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Setterqvist, Eric
Publications (5 of 5) Show all publications
Setterqvist, E. & Forchheimer, R. (2018). Real-Time Communication Systems based on Taut Strings. Journal of Communications and Networks, 20(2), 207-218
Open this publication in new window or tab >>Real-Time Communication Systems based on Taut Strings
2018 (English)In: Journal of Communications and Networks, ISSN 1229-2370, E-ISSN 1976-5541, Vol. 20, no 2, p. 207-218Article in journal (Refereed) Published
Abstract [en]

We consider buffered real-time communication over channels with time-dependent capacities which are known in advance. The real-time constraint is imposed in terms of limited transmission time between sender and receiver. For a network consisting of a single channel it is shown that there is a coding rate strategy, geometrically characterized as a taut string, which minimizes the average distortion with respect to all convex distortionrate functions. Utilizing the taut string characterization further, an algorithm that computes the optimal coding rate strategy is provided. We then consider more general networks with several connected channels in parallel or series with intermediate buffers. It is shown that also for these networks there is a coding rate strategy, geometrically characterized as a taut string, which minimizes the average distortion with respect to all convex distortion-rate functions. The optimal offline strategy provides a benchmark for the evaluation of different coding rate strategies. Further, it guides us in the construction of a simple but rather efficient strategy for channels in the online setting which alternates between a good and a bad state.

Place, publisher, year, edition, pages
IEEE, 2018
Keywords
Buffers; distortion; real-time systems; source coding; taut string
National Category
Telecommunications
Identifiers
urn:nbn:se:liu:diva-148263 (URN)10.1109/JCN.2018.000027 (DOI)000432478900009 ()
Note

Funding Agencies|Research School in Interdisciplinary Mathematics at Linkoping University; Swedish Research Council [2014-6230]

Available from: 2018-06-04 Created: 2018-06-04 Last updated: 2018-06-21
Kruglyak, N. & Setterqvist, E. (2016). Discrete taut strings and real interpolation. Journal of Functional Analysis, 270(2), 671-704
Open this publication in new window or tab >>Discrete taut strings and real interpolation
2016 (English)In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 270, no 2, p. 671-704Article in journal (Refereed) Published
Abstract [en]

Classical taut strings and their multidimensional generalizations appear in a broad range of applications. In this paper we suggest a general approach based on the K-functional of real interpolation that provides a unifying framework of existing theories and extend the range of applications of taut strings. More exactly, we introduce the notion of invariant K-minimal sets, explain their connection to taut strings and characterize all bounded, closed and convex sets in R-n that are invariant K-minimal with respect to the couple (l(1), l(infinity)). (C) 2015 Elsevier Inc. All rights reserved.

Place, publisher, year, edition, pages
ACADEMIC PRESS INC ELSEVIER SCIENCE, 2016
Keywords
Taut strings; Real interpolation; Invariant K-minimal sets
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-124087 (URN)10.1016/j.jfa.2015.10.012 (DOI)000366144300007 ()
Available from: 2016-01-25 Created: 2016-01-19 Last updated: 2017-11-30
Lifshits, M. & Setterqvist, E. (2015). Energy of taut strings accompanying Wiener process. Stochastic Processes and their Applications, 125(2), 401-427
Open this publication in new window or tab >>Energy of taut strings accompanying Wiener process
2015 (English)In: Stochastic Processes and their Applications, ISSN 0304-4149, E-ISSN 1879-209X, Vol. 125, no 2, p. 401-427Article in journal (Refereed) Published
Abstract [en]

Let W be a Wiener process. For r greater than 0 and T greater than 0 let I-W (T, r)(2) denote the minimal value of the energy integral(T)(0) h(t)(2)dt taken among all absolutely continuous functions h(.) defined on [0, T], starting at zero and satisfying W(t) - r less than= h(t) less than= W(t) + r, 0 less than= t less than= T. The function minimizing energy is a taut string, a classical object well known in Variational Calculus, in Mathematical Statistics, and in a broad range of applications. We show that there exists a constant C E (0, infinity) such that for any q greater than 0 r/T-1/2 I-W (T, r) -greater than(Lq) C, as r/T-1/2 -greater than 0, and for any fixed r greater than 0, r/(TIW)-I-1/2 (T, r)-greater than(a.s.) C, as T -greater than infinity. Although precise value of C remains unknown, we give various theoretical bounds for it, as well as rather precise results of computer simulation. While the taut string clearly depends on entire trajectory of W, we also consider an adaptive version of the problem by giving a construction (called Markovian pursuit) of a random function h(t) based only on the values W(s), s less than= t, and having minimal asymptotic energy. The solution, i.e. an optimal pursuit strategy, turns out to be related with a classical minimization problem for Fisher information on the bounded interval.

Place, publisher, year, edition, pages
Elsevier, 2015
Keywords
Gaussian processes; Markovian pursuit; Taut string; Wiener process
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-115334 (URN)10.1016/j.spa.2014.09.020 (DOI)000349501200001 ()
Note

Funding Agencies| [RFBR 13-01-00172]; [SPbSU 6.38.672.2013]

Available from: 2015-03-13 Created: 2015-03-13 Last updated: 2017-12-04
Niyobuhungiro, J. & Setterqvist, E. (2014). ROF model on the graph. Linköping University Electronic Press
Open this publication in new window or tab >>ROF model on the graph
2014 (English)Report (Other academic)
Abstract [en]

In this paper we consider an analogue of the well-known in image processing, Rudin-Osher-Fatemi (ROF) denoising model on a general finite directed and connected graph. We consider the space BV on the graph and show that the unit ball of its dual space can be described as the image of the unit ball of the space ℓinfinity on the graph by a divergence operator. Based on this result, we propose a new fast algorithm to find the exact minimizer for the ROF model. Finally we prove convergence of the algorithm and illustrate its performance on some image denoising test examples.

Place, publisher, year, edition, pages
Linköping University Electronic Press, 2014. p. 25
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2014:06
Keywords
ROF model, Directed graph, L–functional, Image processing, Dual BV, Regularization.
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-106920 (URN)LiTH-MAT-R--2014/06--SE (ISRN)
Available from: 2014-05-26 Created: 2014-05-26 Last updated: 2014-05-27
Kruglyak, N. & Setterqvist, E. (2008). Sharp Estimates of the Identity Minus Hardy Operator on the Cone of Decreasing Functions. Proceedings of the American Mathematical Society, 136(7), 2505-2513
Open this publication in new window or tab >>Sharp Estimates of the Identity Minus Hardy Operator on the Cone of Decreasing Functions
2008 (English)In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 136, no 7, p. 2505-2513Article in journal (Refereed) Published
Abstract [en]

It is shown that if we restrict the identity minus Hardy operator on the cone of nonnegative decreasing functions f in L-p, then we have the sharp estimate

parallel to(I - H)f parallel to(Lp) <= 1/(p - 1)(1/p) parallel to f parallel to(Lp) for p = 2, 3, 4, .... In other words,

parallel to f** - f*parallel to(Lp) <= 1/(p - 1)(1/p) parallel to f parallel to(Lp) for each f is an element of L-p and each integer p >= 2.

It is also shown, via a connection between the operator I - H and Laguerre functions, that

parallel to(1 - alpha)I + Phi(I - H)parallel to(L2 -> L2) = parallel to I - alpha H parallel to(L2 -> L2) = 1 for all a is an element of [ 0, 1].

 

Place, publisher, year, edition, pages
American Mathematical Society (AMS), 2008
Keywords
The Hardy operator, cone of decreasing functions, sharp estimates
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-90402 (URN)10.1090/S0002-9939-08-09200-9 (DOI)
Available from: 2013-03-25 Created: 2013-03-25 Last updated: 2017-12-06
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