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The many benefits of putting stack filters into disjunctive or conjunctive normal form

Marcel Wild
<span title="">2005</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lx7dev2le5anbg6oarljwh7lie" style="color: black;">Discrete Applied Mathematics</a> </i> &nbsp;
Many important properties of stack filters (idempotency, co-idempotency, order relations) can be tested in polynomial time if the DNF and/or CNF of b are known.  ...  Stack filters are nonlinear filters used for image processing (examples: median filters, order statistics).  ...  When is a min-max operator then the topical operator I − is never a min-max operator, but again there is a polynomial algorithm to decide its idempotency (Section 4.2). Theorem 5.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.dam.2004.06.027">doi:10.1016/j.dam.2004.06.027</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6jvv5xrdqncljemqf2m4vkbhgq">fatcat:6jvv5xrdqncljemqf2m4vkbhgq</a> </span>
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Page 2440 of Mathematical Reviews Vol. , Issue 2004c [page]

<span title="">2004</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Ivan Izmestiev (Berlin) 2004c:68153 68U10 06D99 06E30 06F20 Wild, Marcel (SA-STEL; Stellenbosch) Idempotent and co-idempotent stack filters and min-max operators. (English summary) Theoret. Comput.  ...  Then combinatorial criteria on the sets @(p) are given in order to characterize the idempotence of a flat operator ‘P, and its “co-idempotence”, in other words the idempotence of id-‘¥, where id is the  ...
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LULU Theory, Idempotent Stack Filters, and the Mathematics of Vision of Marr [chapter]

Carl H. Rohwer, Marcel Wild
<span title="">2007</span> <i title="Elsevier"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ksd7mieepvestjxbiecsscpfdq" style="color: black;">Advances in Imaging and Electron Physics</a> </i> &nbsp;
Scheidt and E.G. Rohwer for the background and apportunity to apply our ideas.  ...  ACKNOWLEDGMENTS We authors thank Lauretta Adams for excellent word processing and J.P. du Toit for assistance with artwork, and the application in Image Analysis. For the latter, we also thank T.  ...  Within the domain of signal processing we mention stack filter optimization and its quest for idempotency, but new results and perspectives on ordered semigroups, max-min systems, monotone Boolean networks  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s1076-5670(06)46002-x">doi:10.1016/s1076-5670(06)46002-x</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ttcpat5vuzam5mklhtxyoacl5m">fatcat:ttcpat5vuzam5mklhtxyoacl5m</a> </span>
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Fuzzy mathematical morphologies: A comparative study

Isabelle Bloch, Henri Maitre
<span title="">1995</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/jm6w2xclfzguxnhmnmq5omebpi" style="color: black;">Pattern Recognition</a> </i> &nbsp;
We will compare their properties with respect to mathematical morphology and to fuzzy sets and interpret them in terms of logic and decision theory.  ...  and imprecise spatial information Data fusion © © o mo !  ...  As opposed to T-norms and T-conorms, mean operators (in between the "min" and the "max") always satisfy the idempotence property.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0031-3203(94)00312-a">doi:10.1016/0031-3203(94)00312-a</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4cl4rsabhzgpvf6pxycvibwuly">fatcat:4cl4rsabhzgpvf6pxycvibwuly</a> </span>
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LULU Operators and Discrete Pulse Transform for Multidimensional Arrays

Roumen Anguelov, Inger Fabris-Rotelli
<span title="">2010</span> <i title="Institute of Electrical and Electronics Engineers (IEEE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/dhlhr4jqkbcmdbua2ca45o7kru" style="color: black;">IEEE Transactions on Image Processing</a> </i> &nbsp;
Her current research interests include feature detection in images, hierarchical decomposition of images, and noise removal in signals and images.  ...  His current research interests include hierarchical decomposition of images, mathematical morphology, and PDE-based methods of image analysis.  ...  However, and are also min-max operators as defined by Wild [39] . As such, their co-idempotence follows from [39, Corollary 11] . Therefore (25) III.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/tip.2010.2050639">doi:10.1109/tip.2010.2050639</a> <a target="_blank" rel="external noopener" href="https://www.ncbi.nlm.nih.gov/pubmed/20483683">pmid:20483683</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zhlps53nvjdtbm73t5b4mo2wfm">fatcat:zhlps53nvjdtbm73t5b4mo2wfm</a> </span>
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Lattice Image Processing: A Unification of Morphological and Fuzzy Algebraic Systems

Petros Maragos
<span title="">2005</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/3gper64usvcfvj55pdy33eng4a" style="color: black;">Journal of Mathematical Imaging and Vision</a> </i> &nbsp;
The theoretical results of this paper develop foundations for unifying large classes of nonlinear translation-invariant image and signal processing systems of the max or min type.  ...  structure that includes generalized Minkowski operators and lattice fuzzy image operators.  ...  (h) Fuzzy min gradient min[δ(F), 1 − ε(F)]. (i) Fuzzy max gradient max[δ(F), 1 − ε(F)]. Courtesy of [27] .  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s10851-005-4897-z">doi:10.1007/s10851-005-4897-z</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/drydrjzukfdbvbyyspquemzytu">fatcat:drydrjzukfdbvbyyspquemzytu</a> </span>
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Calculating the output distribution of stack filters that are erosion-dilation cascades, in particularLU LU-filters

R. Anguelov, P.W. Butler, C.H. Rohwer, M. Wild
<span title="2015-06-11">2015</span> <i title="Informa UK Limited"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/xlgc3bqqcjgdpg327mauehpp2u" style="color: black;">Quaestiones Mathematicae. Journal of the South African Mathematical Society</a> </i> &nbsp;
Two procedures to compute the output distribution phi_S of certain stack filters S (so called erosion-dilation cascades) are given.  ...  One rests on the disjunctive normal form of S and also yields the rank selection probabilities. The other is based on inclusion-exclusion and e.g. yields phi_S for some important LULU-operators S.  ...  This framework is based on concepts like idempotency, co-idempotency, trend preservation, total variation preservation, consistent decomposition.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2989/16073606.2014.981684">doi:10.2989/16073606.2014.981684</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/yz5ayxlawjdcnk4d2uxjxh4sym">fatcat:yz5ayxlawjdcnk4d2uxjxh4sym</a> </span>
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Morphological Filtering for Image Enhancement and Feature Detection [chapter]

Petros Maragos
<span title="">2005</span> <i title="Elsevier"> Handbook of Image and Video Processing </i> &nbsp;
Acknowledgements Research support during writing of this chapter was provided in part by the European Network of Excellence 'MUSCLE' and by the Greek research programs PENED-2001 of G.S.R.T. and Pythagoras  ...  Replacing min/max with a more general rank leads to rank filters.  ...  idempotent, and extensive (20) where I is an arbitrary index set, idempotence of an operator ψ means that ψ(ψ(f )) = ψ(f ), and anti-extensivity and extensivity of operators α and β means that α(f )  ...
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The Pseudo Projection Operator: Applications of Deep Learning to Projection Based Filtering in Non-Trivial Frequency Regimes [article]

Matthew L. Weiss, Nathan C. Frey, Siddharth Samsi, Randy C. Paffenroth, Vijay Gadepally
<span title="2022-04-13">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Traditional frequency based projection filters, or projection operators (PO), separate signal and noise through a series of transformations which remove frequencies where noise is present.  ...  Based upon these results, we suggest future application of the PPO to filtering problems in the physical and biological sciences.  ...  Research was sponsored by the United States Air Force Research Laboratory and the United States Air Force Artificial Intelligence Accelerator and was accomplished under Cooperative Agreement Number FA8750  ...
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Representations for Morphological Image Operators and Analogies with Linear Operators [chapter]

Petros Maragos
<span title="">2013</span> <i title="Elsevier"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ksd7mieepvestjxbiecsscpfdq" style="color: black;">Advances in Imaging and Electron Physics</a> </i> &nbsp;
The results are operator representations as a supremum or infimum of nonlinear convolutions that are either of the max-plus type or their generalizations in weighted lattices.  ...  and morphological operators, especially of the monotone type.  ...  The work was partially supported by the basic research grant COGNIMUSE which is implemented under the "ARISTEIA Action of the Operational Program Education and Lifelong Learning and is co-funded by the  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/b978-0-12-407702-7.00002-4">doi:10.1016/b978-0-12-407702-7.00002-4</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xeckgeughzemdpi632fqypmi7e">fatcat:xeckgeughzemdpi632fqypmi7e</a> </span>
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Remote Sensing Image Classification Using Attribute Filters Defined Over the Tree of Shapes

Gabriele Cavallaro, Mauro Dalla Mura, Jon Atli Benediktsson, Antonio Plaza
<span title="">2016</span> <i title="Institute of Electrical and Electronics Engineers (IEEE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/4odsbtjobjalfki6xxabjpdu6y" style="color: black;">IEEE Transactions on Geoscience and Remote Sensing</a> </i> &nbsp;
In this work, we considered the definition of min, max, direct and subtractive filter rules for the computation of attribute filters over the tree of shapes representation.  ...  Among those, connected operators (e.g., attribute filters) have proven their effectiveness in processing very high resolution images.  ...  This idea has been formalized as the so-called shape filters [26] , and the operators which are anti-extensive and idempotent are not necessarily increasing.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/tgrs.2016.2530690">doi:10.1109/tgrs.2016.2530690</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4hc45k66sfhy5p25gcc7vcfk7e">fatcat:4hc45k66sfhy5p25gcc7vcfk7e</a> </span>
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Morphological Attribute Profiles for the Analysis of Very High Resolution Images

Mauro Dalla Mura, Jón Atli Benediktsson, Björn Waske, Lorenzo Bruzzone
<span title="">2010</span> <i title="Institute of Electrical and Electronics Engineers (IEEE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/4odsbtjobjalfki6xxabjpdu6y" style="color: black;">IEEE Transactions on Geoscience and Remote Sensing</a> </i> &nbsp;
morphological filters based on a predefined structuring element.  ...  Here, the features extracted by the proposed operators were used for the classification of two very high resolution panchromatic images acquired by Quickbird on the city of Trento, Italy.  ...  ACKNOWLEDGMENT This work was supported in part by the Research Fund of the University of Iceland and of the University of Trento.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/tgrs.2010.2048116">doi:10.1109/tgrs.2010.2048116</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4vndvaolxbbupmflz7bzcn4fwu">fatcat:4vndvaolxbbupmflz7bzcn4fwu</a> </span>
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Algorithms for weighted multi-tape automata [article]

Andre Kempe Xerox Research Centre Europe, France
<span title="2004-06-02">2004</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
This report defines various operations for weighted multi-tape automata (WMTAs) and describes algorithms that have been implemented for those operations in the WFSC toolkit.  ...  The latter will be recalled to make this report more complete and self-standing.  ...  δ cyc , δ max − δ min ) 6 δ max2 ← δ max +δ cyc 7 return (δ max , δ max2 ) MAXDEL(q 1 , j, k, (χ ′ 1 , χ ′ 2 ), (δ ′ max , δ ′ min , δ ′ cyc )) → ( δ max , δ min , δ cyc ) : 8 δ maxmax(δ ′ max , χ ′  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/cs/0406003v1">arXiv:cs/0406003v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gew4p2psjfezva7aramgrvfn4u">fatcat:gew4p2psjfezva7aramgrvfn4u</a> </span>
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Stratified noncommutative geometry [article]

David Ayala, Aaron Mazel-Gee, Nick Rozenblyum
<span title="2021-09-21">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We introduce a theory of stratifications of noncommutative stacks (i.e. presentable stable ∞-categories), and we prove a reconstruction theorem that expresses them in terms of their strata and gluing data  ...  We also provide a suite of fundamental operations for constructing new stratifications from old ones: restriction, pullback, quotient, pushforward, and refinement.  ...  The main results that we use are the definition and basic properties of lax functors, the construction of left-laxification, and the Grothendieck construction for (∞, 2)-categories (which are unconditionally  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1910.14602v3">arXiv:1910.14602v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/d4b6qmn5drhejhiep2pvmg4mqu">fatcat:d4b6qmn5drhejhiep2pvmg4mqu</a> </span>
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The index map in algebraic K-theory

Oliver Braunling, Michael Groechenig, Jesse Wolfson
<span title="2017-11-14">2017</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/jjaooxtuefhdjnk64q3lsthiru" style="color: black;">Selecta Mathematica, New Series</a> </i> &nbsp;
We explain how it arises in analogy with the classical index map of Fredholm operators, and we relate the K-theory torsor to previously studied dimension and determinant torsors.  ...  Saito and IPMU for the pleasant working conditions while this article was being completed. We would like to thank A. Beilinson and V.  ...  Kremnizer for introducing us first to these questions and then to each other. We thank E. Getzler, M. Kapranov, R. Nest and B. Tsygan for helpful conversations. We would like to thank T.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00029-017-0364-0">doi:10.1007/s00029-017-0364-0</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/eewyhk3bdjbxndo3355xu2fpjq">fatcat:eewyhk3bdjbxndo3355xu2fpjq</a> </span>
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