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• 1.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology. Department of Mathematics, School of pure and applied sciences, College of Science and Technology, University of Rwanda.
Characterization of optimal decompositions in real interpolation2014In: Journal of Approximation Theory, ISSN 0021-9045, E-ISSN 1096-0430, Vol. 185, p. 1-11Article in journal (Refereed)

We use duality in convex analysis to characterize optimal decompositions for the $L$-functional. We also describe a geometry of optimal decompositions for the $\small L_{p,1}$-functional for the couple $\small \left( \ell^{p},X \right)$ on $\small \mathbb{R}^{n}$.

• 2.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Subdifferentiability of Infimal Convolution on Banach Couples2015In: Functiones et Approximatio Commentarii Mathematici, ISSN 0208-6573, Vol. 52, no 2, p. 311-326Article in journal (Other academic)

We use duality in convex analysis and particularly the famous Attouch-Brezis theorem to prove subdifferentiability of infimal convolution on Banach couples.

• 3.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
A duality approach for optimal decomposition in real interpolation2013Report (Other academic)

We use our previous results on subdifferentiability and dual characterization of optimal decomposition for an infimal convolution to establish mathematical properties of exact minimizers (optimal decomposition) for the K–,L–, and E– functionals of the theory of real interpolation. We characterize the geometry of optimal decomposition for the couple (p, X) on Rn and provide an extension of a result that we have establshed recently for the couple (2, X) on Rn. We will also apply the Attouch–Brezis theorem to show the existence of optimal decomposition for these functionals for the conjugate couple.

• 4.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Exact Minimizers in Real Interpolation: Characterization and Appliations2015Doctoral thesis, comprehensive summary (Other academic)

The main idea of the thesis is to develop new connections between the theory of real interpolation and applications. Near and exact minimizers for E–, K– and L–functionals of the theory of real interpolation are very important in applications connected to regularization of inverse problems such as image processing. The problem which appears is how to characterize and construct these minimizers. These exact minimizers referred to as optimal decompositions in the thesis, have certain extremal properties that we completely express and characterize in terms of duality. Our characterization generalizes known characterization for a particular Banach couple. The characterization presented in the thesis also makes it possible to understand the geometrical meaning of optimal decomposition for some important particular cases and gives a possibility to construct them. One of the most famous models in image processing is the total variation regularization  published by Rudin, Osher and Fatemi. We propose a new fast algorithm to find the exact minimizer for this model. Optimal decompositions mentioned have some connections to optimization problems which are also pointed out. The thesis is based on results that have been presented in international conferences and have been published in five papers.

In Paper 1, we characterize optimal decomposition for the E–, K– and Lp0,p1 –functional. We also present a geometrical interpretation of optimal decomposition for the Lp,1–functional for the couple (ℓp, X) on Rn. The characterization presented is useful in the sense that it gives insights into the construction of these minimizers.

The characterization mentioned in Paper 1 is based on optimal decomposition for infimal convolution. The operation of infimal convolution is a very important and non–trivial tool in functional analysis and is also very well–known within the context of convex analysis. The L–, K– and E–functionals can be regarded as an infimal convolution of two well–defined functions. Unfortunately tools from convex analysis can not be applied in a straightforward way in this context of couples of spaces. The most important requirement that an infimal convolution would satisfy for a decomposition to be optimal is subdifferentiability. In Paper 2, we have used an approach based on the famous Attouch–Brezis theorem to prove subdifferentiability of infimal convolution on Banach couples.

In Paper 3, we apply result from Paper 1 to the well–known Rudin–Osher–Fatemi (ROF) image denoising model on a general finite directed graph. We define the space BV of functions of bounded variation 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 `¥ 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. Proof of convergence of the algorithm is presented and its performance on image denoising test examples is illustrated.

In Paper 4, we present some extensions of results presented in Paper 1 and Paper 2. First we extend the results from Banach couples to Banach triples. Then we prove that our approach can apply when complex spaces are considered instead of real spaces. Finally we compare the performance of the algorithm that was proposed in Paper 3 with the Split Bregman algorithm which is one of the benchmark algorithms known for the ROF model. We find out that in most cases both algorithms behave in a similar way and that in some cases our algorithm decreases the error faster with the number of iterations.

In Paper 5, we point out some connections between optimal decompositions mentioned in the thesis and optimization problems. We apply the approach used in Paper 2 to two well–known optimization problems, namely convex and linear programming to investigate connections with standard results in the framework of these problems. It is shown that we can derive proofs for duality theorems for these problems under the assumptions of our approach.

1. Characterization of optimal decompositions in real interpolation
Open this publication in new window or tab >>Characterization of optimal decompositions in real interpolation
2014 (English)In: Journal of Approximation Theory, ISSN 0021-9045, E-ISSN 1096-0430, Vol. 185, p. 1-11Article in journal (Refereed) Published
##### Abstract [en]

We use duality in convex analysis to characterize optimal decompositions for the $L$-functional. We also describe a geometry of optimal decompositions for the $\small L_{p,1}$-functional for the couple $\small \left( \ell^{p},X \right)$ on $\small \mathbb{R}^{n}$.

##### Place, publisher, year, edition, pages
Elsevier, 2014
##### Keywords
Optimal decompositions, Real interpolation, Duality
Mathematics
##### Identifiers
urn:nbn:se:liu:diva-107565 (URN)10.1016/j.jat.2014.05.015 (DOI)000340691300001 ()
##### Note

Available from: 2014-06-16 Created: 2014-06-16 Last updated: 2017-12-05
2. Subdifferentiability of Infimal Convolution on Banach Couples
Open this publication in new window or tab >>Subdifferentiability of Infimal Convolution on Banach Couples
2015 (English)In: Functiones et Approximatio Commentarii Mathematici, ISSN 0208-6573, Vol. 52, no 2, p. 311-326Article in journal (Other academic) Published
##### Abstract [en]

We use duality in convex analysis and particularly the famous Attouch-Brezis theorem to prove subdifferentiability of infimal convolution on Banach couples.

##### Place, publisher, year, edition, pages
Wydawnictwo Naukowe Uniwersytetu im. Adama Mickiewicza, 2015
##### Keywords
real interpolation, infimal convolution, subdifferentiability
Mathematics
##### Identifiers
urn:nbn:se:liu:diva-118353 (URN)10.7169/facm/2015.52.2.9 (DOI)
##### Note

At the time for thesis presentation publication was in status: Manuscript

Available from: 2015-05-27 Created: 2015-05-27 Last updated: 2016-08-04Bibliographically approved
3. A New Reiterative Algorithm for the Rudin-Osher-Fatemi Denoising Model on the Graph
Open this publication in new window or tab >>A New Reiterative Algorithm for the Rudin-Osher-Fatemi Denoising Model on the Graph
2014 (English)In: Proceedings of The 2nd International Conference on Intelligent Systems and Image Processing 2014, ICISIP2014, 2014, p. 81-88Conference paper, Published paper (Refereed)
##### Abstract [en]

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. Then we consider the space BV of functions of bounded variation 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 ℓ on the graph by a divergence operator. Based on this result, we propose a new fast reiterative algorithm to find the exact minimizer for the ROF model. Finally we prove convergence of the algorithm and illustrate its performance on some test examples. We would like to note that consideration of an image on the graph is important in the sense that it provides useful insights when instead of rectangular domain we have some manifold, which is a good representation for images arising in applications.

##### Keywords
L–functional, Image processing, Dual BV, Regularization
##### National Category
Mathematics Other Natural Sciences
##### Identifiers
urn:nbn:se:liu:diva-111218 (URN)10.12792/icisip2014.018 (DOI)
##### Conference
The 2nd International Conference on Intelligent Systems and Image Processing 2014, ICISIP2014, September 26 – September 29, 2014, Kitakyushu, Japan
Available from: 2014-10-11 Created: 2014-10-11 Last updated: 2015-05-27Bibliographically approved
4. Exact Minimizers in Real Interpolation: Some additional results
Open this publication in new window or tab >>Exact Minimizers in Real Interpolation: Some additional results
##### Abstract [en]

We present some extensions of results presented in our recent papers. First we extend the characterization of optimal decompositions for a Banach couple to optimal decompositions for a Banach triple. Next we show that our approach can apply when complex spaces are considered instead of real spaces. Finally we compare the performance of the algorithm that we have proposed for the ROF model with the Split Bregman algorithm. The Split Bregman algorithm can in principle be regarded as a benchmark algorithm for the ROF model. We find out that in most cases both algorithms behave in a similar way and that in some cases our algorithm decreases the error faster with the number of iterations.

##### Keywords
Regular Banach triple, Optimal decompositions, Complex Banach couple, Real Interpolation, Convex Duality
Mathematics
##### Identifiers
urn:nbn:se:liu:diva-118355 (URN)
Available from: 2015-05-27 Created: 2015-05-27 Last updated: 2015-05-27Bibliographically approved
5. Optimal decomposition for infimal convolution on Banach Couples
Open this publication in new window or tab >>Optimal decomposition for infimal convolution on Banach Couples
##### Abstract [en]

Infimal convolution of functions defined on a regular Banach couple is considered. By using a theorem due to Attouch and Brezis, we establish sufficient conditions for this infimal convolution to be subdifferentiable. We also provide a result which gives a dual characterization of optimal decomposition for an infimal convolution in general. We plan tu use these results as tools to study the mathematical properties of exact minimizers for the K–, L–, and E– functionals of the theory of real interpolation. This will be done in a separate study. In this study, we apply our approach to two well–known optimization problems, namely convex and linear programming and provide proofs for duality theorems based on infimal convolution.

##### Series
LiTH-MAT-R, ISSN 0348-2960 ; 7
Mathematics
##### Identifiers
urn:nbn:se:liu:diva-94152 (URN)LiTH-MAT-R--2013/07--SE (ISRN)
Available from: 2013-06-17 Created: 2013-06-17 Last updated: 2015-05-27
• 5.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering. Department of Mathematics, School of Pure and Applied Science, College of Science and Technology, University of Rwanda, P.O. Box 3900 Kigali, Rwanda.

We present some extensions of results presented in our recent papers. First we extend the characterization of optimal decompositions for a Banach couple to optimal decompositions for a Banach triple. Next we show that our approach can apply when complex spaces are considered instead of real spaces. Finally we compare the performance of the algorithm that we have proposed for the ROF model with the Split Bregman algorithm. The Split Bregman algorithm can in principle be regarded as a benchmark algorithm for the ROF model. We find out that in most cases both algorithms behave in a similar way and that in some cases our algorithm decreases the error faster with the number of iterations.

• 6.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering. Department of Mathematics, School of Pure and Applied Science, College of Science and Technology, University of Rwanda, Rwanda.

We present some extensions of results presented in our recent papers. First we extend the characterization of optimal decompositions for a Banach couple to optimal decompositions for a Banach triple. Next we show that our approach can apply when complex spaces are considered instead of real spaces. Finally we compare the performance of the algorithm that we have proposed for the ROF model with the Split Bregman algorithm. The Split Bregman algorithm can in principle be regarded as a benchmark algorithm for the ROF model. We find out that in most cases both algorithms behave in a similar way and that in some cases our algorithm decreases the error faster with the number of iterations.

• 7.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
Geometry of Optimal Decomposition for the Couple (2, X) on Rn2013Report (Other academic)

We investigate the geometry of optimal decomposition for the L– functional

$L_{2,1}(t, x; \ell{^2}, X) = \; \inf_{ x =x_0+x_1} \; \left(\frac{1}{2}\left\|x_{0}\right\|^{2}_{\ell^{2}}+t\left\|x_{1}\right\|_{X}\right)\;,$

where space ℓ2 is defined by the standard Euclidean norm $\left\|.\right\|_{2}$ and where X is a Banach space on Rn and t is a given positive parameter. Our proof is based on some geometrical considerations and Yves Meyer’s duality approach which was considered for the couple (L2, BV) in connection with the famous in image processing ROF denoising model. Our goal is also to investigate possibility to extend Meyer’s approach to more general couples than (L2, BV) .

• 8.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
Optimal decomposition for infimal convolution on Banach Couples2013Report (Other academic)

Infimal convolution of functions defined on a regular Banach couple is considered. By using a theorem due to Attouch and Brezis, we establish sufficient conditions for this infimal convolution to be subdifferentiable. We also provide a result which gives a dual characterization of optimal decomposition for an infimal convolution in general. We plan tu use these results as tools to study the mathematical properties of exact minimizers for the K–, L–, and E– functionals of the theory of real interpolation. This will be done in a separate study. In this study, we apply our approach to two well–known optimization problems, namely convex and linear programming and provide proofs for duality theorems based on infimal convolution.

• 9.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
Optimal Decomposition in Real Interpolation and Duality in Convex Analysis2013Licentiate thesis, monograph (Other academic)

This thesis is devoted to the study of mathematical properties of exact minimizers for the K–,L–, and E– functionals of the theory of real interpolation. Recently exact minimizers for these functionals have appeared in important results in image processing.

In the thesis, we present a geometry of optimal decomposition for L– functional for the couple (ℓ2, X), where space ℓ2 is defined by the standard Euclidean norm ǁ · ǁ2 and where X is a Banach space on Rn. The well known ROF denoising model is a special case of an L– functional for the couple (L2, BV) where L2 and BV stand for the space of square integrable functions and the space of functions with bounded variation on a rectangular domain respectively. We provide simple proofs and geometrical interpretation of optimal decomposition by following ideas by Yves Meyer who has used a duality approach to characterize optimal decomposition for ROF denoising model.

The operation of infimal convolution is a very important and non–trivial tool in functional analysis and is also very well–known within the context of convex analysis. The L–, K– and E– functionals can be regarded as an infimal convolution of two well defined functions but unfortunately tools from convex analysis can not be applied in a straigtforward way in this context of couples of spaces. We have considered infimal convolution on Banach couples and by using a theorem due to Attouch and Brezis, we have established sufficient conditions for an infimal convolution on a given Banach couple to be subdifferentiable, which turns out to be the most important requirement that an infimal convolution would satisfy for a decomposition to be optimal. We have also provided a lemma that we have named Key Lemma, which characterizes optimal decomposition for an infimal convolution in general.

The main results concerning mathematical properties of optimal decomposition for L–, K– and E– functionals for the case of general regular Banach couples are presented. We use a duality approach which can be summarized in three steps: First we consider the concerned functional as an infimal convolution and reformulate the infimal convolution at hand as a minimization of a sum of two specific functions on the intersection of the couple. Then we prove that it is subdifferentiable and finally use the characterizaton of its optimal decomposition.

We have also investigated how powerful our approach is by applying it to two well–known optimization problems, namely convex and linear programming. As a result we have obtained new proofs for duality theorems which are central for these problems.

• 10.
Linköping University, Department of Mathematics. Linköping University, Faculty of Science & Engineering. University of Rwanda, Department of Mathematics, School of Pure and Applied Sciences, College of Science and Technology .
Linköping University, Department of Mathematics. Linköping University, Faculty of Science & Engineering.
A New Reiterative Algorithm for the Rudin-Osher-Fatemi Denoising Model on the Graph2014In: Proceedings of The 2nd International Conference on Intelligent Systems and Image Processing 2014, ICISIP2014, 2014, p. 81-88Conference paper (Refereed)

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. Then we consider the space BV of functions of bounded variation 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 ℓ on the graph by a divergence operator. Based on this result, we propose a new fast reiterative algorithm to find the exact minimizer for the ROF model. Finally we prove convergence of the algorithm and illustrate its performance on some test examples. We would like to note that consideration of an image on the graph is important in the sense that it provides useful insights when instead of rectangular domain we have some manifold, which is a good representation for images arising in applications.

• 11.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology. Department of Mathematics, School of pure and applied sciences, College of Science and Technology, University of Rwanda, P.O. Box 117 Butare, Rwanda.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
ROF model on the graph2014Report (Other academic)

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.

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Cite
Citation style
• apa
• harvard1
• ieee
• modern-language-association-8th-edition
• vancouver
• oxford
• Other style
More styles
Language
• de-DE
• en-GB
• en-US
• fi-FI
• nn-NO
• nn-NB
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• Other locale
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• text
• asciidoc
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