@misc{WelkBreussSridhar, author = {Welk, Martin and Breuß, Michael and Sridhar, Vivek}, title = {Matrix Morphology with Extremum Principle}, series = {Mathematical Morphology and Its Applications to Signal and Image Processing : 14th International Symposium, ISMM 2019, Saarbr{\"u}cken, Germany, July 8-10, 2019, Proceedings}, journal = {Mathematical Morphology and Its Applications to Signal and Image Processing : 14th International Symposium, ISMM 2019, Saarbr{\"u}cken, Germany, July 8-10, 2019, Proceedings}, publisher = {Springer}, address = {Cham}, isbn = {978-3-030-20866-0}, doi = {10.1007/978-3-030-20867-7_14}, pages = {177 -- 188}, abstract = {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.}, language = {en} } @misc{SridharBreussKahra, author = {Sridhar, Vivek and Breuß, Michael and Kahra, Marvin}, title = {Fast Approximation of Color Morphology}, series = {Advances in Visual Computing : 16th International Symposium, ISVC 2021, Virtual Event, October 4-6, 2021, Proceedings, Part II}, journal = {Advances in Visual Computing : 16th International Symposium, ISVC 2021, Virtual Event, October 4-6, 2021, Proceedings, Part II}, editor = {Bebis, George}, publisher = {Springer}, address = {Cham}, isbn = {978-3-030-90435-7}, doi = {10.1007/978-3-030-90436-4_39}, pages = {488 -- 499}, abstract = {The basic filters in mathematical morphology are dilation and erosion. They are defined by a flat or non-flat structuring element that is usually shifted pixel-wise over an image and a comparison process that takes place within the corresponding mask. The algorithmic complexity of fast algorithms that realise dilation and erosion for color images usually depends on size and shape of the structuring element. In this paper we propose and investigate an easy and fast way to make use of the fast Fourier transform for an approximate computation of dilation and erosion for color images. Similarly in construction as many other fast algorithms, the method extends a recent scheme proposed for single-channel filtering. It is by design highly flexible, as it can be used with flat and non-flat structuring elements of any size and shape. Moreover, its complexity only depends on the number of pixels in the filtered images. We analyse here some important aspects of the approximation, and we show experimentally that we obtain results of very reasonable quality while the method has very attractive computational properties.}, language = {en} } @misc{SridharBreuss, author = {Sridhar, Vivek and Breuß, Michael}, title = {Sampling of Non-flat Morphology for Grey Value Images}, series = {Computer Analysis of Images and Patterns}, volume = {2021}, journal = {Computer Analysis of Images and Patterns}, publisher = {Springer}, address = {Cham}, isbn = {978-3-030-89130-5}, issn = {978-3-030-89131-2}, doi = {10.1007/978-3-030-89131-2_8}, pages = {88 -- 97}, abstract = {Sampling is a basic operation in image processing. In previous literature, a morphological sampling theorem has been established showing how sampling interacts with image reconstruction by morphological operations. However, while many aspects of morphological sampling have been investigated for binary images in classic works, only some of them have been extended to grey scale imagery. Especially, previous attempts to study the relation between sampling and grey scale morphology are restricted by construction to flat morphological filters. In order to establish a sampling theory for non-flat morphology, we establish an alternative definition for grey scale opening and closing relying on the umbra notion. Making use of this, we prove a sampling theorem about the interaction of sampling with fundamental morphological operations for non-flat morphology. This allows to make precise corresponding relations between sampling and image reconstruction, extending classic results for flat morphology of grey value images.}, language = {en} } @misc{KahraSridharBreuss, author = {Kahra, Marvin and Sridhar, Vivek and Breuß, Michael}, title = {Fast Morphological Dilation and Erosion for Grey Scale Images Using the Fourier Transform}, series = {Scale Space and Variational Methods in Computer Vision}, journal = {Scale Space and Variational Methods in Computer Vision}, publisher = {Springer}, address = {Cham}, isbn = {978-3-030-75548-5}, issn = {978-3-030-75549-2}, doi = {10.1007/978-3-030-75549-2_6}, pages = {65 -- 77}, abstract = {The basic filters in mathematical morphology are dilation and erosion. They are defined by a flat or non-flat structuring element that is usually shifted pixel-wise over an image and a comparison process that takes place within the corresponding mask. Existing fast algorithms that realise dilation and erosion for grey value images are often limited with respect to size or shape of the structuring element. Usually their algorithmic complexity depends on these aspects. Many fast methods only address flat morphology. In this paper we propose a novel way to make use of the fast Fourier transform for the computation of dilation and erosion. Our method is by design highly flexible, as it can be used with flat and non-flat structuring elements of any size and shape. Moreover, its complexity does not depend on size or shape of the structuring element, but only on the number of pixels in the filtered images. We show experimentally that we obtain results of very reasonable quality with the proposed method.}, language = {en} } @misc{SridharBreuss, author = {Sridhar, Vivek and Breuß, Michael}, title = {An Exact Fast Fourier Method for Morphological Dilation and Erosion Using the Umbra Technique}, series = {19th Conference on Robots and Vision (CRV), 2022}, journal = {19th Conference on Robots and Vision (CRV), 2022}, publisher = {IEEE}, isbn = {978-1-6654-9774-9}, doi = {10.1109/CRV55824.2022.00032}, pages = {190 -- 196}, abstract = {In this paper we consider the fundamental operations dilation and erosion of mathematical morphology. It is well known that many powerful image filtering operations can be constructed by their combinations. We propose a fast and novel algorithm based on the Fast Fourier Transform to compute grey-value morphological operations on an image. The novel method may deal with non-flat filters and incorporates no restrictions on shape and size of the filtering window, in contrast to many other fast methods in the field. Unlike fast Fourier techniques from previous works, the novel method gives exact results and is not an approximation. The key aspect which allows to achieve this is to explore here for the first time in this context the umbra formulation of images and filters. We show that the new method is in practice particularly suitable for filtering images with small tonal range or when employing large filter sizes.}, language = {en} } @incollection{SridharBreuss, author = {Sridhar, Vivek and Breuß, Michael}, title = {A Novel Approach for Computation of Morphological Operations Using the Number Theoretic Transform}, series = {Discrete Geometry and Mathematical Morphology : Second International Joint Conference, DGMM 2022, Strasbourg, France, October 24-27, 2022, Proceedings}, booktitle = {Discrete Geometry and Mathematical Morphology : Second International Joint Conference, DGMM 2022, Strasbourg, France, October 24-27, 2022, Proceedings}, publisher = {Springer}, address = {Cham}, isbn = {978-3-031-19896-0}, issn = {0302-9743}, doi = {10.1007/978-3-031-19897-7_15}, pages = {182 -- 192}, abstract = {The fundamental operations of mathematical morphology are dilation and erosion. They are often implemented using a sliding window with the purpose to compute maximum respectively minimum of pixel values within the corresponding mask. We reformulate the problem of morphological dilation respectively erosion of an image with a non-flat filter as a convolution of their umbras. To this end, we propose to make use of the number theoretic transform to compute the convolution in this setting. In contrast to other possible schemes, this transform represents a completely discrete computational approach. It allows exact convolution of sequences made up of integers. Therefore we propose by the combination of umbra framework and number theoretic transform a well-engineered combination. There is no restriction on size or shape of the structuring element, and also flat and non-flat filters can be realised.}, language = {en} } @misc{SridharBreuss, author = {Sridhar, Vivek and Breuß, Michael}, title = {Morphological Sampling Theorem and its Extension to Grey-value Images}, series = {arXiv}, journal = {arXiv}, doi = {10.48550/arXiv.2305.13279}, pages = {1 -- 24}, abstract = {Sampling is a basic operation in image processing. In classic literature, a morphological sampling theorem has been established, which shows how sampling interacts by morphological operations with image reconstruction. Many aspects of morphological sampling have been investigated for binary images, but only some of them have been explored for grey-value imagery. With this paper, we make a step towards completion of this open matter. By relying on the umbra notion, we show how to transfer classic theorems in binary morphology about the interaction of sampling with the fundamental morphological operations dilation, erosion, opening and closing, to the grey-value setting. In doing this we also extend the theory relating the morphological operations and corresponding reconstructions to use of non-flat structuring elements. We illustrate the theoretical developments at hand of examples.}, language = {en} } @misc{SridharShahinBreussetal., author = {Sridhar, Vivek and Shahin, Keyvan and Breuß, Michael and Reichenbach, Marc}, title = {The Polynomial Connection between Morphological Dilation and Discrete Convolution}, series = {arXiv}, journal = {arXiv}, doi = {10.48550/arXiv.2305.03018}, pages = {1 -- 11}, language = {en} }