Remote sensing pushbroom-type imaging systems acquire entire columns of an image with a single detector. As a consequence, the miss-calibration of the detectors produces stripes on the image. In this context, this paper introduces a new self-calibration destriping method based on an affine response model for the detectors, called Statistical Affine Destriping (SAD). In contrast, some previous contributions were limited to a purely linear model, while many others only considered an additive structured noise model. It is based on the maximum a posteriori estimation of the gain and offset parameters attached to each detector given the observed image. Simple statistical prior assumptions are adopted: respectively, a Gaussian white noise model for the gains and offsets, and a first-order, homogeneous Markov model for the observed scene. Based on a simplification of the posterior likelihood, we propose a very efficient optimization scheme based on a constrained Majorize-Minimize principle, allowing us to process large dimension images. Moreover, simple empirical rules are given to tune the hyperparameters of the destriping method for high-resolution PLEIADES-type images. Compared to the performance of a destriping method limited to gain correction, we observe that the new version provides reliable results in a wider range of situations. We also extend the method in two directions. On the one hand, we consider that some detectors may be atypical, with very high or very low gains or offsets. On the other hand, we extend the method to multispectral image destriping.

In this paper the problem of restoration of unsupervised nonnegative sparse signals is addressed in the Bayesian framework. We introduce a new probabilistic hierarchical prior, based on the Generalized Hyperbolic (GH) distribution, which explicitly accounts for sparsity. On the one hand, this new prior allows us to take into account the non-negativity. On the other hand, thanks to the decomposition of GH distributions as continuous Gaussian mean-variance mixture, a partially collapsed Gibbs sampler (PCGS) implementation is made possible, which is shown to be more efficient in terms of convergence time than the classical Gibbs sampler.

In a low-statistics PET imaging context, the positive bias in regions of low activity is a burning issue. To overcome this problem, algorithms without the built-in non-negativity constraint may be used. They allow negative voxels in the image to reduce, or even to cancel the bias. However, such algorithms increase the variance and are difficult to interpret since the resulting images contain negative activities, which do not hold a physical meaning when dealing with radioactive concentration. In this paper, a post-processing approach is proposed to remove these negative values while preserving the local mean activities. Its original idea is to transfer the value of each voxel with negative activity to its direct neighbors under the constraint of preserving the local means of the image. In that respect, the proposed approach is formalized as a linear programming problem with a specific symmetric structure, which makes it solvable in a very efficient way by a dual-simplex-like iterative algorithm. The relevance of the proposed approach is discussed on simulated and on experimental data. Acquired data from an yttrium-90 phantom show that on images produced by a non-constrained algorithm, a much lower variance in the cold area is obtained after the post-processing step, at the price of a slightly increased bias. More specifically, when compared with the classical OSEM algorithm, images are improved, both in terms of bias and of variance.

We propose a joint least squares and least absolute deviations (JOLESALAD) model, show that the proposed model can cover least absolute shrinkage and selection operator (LASSO) and two of its variants, namely the generalized LASSO (gLASSO) and the constrained LASSO (cLASSO), and prove that under a full rank condition, the JOLESALAD can be transformed into cLASSO. Based on this equivalency, rich tools currently available for LASSO models can be applied to solve gLASSO and cLASSO. As an example of illustration, we demonstrate an application of the proposed model to the restoration of a noisy ramp signal that needs proper use of penalty term.

The blind structured illumination microscopy strategy proposed by Mudry et al. is fully re-founded in this paper, unveiling the central role of the sparsity of the illumination patterns in the mechanism that drives super-resolution in the method. A numerical analysis shows that the resolving power of the method can be further enhanced with optimized one-photon or two-photon speckle illuminations. A much improved numerical implementation is provided for the reconstruction problem under the image positivity constraint. This algorithm rests on a new preconditioned proximal iteration faster than existing solutions, paving the way to 3D and real-time 2D reconstruction.

In this communication, a fast reconstruction algorithm is proposed for fluorescence blind structured illumination microscopy (SIM) under the sample positivity constraint. This new algorithm is by far simpler and faster than existing solutions, paving the way to 3D and real-time 2D reconstruction.

This paper proposes a new algorithm for automatic crack detection from 2D pavement images. It strongly relies on the localization of minimal paths within each image, a path being a series of neighboring pixels and its score being the sum of their intensities. The originality of the approach stems from the proposed way to select a set of minimal paths and the two postprocessing steps introduced to improve the quality of the detection. Such an approach is a natural way to take account of both the photometric and geometric characteristics of pavement images. An intensive validation is performed on both synthetic and real images (from five different acquisition systems), with comparisons to five existing methods. The proposed algorithm provides very robust and precise results in a wide range of situations, in a fully unsupervised manner, which is beyond the current state of the art.

This paper proposes a new algorithm for crack detection based on the selection of minimal paths. It takes account of both photometric and geometric characteristics and requires few information a priori. It is validated on synthetic and real images.

Primal-dual interior-point methods are used in image processing to solve inversion problems that can be reduced to constrained convex minimization. Such iterative methods require the solution of a sequence of linear systems which are used to derive descent directions. This approach is very consuming in terms of computing time and memory usage for large-scale problems, unless the involved matrices have a specific structure. This is the case in the spectral unmixing problem where these matrices are block-diagonal when no spatial regularization is considered. Here, we consider the edge-preserving regularized case and we propose to tackle the linear system solving using a majorization-minimization (MM) approach based on separable quadratic majorant functions. The resulting systems have the same structure as in the non-regularized case and can thereby be solved efficiently. The interior-point algorithm is speeded-up while remaining convergent. An example of spectral unmixing is proposed to illustrate the efficiency of this approach.

This paper deals with the estimation of the arrival times of overlapping ultrasonic echoes. We focus on approaches based on discrete sparse deconvolution. Such methods are limited by the time resolution imposed by the model discretization, which is usually considered at the data sampling rate. In order to get closer to the continuous-time model, we propose to increase the time precision by introducing an up-sampling factor in the discrete model. The problem is then recast as a Multiple Input Single Output (MISO) deconvolution problem. Then, we propose to revisit standard sparse deconvolution algorithms for MISO systems. Specific and efficient algorithmic implementation is derived in such setting. Algorithms are evaluated on synthetic data, showing improvements in robustness toward discretization errors and competitive computational time compared to the standard approaches.

This paper deals with the reconstruction of relaxation time distributions in Nuclear Magnetic Resonance (NMR) spectroscopy. This large scale and ill-posed inverse problem is solved by the iterative minimization of a regularized objective function allowing to encode some prior assumptions on the sought distribution. The numerical optimization of the criterion is performed using a primal-dual interior point algorithm allowing to handle the non-negativity constraint. The performances of the proposed approach are illustrated through the processing of real data from a two-dimensional NMR experiment.

We address the exact recovery of a$k$-sparse vector in the noiseless setting when some partial information on the support is available. This partial information takes the form of either a subset of the true support or an approximate subset including wrong atoms as well. We derive a new sufficient and worst-case necessary (in some sense) condition for the success of some procedures based on$\ell _{p}$-relaxation, orthogonal matching pursuit (OMP), and orthogonal least squares (OLS). Our result is based on the coherence$\mu $of the dictionary and relaxes the well-known condition$\mu < 1/(2k-1)$ensuring the recovery of any$k$-sparse vector in the noninformed setup. It reads$\mu < 1/(2k-g+b-1)$when the informed support is composed of$g$good atoms and$b$wrong atoms. We emphasize that our condition is complementary to some restricted-isometry-based conditions by showing that none of them implies the other. Because this mutual coherence condition is common to all procedures, we carry out a finer analysis based on the null space property (NSP) and the exact recovery condition (ERC). Connections are established regarding the characterization of$\ell _{p}$-relaxation procedures and OMP in the informed setup. First, we emphasize that the truncated NSP enjoys an ordering property when$p$is decreased. Second, the partial ERC for OMP (ERC-OMP) implies in turn the truncated NSP for the informed$\ell _{1}$problem, and the truncated NSP for$p< 1$.

Tropp's analysis of orthogonal matching pursuit (OMP) using the exact recovery condition (ERC) is extended to a first exact recovery analysis of orthogonal least squares (OLS). We show that when the ERC is met, OLS is guaranteed to exactly recover the unknown support in at most$k$iterations where$k$denotes the support cardinality. Moreover, we provide a closer look at the analysis of both OMP and OLS when the ERC is not fulfilled. The existence of dictionaries for which some subsets are never recovered by OMP is proved. This phenomenon also appears with basis pursuit where support recovery depends on the sign patterns, but it does not occur for OLS. Finally, numerical experiments show that none of the considered algorithms is uniformly better than the other but for correlated dictionaries, guaranteed exact recovery may be obtained after fewer iterations for OLS than for OMP.

We revisit the positive cone condition given by Efronfor the over-determined least absolute shrinkage and selection operator (LASSO). It is a sufficient condition ensuring that the number of nonzero entries in the solution vector keeps increasing when the penalty parameter decreases, based on which the least angle regression (LARS)and homotopyalgorithms yield the same iterates. We show that the positive cone condition is equivalent to the diagonal dominance of the Gram matrix inverse, leading to a simpler way to check the positive cone condition in practice. Moreover, we elaborate on a connection between the positive cone condition and the mutual coherence condition given by Donoho and Tsaig, ensuring the exact recovery of any$k$-sparse representation using both LARS and homotopy.

We present a new approach for spatially varying blur identification using a single image. Within each local patch in the image, the local blur is selected between a finite set of candidate PSFs by a maximum likelihood approach. We propose to work with a Generalized Likelihood to reduce the number of parameters and we use the Generalized Singular Value De- composition to limit the computing cost, while making proper image boundary hypotheses. The resulting method is fast and demonstrates good performance on simulated and real examples originating from applications such as motion blur identification and depth from defocus.

This letter describes algorithms for nonnegative matrix factorization (NMF) with the β-divergence (β-NMF). The β-divergence is a family of cost functions parameterized by a single shape parameter β that takes the Euclidean distance, the Kullback-Leibler divergence, and the Itakura-Saito divergence as special cases (β = 2, 1, 0 respectively). The proposed algorithms are based on a surrogate auxiliary function (a local majorization of the criterion function). We first describe a majorization-minimization algorithm that leads to multiplicative updates, which differ from standard heuristic multiplicative updates by a β-dependent power exponent. The monotonicity of the heuristic algorithm can, however, be proven for β ∈ (0, 1) using the proposed auxiliary function. Then we introduce the concept of the majorization-equalization (ME) algorithm, which produces updates that move along constant level sets of the auxiliary function and lead to larger steps than MM. Simulations on synthetic and real data illustrate the faster convergence of the ME approach. The letter also describes how the proposed algorithms can be adapted to two common variants of NMF: penalized NMF (when a penalty function of the factors is added to the criterion function) and convex NMF (when the dictionary is assumed to belong to a known subspace).

This paper proposes accelerated subspace optimization methods in the context of image restoration. Subspace optimization methods belong to the class of iterative descent algorithms for unconstrained optimization. At each iteration of such methods, a stepsize vector allowing the best combination of several search directions is computed through a multidimensional search. It is usually obtained by an inner iterative second-order method ruled by a stopping criterion that guarantees the convergence of the outer algorithm. As an alternative, we propose an original multidimensional search strategy based on the majorize-minimize principle. It leads to a closed-form stepsize formula that ensures the convergence of the subspace algorithm whatever the number of inner iterations. The practical efficiency of the proposed scheme is illustrated in the context of edge-preserving image restoration.

This paper deals with the reconstruction of T1-T2 correlation spectra in Nuclear Magnetic Resonance (NMR) spectroscopy. The ill-posed character of this inverse problem and its large size are the main difficulties of the reconstruction. While maximum entropy is retained as an adequate regularization approach, the choice of an efficient optimization algorithm remains a challenging task. Our proposal is to apply a nonlinear conjugate gradient algorithm with two original features. Firstly, a theoretically well stated line search strategy suitable for the entropy function is applied to ensure a monotonic decrease of the criterion. Secondly, an appropriate preconditioning structure based on a truncated singular value decomposition of the forward model matrix is used to speed up the algorithm convergence. The resulting method reveals far more efficient than the classical Skilling and Bryan method and its applicability is illustrated through real NMR data processing.

In this article, we propose to reconstruct the 3D structure of the solar atmosphere, taking into account the temporal evolution of the coronal structures. To do this we develop a new method, which considers that the object of interest consists of several sources each one having different evolution from the other at different time. The data are the Radon transform of an instantaneous mixture of sources. So, we must solve a separation source problem with a linear mixture of sources. To our best knowledge, this problem is still open. To solve this problem of source separation, we have introduced a sparsity information on the sources, and we have chosen to use the variational Bayesian approach for reduce the computation time.

This paper introduces a new self-calibration destriping technique for pushbroom-type satellite imaging systems. Self-calibration means that no specific training data are required. It is based on the statistical estimation of each detector gain from the observed image, assuming a linear response. Both theoretical and practical behaviors are studied. Our technique is shown to outperform simpler techniques based on column averages in terms of gain estimation precision while keeping the computational cost within admissible limits.

In this paper, a novel non-stationary model of functional Magnetic Resonance Imaging (fMRI) time series is proposed. It allows us to account for some putative habituation effect arising in event-related fMRI paradigms that involves the so-called repetition-suppression phenomenon and induces decreasing magnitude responses over successive trials. Akin to , this model is defined over functionnally homogeneous regions-of-interest (ROIs) and embedded in a joint detection-estimation approach of brain activity. Importantly, its non-stationarity character is embodied in the trial-varying nature of the BOLD response magnitude. Habituation and activation maps are then estimated within the Bayesian framework in a fully unsupervised MCMC procedure. On artificial fMRI datasets, we show that habituation effects can be accurately recovered in activating voxels.

Symmetric ising models define the simplest discrete Markov random fields that can be used for segmentation purpose. Unsupervised segmentation requires an automatic setting of the temperature parameter of ising fields. To this end, partition function (PF) estimation becomes a key issue. In this paper, we present a bilinear extrapolation technique for a fast PF estimation of 3D ising field. The proposed method is a two-step procedure that applies to the context where multiple 3D ising fields are involved over different objects (eg, brain regions) of different size and topology. First, a small set of reference PFs is accurately estimated using path sampling. Second, the large remaining set of PFs is computed using a temperature-dependent bilinear extrapolation technique. It is shown that our approach is accurate and computationally efficient to account for topological fluctuations of ising fields on regular and irregular graphs. A convincing application to joint detection-estimation of brain activity in functional MRI is also presented.

Within-subject analysis in event-related functional magnetic resonance imaging (fMRI) first relies on (i) a detection step to localize which parts of the brain are activated by a given stimulus type, and then on (ii) an estimation step to recover the temporal dynamics of the brain response. Recently, we have proposed a Bayesian detection-estimation approach that jointly addresses (i)-(ii). This approach provides both a spatial activity map and an estimate of brain dynamics. Here, we consider an extension that accounts for spatial correlation using a spatial mixture model (SMM) based on a binary Markov random field. It allows us to avoid any spatial smoothing of the data prior to the statistical analysis. Our simulation results support that SMM gives a better control of false positive (specificity) and false negative (sensitivity) rates than independent mixtures.

Our contribution deals with blind deconvolution of sparse spike trains. More precisely, we examine the problem in the Markov chain Monte-Carlo (MCMC) framework, where the unknown spike train is modeled as a Bernoulli-Gaussian process. In this context, we point out that time-shift and scale ambiguities jeopardize the robustness of basic MCMC methods, in quite a similar manner to the label switching effect studied by Stephens (2000) in mixture model identification. Finally, we propose proper modifications of the MCMC approach, in the same spirit as Stephens' contribution.

The classical approach of within-subject analysis in event-related functional magnetic resonance imaging (fMRI) first relies on (i) a detection step to localize which parts of the brain are activated by a given stimulus type, and then on (ii) an estimation step to recover the temporal dynamics of the brain response. To date, specially in region-based analysis, the two questions have been addressed separately while intrinsically connected to each other. This situation motivates the need for new methods in neuroimaging that go beyond this unsatisfactory trade-off. In this paper, we propose a generalization of a region based Bayesian detection-estimation approach that addresses (i)-(ii) simultaneously as a bilinear inverse problem. The proposed extension relies on a 2-class Gamma-Gaussian prior mixture modeling to classify the voxels of the brain region either as activated or unactivated. Our approach provides both a spatial activity map and a HRF estimation using Monte Carlo Markov chain (MCMC) techniques. Results show that this novel mixture model yields lower false positive rates and a better sensitivity in comparison with a 2-class Gaussian mixture

This paper provides original results on the global and local convergence properties of half-quadratic (HQ) algorithms resulting from the Geman and Yang (GY) and Geman and Reynolds (GR) primal-dual constructions. First, we show that the convergence domain of the GY algorithm can be extended with the benefit of an improved convergence rate. Second, we provide a precise comparison of the convergence rates for both algorithms. This analysis shows that the GR form does not benefit from a better convergence rate in general. Moreover, the GY iterates often take advantage of a low cost implementation. In this case, the GY form is usually faster than the GR form from the CPU time viewpoint.

The paper investigates a new method for the extraction of spectral trajectories from nonstationary multicomponent narrowband signals. The main idea is to apply statistical filtering so as to track frequency trajectories in the short term Fourier transform. A nonlinear observation model is defined and a special particle filtering algorithm is developed.

A main issue in source separation is to deal with the indeterminacies. Well known are the ordering and scale ambiguities, but other types of indeterminacies may also occur. In this paper we address these indeterminacies in the case of non-negative sources and non-negative mixing coefficients. On the one hand, we fully develop the case of two sources. On the other hand, in the general case we formulate necessary conditions for the uniqueness of the solution (up to ordering and scale ambiguities).

In this contribution, we propose a Bayesian sampling solution to the problem of noisy blind separation of generalized hyperbolic (GH) signals. GH models, introduced by Barndorff-Nielsen in 1977, represent a parametric family able to cover a wide range of real signal distributions. The alternative construction of these distributions as a normal mean-variance (continuous) mixture leads to an efficient implementation of the MCMC method applied to source separation. The incomplete data structure of the GH distribution is indeed compatible with the hidden variable nature of the source separation problem. Our algorithm involves hyperparameters estimation as well. Therefore, it can be used, independently, to fit the parameters of the GH distribution to real data.

We address the reconstruction of a 3D image from a set of incomplete X-ray tomographic data. In the case where the image is composed of one or several objects lying in a uniform background, we define a sparse parameterization by considering the active voxels, i.e., the voxels that do not lay inside the background. Estimation of the active voxel densities is performed using the maximum a posteriori (MAP) estimator. In order to implement sparse parameter estimation, we design an original multiresolution scheme, which handles coarse to fine resolution images. This scheme affords automatic selection of active voxels at each resolution level, and provides a drastic decrease of the computation time. We finally show the performance of our method on synthetic data.

The images recorded by optical telescopes are often degraded by aberrations that induce phase variations in the pupil plane. Several wavefront sensing techniques have been proposed to estimate aberrated phases. One of them is phase diversity, for which the joint least-square approach introduced by Gonsalves et al. is a reference method to estimate phase coefficients from the recorded images. In this paper, we rely on the asymptotic theory of Toeplitz matrices to show that Gonsalves' technique provides a consistent phase estimator as the size of the images grows. No comparable result is yielded by the classical joint maximum likelihood interpretation (e.g., as found in the work by Paxman et al.). Finally, our theoretical analysis is illustrated through simulated problems.

We develop a method for the formation of Doppler radar images with enhanced features. This problem, when studied as an adaptive spectral estimation problem, is particularly ill-posed because of the small number of data. Our approach is based on a regularized estimation of depth-frequency images which combines a high-resolution Fourier model of the observations with prior information about the nature of the features of interest. We first derive an appropriate model and a regularized criterion for meteorological clutter restoration before addressing the extension to the identification of spot-like targets superimposed on this clutter. We also adapt quasi-Newton algorithms, based on half-quadratic regularization, for the computation of the solution. The practical interest of our approach is validated on simulated and real data.

The paper presents a fast MCMC (Markov chain Monte Carlo) algorithm specially designed for high dimensional models with block structure. Such models are often met in Bayesian inference problems in signal processing, such as spectral estimation, harmonic analysis, blind deconvolution or signal classification. Our algorithm generates samples distributed according to a posterior distribution. We show that sampling the amplitudes together with the remaining model parameters leads to quicker computations than sampling from the marginal posterior, where amplitudes have been integrated out. Simulation results demonstrate the soundness of this approach for high dimensional models.

The main contribution of this communication is the derivation of a generalized form of the Geman & Yang construction for minimization of convex, non-quadratic criteria. The generalization provides a means of obtaining a normal matrix with predefined structure. We show that this property can be used to improve the numerical efficiency of edge preserving MRI reconstruction. The improvement is assessed experimentally using synthetic data. In order to illustrate the practicality of the method, an example of processing large size, 3D real data (the MRI image of a knee) is also provided.

In functional magnetic resonance imaging (fMRI), the hemodynamic response function (HRF) represents the impulse response of the neurovascular system. Its identification is essential for a better understanding of cerebral activity since it provides a typical time course of the response to a stimulus in a given region of interest (ROI). The authors have developed an HRF estimation method based on a single time course (Ciuciu, P. et al., IEEE Trans. Medical Imaging, vol.22, no.10, p.1235-51, 2003). We now propose an extension that takes the spatial homogeneity of the HRF into account. Our hypothesis, based on biological results, is that a ROI can be characterized by a single HRF but varying magnitude in space. Our goal is to estimate those magnitudes that could then be interpreted as a correlate of the neural response. We are thus faced with a semi-blind deconvolution inverse problem since the time arrivals of the neural response are known - they correspond to stimuli timing. To cope with this issue, we introduce specific prior information about the HRF and the neural response. Finally, we develop an MCMC approach to approximate the posterior mean estimates of unknown quantities. Simulation results show the improvement brought by our formulation compared to our earlier approach.

IRLS and RSD algorithms of robust statistics arise as special cases of half-quadratic schemes [1]. Here, we adopt a statistical framework and we show that both algorithms are instances of the EM algorithm. The augmented dataset respectively involves a scale and a location mixture of Gaussians. The su#cient conditions for the construction cover a broad class of already known robust statistics.

This paper deals with the estimation of the blood oxygen level-dependent response to a stimulus, as measured in functional magnetic resonance imaging (fMRI) data. A precise estimation is essential for a better understanding of cerebral activations. The most recent works have used a nonparametric framework for this estimation, considering each brain region as a system characterized by its impulse response, the so-called hemodynamic response function (HRF). However, the use of these techniques has remained limited since they are not well-adapted to real fMRI data. Here, we develop a threefold extension to previous works. We consider asynchronous event-related paradigms, account for different trial types and integrate several fMRI sessions into the estimation. These generalizations are simultaneously addressed through a badly conditioned observation model. Bayesian formalism is used to model temporal prior information of the underlying physiological process of the brain hemodynamic response. By this way, the HRF estimate results from a tradeoff between information brought by the data and by our prior knowledge. This tradeoff is modeled with hyperparameters that are set to the maximum-likelihood estimate using an expectation conditional maximization algorithm. The proposed unsupervised approach is validated on both synthetic and real fMRI data, the latter originating from a speech perception experiment.

Computed tomography (CT) images exhibit a variable amount of noise and blur, depending on the physical characteristics of the apparatus and the selected reconstruction method. Standard algorithms tend to favor reconstruction speed over resolution, thereby jeopardizing applications where accuracy is critical. In this paper, we propose to enhance CT images by applying half-quadratic edge-preserving image restoration (or deconvolution) to them. This approach may be used with virtually any CT scanner, provided the overall point-spread function can be roughly estimated. In image restoration, Markov random fields (MRFs) have proven to be very flexible a priori models and to yield impressive results with edge-preserving penalization, but their implementation in clinical routine is limited because they are often viewed as complex and time consuming. For these practical reasons, we focused on numerical efficiency and developed a fast implementation based on a simple three-dimensional MRF model with convex edge-preserving potentials. The resulting restoration method provides good recovery of sharp discontinuities while using convex duality principles yields fairly simple implementation of the optimization. Further reduction of the computational load can be achieved if the point-spread function is assumed to be separable. Synthetic and real data experiments indicate that the method provides significant improvements over standard reconstruction techniques and compares well with convex-potential Markov-based reconstruction, while being more flexible and numerically efficient.

This study gives original results on the global and local convergence properties of half-quadratic (HQ) algorithms resulting from the Geman and Yang (GY) and Geman and Reynolds (GR) primal-dual constructions. In particular, we show that the: convergence domain of the GY algorithm can be extended with the benefit of an improved convergence rate.

This paper is devoted to adaptive long autoregressive spectral analysis when i) very few data are available and ii) information does exist beforehand concerning the spectral smoothness and time continuity of the analyzed signals. The contribution is founded on two papers by Kitagawa and Gersch (1985). The first one deals with spectral smoothness in the regularization framework, while the second one is devoted to time continuity in the Kalman formalism. The present paper proposes an original synthesis of the two contributions. A new regularized criterion is introduced that takes bath pieces of information into account. The criterion is efficiently optimized by a Kalman smoother. One of the major features of the method is that it is entirely unsupervised. The problem of automatically adjusting the hyperparameters that balance data-based versus prior-based information is solved by maximum likelihood (ML). The improvement is quantified in the field of meteorological radar.

This paper deals with convex half-quadratic criteria and associated minimization algorithms for the purpose of image restoration. It brings a number of original elements within a unified mathematical presentation based on convex duality. Firstly, the Geman and Yang's and Geman and Reynolds's constructions are revisited, with a view to establishing the convexity properties of the resulting half-quadratic augmented criteria, when the original nonquadratic criterion is already convex. Secondly, a family of convex Gibbsian energies that incorporate interacting auxiliary variables is revealed as a potentially fruitful extension of the Geman and Reynolds's construction.

When short data records are available, spectral analysis is basically an undetermined linear inverse problem. One usually considers the theoretical setting of regularization to solve such ill-posed problems. In this paper, we first show that "nonparametric" and "high resolution" are not incompatible in the field of spectral analysis. To this end, we introduce non-quadratic convex penalization functions, like in low level image processing. The spectral amplitudes estimate is then defined as the unique minimizer of a compound convex criterion. An original scheme of regularization to simultaneously retrieve narrow-band and wide-band spectral features is finally proposed.

An acoustic mismatch between a given utterance and a model degrades the performance of the speech recognition process. We choose to model speech by hidden Markov models (HMMs) in the cepstrum domain and the mismatch by a parametric function. In order to reduce the mismatch, one has to estimate the parameters of this function. We present a frame synchronous estimation of these parameters. We show that the parameters can be computed recursively. Thanks to such methods, parameters variations can be tracked. We give general equations and study the particular case of an affine transform. Finally, we report recognition experiments carried out over both PSTN and cellular telephone network to show the efficiency of the method in a real context.

We propose a method for the reconstruction of signals and images observed partially through a linear operator with a large support (e.g., a Fourier transform on a sparse set). This inverse problem is ill-posed and we resolve it by incorporating the prior information that the reconstructed objects are composed of smooth regions separated by sharp transitions. This feature is modeled by a piecewise Gaussian (PG) Markov random field (MRF), known also as the weak-string in one dimension and the weak-membrane in two dimensions. The reconstruction is defined as the maximum a posteriori estimate. The prerequisite for the use of such a prior is the success of the optimization stage. The posterior energy corresponding to a PG MRF is generally multimodal and its minimization is particularly problematic. In this context, general forms of simulated annealing rapidly become intractable when the observation operator extends over a large support. In this paper, global optimization is dealt with by extending the graduated nonconvexity (GNC) algorithm to ill-posed linear inverse problems. GNC has been pioneered by Blake and Zisserman in the field of image segmentation. The resulting algorithm is mathematically suboptimal but it is seen to be very efficient in practice. We show that the original GNC does not correctly apply to ill-posed problems. Our extension is based on a proper theoretical analysis, which provides further insight into the GNC. The performance of the proposed algorithm is corroborated by a synthetic example in the area of diffraction tomography.

This paper provides a complete characterization of stationary Markov random fields on a finite rectangular (non-toroidal) lattice in the basic case of a second-order neighborhood system. Equivalently, it characterizes stationary Markov fields on Z/sup 2/ whose restrictions to finite rectangular subsets are still Markovian (i.e., even on the boundaries). Until now, Pickard random fields formed the only known class of such fields. First, we derive a necessary and sufficient condition for Markov random fields on a finite lattice to be stationary. It is shown that their joint distribution factors in terms of the marginal distribution on a generic (2/spl times/2) cell which must fulfil some consistency constraints. Second, we solve the consistency constraints and provide a complete characterization of such measures in three cases. Symmetric measures and Gaussian measures are shown to necessarily belong to the Pickard class, whereas binary measures belong either to the Pickard class, or to a new nontrivial class which is further studied. In particular, the corresponding fields admit a simple parameterization and may be simulated in a simple, although nonunilateral manner.

We focus on the nonlinear inverse problem of diffraction tomography. We set the problem as one of estimation within the Bayesian framework and define the solution as the maximum a posteriori (MAP) estimate which corresponds to the global minimum of a multimodal criterion. The objective of this paper is to present a new deterministic single site update algorithm specially designed to compute this solution. The term of fidelity to the data, function of one pixel value, can be written as a second order rational fraction. Thus, the 1-D MAP criterion can be evaluated and minimized-at a very low computational cost. Moreover, for certain MRF models the global minimum can even be computed explicitly as roots of a polynomial. The proposed algorithm turns these properties to advantage and moreover performs the updates of intermediate quantities at a particularly low cost compared to the criterion evaluation. Even if not guaranteed to converge towards the global minimum, the algorithm has shown itself to give satisfactory practical results.

We address the problem of identification of sinusoidal components from observed data, which is fundamental for array signal processing and spectral line decomposition. Joint detection and estimation are proposed in a unified Bayesian framework, so that no preliminary estimate of the number of signals is required. All unknown quantities are estimated from a unique regularized "stochastic" likelihood function, including the number of sources and statistical parameters. The impulsive solution is modeled as a continuous Poisson-Gaussian process. A powerful iterative technique is proposed to maximize the posterior likelihood. Simulation results show that the method behaves particularly well for small data sets, even for a single experiment.

We design a model meant to be the equivalent of Blake's weak string but in the probabilistic framework. Indepen-dent line sites delimit piecewise stationary Gaussian au-toregressives AR(1) corrupted with Gaussian white noise. Thanks to the Bayesian interpretation, we dene the joint probability which in turn yields the likelihood. We demon-strate how to make its computation possible in cubic time. This calculation allows the set of parameters to be tested but not estimated due to the complex form of the criterion. Yet the computations done so far provide the materials for an iterative maximization. Indeed, the Expectation Max-imization algorithm happens to match the features of this model and is also easily calculable. When the likelihood is known, the cost of one step of the latter algorithm is negligible in comparison with the previous calculations. 1.

This article presents a non-supervised segmentation method based upon a discrete-level unilateral Markov field model of the image. Such models have been shown to yield numerically efficient algorithms, for segmentation and for hyperparameter estimation as well. Our contribution lies in the derivation of a parsimonious telegraphic parameterization of the unilateral Markov field. On a theoretical level, this parameterization ensures that some important properties of the field (e.g., stationarity) do hold. On a practical level, it reduces the computational complexity of the algorithm used in the segmentation and parameter estimation stages of the procedure. In addition, it decreases the number of hyperparameters that must be estimated, thereby improving the convergence speed and accuracy of the corresponding estimation method.

Electrical impedance tomography of closed conductive media is an ill-posed inverse problem. Using the finite elements method to solve the corresponding direct problem allows one to preserve the nonlinear dependence of the observation set upon the conductivity distribution. We show that the Bayesian approach presented by Demoment (1989) for linear inverse imaging problems is still valid for such a nonlinear inverse problem. Our contribution is based on an edge-preserving Markov model as a prior for conductivity distribution. Maximum a posteriori reconstruction results from 40-dB noisy measurements (simulated with a finer mesh) yield significant resolution improvement compared to classical methods.

We present optimized algorithms for image deblurring in the case of a separable point spread function (PSF). Our work is in the usual context of Bayesian estimation with Gibbs random fields (GRF). The derived algorithms fall into the class of single site update algorithms (SSUAs), which exhibit a high convergence rate per iteration and small memory requirements, while hard domain constraints such as positivity are easily introduced. On the other hand, standard forms of SSUAs rapidly become intractable when the size of the PSF is large. We show how PSF separability can benefit the SSUAs, in order to reduce the cost of each pixel update from O(2pq) to O(p+q) (p/spl times/q is the size of the PSF). We show that the resulting deterministic SSUA compares very favorably with global update algorithms (GUAs). The new separable form can also benefit other SSUAs, especially stochastic versions such as simulated annealing (SA) and Monte Carlo Markov chain (MCMC) algorithms.

A Bayesian estimator of the magnetic resonance (MR) velocity image is proposed. It is based on a Markov model accounting for the spatial structure of the flow velocity. On the other hand, low MR signal intensity yields high uncertainty on the velocity. Such an important property is taken into account through the observation model. The resulting posterior likelihood is optimized using an iterative coordinate descent (ICD) algorithm. Compared to the usual least squares solution, simulation results on flows with parabolic and flat profiles demonstrate a significant gain of in the mean square error.

Because true maximum likelihood (ML) is too expensive, the dominant approach in Bernoulli-Gaussian (BG) myopic deconvolution consists in the joint maximization of a single generalized likelihood with respect to the input signal and the hyperparameters. This article assesses the theoretical properties of a related maximum generalized marginal likelihood (MGML) estimator in a simplified framework: the filter is reduced to identity, so that the output data is a mixture of Gaussian populations. Our results are three-fold: first, exact MGML estimates can be efficiently computed; second, this estimator performs better than ML in the short sample case whereas it is drastically less expensive; third, asymptotic estimates are significant although biased.

We propose a method for the reconstruction of an image, only partially observed through a linear integral operator. As such an inverse problem is ill-posed, prior information must be introduced. We consider the case of a compound Markov random field with a non-interacting line process. In order to maximise the posterior likelihood function, we propose an extension of the graduated non convexity principle pioneered by Blake and Zisserman (1987) which allows its use for ill-posed linear inverse problems. We discuss the role of the observation scale and some aspects of the implemented algorithm. Finally, we present an application of the method to a diffraction tomography imaging problem.>

Deals with Bayesian estimation of 2D stratified structures from echosounding signals. This problem is of interest in seismic exploration, but also for nondestructive testing or medical imaging. The proposed approach consists of a multichannel Bayesian deconvolution method of the 2D reflectivity based upon a theoretically sound prior stochastic model. The Markov-Bernoulli random field representation introduced by Idier et al. (1993) is used to model the geometric properties of the reflectivity, and emphasis is placed on representation of the amplitudes and on deconvolution algorithms. It is shown that the algorithmic structure and computational complexity of the proposed multichannel methods are similar to those of single-channel B-G deconvolution procedures, but that explicit modeling of the stratified structure results in significantly better performances. Simulation results and examples of real-data processing illustrate the performances and the practicality of the multichannel approach.>

Bayesian estimation of two-dimensional stratified structures is described. The major point addressed is the derivation of a statistical prior model that adequately describes such layered media. This problem is of interest in applications in which the data are generally processed in one dimension only. In order to take local interactions into account, a Markovian description is used. The model is derived so as to fulfill a set of constraints that summarize physical and geometrical characteristics of the problem as well as practical requirements. The resulting class of Markov random fields presents a unilateral structure on a nonrectangular lattice and a hierarchical organization which involves a line process. In addition, it is shown to be an extension of one-dimensional models already in use. The properties of the model are investigated, and its practicality is demonstrated by an application to seismic deconvolution. Simulation results show significant improvements with respect to the usual one-dimensional methods.>

An iterative algorithm for deconvolution of Bernoulli-Gaussian processes is presented. This detection-estimation problem is formulated as that of a change of initial conditions in linear least-squares estimation. An algorithm with a very simple structure is obtained. It allows the evaluation of either marginal or joint likelihood criteria without any approximation; the resulting method is easy to implement and computationally inexpensive and remains nearly optimal. >

Two dimensional (2D) Bayesian restoration of layered media and its application to seismic deconvolution are addressed. The Bernoulli-Gaussian (B-G) approach is suboptimal in a 2D setting, since it does not account for spatial continuity. To compensate for this deficiency, a class of Markov random fields which suitably extends 1D Bernoulli processes to two dimensions is built. It provides a manageable model for 2D stratified structures. A maximum a posteriori detection-estimation algorithm is proposed. Compared to standard B-G algorithms, it performs much better, without an increase in computational load.>