Computing the output distribution and selection probabilities of a stack filter from the DNF of its positive Boolean function

Many nonlinear filters used in practise are stack filters. An algorithm is presented which calculates the output distribution of an arbitrary stack filter S from the disjunctive normal form (DNF) of its underlying positive Boolean function. The so called selection probabilities can be computed along the way.Comment: This is the version published in Journal of Mathematical Imaging and Vision, online first, 1 august 201

A Sparse Multi-Scale Algorithm for Dense Optimal Transport

Discrete optimal transport solvers do not scale well on dense large problems since they do not explicitly exploit the geometric structure of the cost function. In analogy to continuous optimal transport we provide a framework to verify global optimality of a discrete transport plan locally. This allows construction of an algorithm to solve large dense problems by considering a sequence of sparse problems instead. The algorithm lends itself to being combined with a hierarchical multi-scale scheme. Any existing discrete solver can be used as internal black-box.Several cost functions, including the noisy squared Euclidean distance, are explicitly detailed. We observe a significant reduction of run-time and memory requirements.Comment: Published "online first" in Journal of Mathematical Imaging and Vision, see DO

Digital shy maps

[EN] We study properties of shy maps in digital topology.Boxer, L. (2017). Digital shy maps. Applied General Topology. 18(1):143-152. doi:10.4995/agt.2017.6663.SWORD143152181C. Berge, Graphs and Hypergraphs, 2nd edition, North-Holland, Amsterdam, 1976. https://doi.org/10.1016/0167-8655(94)90012-4Boxer, L. (1994). Digitally continuous functions. Pattern Recognition Letters, 15(8), 833-839. doi:10.1016/0167-8655(94)90012-4L. Boxer, A classical construction for the digital fundamental group, Pattern Recognition Letters 10 (1999), 51-62. https://doi.org/10.1007/s10851-005-4780-y https://doi.org/10.1007/s10851-006-9698-5Boxer, L. (2005). Properties of Digital Homotopy. Journal of Mathematical Imaging and Vision, 22(1), 19-26. doi:10.1007/s10851-005-4780-yBoxer, L. (2006). Digital Products, Wedges, and Covering Spaces. Journal of Mathematical Imaging and Vision, 25(2), 159-171. doi:10.1007/s10851-006-9698-5L. Boxer, Remarks on digitally continuous multivalued functions, Journal of Advances in Mathematics 9, no. 1 (2014), 1755-1762.L. Boxer and I. Karaca, Fundamental groups for digital products, Advances and Applications in Mathematical Sciences 11, no. 4 (2012), 161-180.Boxer, L., & Staecker, P. C. (2016). Connectivity Preserving Multivalued Functions in Digital Topology. Journal of Mathematical Imaging and Vision, 55(3), 370-377. doi:10.1007/s10851-015-0625-5Escribano, C., Giraldo, A., & Sastre, M. A. (s. f.). Digitally Continuous Multivalued Functions. Lecture Notes in Computer Science, 81-92. doi:10.1007/978-3-540-79126-3_9Escribano, C., Giraldo, A., & Sastre, M. A. (2011). Digitally Continuous Multivalued Functions, Morphological Operations and Thinning Algorithms. Journal of Mathematical Imaging and Vision, 42(1), 76-91. doi:10.1007/s10851-011-0277-zGiraldo, A., & Sastre, M. A. (2015). On the Composition of Digitally Continuous Multivalued Functions. Journal of Mathematical Imaging and Vision, 53(2), 196-209. doi:10.1007/s10851-015-0570-3HAN, S. (2005). Non-product property of the digital fundamental group. Information Sciences, 171(1-3), 73-91. doi:10.1016/j.ins.2004.03.018V. A. Kovalevsky, A new concept for digital geometry, shape in picture, Springer, New York (1994). https://doi.org/10.1016/0167-8655(86)90017-6Rosenfeld, A. (1986). ‘Continuous’ functions on digital pictures. Pattern Recognition Letters, 4(3), 177-184. doi:10.1016/0167-8655(86)90017-6Tsaur, R., & Smyth, M. B. (2001). «Continuous» Multifunctions in Discrete Spaces with Applications to Fixed Point Theory. Lecture Notes in Computer Science, 75-88. doi:10.1007/3-540-45576-0_

On a fast bilateral filtering formulation using functional rearrangements

We introduce an exact reformulation of a broad class of neighborhood filters, among which the bilateral filters, in terms of two functional rearrangements: the decreasing and the relative rearrangements. Independently of the image spatial dimension (one-dimensional signal, image, volume of images, etc.), we reformulate these filters as integral operators defined in a one-dimensional space corresponding to the level sets measures. We prove the equivalence between the usual pixel-based version and the rearranged version of the filter. When restricted to the discrete setting, our reformulation of bilateral filters extends previous results for the so-called fast bilateral filtering. We, in addition, prove that the solution of the discrete setting, understood as constant-wise interpolators, converges to the solution of the continuous setting. Finally, we numerically illustrate computational aspects concerning quality approximation and execution time provided by the rearranged formulation.Comment: 29 pages, Journal of Mathematical Imaging and Vision, 2015. arXiv admin note: substantial text overlap with arXiv:1406.712

Fundamental groups and Euler characteristics of sphere-like digital images

[EN] The current paper focuses on fundamental groups and Euler characteristics of various digital models of the 2-dimensional sphere. For all models that we consider, we show that the fundamental groups are trivial, and compute the Euler characteristics (which are not always equal). We consider the connected sum of digital surfaces and investigate how this operation relates to the fundamental group and Euler characteristic. We also consider two related but dierent notions of a digital image having "no holes," and relate this to the triviality of the fundamental group. Many of our results have origins in the paper [15] by S.-E. Han, which contains many errors. We correct these errors when possible, and leave some open questions. We also present some original results.Boxer, L.; Staecker, PC. (2016). Fundamental groups and Euler characteristics of sphere-like digital images. Applied General Topology. 17(2):139-158. doi:10.4995/agt.2016.4624.SWORD139158172Boxer, L. (1994). Digitally continuous functions. Pattern Recognition Letters, 15(8), 833-839. doi:10.1016/0167-8655(94)90012-4Boxer, L. (2005). Properties of Digital Homotopy. Journal of Mathematical Imaging and Vision, 22(1), 19-26. doi:10.1007/s10851-005-4780-yBoxer, L. (2006). Homotopy Properties of Sphere-Like Digital Images. Journal of Mathematical Imaging and Vision, 24(2), 167-175. doi:10.1007/s10851-005-3619-xBoxer, L. (2006). Digital Products, Wedges, and Covering Spaces. Journal of Mathematical Imaging and Vision, 25(2), 159-171. doi:10.1007/s10851-006-9698-5Boxer, L. (2010). Continuous Maps on Digital Simple Closed Curves. Applied Mathematics, 01(05), 377-386. doi:10.4236/am.2010.15050Chen, L., & Zeng, T. (2014). A Convex Variational Model for Restoring Blurred Images with Large Rician Noise. Journal of Mathematical Imaging and Vision, 53(1), 92-111. doi:10.1007/s10851-014-0551-yHan, S.-E. (2007). Digital fundamental group and Euler characteristic of a connected sum of digital closed surfaces. Information Sciences, 177(16), 3314-3326. doi:10.1016/j.ins.2006.12.013Han, S.-E. (2008). Equivalent (k0,k1)-covering and generalized digital lifting. Information Sciences, 178(2), 550-561. doi:10.1016/j.ins.2007.02.004Kong, T. Y. (1989). A digital fundamental group. Computers & Graphics, 13(2), 159-166. doi:10.1016/0097-8493(89)90058-7Rosenfeld, A. (1979). Digital Topology. The American Mathematical Monthly, 86(8), 621. doi:10.2307/2321290Rosenfeld, A. (1986). ‘Continuous’ functions on digital pictures. Pattern Recognition Letters, 4(3), 177-184. doi:10.1016/0167-8655(86)90017-