = {Cham}, isbn = {978-3-030-22367-0}, doi = {10.1007/978-3-030-22368-7_7}, pages = {79 -- 91}, abstract = {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.}, language = {en} } @misc{HoeltgenQueauBreussetal., author = {Hoeltgen, Laurent and Queau, Yvain and Breuß, Michael and Radow, Georg}, title = {Optimised photometric stereo via non-convex variational minimisation}, publisher = {BMVA Press}, address = {York, UK}, pages = {1}, language = {en} }