OPTIMED: Optimisation Iterative pour la Résolution de Problèmes Inverses de Grande Taille

OPTIMED: Optimisation Iterative pour la Résolution de Problèmes Inverses de Grande Taille

Lotfi Chaari Emilie Chouzenoux  Nelly Pustelnik  Caroline Chaux  Saïd Moussaoui 

INRIA Rhône-Alpes, Projet MISTIS

Université Paris-Est, IGM (UMR CNRS 8049)

Laboratoire de Physique, ENS Lyon (UMR CNRS 5672)

LUNAM Université, Ecole Centrale Nantes, IRCCyN (UMR CNRS 6597)

30 June 2011
| Citation



This paper deals with image reconstruction using iterative algorithms. The framework under consideration involves large amount of data and/or large observation operator. We proposed here algorithms for which we were able to state convergence proofs. The reliability and accuracy of the proposed approaches are demonstrated via three applications: the positron emission tomography (PET), the parallel magnetic resonance imaging (MRI) and twodimensional nuclear magnetic resonance (2D NMR).

Extended Abstract

The main objective of the ANR project OPTIMED (MDMSA program) was to develop new reconstruction methods to solve large-scale inverse problems arising in medical imaging. Two different modalities have been studied: the Positron Emission Tomography (PET) and the Magnetic Resonance Imaging (MRI). Furthermore, the application area has been extended to two dimensional Nuclear Magnetic Resonance (2D NMR) relaxometry. In these problems, existing convex optimization methods were unable to deal efficiently with the large amount of data and constraints, or to deliver reliable numerical solutions in reasonable time.

From a methodological point of view, the goal was to bring solution to large scale inverse problems (3D, 3D+t...) by solving optimization problems where the objective function accounts for the physical properties of the acquisition system as well as any prior information one can have on the imaged object. Two main avenues of research were therefore explored:

– Improve the reconstructed image quality by considering an adequate criterion in accordance with the noise statistical property (Poissonian or Gaussian) and by introducing relevant prior knowledge. Among all the possible priors, we favor the one promoting sparsity of the data after some linear transform (e.g. Wavelet frame decomposition), which appears to be a judicious choice when dealing with large amount of data (reduction of the memory load/swap time and introduction of regularity constraints in a more tractable fashion). Furthermore, the Lagrangian formulation of the constraints that have to be fulfilled by the solution, require the addition of a Barrier function in the criterion. This can lead to non-smooth optimization problems involving non-differentiable objective and/or constraint functions.

– Reduce the computational time that can be crucial for medical diagnosis. That is why, it was necessary to develop and implement iterative algorithms that had a convergence speed as high as possible. A compromise between convergence speed and iteration cost has thus to be made. Furthermore, in order to take into account the new available parallel architectures, we proposed splitting optimization algorithms well adapted to a parallelization process. In order to explore these avenues, two main families of optimization approaches have been explored during this study: proximal and majorization-minimization (MM) methods.

Proximal approaches

The proximal methods refer to iterative algorithms enable to minimize a sum of convex functions non-necessarily differentiable and that can involve linear perators. This class of algorithms is based on proximity operators that is a generalization of the projection onto a convex set. The relevance of these algorithms is mainly due to their strong convergence properties, their robustness to numerical errors, and their good numerical behavior. In the recent literature related to signal and image restoration, numerous works have considered proximal algorithms in order to efficiently deal with 1 -prior, wavelet-frame representations, non-necessarily Lipschitz differentiable data fidelity term, or additional constraints such as the data positivity.


At each iteration of a MM algorithm, a tangent majorant function, that majorizes the objective function and is equal to it at the current iterate, is constructed. The next iterate is obtained by minimizing this tangent majorant function, resulting in a sequence of simple iterates that reduces the cost function monotonically. Recently, an efficient MM quadratic line search procedure for descent algorithms has been introduced. In this strategy, the stepsize value results from successive minimizations of quadratic tangent majorant functions for the restriction of the cost function along the search direction. This method benefits from strong convergence results. We prove that it generalizes into a multi-dimensional search that ensures the convergence of several subspace algorithms. Quadratic MM strategies exclude the case of barrier criteria. Such functions are encountered for example in maximum entropy reconstruction and Positron Emission Tomography. We propose an MM line search algorithm for barrier functions, based on a quadratic majorant function augmented with a logarithmic term. This leads to a simple line search that ensures the convergence of several classical descent optimization strategies.

In summary, the ANR project OPTIMED offered an opportunity to apply modern tools of convex analysis and the theory of monotonic operators to solve large scale optimization problems. All these developments have been efficiently applied to real problems on image reconstruction in PET, parallel MRI and 2D NMR relaxometry.


Cet article s’intéresse au développement d’algorithmes itératifs pour la reconstruction d’images. La particularité de ces problèmes réside dans la taille élevée des images et la quantité importante de données générée par le processus de mesure. Outre le fait de surmonter la difficulté liée à la grande taille du problème, les algorithmes proposés se fondent sur des résultats de convergence théoriquement établis. Nous illustrons ces développements à l’aide de trois modalités d’imagerie distinctes : la tomographie par émission de positrons (TEP), l’imagerie par résonance magnétiques nucléaire (IRM) parallèle et la résonance magnétique nucléaire bidimentionnelle (RMN 2D).


convex optimization, iterative algorithms, PET, MRI, 2D-NMR.


optimisation convexe, algorithmes itératifs, TEP, IRM, RMN-2D.

1. Introduction
2. Quelques Schémas Itératifs pour la Résolution du Problème d’Optimisation
3. Reconstruction d’Images en Tomographie par Emission de Positrons
4. Reconstruction Régularisée en IRM Parallèle
5. Reconstruction de Spectres en Relaxométrie RMN Bidimensionnelle
6. Conclusion

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