Transductive few-shot learning has recently triggered wide attention in computer vision. Yet, current methods introduce key hyper-parameters, which control the prediction statistics of the test batches, such as the level of class balance, affecting performances significantly. Such hyper-parameters are empirically grid-searched over validation data, and their configurations may vary substantially with the target dataset and pre-training model, making such empirical searches both sub-optimal and computationally intractable. In this work, we advocate and introduce the unrolling paradigm, also referred to as "learning to optimize", in the context of few-shot learning, thereby learning efficiently and effectively a set of optimized hyperparameters. Specifically, we unroll a generalization of the ubiquitous Expectation-Maximization (EM) optimizer into a neural network architecture, mapping each of its iterates to a layer and learning a set of key hyper-parameters over validation data. Our unrolling approach covers various statistical feature distributions and pre-training paradigms, including recent foundational vision-language models and standard vision-only classifiers. We report comprehensive experiments, which cover a breadth of fine-grained downstream image classification tasks, showing significant gains brought by the proposed unrolled EM algorithm over iterative variants. The achieved improvements reach up to 10% and 7.5% on vision-only and vision-language benchmarks, respectively. The source code and learned parameters are available at https://github.com/ZhouLong0/UNEM-Transductive.

Abstract. This article introduces a novel approach to learning monotone neural networks (NNs) through a newly defined penalization loss. The proposed method is particularly effective in solving classes of variational problems, specifically monotone inclusion problems, commonly encountered in image processing tasks. The forward-backward-forward (FBF) algorithm is employed to address these problems, offering a solution even when the Lipschitz constant of the NN is unknown. Notably, the FBF algorithm provides convergence guarantees under the condition that the learned operator is monotone. Building on plug-and-play methodologies, our objective is to apply these newly learned operators to solving nonlinear inverse problems. To achieve this, we initially formulate the problem as a variational inclusion problem. Subsequently, we train a monotone NN to approximate an operator that may not inherently be monotone. Leveraging the FBF algorithm, we then show simulation examples where the nonlinear inverse problem is successfully solved.

Transductive inference has been widely investigated in few-shot image classification, but completely overlooked in the recent, fast growing literature on adapting vision-langage models like CLIP. This paper addresses the transductive zero-shot and few-shot CLIP classification challenge, in which inference is performed jointly across a mini-batch of unlabeled query samples, rather than treating each instance independently. We initially construct informative vision-text probability features, leading to a classification problem on the unit simplex set. Inspired by Expectation-Maximization (EM), our optimization-based classification objective models the data probability distribution for each class using a Dirichlet law. The minimization problem is then tackled with a novel block Majorization-Minimization algorithm, which simultaneously estimates the distribution parameters and class assignments. Extensive numerical experiments on 11 datasets underscore the benefits and efficacy of our batch inference approach. On zero-shot tasks with test batches of 75 samples, our approach yields near 20% improvement in ImageNet accuracy over CLIP's zero-shot performance. Additionally, we outperform state-of-the-art methods in the few-shot setting. The code is available at: https://github.com/SegoleneMartin/transductive-CLIP.

This paper introduces a novel optimization approach for stain separation in digital histopathological images. Our stain separation cost function incorporates a smooth total variation regularization and is minimized by using a projected gradient algorithm. To enhance computational efficiency and enable supervised learning of the hyperparameters, we further unroll our algorithm into a neural network. The unrolled architecture is not only more efficient for solving the stain separation problem, but also allows to design a highly interpretable and flexible method. Experimental results demonstrate the effectiveness of the proposed unrolled projected gradient algorithm in achieving accurate and visually consistent stain separation.

The optimization of prediction and update operators plays a prominent role in lifting-based image coding schemes. In this paper, we focus on learning the prediction and update models involved in a recent Fully Connected Neural Network (FCNN)-based lifting structure. While a straightforward approach consists in separately learning the different FCNN models by optimizing appropriate loss functions, jointly learning those models is a more challenging problem. To address this problem, we first consider a statistical model-based entropy loss function that yields a good approximation to the coding rate. Then, we develop a multi-scale optimization technique to learn all the FCNN models simultaneously. For this purpose, two loss functions defined across the different resolution levels of the proposed representation are investigated. While the first function combines standard prediction and update loss functions, the second one aims to obtain a good approximation to the rate-distortion criterion. Experimental results carried out on two standard image datasets, show the benefits of the proposed approaches in the context of lossy and lossless compression.

This article introduces a novel approach for building a robust Automatic Gesture Recognition system based on Surface Electromyographic (sEMG) signals, acquired at the forearm level. Our main contribution is to propose new constrained learning strategies that ensure robustness against adversarial perturbations by controlling the Lipschitz constant of the classifier. We focus on nonnegative neural networks for which accurate Lipschitz bounds can be derived, and we propose different spectral norm constraints offering robustness guarantees from a theoretical viewpoint. Experimental results on four publicly available datasets highlight that a good tradeoff in terms of accuracy and performance is achieved. We then demonstrate the robustness of our models, compared with standard trained classifiers in four scenarios, considering both white-box and black-box attacks.

Classification has been the focal point of research on adversarial attacks, but only a few works investigate methods suited to denser prediction tasks, such as semantic segmentation. The methods proposed in these works do not accurately solve the adversarial segmentation problem and, therefore, overestimate the size of the perturbations required to fool models. Here, we propose a white-box attack for these models based on a proximal splitting to produce adversarial perturbations with much smaller$\ell_{\infty}$norms. Our attack can handle large numbers of constraints within a nonconvex minimization framework via an Augmented Lagrangian approach, coupled with adaptive constraint scaling and masking strategies. We demonstrate that our attack significantly outperforms previously proposed ones, as well as classification attacks that we adapted for segmentation, providing a first comprehensive benchmark for this dense task.

Neural networks have become ubiquitous tools for solving signal and image processing problems, and they often outperform standard approaches. Nevertheless, training the layers of a neural network is a challenging task in many applications. The prevalent training procedure consists of minimizing highly non-convex objectives based on data sets of huge dimension. In this context, current methodologies are not guaranteed to produce global solutions. We present an alternative approach which foregoes the optimization framework and adopts a variational inequality formalism. The associated algorithm guarantees convergence of the iterates to a true solution of the variational inequality and it possesses an efficient block-iterative structure. A numerical application is presented.

Independent vector analysis (IVA) generalizes independent component analysis (ICA) to multiple datasets, and when used with a multivariate Gaussian model (IVA-G), provides a powerful tool for joint analysis of multiple datasets in an array of applications. While IVA-G enjoys uniqueness guarantees, the current solution to the problem exhibits significant variability across runs necessitating the use of a scheme for selecting the most consistent one, which is costly. In this paper, we present a penalized maximum-likelihood framework for the problem, which enables us to derive a non-convex cost function that depends on the precision matrices of the source component vectors, the main mechanism by which IVA-G leverages correlation across the datasets. By adding a quadratic regularization, a block-coordinate proximal algorithm is shown to offer a suitable solution to this minimization problem. The proposed method also provides convergence guarantees that are lacking in other state-of-the-art approaches to the problem. This also allows us to obtain overall slightly better performance, and in particular, we show that our method yields better estimation in average than the current IVA-G algorithm for various source numbers, datasets, and degrees of correlation across the data.

Abstract. Most optimization problems arising in imaging science involve high-dimensional linear operators and their adjoints. In the implementations of these operators, changes may be introduced for various practical considerations (e.g., memory limitation, computational cost, convergence speed), leading to an adjoint mismatch. This occurs for the X-ray tomographic inverse problems found in computed tomography (CT), where a surrogate operator often replaces the adjoint of the measurement operator (called the projector). The resulting adjoint mismatch can jeopardize the convergence properties of iterative schemes used for image recovery. In this paper, we study the theoretical behavior of a panel of primal-dual proximal algorithms, which rely on forward-backward-(forward) splitting schemes when an adjoint mismatch occurs. We analyze these algorithms by focusing on the resolution of possibly nonsmooth convex penalized minimization problems in an infinite-dimensional setting. Using tools from fixed point theory, we show that they can solve monotone inclusions beyond minimization problems. Such findings indicate that these algorithms can be seen as a generalization of classical primal-dual formulations. The applicability of our findings is also demonstrated through two numerical experiments in the context of CT image reconstruction.

In this paper, we introduce a variational Bayesian algorithm (VBA) for image blind deconvolution. Our VBA generic framework incorporates smoothness priors on the unknown blur/image and possible affine constraints (e.g., sum to one) on the blur kernel, integrating the VBA within a neural network paradigm following an unrolling methodology. The proposed architecture is trained in a supervised fashion, which allows us to optimally set two key hyperparameters of the VBA model and leads to further improvements in terms of resulting visual quality. Various experiments involving grayscale/color images and diverse kernel shapes, are performed. The numerical examples illustrate the high performance of our approach when compared to state-of-the-art techniques based on optimization, Bayesian estimation, or deep learning.

In this paper, we propose a novel variational approach for supervised classification based on transform learning. Our approach consists of formulating an optimization problem on both the transform matrix and the centroids of the classes in a low-dimensional transformed space. The loss function is based on the distance to the centroids, which can be chosen in a flexible manner. To avoid trivial solutions or highly correlated clusters, our model incorporates a penalty term on the centroids, which encourages them to be separated. The resulting non-convex and non-smooth minimization problem is then solved by a primal-dual alternating minimization strategy. We assess the performance of our method on a bunch of supervised classification problems and compare it to state-of-the-art methods.

The multivariate power exponential (MEP) distribution can model a broad range of signals. In noisy scenarios, the robust estimation of the MEP parameters has been traditionally addressed by a fixed-point approach associated with a nonconvex optimization problem. Establishing convergence properties for this approach when the distribution mean is unknown is still an open problem. As an alternative, this paper presents a novel convex formulation for robustly estimating MEP parameters in the presence of multiplicative perturbations. The proposed approach is grounded on a re-parametrization of the original likelihood function in a way that ensures convexity. We also show that this property is preserved for several typical regularization functions. Compared with the robust Tyler’s estimator, the proposed method shows a more accurate precision matrix estimation, with similar mean and covariance estimation performance.

Positron emission tomography (PET) is a quantitative imaging modality widely used in oncology, neurology, and pharmacology. The data acquired by a PET scanner correspond to projections of the concentration activity, assumed to follow a Poisson distribution. The reconstruction of images from tomographic projections corrupted by Poisson noise is a challenging ill-posed large-scale inverse problem. Several available solvers use the majorization-minimization (MM) principle, though relying on various construction strategies with a lack of unifying framework. This work fills the gap by introducing the concept of Bregman majorization. This leads to a unified view of MM-based methods for image reconstruction in the presence of Poisson noise. From this general approach, we exhibit three algorithmic solutions and compare their computational efficiency on a problem of dynamic PET image reconstruction, either using GPU or CPU processing.

Standard few-shot benchmarks are often built upon simplifying assumptions on the query sets, which may not always hold in practice. In particular, for each task at testing time, the classes effectively present in the unlabeled query set are known a priori, and correspond exactly to the set of classes represented in the labeled support set. We relax these assumptions and extend current benchmarks, so that the query-set classes of a given task are unknown, but just belong to a much larger set of possible classes. Our setting could be viewed as an instance of the challenging yet practical problem of extremely imbalanced $K$-way classification, $K$ being much larger than the values typically used in standard benchmarks, and with potentially irrelevant supervision from the support set. Expectedly, our setting incurs drops in the performances of state-of-the-art methods. Motivated by these observations, we introduce a \textbf{P}rim\textbf{A}l \textbf{D}ual Minimum \textbf{D}escription \textbf{LE}ngth (\textbf{PADDLE}) formulation, which balances data-fitting accuracy and model complexity for a given few-shot task, under supervision constraints from the support set. Our constrained MDL-like objective promotes competition among a large set of possible classes, preserving only effective classes that befit better the data of a few-shot task. It is hyper-parameter free, and could be applied on top of any base-class training. Furthermore, we derive a fast block coordinate descent algorithm for optimizing our objective, with convergence guarantee, and a linear computational complexity at each iteration. Comprehensive experiments over the standard few-shot datasets and the more realistic and challenging \textit{i-Nat} dataset show highly competitive performances of our method, more so when the numbers of possible classes in the tasks increase. Our code is publicly available at \url{https://github.com/SegoleneMartin/PADDLE}.

This work addresses the resolution of penalized least-squares problems using the proximal gradient algorithm (PGA). PGA can be accelerated by preconditioning strategies. However, typical effective choices of preconditioners may correspond to intricate matrices that are not easily inverted, leading to increased complexity in the computation of the proximity step. To relax these requirements, we propose an unmatched preconditioning approach where two metrics are used in the gradient step and the proximity step. We provide convergence conditions for this new iterative scheme and characterize its limit point. Simulations for tomographic image reconstruction from undersampled measurements show the benefits of our approach for various simple choices of metrics.

Lifting-based wavelet transform has been extensively used for efficient compression of various types of visual data. Generally, the performance of such coding schemes strongly depends on the lifting operators used, namely the prediction and update filters. Unlike conventional schemes based on linear filters, we propose, in this paper, to learn these operators by exploiting neural networks. More precisely, a classical Fully Connected Neural Network (FCNN) architecture is firstly employed to perform the prediction and update. Then, we propose to improve this FCNN-based Lifting Scheme (LS) in order to better take into account the input image to be encoded. Thus, a novel dynamical FCNN model is developed, making the learning process adaptive to the input image contents for which two adaptive learning techniques are proposed. While the first one resorts to an iterative algorithm where the computation of two kinds of variables is performed in an alternating manner, the second learning method aims to learn the model parameters directly through a reformulation of the loss function. Experimental results carried out on various test images show the benefits of the proposed approaches in the context of lossy and lossless image compression.

This paper presents a quantitative approach to demonstrate the robustness of neural networks for tabular data. These data form the backbone of the data structures found in most industrial applications. We analyse the effect of various widely used techniques we encounter in neural network practice, such as regularization of weights, addition of noise to the data, and positivity constraints. This analysis is performed by using three state-of-the-art techniques, which provide mathematical proofs of robustness in terms of Lipschitz constant for feed-forward networks. The experiments are carried out on two prediction tasks and one classification task. Our work brings insights into building robust neural network architectures for safety critical systems that require certification or approval from a competent authority.

Based on its great successes in inference and denosing tasks, Dictionary Learning (DL) and its related sparse optimization formulations have garnered a lot of research interest. While most solutions have focused on single layer dictionaries, the recently improved Deep DL methods have also fallen short on a number of issues. We hence propose a novel Deep DL approach where each DL layer can be formulated and solved as a combination of one linear layer and a Recurrent Neural Network, where the RNN is flexibly regraded as a layer-associated learned metric. Our proposed work unveils new insights between the Neural Networks and Deep DL, and provides a novel, efficient and competitive approach to jointly learn the deep transforms and metrics. Extensive experiments are carried out to demonstrate that the proposed method can not only outperform existing Deep DL, but also state-of-the-art generic Convolutional Neural Networks.

The objective of this paper is to investigate techniques for learning Fully Connected Network (FCN) models in a lifting based image coding scheme. More precisely, based on a 2D non separable lifting structure composed of three FCN-based prediction stages followed by an FCN-based update one, we first propose to resort to an $\ell_{p}$ loss function, with $p\in\{1,2\}$, to learn the three FCN prediction models. While the latter are separately learned in the first approach, a novel joint learning approach is then developed by minimizing a weighted $\ell_{p}$ loss function related to the global prediction error. Experimental results, carried out on the standard Challenge Learned Image Compression (CLIC) dataset, show the benefits of the proposed techniques in terms of rate-distortion performance.

Image restoration has long been one of the key research topics in image processing. Many mathematical approaches have been developed to solve this problem, e.g., variational methods, wavelet techniques, or Bayesian methods. With the widespread of neural network (NN) models in all the subdomains of data science, the performance limits of these methods are further pushed. One of the most successful strategies consists of plugging NNs in existing optimization algorithms. However, so doing raises several mathematical and practical challenges. One of the main issues is to secure the convergence of the resulting iterative scheme. Further questions concerning the characterization of the reached limit are also worth being addressed. In this paper, we show that the theory of maximally monotone operators allows us to bring insightful answers to these problems and to design firmly nonexpansive NNs; combining these with postprocessing NNs leads to excellent global restoration quality.

Sparsifying deep neural networks is of paramount interest in many areas, especially when those networks have to be implemented on low-memory devices. In this article, we propose a new formulation of the problem of generating sparse weights for a pre-trained neural network. By leveraging the properties of standard nonlinear activation functions, we show that the problem is equivalent to an approximate subdifferential inclusion problem. The accuracy of the approximation controls the sparsity. We show that the proposed approach is valid for a broad class of activation functions (ReLU, sigmoid, softmax). We propose an iterative optimization algorithm to induce sparsity whose convergence is guaranteed. Because of the algorithm flexibility, the sparsity can be ensured from partial training data in a minibatch manner. To demonstrate the effectiveness of our method, we perform experiments on various networks in different applicative contexts: image classification, speech recognition, natural language processing, and time-series forecasting.

We introduce a new paradigm for solving regularized variational problems. These are typically formulated to address ill-posed inverse problems encountered in signal and image processing. The objective function is traditionally defined by adding a regularization function to a data fit term, which is subsequently minimized by using iterative optimization algorithms. Recently, several works have proposed to replace the operator related to the regularization by a more sophisticated denoiser. These approaches, known as plug-and-play (PnP) methods, have shown excellent performance. Although it has been noticed that, under some Lipschitz properties on the denoisers, the convergence of the resulting algorithm is guaranteed, little is known about characterizing the asymptotically delivered solution. In the current article, we propose to address this limitation. More specifically, instead of employing a functional regularization, we perform an operator regularization, where a maximally monotone operator (MMO) is learned in a supervised manner. This formulation is flexible as it allows the solution to be characterized through a broad range of variational inequalities, and it includes convex regularizations as special cases. From an algorithmic standpoint, the proposed approach consists in replacing the resolvent of the MMO by a neural network (NN). We present a universal approximation theorem proving that nonexpansive NNs are suitable models for the resolvent of a wide class of MMOs. The proposed approach thus provides a sound theoretical framework for analyzing the asymptotic behavior of first-order PnP algorithms. In addition, we propose a numerical strategy to train NNs corresponding to resolvents of MMOs. We apply our approach to image restoration problems and demonstrate its validity in terms of both convergence and quality.

Electrical motors are the most important source of mechanical energy in the industrial world. Their modeling traditionally relies on a physics-based approach, which aims at taking their complex internal dynamics into account. In this paper, we explore the feasibility of modeling the dynamics of an electrical motor by following a data-driven approach, which uses only its inputs and outputs and does not make any assumption on its internal behaviour. We propose a novel encoder-decoder architecture which benefits from recurrent skip connections. We also propose a novel loss function that takes into account the complexity of electrical motor quantities and helps in avoiding model bias. We show that the proposed architecture can achieve a good learning performance on our high-frequency high-variance datasets. Two datasets are considered: the first one is generated using a simulator based on the physics of an induction motor and the second one is recorded from an industrial electrical motor. We benchmark our solution using variants of traditional neural networks like feedforward, convolutional, and recurrent networks. We evaluate various design choices of our architecture and compare it to the baselines. We show the domain adaptation capability of our model to learn dynamics just from simulated data by testing it on the raw sensor data. We finally show the effect of signal complexity on the proposed method ability to model temporal dynamics.

This paper presents a new scheme to perform the canonical polyadic decomposition (CPD) of a symmetric tensor. We first formulate the CPD problem as a truncated moment problem, where a measure has to be recovered knowing some of its moments. The support of the measure is discrete and encodes the CPD. The support is then retrieved by solving a polynomial system. Using algebraic results, our method resorts only to classical linear algebra operations (eigenvalue method and Schur reordered factorization). This new viewpoint offers theoretical guarantees on the retrieved decomposition. Finally experimental results show the validity of our method and a better reconstruction accuracy compared to classic CPD algorithms.

This work proposes a new learning strategy for training a feedforward neural network subject to spectral norm and nonnegativity constraints. Our primary goal is to control the Lipschitz constant of the network in order to make it robust against adversarial perturbations of its inputs. We propose a stochastic projected gradient descent algorithm which allows us to adjust this constant in the training process. The algorithm is evaluated in the context of designing a fully connected network for Automatic Gesture Recognition based on EMG signals. We perform a comparison with the same architecture trained either in a standard manner or with simpler constraints. The obtained results highlight that a good accuracy-robustness trade-off can be achieved.

Building nonexpansive Convolutional Neural Networks (CNNs) is a challenging problem that has recently gained a lot of attention from the image processing community. In particular, it appears to be the key to obtain convergent Plugand-Play algorithms. This problem, which relies on an accurate control of the the Lipschitz constant of the convolutional layers, has also been investigated for Generative Adversarial Networks to improve robustness to adversarial perturbations. However, to the best of our knowledge, no efficient method has been developed yet to build nonexpansive CNNs. In this paper, we develop an optimization algorithm that can be incorporated in the training of a network to ensure the nonexpansiveness of its convolutional layers. This is shown to allow us to build firmly nonexpansive CNNs. We apply the proposed approach to train a CNN for an image denoising task and show its effectiveness through simulations.

Wavelet decompositions based on lifting schemes have been widely used in image coding. Generally, the efficiency of such compression methods strongly depends on the design of the lifting operators, namely the prediction and update filters. To improve their performance, we propose in this paper to optimize these filters by resorting to two learning strategies. In the first one, classical Fully Connected Networks (FCNs) are exploited to perform the prediction and update. In the second approach, we develop an adaptive learning method that takes into account the input image, yielding a dynamical model of FCN. Experimental results, carried out on the standard Challenge Learned Image Compression (CLIC) dataset, show the benefits that can be drawn from the proposed approaches compared to conventional ones.

This paper focuses on the quantitative analysis of deep neural networks used in data-driven modeling of induction motor dynamics. With the availability of a large amount of data generated by industrial sensor networks, it is now possible to train deep neural networks. Recently researchers have started exploring the usage of such networks for physics modeling, online control, monitoring, and fault prediction in induction motor operations. We consider the problem of estimating speed and torque from currents and voltages of an induction motor. Neural networks provide quite good performance for this task when analysed from a machine learning perspective using standard metrics. We show, however, that there are some caveats in using machine learning metrics to analyze a neural network model when applied to induction motor problems. Given the mission- critical nature of induction motor operations, the performance of neural networks has to be validated from an electrical engineering point of view. To this end, we evaluate several traditional neural network architectures and recent state of the art architectures on dynamic and quasi-static benchmarks using electrical engineering metrics.

We study a variational formulation for reconstructing nonlinearly distorted signals corrupted with a Poisson-Gaussian noise. In this situation, the data fidelity term consists of a sum of a weighted least squares term and a logarithmic one. Both of them are precomposed by a nonlinearity, modelling a clipping effect, which is assumed to be rational. A regularization term, being a piecewise rational approximation of the ℓ0function provides a suitable sparsity measure with respect to a preset linear operator. We propose a global optimization approach for such a problem. More specifically, it is first transformed into a generalized moment problem by introducing some auxiliary variables. Then, a hierarchy of semidefinite programming relaxations is built. Numerical examples show the good performance of the proposed approach.

For dealing with sparse models, a large number of continuous approximations of the ℓ0penalization have been proposed. However, the most accurate ones lead to non-convex opti-mization problems. In this paper, by observing that many such approximations are piecewise rational functions, we show that the original optimization problem can be recast as a multivariate polynomial problem. The latter is then globally solved by using recent optimization methods which consist of building a hierarchy of convex problems. Finally, experimental results illustrate that our method always provides a global optimum of the initial problem for standard ℓ0approximations. This is in contrast with existing local algorithms whose results depend on the initialization.

Image restoration is frequently addressed by resorting to variational methods which account for some prior knowledge about the solution. The success of these methods, however, heavily depends on the estimation of a set of hyperparameters. Deep learning architectures are, on the contrary, very generic and efficient, but they offer limited control over their output. In this paper, we present iRestNet, a neural network architecture which combines the benefits of both approaches. iRestNet is obtained by unfolding a proximal interior point algorithm. This enables enforcing hard constraints on the pixel range of the restored image thanks to a logarithmic barrier strategy, without requiring any parameter setting. Explicit expressions for the involved proximity operator, and its differential, are derived, which allows training iRestNet with gradient descent and backpropagation. Numerical experiments on image deblurring show that the proposed approach provides good image quality results compared to state-of-the-art variational and machine learning methods.

Joint deconvolution and segmentation of ultrasound images is a challenging problem in medical imaging. By adopting a hierarchical Bayesian model, we propose an accelerated Markov chain Monte Carlo scheme where the tissue reflectivity function is sampled thanks to a recently introduced proximal unadjusted Langevin algorithm. This new approach is combined with a forward-backward step and a preconditioning strategy to accelerate the convergence, and with a method based on the majorization-minimization principle to solve the inner nonconvex minimization problems. As demonstrated in numerical experiments conducted on both simulated and in vivo ultrasound images, the proposed method provides high-quality restoration and segmentation results and is up to six times faster than an existing Hamiltonian Monte Carlo method.

In recent years, there has been a growing interest in problems in graph estimation and model selection, which all share very similar matrix variational formulations, the most popular one being probably GLASSO. Unfortunately, the standard GLASSO formulation does not take into account noise corrupting the data: this shortcoming leads us to propose a novel criterion, where the regularization function is decoupled in two terms, one acting only on the eigenvalues of the matrix and the other on the matrix elements. Incorporating noise information into the model has the side-effect to make the cost function non-convex. To overcome this difficulty, we adopt a majorization-minimization approach, where at each iteration a convex approximation of the original cost function is minimized via the Douglas-Rachford procedure. The achieved results are very promising w.r.t. classical approaches.

Interior point methods have been known for decades to be useful for the resolution of small to medium size constrained optimization problems. These approaches have the benefit of ensuring feasibility of the iterates through a logarithmic barrier. We propose to incorporate a proximal forward-backward step in the resolution of the barrier subproblem to account for non-necessarily differentiable terms arising in the objective function. The combination of this scheme with a novel line-search strategy gives rise to the so-called Proximal Interior Point Algorithm (PIPA) suitable for the minimization of the sum of a smooth convex function and a non-smooth convex one under general convex constraints. The convergence of PIPA is secured under mild assumptions. As demonstrated by numerical experiments carried out on a large-scale hyperspectral image unmixing application, the proposed method outperforms the state-of-the-art.

Texture segmentation still constitutes an on-going challenge, especially when processing large-size images. Recently, procedures integrating a scale-free (or fractal) wavelet-leader model allowed the problem to be reformulated in a convex optimization framework by including a TV penalization. In this case, the TV penalty plays a prominent role with respect to the data fidelity term, which makes the approach costly in terms of memory and computation cost. The present contribution aims to investigate the potential of recent block-coordinate dual and primal-dual proximal algorithms for overcoming this numerical issue. Our study shows that a key ingredient in the success of the proposed block-coordinate approaches lies in the design of the blocks of variables which are updated at each iteration. Numerical experiments conducted over synthetic textures having piece-wise constant fractal properties confirm our theoretical analysis. The proposed lattice block design strategy is shown to yield significantly lower memory and computational requirements.

In this paper, we present a new distributed algorithm for minimizing a sum of non-necessarily differentiable convex functions composed with arbitrary linear operators. The overall cost function is assumed strongly convex. Each involved function is associated with a node of a hypergraph having the ability to communicate with neighboring nodes sharing the same hyperedge. Our algorithm relies on a primal-dual splitting strategy with established convergence guarantees. We show how it can be efficiently implemented to take full advantage of a multicore architecture. The good numerical performance of the proposed approach is illustrated in a problem of video sequence denoising, where a significant speedup is achieved.

Discovering meaningful gene interactions is crucial for the identification of novel regulatory processes in cells. Building accurately the related graphs remains challenging due to the large number of possible solutions from available data. Nonetheless, enforcing a priori on the graph structure, such as modularity, may reduce network indeterminacy issues. BRANE Clust (Biologically-Related A priori Network Enhancement with Clustering) refines gene regulatory network (GRN) inference thanks to cluster information. It works as a post-processing tool for inference methods (i.e., CLR, GENIE3). In BRANE Clust, the clustering is based on the inversion of a system of linear equations involving a graph-Laplacian matrix promoting a modular structure. Our approach is validated on DREAM4 and DREAM5 datasets with objective measures, showing significant comparative improvements. We provide additional insights on the discovery of novel regulatory or co-expressed links in the inferred Escherichia coli network evaluated using the STRING database. The comparative pertinence of clustering is discussed computationally (SIMoNe, WGCNA, X-means) and biologically (RegulonDB). BRANE Clust software is available at: http://www-syscom.univ-mlv.fr/~pirayre/Codes-GRN-BRANE-clust.html.

While φ-divergences have been extensively studied in convex analysis, their use in optimization problems often remains challenging. In this regard, one of the main shortcomings of existing methods is that the minimization of φ-divergences is usually performed with respect to one of their arguments, possibly within alternating optimization techniques. In this paper, we overcome this limitation by deriving new closed-form expressions for the proximity operator of such two-variable functions. This makes it possible to employ standard proximal methods for efficiently solving a wide range of convex optimization problems involving φ-divergences. In addition, we show that these proximity operators are useful to compute the epigraphical projection of several functions. The proposed proximal tools are numerically validated in the context of optimal query execution within database management systems, where the problem of selectivity estimation plays a central role. Experiments are carried out on small to large scale scenarios.

Variational Bayesian approaches have been successfully applied to image segmentation. They usually rely on a Potts model for the hidden label variables and a Gaussian assumption on pixel intensities within a given class. Such models may however be limited, especially in the case of multicomponent images. We overcome this limitation with HOGMep, a Bayesian formulation based on a higher-order graphical model (HOGM) on labels and a Multivariate Exponential Power (MEP) prior for intensities in a class. Then, we develop an efficient statistical estimation method to solve the associated problem. Its flexibility accommodates to a broad range of applications, demonstrated on multicomponent image segmentation and restoration.

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In the field of 3D image recovery, huge amounts of data need to be processed. Parallel optimization methods are then of main interest since they allow to overcome memory limitation issues, while benefiting from the intrinsic acceleration provided by recent multicore computing architectures. In this context, we propose a Block Parallel Majorize-Minimize Memory Gradient (BP3MG) algorithm for solving large scale optimization problems. This algorithm combines a block coordinate strategy with an efficient parallel update. The proposed method is applied to a 3D microscopy image restoration problem involving a depth-variant blur, where it is shown to lead to significant computational time savings with respect to a sequential approach.

State-of-the-art methods for solving smooth optimization problems are nonlinear conjugate gradient, low memory BFGS, and majorize-minimize (MM) subspace algorithms. The MM subspace algorithm that has been introduced more recently has shown good practical performance when compared with other methods on various optimization problems arising in signal and image processing. However, to the best of our knowledge, no general result exists concerning the theoretical convergence rate of the MM subspace algorithm. This paper aims at deriving such convergence rates both for batch and online versions of the and in particular, discusses the influence of the choice of the subspace.

With the advent of microarrays, arose the need to analyze gene expression data. Tools for building gene regulation networks are indeed of high interest for regulatory relationship sketching and gene interaction prediction. Given all pairwise gene regulation information available, we propose to determine the presence of edges in the final gene regulatory network by adopting a convex optimization formulation. Our energy minimization strategy includes a regularization term accounting for the difference of connectivity of particular genes (i.e. transcription factors), and we employ proximal methods to compute the optimal solution. The resulting algorithm, called “Brane relax”, outperforms state-of-the-art methods while keeping a reduced computational cost.

Primal-dual proximal optimization methods have recently gained much interest for dealing with very large-scale data sets encoutered in many application fields such as machine learning, computer vision and inverse problems [1-3]. In this work, we propose a novel random block-coordinate version of such algorithms allowing us to solve a wide array of convex variational problems. One of the main advantages of the proposed algorithm is its ability to solve composite problems involving large-size matrices without requiring any inversion. In addition, the almost sure convergence to an optimal solution to the problem is guaranteed. We illustrate the good performance of our method on a mesh denoising application.

Inverse problems encountered in video processing often require to minimize criteria involving a high number of variables. Among available optimization techniques, proximal methods have shown their efficiency in solving large-scale possibly nonsmooth problems. When some of the proximity operators involved in these methods do not have closed form expressions, they may constitute a bottleneck in terms of computational complexity and memory requirements. In this paper, we address this problem and propose accelerated techniques for solving it. A new dual block-coordinate forward-backward algorithm computing the proximity operator of a sum of convex functions composed with linear operators is proposed and theoretically analyzed. The numerical performance of the approach is assessed through an application to deconvolution and super-resolution of interlaced video sequences.

When acquiring high-speed video (more than 100 frames per second), artificial lighting can cause severe non-uniform luminosity and chroma variation between frames, commonly labeled periodic flicker. Non-uniform periodic flicker is not easy to correct in the presence of general motion, since its estimation requires background and object tracking, and most tracking techniques assume consistent illumination. In this paper, we propose a joint tracking/color correction scheme using a block matching technique paired with color variation estimation. We introduce a robust method for stabilizing brightness variations in image sequences. A post-processing step is also proposed in order to deal with blocking artifacts. We demonstrate the efficacy of our method both on simulated and real data.

This paper proposes a novel approach for echo-like multiple removal in two-dimensional seismic images. It is based on constrained adaptive filtering associated with geometric wavelets. Approximate templates of multiple reflections are assumed to be available and they are matched to multiple reflections throughout estimated finite impulse response filters. The problem is formulated under a constrained convex optimization form where the data of interest and filters are estimated jointly. Proximal approaches are used to perform the minimization of the derived criterion. The effectiveness of the proposed approach is demonstrated with various noise levels on realistic simulated data and on field seismic data.

BACKGROUND: Inferring gene networks from high-throughput data constitutes an important step in the discovery of relevant regulatory relationships in organism cells. Despite the large number of available Gene Regulatory Network inference methods, the problem remains challenging: the underdetermination in the space of possible solutions requires additional constraints that incorporate a priori information on gene interactions. METHODS: Weighting all possible pairwise gene relationships by a probability of edge presence, we formulate the regulatory network inference as a discrete variational problem on graphs. We enforce biologically plausible coupling between groups and types of genes by minimizing an edge labeling functional coding for a priori structures. The optimization is carried out with Graph cuts, an approach popular in image processing and computer vision. We compare the inferred regulatory networks to results achieved by the mutual-information-based Context Likelihood of Relatedness (CLR) method and by the state-of-the-art GENIE3, winner of the DREAM4 multifactorial challenge. RESULTS: Our BRANE Cut approach infers more accurately the five DREAM4 in silico networks (with improvements from 6% to 11%). On a real Escherichia coli compendium, an improvement of 11.8% compared to CLR and 3% compared to GENIE3 is obtained in terms of Area Under Precision-Recall curve. Up to 48 additional verified interactions are obtained over GENIE3 for a given precision. On this dataset involving 4345 genes, our method achieves a performance similar to that of GENIE3, while being more than seven times faster. The BRANE Cut code is available at: http://www-syscom.univ-mlv.fr/~pirayre/Codes-GRN-BRANE-cut.html. CONCLUSIONS: BRANE Cut is a weighted graph thresholding method. Using biologically sound penalties and data-driven parameters, it improves three state-of-the art GRN inference methods. It is applicable as a generic network inference post-processing, due to its computational efficiency.

The Poisson--Gaussian model can accurately describe the noise present in a number of imaging systems. However most existing restoration methods rely on approximations of the Poisson--Gaussian noise statistics. We propose a convex optimization strategy for the reconstruction of images degraded by a linear operator and corrupted with a mixed Poisson--Gaussian noise. The originality of our approach consists of considering the exact, mixed continuous-discrete model corresponding to the data statistics. After establishing the Lipschitz differentiability and convexity of the Poisson--Gaussian neg-log-likelihood, we derive a primal-dual iterative scheme for minimizing the associated penalized criterion. The proposed method is applicable to a large choice of convex penalty terms. The robustness of our scheme allows us to handle computational difficulties due to infinite sums arising from the computation of the gradient of the criterion. We propose finite bounds for these sums, that are dependent on the current image estimate, and thus adapted to each iteration of our algorithm. The proposed approach is validated on image restoration examples. Then, the exact data fidelity term is used as a reference for studying some of its various approximations. We show that in a variational framework the shifted Poisson and exponential approximations lead to very good restoration results.

The ℓ1/ℓ2ratio regularization function has shown good performance for retrieving sparse signals in a number of recent works, in the context of blind deconvolution. Indeed, it benefits from a scale invariance property much desirable in the blind context. However, the ℓ1/ℓ2function raises some difficulties when solving the nonconvex and nonsmooth minimization problems resulting from the use of such a penalty term in current restoration methods. In this paper, we propose a new penalty based on a smooth approximation to the ℓ1/ℓ2function. In addition, we develop a proximal-based algorithm to solve variational problems involving this function and we derive theoretical convergence results. We demonstrate the effectiveness of our method through a comparison with a recent alternating optimization strategy dealing with the exact ℓ1/ℓ2term, on an application to seismic data blind deconvolution.

Sparsity inducing penalizations are useful tools in variational methods for machine learning. In this paper, we design a learning algorithm for multiclass support vector machines that allows us to enforce sparsity through various nonsmooth regularizations, such as the mixed ℓ1, p-norm with p ≥ 1. The proposed constrained convex optimization approach involves an epigraphical constraint for which we derive the closed-form expression of the associated projection. This sparse multiclass SVM problem can be efficiently implemented thanks to the flexibility offered by recent primal-dual proximal algorithms. Experiments carried out for handwritten digits demonstrate the interest of considering nonsmooth sparsity-inducing regularizations and the efficiency of the proposed epigraphical projection method.

The objective of this paper is to study rate-distortion properties of a quantized Bernoulli-Generalized Gaussian source. Such source model has been found to be well-adapted for signals having a sparse representation in a transformed domain. We provide here accurate approximations of the entropy and the distortion functions evaluated through a p-th order error measure. These theoretical results are then validated experimentally. Finally, the benefit that can be drawn from the proposed approximations in bit allocation problems is illustrated for a wavelet-based compression scheme.

Random and structured noise both affect seismic data, hiding the reflections of interest (primaries) that carry meaningful geophysical interpretation. When the structured noise is composed of multiple reflections, its adaptive cancellation is obtained through time-varying filtering, compensating inaccuracies in given approximate templates. The under-determined problem can then be formulated as a convex optimization one, providing estimates of both filters and primaries. Within this framework, the criterion to be minimized mainly consists of two parts: a data fidelity term and hard constraints modeling a priori information. This formulation may avoid, or at least facilitate, some parameter determination tasks, usually difficult to perform in inverse problems. Not only classical constraints, such as sparsity, are considered here, but also constraints expressed through hyperplanes, onto which the projection is easy to compute. The latter constraints lead to improved performance by further constraining the space of geophysically sound solutions.

Many inverse problems require to minimize a criterion being the sum of a non necessarily smooth function and a Lipschitz differentiable function. Such an optimization problem can be solved with the Forward-Backward algorithm which can be accelerated thanks to the use of variable metrics derived from the Majorize-Minimize principle. The convergence of this approach is guaranteed provided that the criterion satisfies some additional technical conditions. Combining this method with an alternating minimization strategy will be shown to allow us to address a broad class of optimization problems involving large-size signals. An application example to a nonconvex spectral unmixing problem will be presented.

A wide array of image recovery problems can be abstracted into the problem of minimizing a sum of composite convex functions in a Hilbert space. To solve such problems, primal-dual proximal approaches have been developed which provide efficient solutions to large-scale optimization problems. The objective of this paper is to show that a number of existing algorithms can be derived from a general form of the forward-backward algorithm applied in a suitable product space. Our approach also allows us to develop useful extensions of existing algorithms by introducing a variable metric. An illustration to image restoration is provided.

This paper deals with noise parameter estimation from a single image under Poisson-Gaussian noise statistics. The problem is formulated within a mixed discrete-continuous optimization framework. The proposed approach jointly estimates the signal of interest and the noise parameters. This is achieved by introducing an adjustable regularization term inside an optimized criterion, together with a data fidelity error measure. The optimal solution is sought iteratively by alternating the minimization of a label field and of a noise parameter vector. Noise parameters are updated at each iteration using an Expectation-Maximization approach. The proposed algorithm is inspired from a spatial regularization approach for vector quantization. We illustrate the usefulness of our approach on macroconfocal images. The identified noise parameters are applied to a denoising algorithm, so yielding a complete denoising scheme.

With the development of new imaging systems delivering large-size data sets, phase retrieval has become recently the focus of much attention. The problem is especially challenging due to its intrinsically nonconvex formulation. In addition, the applicability of many existing solutions may be limited either by their estimation performance or by their computational cost, especially in the case of non-Fourier measurements. In this paper, we propose a novel phase retrieval approach, which is based on a smooth nonconvex approximation of the standard data fidelity term. In addition, the proposed method allows us to employ a wide range of convex separable regularization functions. The optimization process is performed by a block coordinate proximal algorithm which is amenable to solving large-scale problems. An application of this algorithm to an image reconstruction problem shows that it may be very competitive with respect to state-of-the-art methods.

Nonlocal total variation (NLTV) has emerged as a useful tool in variational methods for image recovery problems. In this paper, we extend the NLTV-based regularization to multicomponent images by taking advantage of the structure tensor (ST) resulting from the gradient of a multicomponent image. The proposed approach allows us to penalize the nonlocal variations, jointly for the different components, through various l(1, p)-matrix-norms with p ≥ 1. To facilitate the choice of the hyperparameters, we adopt a constrained convex optimization approach in which we minimize the data fidelity term subject to a constraint involving the ST-NLTV regularization. The resulting convex optimization problem is solved with a novel epigraphical projection method. This formulation can be efficiently implemented because of the flexibility offered by recent primal-dual proximal algorithms. Experiments are carried out for color, multispectral, and hyperspectral images. The results demonstrate the interest of introducing a nonlocal ST regularization and show that the proposed approach leads to significant improvements in terms of convergence speed over current state-of-the-art methods, such as the alternating direction method of multipliers.

In this paper, we develop an efficient bit allocation strategy for subband-based image coding systems. More specifically, our objective is to design a new optimization algorithm based on a rate-distortion optimality criterion. To this end, we consider the uniform scalar quantization of a class of mixed distributed sources following a Bernoulli-generalized Gaussian distribution. This model appears to be particularly well-adapted for image data, which have a sparse representation in a wavelet basis. In this paper, we propose new approximations of the entropy and the distortion functions using piecewise affine and exponential forms, respectively. Because of these approximations, bit allocation is reformulated as a convex optimization problem. Solving the resulting problem allows us to derive the optimal quantization step for each subband. Experimental results show the benefits that can be drawn from the proposed bit allocation method in a typical transform-based coding application.

TV-like constraints/regularizations are useful tools in variational methods for multicomponent image restoration. In this paper, we design more sophisticated non-local TV constraints which are derived from the structure tensor. The proposed approach allows us to measure the non-local variations, jointly for the different components, through various ℓ1,pmatrix norms with p ≥ 1. The related convex constrained optimization problems are solved through a novel epigraphical projection method. This formulation can be efficiently implemented thanks to the flexibility offered by recent primal-dual proximal algorithms. Experiments carried out for color images demonstrate the interest of considering a Non-Local Structure Tensor TV and show that the proposed epigraphical projection method leads to significant improvements in terms of convergence speed over existing numerical solutions.

Convex optimization problems involving information measures have been extensively investigated in source and channel coding. These measures can also be successfully used in inverse problems encountered in signal and image processing. The related optimization problems are often challenging due to their large size. In this paper, we derive closed-form expressions of the proximity operators of Kullback-Leibler and Jeffreys-Kullback divergences. Building upon these results, we develop an efficient primal-dual proximal approach. This allows us to address a wide range of convex optimization problems whose objective function expression includes one of these divergences. An image registration application serves as an example for illustrating the good performance of the proposed method.

Both random and structured perturbations affect seismic data. Their removal, to unveil meaningful geophysical information, requires additional priors. Seismic multiples are one form of structured perturbations related to wave-field bouncing. In this paper, we model these undesired signals through a time-varying filtering process accounting for inaccuracies in amplitude, time-shift and average frequency of available templates. We recast the problem of jointly estimating the filters and the signal of interest (primary) in a new convex variational formulation, allowing the incorporation of knowledge about the noise statistics. By making some physically plausible assumptions about the slow time variations of the filters, and by adopting a potential promoting the sparsity of the primary in a wavelet frame, we design a primal-dual algorithm which yields good performance in the provided simulation examples.

In multichannel imaging, several observations of the same scene acquired in different spectral ranges are available. Very often, the spectral components are degraded by a blur modelled by a linear operator and an additive noise. In this paper, we address the problem of recovering the image components in a wavelet domain by adopting a variational approach. Our contribution is twofold. First, an appropriate multivariate penalty function is derived from a novel joint prior model of the probability distribution of the wavelet coefficients located at the same spatial position in a given subband through all the channels. Secondly, we address the challenging issue of computing the Maximum A Posteriori estimate by using a Majorize-Minimize optimization strategy. Simulation tests carried out on multispectral satellite images show that the proposed method outperforms conventional techniques.

In this work, we consider a class of differentiable criteria for sparse image computing problems, where a nonconvex regularization is applied to an arbitrary linear transform of the target image. As special cases, it includes edge-preserving measures or frame-analysis potentials commonly used in image processing. As shown by our asymptotic results, the $\ell_2-\ell_0$ penalties we consider may be employed to provide approximate solutions to $\ell_0$-penalized optimization problems. One of the advantages of the proposed approach is that it allows us to derive an efficient majorize-minimize subspace algorithm. The convergence of the algorithm is investigated by using recent results in nonconvex optimization. The fast convergence properties of the proposed optimization method are illustrated through image processing examples. In particular, its effectiveness is demonstrated on several data recovery problems.

Algorithms based on total variation (TV) minimization are prevalent in image processing. They play a key role in a variety of applications such as image denoising, compressive sensing, and inverse problems in general. In this work, we extend the TV dual framework that includes Chambolle's and Gilboa and Osher's projection algorithms for TV minimization. We use a flexible graph data representation that allows us to generalize the constraint on the projection variable. We show how this new formulation of the TV problem may be solved by means of fast parallel proximal algorithms. In denoising and deblurring examples, the proposed approach is shown not only to perform better than recent TV-based approaches, but also to perform well on arbitrary graphs instead of regular grids. The proposed method consequently applies to a variety of other inverse problems including image fusion and mesh filtering.

The concept of cosparsity has been recently introduced in the arena of compressed sensing. In cosparse modelling, the ℓ0(or ℓ1) cost of an analysis-based representation of the target signal isminimized under a data fidelity constraint. By taking benefit from recent advances in proximal algorithms, we show that it is possible to efficiently address a more general framework where a convex block sparsity measure is minimized under various convex constraints. The main contribution of this work is the introduction of a new epigraphical projection technique, which allows us to consider more flexible data fidelity constraints than the standard linear or quadratic ones. The validity of our approach is illustrated through an application to an image reconstruction problem in the presence of Poisson noise.

A Poisson-Gaussian model accurately describes the noise present in many imaging systems such as CCD cameras or fluorescence microscopy. However most existing restoration strategies rely on approximations of the Poisson-Gaussian noise statistics. We propose a convex optimization algorithm for the reconstruction of signals degraded by a linear operator and corrupted with mixed Poisson-Gaussian noise. The originality of our approach consists of considering the exact continuous-discrete model corresponding to the data statistics. After establishing the Lipschitz differentiability of the Poisson-Gaussian log-likelihood, we derive a primal-dual iterative scheme for minimizing the associated penalized criterion. The proposed method is applicable to a large choice of penalty terms. The robustness of our scheme allows us to handle computational difficulties due to infinite sums arising from the computation of the gradient of the criterion. The proposed approach is validated on image restoration examples.

Many existing works related to lossy-to-lossless image compression are based on the lifting concept. In this paper, we present a sparse optimization technique based on recent convex algorithms and applied to the prediction filters of a two-dimensional non separable lifting structure. The idea consists of designing these filters, at each resolution level, by minimizing the sum of the ℓ1-norm of the three detail subbands. Extending this optimization method in order to perform a global minimization over all resolution levels leads to a new optimization criterion taking into account linear dependencies between the generated coefficients. Simulations carried out on still images show the benefits which can be drawn from the proposed optimization techniques.

The objective of this paper is to design an efficient bit allocation algorithm in the subband coding context based on an analytical approach. More precisely, we consider the uniform scalar quantization of subband coefficients modeled by a Generalized Gaussian distribution. This model appears to be particularly well-adapted for data having a sparse representation in the wavelet domain. Our main contribution is to reformulate the bit allocation problem as a convex programming one. For this purpose, we firstly define new convex approximations of the entropy and distortion functions. Then, we derive explicit expressions of the optimal quantization parameters. Finally, we illustrate the application of the proposed method to wavelet-based coding systems.

Algorithms based on the minimization of the Total Variation are prevalent in computer vision. They are used in a variety of applications such as image denoising, compressive sensing and inverse problems in general. In this work, we extend the TV dual framework that includes Chambolle's and Gilboa Osher's projection algorithms for TV minimization in a flexible graph data representation by generalizing the constraint on the projection variable. We show how this new formulation of the TV problem may be solved by means of a fast parallel proximal algorithm, which performs better than the classical TV approach for denoising, and is also applicable to inverse problems such as image deblurring.

Stereo matching is an active area of research in image processing. In a recent work, a convex programming approach was developed in order to generate a dense disparity field. In this paper, we address the same estimation problem and pro pose to solve it in a more general convex optimization frame work based on proximal methods. More precisely, unlike previous works where the criterion must satisfy some restrictive conditions in order to be able to numerically solve the minimization problem, this work offers a great flexibility in the choice of the involved criterion. The method is validated in a stereo image coding framework, and the results demonstrate the good performance of the proposed parallel proximal algorithm.

In this paper, we consider a class of differentiable criteria for sparse image recovery problems. The regularization is applied to a linear transform of the target image. As special cases, it includes edge preserving measures or frame analysis potentials. As shown by our asymptotic results, the considered ℓ2- ℓ0penalties may be employed to approximate solutions to ℓ0penalized optimization problems. One of the advantages of the approach is that it allows us to derive an efficient Majorize-Minimize Memory Gradient algorithm. The fast convergence properties of the proposed optimization algorithm are illustrated through image restoration examples.

Regularization approaches have demonstrated their effectiveness for solving ill-posed problems. However, in the context of variational restoration methods, a challenging question remains, namely how to find a good regularizer. While total variation introduces staircase effects, wavelet-domain regularization brings other artefacts, e.g., ringing. However, a tradeoff can be made by introducing a hybrid regularization including several terms not necessarily acting in the same domain (e.g., spatial and wavelet transform domains). While this approach was shown to provide good results for solving deconvolution problems in the presence of additive Gaussian noise, an important issue is to efficiently deal with this hybrid regularization for more general noise models. To solve this problem, we adopt a convex optimization framework where the criterion to be minimized is split in the sum of more than two terms. For spatial domain regularization, isotropic or anisotropic total variation definitions using various gradient filters are considered. An accelerated version of the Parallel Proximal Algorithm is proposed to perform the minimization. Some difficulties in the computation of the proximity operators involved in this algorithm are also addressed in this paper. Numerical experiments performed in the context of Poisson data recovery, show the good behavior of the algorithm as well as promising results concerning the use of hybrid regularization techniques.

Many existing works related to lossy-to-lossless image compression are based on the lifting concept. However, it has been observed that the separable lifting scheme structure presents some limitations because of the separable processing performed along the image lines and columns. In this paper, we propose to use a 2D non separable lifting scheme decomposition that enables progressive reconstruction and exact decoding of images. More precisely, we focus on the optimization of all the involved decomposition operators. In this respect, we design the prediction filters by minimizing the variance of the detail signals. Concerning the update filters, we propose a new optimization criterion which aims at reducing the inherent aliasing artefacts. Simulations carried out on still and stereo images show the benefits which can be drawn from the proposed optimization of the lifting operators.

We consider a variational formulation of blind image recovery problems. A novel iterative proximal algorithm is proposed to solve the associated nonconvex minimization problem. Under suitable assumptions, this algorithm is shown to have better convergence properties than standard alternating minimization techniques. The objective function includes a smooth convex data fidelity term and nonsmooth convex regularization terms modeling prior information on the data and on the unknown linear degradation operator. A novelty of our approach is to bring into play recent nonsmooth analysis results. The pertinence of the proposed method is illustrated in an image restoration example.

We propose a variational method for decomposing an image into a geometry and a texture component. Our model involves the sum of two functions promoting separately properties of each component, and of a coupling function modeling the interaction between the components. None of these functions is required to be differentiable, which significantly broadens the range of decompositions achievable through variational approaches. The convergence of the proposed proximal algorithm is guaranteed under suitable assumptions. Numerical examples are provided that show an application of the algorithm to image decomposition and restoration in the presence of Poisson noise.

Quantization, defined as the act of attributing a finite number of grey-levels to an image, is an essential task in image acquisition and coding. It is also intricately linked to various image analysis tasks, such as denoising and segmentation. In this paper, we investigate quantization combined with regularity constraints, a little-studied area which is of interest, in particular, when quantizing in the presence of noise or other acquisition artifacts. We present an optimization approach to the problem involving a novel two-step, iterative, flexible, joint quantizing-regularization method featuring both convex and combinatorial optimization techniques. We show that when using a small number of grey-levels, our approach can yield better quality images in terms of SNR, with lower entropy, than conventional optimal quantization methods.

A broad range of signal recovery problems can be abstracted into the problem of minimizing the sum of several convex functions in a Hilbert space. We propose a proximal decomposition algorithm which, under mild conditions, provides a solution to such a problem. A significant improvement over the methods currently in use in the area of signal recovery is that it is not limited to two nondifferentiable functions. An application to image restoration is demonstrated.

Iterative optimization algorithms such as the forward-backward and Douglas-Rachford algorithms have gained much popularity since they provide efficient solutions to a wide class of non-smooth convex minimization problems arising in signal/image recovery. However, when images are degraded by a convolution operator and a Poisson noise, a particular attention must be paid to the associated minimization problem. To solve it, we propose a new optimization method which consists of two nested iterative steps. The effectiveness of the proposed method is demonstrated via numerical comparisons.

Parallel magnetic resonance imaging (pMRI) relying on multiple receiver coils has emerged as a powerful 3D imaging technique for reducing scanning time or increasing spatial or temporal resolution. The acquired k-space is subsampled, and full field of view (FoV) images are then reconstructed from the acquired aliased data by applying methods such as the SENSE algorithm. However, reconstructed images using SENSE may suffer from several kinds of artifacts mainly because of noise and inaccurate sensitivity profiles. In this paper, we propose a regularized SENSE reconstruction method in which the regularization takes place in the wavelet transform domain. More precisely, a Bayesian strategy is adopted by introducing a bivariate prior to model the complex-valued signal. Experiments on synthetic data and real T1-weighted MRI images at 1.5 Tesla magnetic field show that the proposed method provides improved reconstruction.

Research in stereo image coding has focused on the disparity estimation/compensation process to exploit the cross-view redundancies. Most of the reported methods use a classical block-based technique in order to estimate the disparity field. However, this estimation technique does not always provide an accurate disparity map, which may affect the disparity compensation step. In this paper, we propose to use an estimation method that produces a dense and smooth disparity map. Then, on the one hand, this map is segmented and efficiently coded by exploiting the high correlation between neighboring disparity values. On the other hand, we integrate the disparity information into a vector lifting scheme for stereo image coding. Experimental results indicate that the proposed coding scheme outperforms the conventional methods employing a block-based disparity estimation.

The objective of this paper is to develop methods for solving image recovery problems subject to constraints on the solution. More precisely, we will be interested in problems which can be formulated as the minimization over a closed convex constraint set of the sum of two convex functions f and g, where f may be nonsmooth and g is differentiable with a Lipschitz-continuous gradient. To reach this goal, we derive two types of algorithms that combine forward-backward and Douglas–Rachford iterations. The weak convergence of the proposed algorithms is proved. In the case when the Lipschitz-continuity property of the gradient of g is not satisfied, we also show that, under some assumptions, it remains possible to apply these methods to the considered optimization problem by making use of a quadratic extension technique. The effectiveness of the algorithms is demonstrated for two wavelet-based image restoration problems involving a signal-dependent Gaussian noise and a Poisson noise, respectively.

Many research efforts have been devoted to the improvement of stereo image coding techniques for storage or transmission. In this paper, we are mainly interested in lossy-to-lossless coding schemes for stereo images allowing progressive reconstruction. The most commonly used approaches for stereo compression are based on disparity compensation techniques. The basic principle involved in this technique first consists of estimating the disparity map. Then, one image is considered as a reference and the other is predicted in order to generate a residual image. In this paper, we propose a novel approach, based on vector lifting schemes (VLS), which offers the advantage of generating two compact multiresolution representations of the left and the right views. We present two versions of this new scheme. A theoretical analysis of the performance of the considered VLS is also conducted. Experimental results indicate a significant improvement using the proposed structures compared with conventional methods.

Illumination changes cause serious problems in many computer vision applications. We present a new method for addressing robust depth estimation from a stereo pair under varying illumination conditions. First, a spatially varying multiplicative model is developed to account for brightness changes induced between left and right views. The depth estimation problem, based on this model, is then formulated as a constrained optimization problem in which an appropriate convex objective function is minimized under various convex constraints modelling prior knowledge and observed information. The resulting multiconstrained optimization problem is finally solved via a parallel block iterative algorithm which offers great flexibility in the incorporation of several constraints. Experimental results on both synthetic and real stereo pairs demonstrate the good performance of our method to efficiently recover depth and illumination variation fields, simultaneously.

We consider the uniform scalar quantization of a class of mixed distributed memoryless sources, namely sources having a Bernoulli Generalized Gaussian (BGG) distribution. Both for low and high resolutions, asymptotic expressions of the distortion for a pth-order moment error measure, and close approximations of the entropy are provided for these sources. Operational rate-distortion functions at high bit rate and their slope factors at low bit rate are derived. The dependence of these results on p and the distribution parameters as well as the relation to the Shannon optimal rate-distortion bound are then discussed. The application of these results to transform coding in two simple cases is finally highlighted.

In this paper, we provide operational rate-distortion results for memoryless generalized Gaussian sources. Close approximations of the entropy are provided for these sources, after a uniform scalar quantization at low/high resolution. Asymptotic expressions of the distortion for an arbitrary p-th order error measure are also given. The resulting approximations at low/high bitrate of the operational rate-distortion function are thus compared with the Shannon optimal bound showing the overall good performance of uniform quantization rules.

Abstract—This paper addresses the problem of dense disparity estimation from a pair of color stereo images. Based on a convex set theoretic formulation, the stereo matching problem is cast as a convex programming problem in which a color-based objective function is minimized under specific convex constraints. These constraints arise from prior knowledge and rely on various properties of the disparity field to be estimated. The resulting multi-constrained optimization problem is solved via an efficient parallel block-iterative algorithm. Four different color spaces have been tested in order to evaluate their suitability for stereo matching. Experiments on standard stereo images show that the matching results have been efficiently improved when using color information instead of grey values. I.

Dual-tree wavelet transforms have recently gained popularity since they provide low-redundancy directional analyses of images. In our recent work, dyadic real dual-tree decompositions have been extended to the M-band case, so adding much flexibility to this analysis tool. In this work, we propose to further extend this framework on two fronts by considering (i) biorthogonal and (ii) complex M-band dual-tree decompositions. Denoising results are finally provided to demonstrate the validity of the proposed design rules.

When an oversampled FIR filter bank structure is used for signal analysis, a main problem is to guarantee its invertibility and to be able to determine an inverse synthesis filter bank. As the analysis scheme corresponds to a redundant decomposition, there is no unique inverse filter bank and some of the solutions can lead to artifacts in textured image filtering applications. In this paper, the flexibility in the choice of the inverse filter bank is exploited to find the best-localized impulse responses. The design is performed by solving a constrained optimization problem which is reformulated in a smaller dimensional space. Application to seismic data clearly shows the improvements brought by the optimization process.

In this paper, we consider a two description coding scheme based on a general frame synthesis operator. Through some approximations of the original rate-distortion problem, the design of the efficiently encoded coefficients is formulated as a convex optimization problem. We also show that there exists a close link between the proposed coding strategy and compressed sensing problems. Simulations results are provided to show the validity of our approach.

Dual-tree wavelet decompositions have recently gained much popularity, mainly due to their ability to provide an accurate directional analysis of images combined with a reduced redundancy. When the decomposition of a random process is performed-which occurs in particular when an additive noise is corrupting the signal to be analyzed-it is useful to characterize the statistical properties of the dual-tree wavelet coefficients of this process. As dual-tree decompositions constitute over-complete frame expansions, correlation structures are introduced among the coefficients, even when a white noise is analyzed. In this paper, we show that it is possible to provide an accurate description of the covariance properties of the dual-tree coefficients of a wide-sense-stationary process. The expressions of the (cross-) covariance sequences of the coefficients are derived in the one- and two-dimensional cases. Asymptotic results are also provided, allowing to predict the behavior of the second-order moments for large lag values or at coarse resolution. In addition, the cross-correlations between the primal and dual wavelets, which play a primary role in our theoretical analysis, are calculated for a number of classical wavelet families. Simulation results are finally provided to validate these results.

Under consideration is the large body of signal recovery problems that can be formulated as the problem of minimizing the sum of two (not necessarily smooth) proper lower semicontinuous convex functions in a real Hilbert space. This generic problem is analyzed and a decomposition method is proposed to solve it. The convergence of the method, which is based on an extension of the Douglas-Rachford algorithm for monotone operators splitting, is established under general conditions. Various signal recovery applications are discussed and numerical results are provided

We propose a new estimator for image denoising using a 2D dual-tree M-band wavelet transform. Our work extends existing block-based wavelet thresholding methods by exploiting simultaneously coefficients in the two M-band wavelet trees. The contributions of this paper are two-fold. Firstly, we perform a statistical analysis of the noise in the considered redundant decomposition. Secondly, we propose an efficient method to remove the noise. Our approach relies on an extension of Stein's formula which allows us to take into account the specific correlations of the noise components. Simulation results are then presented to validate the proposed method

In a previous work, we proposed a relatively simple method to build non separable perfect reconstruction oversampled lapped transforms. The main drawback of this method was that the redundancy factor was constrained to be equal to the overlapping one. This constitutes a strong limitation for applications such as seismic processing involving three-dimensional data sets. The memory requirements may indeed become hard to meet if the redundancy is not reduced. In this paper, we propose an approach to guarantee that a given lapped transform is invertible by a finite length filter bank. We show how to compute a corresponding synthesis filter bank. The proposed analysis/synthesis filter bank system is applied to directional filtering of noisy three-dimensional seismic data

We propose a two-dimensional generalization to the M-band case of the dual-tree decomposition structure (initially proposed by Kingsbury and further investigated by Selesnick) based on a Hilbert pair of wavelets. We particularly address: 1) the construction of the dual basis and 2) the resulting directional analysis. We also revisit the necessary pre-processing stage in the M-band case. While several reconstructions are possible because of the redundancy of the representation, we propose a new optimal signal reconstruction technique, which minimizes potential estimation errors. The effectiveness of the proposed M-band decomposition is demonstrated via denoising comparisons on several image types (natural, texture, seismics), with various M-band wavelets and thresholding strategies. Significant improvements in terms of both overall noise reduction and direction preservation are observed.

The paper deals with blind source separation by contrast function maximization. A general class of separation criteria valid for both i.i.d. and non i.i.d. sources is exhibited; it is based on third or higher order cross-cumulants between the separator outputs and fixed signals, called references. We show that this approach is applicable not only in a semi-blind context, but also in a completely blind scenario. The most interesting feature concerning our criteria is their quadratic form. It follows a highly simplified optimization procedure which we describe. Simulation results illustrate the validity of our approach, and the appeal of this new class of contrast functions.

We propose a 2D generalization to the M-band case of the dual-tree structure (initially proposed by N.G. Kingsbury and further investigated by I.W. Selesnick) based on a Hilbert pair of wavelets. We particularly address the construction of the dual basis and the resulting directional analysis. We revisit the necessary preprocessing stage in the M-band case. While several reconstructions are possible because of the redundancy of the representation, we propose a new optimal signal reconstruction technique, which minimizes potential estimation errors. The effectiveness of the proposed M-band decomposition is demonstrated via image denoising comparisons.

Multiple description coding (MDC) recently appeared as a joint source-channel coding technique specifically designed for real-time multimedia applications over best effort switched packet networks such as the Internet, in order to cope with packet losses due to transmission errors or network congestion. In this paper we compare several redundant wavelet decompositions within the framework of multiple description of scalable video coding. Special attention is paid to the optimal design of the central decoder. Simulation results are provided for motion-compensated filter banks so as to evaluate the efficiency of the central and side decoding strategies. Compared with other techniques, a key factor of the proposed analysed schemes is their reduced redundancy factor.

We consider the problem of image transmission on error-prone networks with little or no error protection. To this end, we build a multiple description scheme based on classical biorthogonal filter banks, that achieves good reconstruction even at low bitrates. The novelty of the proposed approach mainly consists in building a 2D wavelet frame representation with low redundancy. Another contribution of this work is the use of a convex optimization approach at the decoder end in order to best take advantage of all the received information. The quantization constraints define convex sets, which allow us to apply fast iterative projection techniques to find a feasible solution of the decoding problem.

Multichannel imaging systems provide several observations of the same scene which are often corrupted by noise. In this paper, we are interested in multispectral image denoising in the wavelet domain. We adopt a multivariate statistical approach in order to exploit the correlations existing between the different spectral components. Our main contribution is the application of Stein's principle to build a new estimator for arbitrary multichannel images embedded in additive Gaussian noise. Simulation tests carried out on optical satellite images show that the proposed method outperforms conventional wavelet shrinkage techniques.

Multichannel imaging systems provide several observations of the same scene which are often corrupted by additive noise. We are interested in multispectral image denoising in the wavelet domain. We adopt a multivariate approach in order to exploit the correlations existing between the different spectral components. Our main contribution is the application of Stein's principle to build a new estimator for arbitrary multichannel images embedded in Gaussian noise. Simulation tests carried out on multispectral satellite images show that the proposed method outperforms conventional wavelet shrinkage techniques.

The paper is concerned with blind separation of convolutive mixtures of mutually independent signals. We consider the MISO extraction of one source signal based on the maximization of a contrast function (CF); a new, so-called "reference" CF is proposed, which is based on cross-statistics between the estimated output and a reference signal. The proposed CF is valid both for i.i.d. and non i.i.d. sources. It presents the advantage over other CFs to be a quadratic function, which makes its optimization much easier to realize. Finally, simulations demonstrate the validity of this CF and show that it leads to improved separation performances.

Convex set theoretic estimation methods have been shown to be effective in numerous signal recovery problems due to their ability to incorporate a wide range of deterministic and probabilistic information in the form of constraints on the solution. To date, probabilistic information has been used exclusively to constrain statistics of the estimation residual to be consistent with known properties of the noise. In this paper, we propose a new technique to construct constraint sets from probabilistic information based on Stein's identity. In this framework, probabilistic attributes of the signal to be recovered are estimated from the data. The proposed approach is applicable to signal formation models involving additive Gaussian noise and it leads to geometrically simple sets that can easily be handled via projection methods. An application to image denoising is demonstrated.

First-order finite-difference information has been exploited in a variety of image and signal restoration settings. These approaches typically require - implicitly or explicitly - that certain attributes of the finite-difference images be known a priori. We propose a new statistical framework in which such attributes are estimated a posteriori from the observed data under the assumption that the noise is additive and Gaussian. Our analysis can be directly applied to the construction of property sets in set theoretic estimation methods. The proposed framework is illustrated through an application to image denoising.

Multiple description coding (MDC) is a joint source-channel coding technique specifically designed for real-time multimedia applications over best effort switched packet networks (such as Internet), in order to cope with packet losses due to transmission errors or network congestion. Error resilience of transmitted bitstreams is thus significantly increased, but this does not solve problems like bitstream adaptation to bandwidth variations or receiver characteristics, which are in turn addressed by scalable coding techniques. In this paper, we present a new method of multiple description coding of scalable video, combining the scalability features with MDC. We propose a redundant motion-compensated temporal scheme related to Haar multiresolution analysis. We also present an equivalent lifting implementation leading to simple central and lateral decoders.

Total variation has proven to be a valuable concept in connection with the recovery of images featuring piecewise smooth components. So far, however, it has been used exclusively as an objective to be minimized under constraints. In this paper, we propose an alternative formulation in which total variation is used as a constraint in a general convex programming framework. This approach places no limitation on the incorporation of additional constraints in the restoration process and the resulting optimization problem can be solved efficiently via block-iterative methods. Image denoising and deconvolution applications are demonstrated.

In this paper, we propose wavelet-thresholding estimators for spectrum analysis of a zero-mean cyclostationary signal. In the case of Gaussian regression, it is known that wavelet estimators outperform traditional linear methods if the regularity of the function to be estimated varies substantially over its domain of definition. The goal of this paper is to extend these wavelet methods to the estimation of cyclospectra. In this context, we show both theoretically and through a simulation example that wavelet-thresholding estimators lead to improved performance compared to kernel methods.

Total variation has proven to be a valuable concept in connection with the recovery of images featuring piecewise smooth components. So far, however, it has been used exclusively as an objective to be minimized under a single constraint. In this paper, we propose an alternative framework in which total variation is used as a constraint in a general quadratic programming context. The advantage of this approach is that it allows for a wider range of constraints to be easily incorporated in the recovery process.

In this letter, we revisit a number of concepts that have recently proven to be useful in multiresolution signal analysis, specifically by replacing the now classical linear-scale transition operators by nonlinear ones. More precisely, we address the problem of designing appropriate operators associated to nonlinear filter banks using multiscale analysis. We first establish a connection between nonlinear filter banks and partial differential equations operators used in scale-space theory. Toward this end, we propose specific structures of nonlinear three-band decompositions ensuring a perfect reconstruction. The behavior of the proposed structures is analyzed for a step-like signal in a high SNR scenario, and a simulation is proposed for a more complex scenario.

In this paper, we are interested in using multiwavelet transforms in the context of progressive and lossless coding of multi component images. More precisely, multiwavelet decompositions that map integers to integers are considered since they guarantee a perfect reconstruction in the absence of quantizers. Generally, these decompositions are performed separately on each spectral component of a multicomponent image. Therefore, they fail to exploit the spectral redundancies. Our main contribution in this paper consists in modifying multiwavelet decompositions in order to take into account simultaneously the spatial and the spectral redundancies contained in a multicomponent image. Simulation tests carried out on natural multicomponent images show that the the generalized interband decomposition outperforms the state-of-art lossless coders.

In this paper, we consider the blind signal separation problem in the convolutive case. More precisely, we present a generalization of classical contrast functions to more flexible asymmetric forms and give examples of these new criteria. We also realize a statistical study of the proposed source separation approach, including both the consistency and the asymptotic normality aspects.

In this paper, a nonlinear subband decomposition scheme with perfect reconstruction is proposed for lossless and progressive coding of multispectral images. The merit of this new scheme is to exploit efficiently the spatial and the spectral redundancies contained in the multispectral images related to a scene of interest. Besides, the proposed method is suitable for telebrowsing applications. Experiments carried out on real scenes allow to assess its performances. The simulation results demonstrate that our approach leads to improved compression performances compared with currently used lossless coders.

A nonlinear subband decomposition scheme with perfect reconstruction is proposed for lossless coding of multispectral images. The merit of this new scheme is to exploit efficiently both the spatial and the spectral redundancies contained in a multispectral image sequence. Besides, it is suitable for progressive coding, which constitutes a desirable feature for telebrowsing applications. Simulation tests performed on real scenes allow assessment of the performances of this new multiresolution coding algorithm. The achieved compression ratios are higher than those obtained with currently used lossless coders.

This paper addresses the problem of blind separation of convolutive mixtures via contrast maximization. New frequency-domain contrast functions are constructed based on second and higher-order spectra of the observations. They allow one to separate mixtures of sources which are spatially independent, and temporally possibly non i.i.d. linear or non-linear processes. The proposed criteria provide a framework for extending to the convolutive case contrasts that have been proposed in the context of instantaneous mixtures.

Traffic flow in high-speed data network systems is often impulsive and long-range dependent. Impulsiveness implies a heavy-tailed marginal distribution, thus lack of finite second-order statistics. Hence, traditional methods for quantifying the long-range dependence of traffic based on its second-order statistics are not applicable. Long-range dependence and self-similarity play an important role in traffic engineering. We have recently shown that the generalized codifference can quantify the dependence structure of impulsive self-similar processes, such as high-speed network traffic. We propose an estimator for the generalized codifference and provide the conditions for it to be asymptotically consistent. We show that these conditions are satisfied for the EAFRP which is a process proposed for modeling high-speed network traffic. We provide simulation results to demonstrate the properties of the proposed estimator, and show how it can be a useful tool in maintaining fairness among users sharing limited network resources.

We propose a separable decomposition approximating the Karhunen-Loeve transform for random fields. We show that this problem is related to a joint singular value decomposition of a set of matrices and we provide an efficient algorithm to compute it. Finally, we illustrate the interest of this new tool for image representation and approximation.

This paper deals with independence measures for linear mixtures of mutually independent random variables. Such measures, also known as contrasts, constitute useful criteria in solving blind source separation problems. By making use of the Schur convexity properties, we show that it is possible to define a wide-ranging class of contrast functions based on the auto-cumulants of the components of the random vector being considered. Among the most appealing characteristics of these new contrast functions is that they can be used to combine cumulants of different orders in a flexible way. Furthermore, extensions of existing cross-cumulant-base contrasts are proposed. Finally, some particularization of our approach to measures of decorrelation is considered. A general characterization of these decorrelation measures using strictly Schur convex functions is provided.

This paper introduces a novel nonlinear low-level representation of an image with signal-dependent noise. For multiplicative noisy image, we introduce an algorithm called multiplicative matching pursuit decomposition (MMPD), that decomposes the signal containing the intrinsic variation into a nonlinear expansion of waveforms that are selected from a redundant dictionary of functions. These waveforms are chosen in order to best match the signal local structures. The convergence of this new multiplicative decomposition has been proved and tested in practice. An application to speckle reduction in SAR images is described.

In many applications it is necessary to characterize the statistical properties of the wavelet/wavelet packet coefficients of a stationary random signal. In particular, in a stationary non-Gaussian noise scenario it may be useful to determine the high-order statistics of the wavelet packet coefficients. In this work we prove that this task may be performed through multidimensional filter banks. In particular, we show how the cumulants of the M-band wavelet packet coefficients of a strictly stationary signal are derived from those of the signal and we provide scale-recursive decomposition and reconstruction formulae to compute these cumulants. High-order wavelet packets, associated with these multidimensional filter banks, are presented along with some of their properties. It is proved that under some conditions these high-order wavelet packets allow us to define frame multiresolution analyses. Finally, the asymptotic normality of the coefficients is studied by showing the geometric decay of their polyspectra/cumulants (of order greater than two) with respect to the resolution level.

Some wavelet-based methods for signal estimation in the presence of noise are reviewed in the context of the parsimonious representation of the underlying signal. Three approaches are considered. The first is based on the application of the minimum description length (MDL) principle. The robustness of this method is improved in the second approach, by relaxing the assumption of known noise distribution following Huber's (1967) work. In the third approach, a Bayesian strategy is adopted in order to incorporate prior information pertaining to the signal of interest; this method is especially useful at low signal-to-noise ratios.

A standard wavelet multiresolution analysis can be defined via a sequence of projectors onto a monotone sequence of closed vector subspaces possessing certain properties. We propose a nonlinear extension of this framework in which the vector subspaces are replaced by convex subsets. These sets are chosen so as to provide a recursive, monotone approximation scheme that allows for various image features to be investigated. Several classes of convex multiresolution analyzes are discussed and numerical applications to image analysis are demonstrated.

A standard wavelet multiresolution analysis can be defined via a sequence of projectors onto a monotone sequence of closed vector subspaces possessing certain properties. We propose a nonlinear extension of this framework in which the vector subspaces are replaced by convex subsets. These sets are chosen so as to provide a recursive, monotone approximation scheme that allows for various signal and image features to be investigated. Several classes of convex multiresolution analyses are discussed and numerical applications to signal and image-processing problems are demonstrated.

We investigate nonlinear multirate filterbanks with maximal decimation and perfect reconstruction. Definitions of the desired properties of such structures are given for general nonlinear filterbanks. We then consider a triangular representation of linear filterbanks and see that it may be easily extended to the nonlinear case. Furthermore, general nonlinear filterbanks are presented, for which perfect reconstruction is either inherently guaranteed or ensured subject to an easily verified condition. Extensions to bidimensional filters are also discussed and an application for nonlinear multiresolution schemes to feature sieves is shown.

Extended forms of contrast functions are introduced to provide statistical measures of independence for orthogonal mixtures. We also define semicontrasts based on second-order statistics which, in some cases, may be sufficient to separate the mixed sources. The corresponding criteria are then used to obtain an optimized representation of a stochastic process in an orthonormal basis of wavelet packets or local cosines.

Two contrasts for the problem of multichannel blind deconvolution have been given and theoretically studied by Comon [1996]. The maximization of these criteria allows us to solve the problem of multi-input/multi-output (MIMO) blind deconvolution. In this paper, we show that many other contrast functions may be considered. The two aforementioned criteria are proved to be included in the wide class of contrast functions, which is here defined through simple conditions.

Automatic segmentation of the electrocardiogram (ECG) is important in both clinical and research settings. Past algorithms have relied on incorporation of detailed heuristics. We avoid heuristics by employing a best-basis algorithm. As large variability of the local SNR causes the standard entropy criterion to produce an overly-fine segmentation, we introduce a novel optimality criterion which is based on a linear combination of the entropy measure and a function of a smoothness measure, and is quite general in form. We tested the algorithm on the MIT-BIH arrythmia database and body surface potential maps.

Multirate filter banks are of great interest for many applications in both signal and image processing. In particular, filter banks with critical subsampling and perfect reconstruction, have received special attention. Often such decompositions are restricted to linear analysis/synthesis filters possibly with some intermediate nonlinear treatment. We introduce a new structure which generates a variety of linear and nonlinear subband decompositions with critical subsampling. The structure is such that perfect reconstruction from the subband coefficients is guaranteed. As an example we consider applying it to image coding.

Wavelet packets and local trigonometric bases provide an efficient framework and fast algorithms to obtain a "best basis" or "best representation" of deterministic signals. Applying these deterministic techniques to stochastic processes may, however, lead to variable results. We revisit this problem and introduce a prior model on the underlying signal in noise and account for the contaminating noise model as well. We thus develop a Bayesian-based approach to the best basis problem, while preserving the classical tree search efficiency.

We consider motion estimation between two images in the presence of illumination variations in the scene. The standard motion model is extended and an autoregressive prediction model obtained. The added prediction coefficient is interpreted as an illumination variation parameter. A pel-recursive motion estimator is adapted to this motion model in order to estimate both motion and illumination variation fields. We present coding experiments on real images containing localised illumination variations and note that the proposed approach allows the prediction error to be greatly reduced compared to the standard pel-recursive algorithm. This algorithm may also be used in other applications such as frame interpolation.

We investigate a M-band wavelet decomposition of second order random processes. In particular, we propose an extension of results which are known for the dyadic wavelet transform. The statistical properties of the M-band wavelet coefficients are listed and recursive relations are derived and used to compute their multiscale characteristics. Special attention is also paid to the multiscale analysis of linear parametric models.

In image sequence compression schemes based on motion compensation, some new information contained in the displaced frame difference image must be coded. As this is expensive to transmit, it is important, particularly at low bit rates, to code only the most important information. We propose a segmentation algorithm based on morphological operations to extract the visually important regions of the displaced frame difference image. This segmentation is combined with a coding scheme which is automatically optimized for a given bit rate constraint. As demonstrated by examples, this method is well adapted to low resolution (QCIF format) sequences.

Processing nonstationary signals is an important and challenging problem. We focus on the class of nonstationary processes with stationary increments of an arbitrary order, and place them in a multiscale framework. Unlike other related studies, we concentrate on the discrete-time analysis and derive a number of new results in addition to placing the related existing ones in the same framework. We extend the study to various parametric models for which we derive the resulting multiresolution description. We show that wide-sense stationarity may be achieved by adequately selecting the analysis wavelet. After generalizing the study to wavelet packet analysis, we show that the latter possesses additional properties which are useful in the presence of other types of nonstationarities.>

The author deals with the problems encountered when wavelet transforms of finite sequences are considered. It is shown that there are two main drawbacks. First, there is no conservation of the supports. This means, in particular, that the transform generates more coefficients than samples in the signal. Second, the statistical properties of the coefficients are not homogeneous, because of boundary effects. Two methods are proposed to construct bases devoted to finite sequences. The first one is based on a periodization of the signal and makes use of a periodic multiresolution analysis. The second one introduces further symmetrizations in order to obtain a smoother extrapolation of the signal.>

The robustness of adaptive differential pulse code modulation (ADPCM) systems versus transmission errors is addressed. To secure the stability of the decoder, it is necessary to modify the form of the LMS (least-mean-square) algorithm used to adapt the predictor. Solutions introducing soft constraints are investigated. The leakage algorithm is proved to be not fully satisfactory, and a new stabilizing algorithm is presented that makes it possible to achieve good performances. Compared to existing methods, the main advantage of this algorithm is its low computational complexity. Form a theoretical point of view, the effect of transmission errors is described by a set of nonlinear recurrent equations. An analysis is carries out in the deterministic second-order case.>

A hybrid image predictive coding method is presented. The intraframe predictor is an adaptive FIR (finite impulse response) filter that uses the LMS (least mean squares) algorithm to track spatial local characteristics of the intensity continuously. The interframe predictor is motion-adaptive, using a pel-recursive method to estimate the displacement vector. A weight coefficient is adapted continuously in order to favor the prediction mode that performs better. For the sequence examined a significant improvement is obtained in comparison with either mode alone. A crucial problem in predictive coding, particularly with adaptive techniques, is that of sensitivity to transmission errors. A method ensuring the autoadjustment of the decoder in the presence of isolated transmission errors is proposed for the intraframe mode. Neither overhead information nor an error-correcting code are needed.>