22 results on '"Escande, Paul"'
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2. Accelerating [formula omitted] deblurring using wavelet expansions of operators
- Author
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Escande, Paul and Weiss, Pierre
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- 2018
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3. Contrast Invariant SNR and Isotonic Regressions
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Weiss, Pierre, Escande, Paul, Bathie, Gabriel, and Dong, Yiqiu
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- 2019
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4. On the Concentration of the Minimizers of Empirical Risks
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Escande, Paul, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)
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[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST] ,FOS: Mathematics ,Mathematics - Statistics Theory ,Statistics Theory (math.ST) ,Concentration inequalities ,Empirical risk minimization - Abstract
Obtaining guarantees on the convergence of the minimizers of empirical risks to the ones of the true risk is a fundamental matter in statistical learning. Instead of deriving guarantees on the usual estimation error, the goal of this paper is to provide concentration inequalities on the distance between the sets of minimizers of the risks for a broad spectrum of estimation problems. In particular, the risks are defined on metric spaces through probability measures that are also supported on metric spaces. A particular attention will therefore be given to include unbounded spaces and non-convex cost functions that might also be unbounded. This work identifies a set of assumptions allowing to describe a regime that seem to govern the concentration in many estimation problems, where the empirical minimizers are stable. This stability can then be leveraged to prove parametric concentration rates in probability and in expectation. The assumptions are verified, and the bounds showcased, on a selection of estimation problems such as barycenters on metric space with positive or negative curvature, subspaces of covariance matrices, regression problems and entropic-Wasserstein barycenters.
- Published
- 2023
5. Full inference for the anisotropic fractional Brownian field
- Author
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Escande, Paul, Richard, Frédéric, Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), and RICHARD, Frédéric
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[STAT.AP]Statistics [stat]/Applications [stat.AP] ,[STAT.ME] Statistics [stat]/Methodology [stat.ME] ,[STAT.AP] Statistics [stat]/Applications [stat.AP] ,[STAT.ME]Statistics [stat]/Methodology [stat.ME] ,[SPI.SIGNAL]Engineering Sciences [physics]/Signal and Image processing ,[SPI.SIGNAL] Engineering Sciences [physics]/Signal and Image processing - Abstract
The anisotropic fractional Brownian field (AFBF) is a non-stationary Gaussian random field which has been used for the modeling of textured images. In this paper, we address the open issue of estimating the functional parameters of this field, namely the topothesy and Hurst functions. We propose an original method which fits the empirical semi-variogram of an image to the semi-variogram of a turning-band field that approximates the AFBF. Expressing the fitting criterion in terms of a separable non-linear least square criterion, we design a minimization algorithm inspired from the variable projection approach. This algorithm also includes a coarse-to-fine multigrid strategy based on approximations of functional parameters. Compared to existing methods, the new method enables to estimate both functional parameters on their whole definition domain. On simulated textures, we show that it has a low estimation error, even when the parameters are approximated with a high precision. We also apply the method to characterize mammograms and sample images with synthetic parenchymal patterns.
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- 2023
6. Approximation of Integral Operators Using Product-Convolution Expansions
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Escande, Paul and Weiss, Pierre
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- 2017
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7. A Variational Model for Multiplicative Structured Noise Removal
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Escande, Paul, Weiss, Pierre, and Zhang, Wenxing
- Published
- 2017
- Full Text
- View/download PDF
8. Random walk informed heterogeneity detection reveals how the lymph node conduit network influences T cells collective exploration behavior.
- Author
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Song, Solène, Senoussi, Malek, Escande, Paul, and Villoutreix, Paul
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RANDOM walks ,LYMPH nodes ,COLLECTIVE behavior ,T cells ,CELL motility ,STOCHASTIC processes ,T cell receptors - Abstract
Random walks on networks are widely used to model stochastic processes such as search strategies, transportation problems or disease propagation. A prominent example of such process is the dynamics of naive T cells within the lymph node while they are scanning for antigens. The observed T cells trajectories in small sub-volumes of the lymph node are well modeled as a random walk and they have been shown to follow the lymphatic conduit network as substrate for migration. One can then ask how does the connectivity patterns of the lymph node conduit network affect the T cells collective exploration behavior. In particular, does the network display properties that are uniform across the whole volume of the lymph node or can we distinguish some heterogeneities? We propose a workflow to accurately and efficiently define and compute these quantities on large networks, which enables us to characterize heterogeneities within a very large published dataset of Lymph Node Conduit Network. To establish the significance of our results, we compared the results obtained on the lymph node to null models of varying complexity. We identified significantly heterogeneous regions characterized as "remote regions" at the poles and next to the medulla, while a large portion of the network promotes uniform exploration by T cells. Author summary: Lymph nodes are organs in which actors of the immune system meet. In particular, the encounter between the naive T cells and their specific antigens occurs in lymph nodes. This event triggers the adaptive immune response. T cells movement has been shown to be well described as a random walk, at least when they are measured on small sub-volumes of the lymph node. In parallel, it was shown that T-cells migrate following the lymphatic conduit network that span the lymph nodes. In this study, we ask, how does the connectivity pattern of the conduit network on which T cells move influences their collective exploration behavior? Are there regions in the lymph node conduit network which have distinct random walk related properties? The topological reconstruction of the lymph node conduit network was recently made available. The network is very large (about 200 000 nodes) and appears very regular, with most nodes being connected to three neighbours. We propose a workflow to detect heterogeneities in such large and quasi-regular networks, building on random walk on network tools, and the measure of two features which we interpret using a series of generated null models for comparison. We show that the lymph node conduit network displays remotely accessible regions at both poles and near medulla, with however most of the network promoting uniform exploration. [ABSTRACT FROM AUTHOR]
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- 2023
- Full Text
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9. Fast wavelet decomposition of linear operators through product-convolution expansions.
- Author
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Escande, Paul and Weiss, Pierre
- Subjects
INVERSE problems ,INTEGRAL operators ,WAVES (Fluid mechanics) ,IMPULSE response ,WORK design - Abstract
Wavelet decompositions of integral operators have proven their efficiency in reducing computing times for many problems, ranging from the simulation of waves or fluids to the resolution of inverse problems in imaging. Unfortunately, computing the decomposition is itself a hard problem which is oftentimes out of reach for large-scale problems. The objective of this work is to design fast decomposition algorithms based on another representation called product-convolution expansion. This decomposition can be evaluated efficiently, assuming that a few impulse responses of the operator are available, but it is usually less efficient than the wavelet decomposition when incorporated in iterative methods. The proposed decomposition algorithms, run in quasi-linear time and we provide some numerical experiments to assess its performance for an imaging problem involving space-varying blurs. [ABSTRACT FROM AUTHOR]
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- 2022
- Full Text
- View/download PDF
10. A scalable estimator of sets of integral operators
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Debarnot, Valentin, Escande, Paul, Weiss, Pierre, Institut des Technologies Avancées en sciences du Vivant (ITAV), Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées, Institut de Mathématiques de Marseille (I2M), Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU), FRM grant number ECO20170637521, and ANR-17-CE23-0013,OMS,Optimisation sur des espaces de mesures(2017)
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[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST] ,[SPI.SIGNAL]Engineering Sciences [physics]/Signal and Image processing ,[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] ,[INFO.INFO-AI]Computer Science [cs]/Artificial Intelligence [cs.AI] - Abstract
International audience; The main objective of this work is to estimate a low dimensional subspace of operators in order to improve the identifiability of blind inverse problems.We propose a scalable method to find a subspace $\widehat \H$ of low-rank tensors that simultaneously approximates a set of integral operators. The method can be seen as a generalization of tensor decomposition models, which was never used in this context. In addition, we propose to construct a convex subset of $\widehat \H$ in order to further reduce the search space. We provide theoretical guarantees on the estimators and a few numerical results.
- Published
- 2019
11. Accelerating ℓ 1 − ℓ 2 deblurring using wavelet expansions of operators
- Author
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Escande, Paul, Weiss, Pierre, Département d'Ingénierie des Systèmes Complexes (DISC), Institut Supérieur de l'Aéronautique et de l'Espace (ISAE-SUPAERO), Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), PRIMO (ITAV), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut des Technologies Avancées en sciences du Vivant (ITAV), Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS), Regenerative Medicine and Skeleton research lab (RMeS), Ecole Nationale Vétérinaire, Agroalimentaire et de l'alimentation Nantes-Atlantique (ONIRIS)-Centre hospitalier universitaire de Nantes (CHU Nantes)-Université de Nantes - UFR de Médecine et des Techniques Médicales (UFR MEDECINE), and Université de Nantes (UN)-Université de Nantes (UN)-Institut National de la Santé et de la Recherche Médicale (INSERM)
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preconditioning ,inverse problems ,[INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV] ,image de- blurring ,ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION ,GPU programming ,[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] ,sparse wavelet expansion ,[SPI.SIGNAL]Engineering Sciences [physics]/Signal and Image processing ,ComputingMilieux_MISCELLANEOUS ,[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] - Abstract
International audience; Image deblurring is a fundamental problem in imaging, usually solved with compu-tationally intensive optimization procedures. The goal of this paper is to provide new efficient strategies to reduce computing times for simple 1 − 2 deblurring models. We show that the minimization can be significantly accelerated by leveraging the fact that images and blur operators are compressible in the same orthogonal wavelet basis. The proposed methodology consists of three ingredients: i) a sparse approximation of the blur operator in wavelet bases, ii) a diagonal preconditioner and iii) an implementation on massively parallel architectures. Combing the three ingredients leads to acceleration factors ranging from 30 to 250 on a typical workstation. For instance, a 1024 × 1024 image can be deblurred in 0.15 seconds.
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- 2018
12. Accelerating l1-l2 deblurring using wavelet expansions of operators
- Author
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Escande, Paul, Weiss, Pierre, Centre National de la Recherche Scientifique - CNRS (FRANCE), Institut National des Sciences Appliquées de Toulouse - INSA (FRANCE), Institut Supérieur de l'Aéronautique et de l'Espace - ISAE-SUPAERO (FRANCE), Université Toulouse III - Paul Sabatier - UT3 (FRANCE), Université Toulouse - Jean Jaurès - UT2J (FRANCE), and Université Toulouse 1 Capitole - UT1 (FRANCE)
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Inverse problems ,Autre ,Computer Science::Computer Vision and Pattern Recognition ,ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION ,Preconditioning ,Image deblurring ,GPU programming ,Sparse wavelet expansion - Abstract
Image deblurring is a fundamental problem in imaging, usually solved with computationally intensive optimization procedures. The goal of this paper is to provide new efficient strategies to reduce computing times for simple deblurring models regularized using orthogonal wavelet transforms. We show that the minimization can be significantly accelerated by leveraging the fact that images and blur operators are compressible in the same orthogonal wavelet basis. The proposed methodology consists of three ingredients: (i) a sparse approximation of the blur operator in wavelet bases, (ii) a diagonal preconditioner and (iii) an implementation on massively parallel architectures. Combining the three ingredients leads to acceleration factors ranging from 4 to 250 on a typical workstation. For instance, a 1024 1024 image can be deblurred in 0.15 s.
- Published
- 2018
13. Compression et inférence des opérateurs intégraux : applications à la restauration d’images dégradées par des flous variables
- Author
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Escande, Paul, Institut Supérieur de l'Aéronautique et de l'Espace, Bigot, Jérémie, and Weiss, Pierre
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Problème inverse ,Astronomy ,Deconvolution ,Décomposition multi-échelle ,Fast algorithms ,Structured multiplicative noise ,High-dimension ,Défloutage ,Similarity measure ,Grande dimension ,Spatially varying blur ,Déconvolution ,Microscopie ,Approximation ,Restauration ,Integral operators ,Curse of dimensionality ,Microscopy ,Scattered data interpolation ,Parcimonie ,Multi-scale approximation ,Opérateurs intégraux ,Astronomie ,Interpolation de données éparpillées ,Product-convolution ,Mesure de similarité ,Restoration ,Deblurring ,Inverse problem ,Flou variable ,Bruit multiplicatif structuté ,Fléau de la dimension ,Produit-convolution ,Estimation ,Sparsity ,Algorithmes rapides - Abstract
Le problème de restauration d'images dégradées par des flous variables connaît un attrait croissant et touche plusieurs domaines tels que l'astronomie, la vision par ordinateur et la microscopie à feuille de lumière où les images sont de taille un milliard de pixels. Les flous variables peuvent être modélisés par des opérateurs intégraux qui associent à une image nette u, une image floue Hu. Une fois discrétisé pour être appliqué sur des images de N pixels, l'opérateur H peut être vu comme une matrice de taille N x N. Pour les applications visées, la matrice est stockée en mémoire avec un exaoctet. On voit apparaître ici les difficultés liées à ce problème de restauration des images qui sont i) le stockage de ce grand volume de données, ii) les coûts de calculs prohibitifs des produits matrice-vecteur. Ce problème souffre du fléau de la dimension. D'autre part, dans beaucoup d'applications, l'opérateur de flou n'est pas ou que partialement connu. Il y a donc deux problèmes complémentaires mais étroitement liés qui sont l'approximation et l'estimation des opérateurs de flou. Cette thèse a consisté à développer des nouveaux modèles et méthodes numériques permettant de traiter ces problèmes. The restoration of images degraded by spatially varying blurs is a problem of increasing importance. It is encountered in many applications such as astronomy, computer vision and fluorescence microscopy where images can be of size one billion pixels. Variable blurs can be modelled by linear integral operators H that map a sharp image u to its blurred version Hu. After discretization of the image on a grid of N pixels, H can be viewed as a matrix of size N x N. For targeted applications, matrices is stored with using exabytes on the memory. This simple observation illustrates the difficulties associated to this problem: i) the storage of a huge amount of data, ii) the prohibitive computation costs of matrix-vector products. This problems suffers from the challenging curse of dimensionality. In addition, in many applications, the operator is usually unknown or only partially known. There are therefore two different problems, the approximation and the estimation of blurring operators. They are intricate and have to be addressed with a global overview. Most of the work of this thesis is dedicated to the development of new models and computational methods to address those issues.
- Published
- 2016
14. Approximation and estimation of integral operatorsApplications to the restoration of images degraded by spatially varying blurs
- Author
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Escande , Paul, Institut Supérieur de l'Aéronautique et de l'Espace (ISAE-SUPAERO), Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), PRES of Toulouse University and Midi-Pyrénées region, ISAE - Institut Supérieur de l'Aéronautique et de l'Espace, Jérémie Bigot, Pierre Weiss, MODIM project funded by the PRES of Toulouse University and Midi-Pyrénées region, Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut Supérieur de l'Aéronautique et de l'Espace ( ISAE-SUPAERO ), Institut de Mathématiques de Toulouse UMR5219 ( IMT ), Centre National de la Recherche Scientifique ( CNRS ) -Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), and Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -PRES Université de Toulouse-Université Paul Sabatier - Toulouse 3 ( UPS ) -Université Toulouse - Jean Jaurès ( UT2J ) -Université Toulouse 1 Capitole ( UT1 )
- Subjects
[ INFO.INFO-TS ] Computer Science [cs]/Signal and Image Processing ,[ INFO.INFO-NA ] Computer Science [cs]/Numerical Analysis [cs.NA] ,numerical analysis ,deblurring ,fast algorithms ,operateur integral ,ondelettes ,integral operator ,imaging processing ,analyse harmonique ,compression de matrice ,algorithmes rapides ,[ INFO.INFO-DC ] Computer Science [cs]/Distributed, Parallel, and Cluster Computing [cs.DC] ,[INFO.INFO-TS]Computer Science [cs]/Signal and Image Processing ,defloutage ,wavelet ,[ INFO.INFO-TI ] Computer Science [cs]/Image Processing ,traitement d'images ,flous variables ,estimation ,matrix compression ,spatially varying blurs ,[ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA] ,[INFO.INFO-NA]Computer Science [cs]/Numerical Analysis [cs.NA] ,analyse numerique ,[INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV] ,harmonic analysis ,[INFO.INFO-DC]Computer Science [cs]/Distributed, Parallel, and Cluster Computing [cs.DC] ,[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] - Abstract
The restoration of images degraded by spatially varying blurs is a problem of increasing importance. It is encountered in many applications such as astronomy, computer vision and fluorescence microscopy where images can be of size $1000 \times 1000 \times 1000$ pixels. Variable blurs can be modelled by linear integral operators $H$ that map a sharp image $u$ to its blurred version $Hu$ defined by\[ Hu(x) = \int_{\Omega} K(x,y) u(y) dy, \quad \forall x \in \Omega = [0,1]^d\]where $K : \Omega \times \Omega \to \mathbb{R}$ is called the kernel. After discretization of the image on a grid of $N$ pixels, $H$ can be viewed as a matrix of size $N \times N$. For targeted applications, matrices contain $10^{18}$ coefficients. This simple observation illustrates the difficulties associated to this problem: i) the storage of a huge amount of data, ii) the prohibitive computation costs of matrix-vector products. This problems suffers from the challenging curse of dimensionality. In addition, in many applications, the operator is usually unknown or only partially known. There are therefore two different problems, the approximation and the estimation of blurring operators. They are intricate \emph{and} have to be addressed with a global overview. Most of the work of this thesis is dedicated to the development of new models and computational methods to address those issues.In a first part, this work studied the approximation methods of integral operators. Existing approaches in the literature can be separated in two classes. The most developed approaches consists in constructing a low rank decomposition of the kernel $K$. Similarly to the seminal work of Beylkin, Coifman and Rokhlin, we studied the representation of blurring operators in wavelet bases. We showed that matrix-vector products can be rapidly computed in $O(N\epsilon^{-M/d})$ with a precision $\epsilon$ in spectral norm where $M$ is a scalar describing the regularity of the kernel. This type of approximations can be used to design restoration algorithms that are two orders of magnitude faster than current methods.The second class contains methods commonly used in the imaging community. They consist in constructing a low rank decomposition of the Time Varying Impulse Response (TVIR) $T : \Omega \times \Omega \to \mathbb{R}$ defined by $T(x,y) = K(x+y,y)$. Until now, those methods were partially studied and this work bridges the gap in the comprehension of their performances. Moreover, it allowed the identification of a representation that ``super'' compresses the operator. This representation appeals for the development of new identification strategies.In a second part, this work addresses the challenging problem of the estimation of operators. Recent theoretical works studied this problem but none of them can be implemented in targeted applications. In the specific case where some scattered impulse responses are observed, this work proposes the construction of an estimator of the operator that can be evaluated numerically in large dimensions. Theoretical guarantees on the performance of the estimator are also provided. \\Finally, this thesis studied other imaging problems. Images from light-sheet microscopy are degraded by stripe shaped attenuations. These phenomena can be modelled by multiplicative structured noises. This work proposes to solve a convex optimization and convincing results are obtained on real data.The development of quantitative indices that measure the similarity of images is a challenging question in imaging.The illumination of a scene can vary between two moments. Most indices are non invariant to these variations and will fail to assess the similarity of the same scene between the two instants. We proposed a similarity measure that is invariant to illumination changes.; Le problème de restauration d'images dégradées par des flous variables connaît un attrait croissant et touche plusieurs domaines tels que l'astronomie, la vision par ordinateur et la microscopie à feuille de lumière où les images sont de taille $1000 \times 1000 \times 1000$ pixels. Les flous variables peuvent être modélisés par des opérateurs intégraux qui associent à une image nette $u$, une image floue $Hu$ définie par\[ Hu(x) = \int_{\Omega} K(x,y) u(y) dy, \quad \forall x \in \Omega = [0,1]^d\]où $K : \Omega \times \Omega \to \mathbb{R}$ est appelé le noyau. Une fois discrétisé pour être appliqué sur des images de $N$ pixels, l'opérateur $H$ peut être vu comme une matrice de taille $N \times N$. Pour les applications visées, la matrice contient $10^{18}$ coefficients. On voit apparaître ici les difficultés liées à ce problème de restauration des images qui sont i) le stockage de ce grand volume de données, ii) les coûts de calculs prohibitifs des produits matrice-vecteur. Ce problème souffre du fléau de la dimension. D'autre part, dans beaucoup d'applications, l'opérateur de flou n'est pas ou que partialement connu. Il y a donc deux problèmes complémentaires mais étroitement liés qui sont l'approximation et l'estimation des opérateurs de flou.Cette thèse a consisté à développer des nouveaux modèles et méthodes numériques permettant de traiter ces problèmes. Dans une première partie, ce travail a étudié les méthodes d'approximation des opérateurs intégraux. Elles peuvent être distinguées en deux groupes. Le plus étoffé contient les méthodes qui construisent une décomposition de rang faible du noyau $K$. Similairement aux travaux fondateurs du groupe de Beylkin, Coifman et Rokhlin, nous avons étudié les représentations des opérateurs de flou dans des bases d'ondelettes. Nous avons montré que les produits matrice-vecteur $Hu$ peuvent être appliqués en $O(N\epsilon^{-M/d})$ avec une précision $\epsilon$ où $M$ est un scalaire décrivant la régularité du noyau. Ces approximations peuvent donner lieu à des temps de restauration des images réduits de deux ordres de grandeurs.Les méthodes du deuxième groupe, couramment utilisées dans la communauté du traitement d'image, construisent une décomposition de rang faible de la Time Varying Impulse Response (TVIR) $T : \Omega \times \Omega \to \mathbb{R}$ définie par $T(x,y) = K(x+y,y)$. Elles sont restées jusqu'à présent non-étudiées et ce travail a permis de combler un manque dans la compréhension de leurs performances. De plus, il a permis d'identifier des représentations qui compressent considérablement les opérateurs et rendent possible l'estimation de l'opérateur.Dans une deuxième partie, ce travail aborde le délicat problème d'estimation des opérateurs. Récemment, de nouveaux travaux théoriques s'y sont intéressés, cependant aucun d'eux ne peut être implémenté en pratique. Dans le cas où quelques réponses impulsionnelles arbitrairement réparties dans l'espace sont connues, ce travail propose une construction d'un estimateur de l'opérateur qui soit numériquement applicable en grande dimension. Nous donnons aussi des garanties théoriques sur ses performances.Dans un troisième temps, cette thèse étudie d'autres types de dégradation. Les images issues de la microscopie à feuille de lumière sont altérées par des atténuations en forme de raies. Ce phénomène peut-être modélisé par un bruit multiplicatif structuré. Ce travail propose un modèle de résolution convexe qui rend le problème soluble aisément et qui donne des résultats probants sur des données réelles.Les indices de comparaison de deux images consistent une question délicate. Entre deux instants, une scène peut être illuminée différemment. Beaucoup d'indices sont sensibles à ces variations et échoueront à reconnaître la similarité de la scène entre deux instants. Nous avons proposé une mesure de similarité des images qui est invariante aux changements d'illumination.
- Published
- 2016
15. Contrast Invariant SNR
- Author
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Weiss, Pierre, Escande, Paul, Dong, Yiqiu, Institut de Mathématiques de Toulouse UMR5219 (IMT), Centre National de la Recherche Scientifique (CNRS)-PRES Université de Toulouse-Université Toulouse III - Paul Sabatier (UPS), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Institut des Technologies Avancées en sciences du Vivant (ITAV), Université Toulouse III - Paul Sabatier (UPS), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Département de Mathématiques, Informatique, Automatique (DMIA), Institut Supérieur de l'Aéronautique et de l'Espace (ISAE-SUPAERO), Department of Applied Mathematics and Computer Science [Lyngby] (DTU Compute), Technical University of Denmark [Lyngby] (DTU), and RITC OPTIMUS
- Subjects
convex optimization ,topographic map ,image quality measure ,[INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV] ,signal-to-noise-ratio ,ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION ,isotonic regression ,[INFO.INFO-CV]Computer Science [cs]/Computer Vision and Pattern Recognition [cs.CV] ,Local contrast change ,[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] ,illumination invariance ,[SPI.SIGNAL]Engineering Sciences [physics]/Signal and Image processing - Abstract
We design an image quality measure independent of local contrast changes, which constitute simple models of illumination changes. Given two images, the algorithm provides the image closest to the first one with the component tree of the second. This problem can be cast as a specific convex program called isotonic regression. We provide a few analytic properties of the solutions to this problem. We also design a tailored first order optimization procedure together with a full complexity analysis. The proposed method turns out to be practically more efficient and reliable than the best existing algorithms based on interior point methods. The algorithm has potential applications in change detection, color image processing or image fusion. A Matlab implementation is available at http://www.math.univ-toulouse.fr/ ∼ weiss/PageCodes.html.
- Published
- 2016
16. Approximation of integral operators using convolution-product expansions
- Author
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Escande, Paul and Weiss, Pierre
- Subjects
FOS: Mathematics ,Numerical Analysis (math.NA) ,Mathematics - Numerical Analysis - Abstract
We consider a class of linear integral operators with impulse responses varying regularly in time or space. These operators appear in a large number of applications ranging from signal/image processing to biology. Evaluating their action on functions is a computation-ally intensive problem necessary for many practical problems. We analyze a technique called convolution-product expansion: the operator is locally approximated by a convolution, allowing to design fast numerical algorithms based on the fast Fourier transform. We design various types of expansions, provide their explicit rates of approximation and their complexity depending on the time varying impulse response smoothness. This analysis suggests novel wavelet based implementations of the method with numerous assets such as optimal approximation rates, low complexity and storage requirements as well as adaptivity to the kernels regularity. The proposed methods are an alternative to more standard procedures such as panel clustering, cross approximations, wavelet expansions or hierarchical matrices.
- Published
- 2016
17. Approximation et Estimation des Opérateurs de Flou Variable
- Author
-
Escande, Paul, Weiss, Pierre, Centre National de la Recherche Scientifique - CNRS (FRANCE), Institut National des Sciences Appliquées de Toulouse - INSA (FRANCE), and Institut Supérieur de l'Aéronautique et de l'Espace - ISAE-SUPAERO (FRANCE)
- Subjects
Sparse approximation ,Autre ,Computer Science::Computer Vision and Pattern Recognition ,Deblurring ,ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION ,Integral operator ,Estimation - Abstract
Restoring images degraded by spatially varying blur is a problem encountered in many disciplines such as astrophysics, computer vision or biomedical imaging. Blurring operators are modelled using integral operators with some regularity and decrease conditions on the kernel. Recently, we studied the approximation of these operators in wavelet bases in which operators are highly compressible. They also allow to fastly compute matrix-vector products with a complexity $O(N\epsilon^{-d/M})$ for a precision $\epsilon$ in spectral norm, where N is the number of pixels of a d-dimensional image and M describes the kernel regularity. Additionnaly, we have shown that the sparsity pattern of the matrix can be pre-defined. We exploit these results to study the estimation/reconstruction of the operator from the knwoledge of few point spread functions located at arbitrary positions in the image domain. We propose an original formulation directly in the wavelet domain and a fast algorithm.
- Published
- 2016
18. Real-time $\ell^1$ − $\ell^2$ deblurring using wavelet expansions of operators
- Author
-
Escande, Paul, Weiss, Pierre, Département de Mathématiques, Informatique, Automatique (DMIA), Institut Supérieur de l'Aéronautique et de l'Espace (ISAE-SUPAERO), Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Institut des Technologies Avancées en sciences du Vivant (ITAV), Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées, Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
inverse problems ,ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION ,Numerical Analysis (math.NA) ,GPU programming ,[INFO.INFO-TS]Computer Science [cs]/Signal and Image Processing ,Optimization and Control (math.OC) ,preconditioning ,[INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV] ,FOS: Mathematics ,[INFO.INFO-IM]Computer Science [cs]/Medical Imaging ,Mathematics - Numerical Analysis ,[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] ,Mathematics - Optimization and Control ,sparse wavelet expansion ,[SPI.SIGNAL]Engineering Sciences [physics]/Signal and Image processing ,[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] ,image de-blurring - Abstract
Image deblurring is a fundamental problem in imaging, usually solved with com-putationally intensive optimization procedures. We show that the minimization can be significantly accelerated by leveraging the fact that images and blur operators are compressible in the same orthogonal wavelet basis. The proposed methodology consists of three ingredients: i) a sparse approximation of the blur operator in wavelet bases, ii) a diagonal preconditioner and iii) an implementation on massively parallel architectures. Combing the three ingredients leads to acceleration factors ranging from 30 to 250 on a typical workstation. For instance, a 1024 x 1024 image can be deblurred in 0.15 seconds, which corresponds to real-time.
- Published
- 2015
19. High-resolution in-depth imaging of optically cleared thick samples using an adaptive SPIM
- Author
-
Masson, Aurore, Escande, Paul, Frongia, Céline, Clouvel, Grégory, Ducommun, Bernard, Lorenzo, Corinne, Institut des Technologies Avancées en sciences du Vivant (ITAV), Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Département d'Ingénierie des Systèmes Complexes (DISC), Institut Supérieur de l'Aéronautique et de l'Espace (ISAE-SUPAERO), Imagine Optic, CHU Toulouse [Toulouse], Centre National de la Recherche Scientifique - CNRS (FRANCE), Institut National des Sciences Appliquées de Toulouse - INSA (FRANCE), Institut Supérieur de l'Aéronautique et de l'Espace - ISAE-SUPAERO (FRANCE), Centre Hospitalier Universitaire de Toulouse - CHU Toulouse (FRANCE), and Imagine Optic (FRANCE)
- Subjects
[PHYS.PHYS.PHYS-OPTICS]Physics [physics]/Physics [physics]/Optics [physics.optics] ,LSFM ,Blind deconvolution ,[SDV.BC]Life Sciences [q-bio]/Cellular Biology ,Optical aberrations ,Article ,Refractometry ,Imaging, Three-Dimensional ,Depth imaging ,Microscopy, Fluorescence ,Autre ,Cell Line, Tumor ,[INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV] ,Image Processing, Computer-Assisted ,Humans ,[SDV.BDD]Life Sciences [q-bio]/Development Biology ,ComputingMilieux_MISCELLANEOUS ,Sample clearing - Abstract
Today, Light Sheet Fluorescence Microscopy (LSFM) makes it possible to image fluorescent samples through depths of several hundreds of microns. However, LSFM also suffers from scattering, absorption and optical aberrations. Spatial variations in the refractive index inside the samples cause major changes to the light path resulting in loss of signal and contrast in the deepest regions, thus impairing in-depth imaging capability. These effects are particularly marked when inhomogeneous, complex biological samples are under study. Recently, chemical treatments have been developed to render a sample transparent by homogenizing its refractive index (RI), consequently enabling a reduction of scattering phenomena and a simplification of optical aberration patterns. One drawback of these methods is that the resulting RI of cleared samples does not match the working RI medium generally used for LSFM lenses. This RI mismatch leads to the presence of low-order aberrations and therefore to a significant degradation of image quality. In this paper, we introduce an original optical-chemical combined method based on an adaptive SPIM and a water-based clearing protocol enabling compensation for aberrations arising from RI mismatches induced by optical clearing methods and acquisition of high-resolution in-depth images of optically cleared complex thick samples such as Multi-Cellular Tumour Spheroids.
- Published
- 2015
20. Sparse Wavelet Representations of Spatially Varying Blurring Operators.
- Author
-
Escande, Paul and Weiss, Pierre
- Subjects
IMAGE reconstruction ,WAVELETS (Mathematics) ,WAVELET transforms ,ALGORITHMS ,MATHEMATICAL convolutions - Abstract
Restoring images degraded by spatially varying blur is a problem encountered in many disciplines such as astrophysics, computer vision, and biomedical imaging. One of the main challenges in performing this task is to design efficient numerical algorithms to approximate integral operators. We introduce a new method based on a sparse approximation of the blurring operator in the wavelet domain. This method requires O(Nϵ
-d/M ) operations to provide ϵ-approximations, where N is the number of pixels of a d-dimensional image and M ≥ 1 is a scalar describing the regularity of the blur kernel. In addition, we propose original methods to define sparsity patterns when only the operator regularity is known. Numerical experiments reveal that our algorithm provides a significant improvement compared to standard methods based on windowed convolutions. [ABSTRACT FROM AUTHOR]- Published
- 2015
- Full Text
- View/download PDF
21. Image restoration using sparse approximations of spatially varying blur operators in the wavelet domain.
- Author
-
Escande, Paul, Weiss, Pierre, and Malgouyres, François
- Published
- 2013
- Full Text
- View/download PDF
22. High-resolution in-depth imaging of optically cleared thick samples using an adaptive SPIM.
- Author
-
Masson A, Escande P, Frongia C, Clouvel G, Ducommun B, and Lorenzo C
- Subjects
- Cell Line, Tumor, Humans, Image Processing, Computer-Assisted methods, Microscopy, Fluorescence standards, Refractometry, Imaging, Three-Dimensional methods, Microscopy, Fluorescence methods
- Abstract
Today, Light Sheet Fluorescence Microscopy (LSFM) makes it possible to image fluorescent samples through depths of several hundreds of microns. However, LSFM also suffers from scattering, absorption and optical aberrations. Spatial variations in the refractive index inside the samples cause major changes to the light path resulting in loss of signal and contrast in the deepest regions, thus impairing in-depth imaging capability. These effects are particularly marked when inhomogeneous, complex biological samples are under study. Recently, chemical treatments have been developed to render a sample transparent by homogenizing its refractive index (RI), consequently enabling a reduction of scattering phenomena and a simplification of optical aberration patterns. One drawback of these methods is that the resulting RI of cleared samples does not match the working RI medium generally used for LSFM lenses. This RI mismatch leads to the presence of low-order aberrations and therefore to a significant degradation of image quality. In this paper, we introduce an original optical-chemical combined method based on an adaptive SPIM and a water-based clearing protocol enabling compensation for aberrations arising from RI mismatches induced by optical clearing methods and acquisition of high-resolution in-depth images of optically cleared complex thick samples such as Multi-Cellular Tumour Spheroids.
- Published
- 2015
- Full Text
- View/download PDF
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