Multifractal analysis (MFA) provides a framework for the global characterization of image textures by describing the spatial fluctuations of their local regularity based on the multifractal spectrum. Several works have shown the interest of using MFA for the description of homogeneous textures in images. Nevertheless, natural images can be composed of several textures and, in turn, multifractal properties associated with those textures. This paper introduces an unsupervised Bayesian multifractal segmentation method to model and segment multifractal textures by jointly estimating the multifractal parameters and labels on images, at the pixel-level. For this, a computationally and statistically efficient multifractal parameter estimation model for wavelet leaders is firstly developed, defining different multifractality parameters for different regions of an image. Then, a multiscale Potts Markov random field is introduced as a prior to model the inherent spatial and scale correlations (referred to as cross-scale correlations) between the labels of the wavelet leaders. A Gibbs sampling methodology is finally used to draw samples from the posterior distribution of the unknown model parameters. Numerical experiments are conducted on synthetic multifractal images to evaluate the performance of the proposed segmentation approach. The proposed method achieves superior performance compared to traditional unsupervised segmentation techniques as well as modern deep learning-based approaches, showing its effectiveness for multifractal image segmentation.

Time-correlated single-photon technology is emerging as an important approach to 3D Imaging. This paper presents a reconstruction algorithm that exploits data statistics and multi-scale information to deliver clean depth and reflectivity images together with associated uncertainty maps. The statistical method has been implemented to run on graphics processing units (GPUs) that enable real-time reconstruction of moving scenes at more than 1000 depth frames per second on the 32 × 64 pixels real Quantic4x4 SPAD sensor array data. Comparisons with state-of-the-art algorithms on simulated and real data demonstrate the robust and efficient performance of the proposed method.

This paper presents a new Bayesian algorithm for the robust reconstruction of multispectral single-photon Lidar data acquired in extreme conditions. We focus on imaging through obscurants (i.e., fog, water) leading to high and possibly non-uniform background noise. The proposed hierarchical Bayesian method accounts for multiscale information to provide distribution estimates for the target’s depth and reflectivity, i.e., point and uncertainty measures of the estimates to improve decision making. The correlations between variables are enforced using a weighting scheme that allows the incorporation of guide information available from other sensors or state-of-the-art algorithms. Results on synthetic and real data show improved reconstruction of the scene in extreme conditions when compared to the state-of-the-art algorithms.

This paper presents a new algorithm for the learning of spatial correlation and non-local restoration of single-photon 3-Dimensional Lidar images acquired in the photon starved regime (fewer or less than one photon per pixel) or with a reduced number of scanned spatial points (pixels). The algorithm alternates between three steps: (i) extract multi-scale information, (ii) build a robust graph of non-local spatial correlations between pixels, and (iii) the restoration of depth and reflectivity images. A non-uniform sampling approach, which assigns larger patches to homogeneous regions and smaller ones to heterogeneous regions, is adopted to reduce the computational cost associated with the graph. The restoration of the 3D images is achieved by minimizing a cost function accounting for the multi-scale information and the non-local spatial correlation between patches. This minimization problem is efficiently solved using the alternating direction method of multipliers (ADMM) that presents fast convergence properties. Various results based on simulated and real Lidar data show the benefits of the proposed algorithm that improves the quality of the estimated depth and reflectivity images, especially in the photon-starved regime or when containing a reduced number of spatial points.

The estimation of the hemodynamic response function (HRF) in functional magnetic resonance imaging (fMRI) is critical to deconvolve a time-resolved neural activity and get insights on the underlying cognitive processes. Existing methods propose to estimate the HRF using the experimental paradigm (EP) in task fMRI as a surrogate of neural activity. These approaches induce a bias as they do not account for latencies in the cognitive responses compared to EP and cannot be applied to resting-state data as no EP is available. In this work, we formulate the joint estimation of the HRF and neural activation signal as a semi blind deconvolution problem. Its solution can be approximated using an efficient alternate minimization algorithm. The proposed approach is applied to task fMRI data for validation purpose and compared to a state-of-the-art HRF estimation technique. Numerical experiments suggest that our approach is competitive with others while not requiring EP information.

This paper presents a novel nonlinear hyperspectral mixture model and its associated supervised unmixing algorithm. The model assumes a linear mixing model corrupted by an additive term which accounts for multiple scattering nonlinearities (NL). The proposed model generalizes bilinear models by taking into account higher order interaction terms. The inference of the abundances and nonlinearity coefficients of this model is formulated as a convex optimization problem suitable for fast estimation algorithms. This formulation accounts for constraints such as the sum-to-one and nonnegativity of the abundances, the non-negativity of the nonlinearity coefficients, and the spatial sparseness of the residuals. The resulting convex problem is solved using the alternating direction method of multipliers (ADMM) whose convergence is ensured theoretically. The proposed mixture model and its unmixing algorithm are validated on both synthetic and real images showing competitive results regarding the quality of the inference and the computational complexity when compared to the state-of-the-art algorithms.

In hyperspectral images, some spectral bands suffer from low signal-to-noise ratio due to noisy acquisition and atmospheric effects, thus requiring robust techniques for the unmixing problem. This paper presents a robust supervised spectral unmixing approach for hyperspectral images. The robustness is achieved by writing the unmixing problem as the maximization of the correntropy criterion subject to the most commonly used constraints. Two unmixing problems are derived: the first problem considers the fully constrained unmixing, with both the nonnegativity and sum-to-one constraints, while the second one deals with the nonnegativity and the sparsity promoting of the abundances. The corresponding optimization problems are solved using an alternating direction method of multipliers (ADMM) approach. Experiments on synthetic and real hyperspectral images validate the performance of the proposed algorithms for different scenarios, demonstrating that the correntropy-based unmixing with ADMM is particularly robust against highly noisy outlier bands.

The correntropy provides a robust criterion for outlier-insensitive machine learning, and its maximisation has been increasingly investigated in signal and image processing. In this paper, we investigate the problem of unmixing hyperspectral images, namely decomposing each pixel/spectrum of a given image as a linear combination of other pixels/spectra called endmembers. The coefficients of the combination need to be estimated subject to the nonnegativity and the sum-to-one constraints. In practice, some spectral bands suffer from low signal-to-noise ratio due to acquisition noise and atmospheric effects, thus requiring robust techniques for the unmixing problem. In this work, we cast the unmixing problem as the maximization of a correntropy criterion, and provide a relevant solution using the alternating direction method of multipliers (ADMM) method. Finally, the relevance of the proposed approach is validated on synthetic and real hyperspectral images, demonstrating that the correntropy-based unmixing is robust to outlier bands.

Linear mixture models are commonly used to represent a hyperspectral data cube as linear combinations of endmember spectra. However, determining the number of endmembers for images embedded in noise is a crucial task. This paper proposes a fully automatic approach for estimating the number of endmembers in hyperspectral images. The estimation is based on recent results of random matrix theory related to the so-called spiked population model. More precisely, we study the gap between successive eigenvalues of the sample covariance matrix constructed from high-dimensional noisy samples. The resulting estimation strategy is fully automatic and robust to correlated noise owing to the consideration of a noise-whitening step. This strategy is validated on both synthetic and real images. The experimental results are very promising and show the accuracy of this algorithm with respect to state-of-the-art algorithms.

This paper proposes a new Bayesian strategy for the smooth estimation of altimetric parameters. The altimetric signal is assumed to be corrupted by a thermal and speckle noise distributed according to an independent and non-identically Gaussian distribution. We introduce a prior enforcing a smooth temporal evolution of the altimetric parameters which improves their physical interpretation. The posterior distribution of the resulting model is optimized using a gradient descent algorithm which allows us to compute the maximum a posteriori estimator of the unknown model parameters. This algorithm has a low computational cost that is suitable for real-time applications. The proposed Bayesian strategy and the corresponding estimation algorithm are evaluated using both synthetic and real data associated with conventional and delay/Doppler altimetry. The analysis of real Jason-2 and CryoSat-2 waveforms shows an improvement in parameter estimation when compared to state-of-the-art estimation algorithms.

This paper presents three hyperspectral mixture models jointly with Bayesian algorithms for supervised hyperspectral unmixing. Based on the residual component analysis model, the proposed general formulation assumes the linear model to be corrupted by an additive term whose expression can be adapted to account for nonlinearities (NLs), endmember variability (EV), or mismodeling effects (MEs). The NL effect is introduced by considering a polynomial expression that is related to bilinear models. The proposed new formulation of EV accounts for shape and scale endmember changes while enforcing a smooth spectral/spatial variation. The ME formulation considers the effect of outliers and copes with some types of EV and NL. The known constraints on the parameter of each observation model are modeled via suitable priors. The posterior distribution associated with each Bayesian model is optimized using a coordinate descent algorithm, which allows the computation of the maximum a posteriori estimator of the unknown model parameters. The proposed mixture and Bayesian models and their estimation algorithms are validated on both synthetic and real images showing competitive results regarding the quality of the inferences and the computational complexity, when compared with the state-of-the-art algorithms.

This paper presents an unsupervised Bayesian algorithm for hyperspectral image unmixing accounting for endmember variability. Each image pixel is modeled by a linear combination of random endmembers to take into account endmember variability in the image. The coefficients of this linear combination (referred to as abundances) allow the proportions of each material (endmembers) to be quantified in the image pixel. An additive noise is also considered in the proposed model generalizing the normal compositional model. The proposed Bayesian algorithm exploits spatial correlations between adjacent pixels of the image and provides spectral information by achieving a spectral unmixing. It estimates both the mean and the covariance matrix of each endmember in the image. A spatial classification is also obtained based on the estimated abundances. Simulations conducted with synthetic and real data show the potential of the proposed model and the unmixing performance for the analysis of hyperspectral images.

This paper proposes a new Bayesian strategy for the estimation of smooth parameters from nonlinear models. The observed signal is assumed to be corrupted by an independent and non identically (colored) Gaussian distribution. A prior enforcing a smooth temporal evolution of the model parameters is considered. The joint posterior distribution of the unknown parameter vector is then derived. A Gibbs sampler coupled with a Hamiltonian Monte Carlo algorithm is proposed which allows samples distributed according to the posterior of interest to be generated and to estimate the unknown model parameters/hyperparameters. Simulations conducted with synthetic and real satellite altimetric data show the potential of the proposed Bayesian model and the corresponding estimation algorithm for nonlinear regression with smooth estimated parameters.

Delay/Doppler altimetry (DDA) aims at reducing the measurement noise and increasing the along-track resolution in comparison with conventional pulse-limited altimetry. In a previous paper, we have proposed a semi-analytical model for DDA, which considers some simplifications as the absence of mispointing antenna. This paper first proposes a new analytical expression for the flat surface impulse response (FSIR), considering antenna mispointing angles, a circular antenna pattern, no vertical speed effect, and uniform scattering. The 2-D delay/Doppler map is then obtained by a numerical computation of the convolution between the proposed analytical function, the probability density function of the heights of the specular scatterers, and the time/frequency point target response of the radar. The approximations used to obtain the semi-analytical model are analyzed, and the associated errors are quantified by analytical bounds for these errors. The second contribution of this paper concerns the estimation of the parameters associated with the multilook semi-analytical model. Two estimation strategies based on the least squares procedure are proposed. The proposed model and algorithms are validated on both synthetic and real waveforms. The obtained results are very promising and show the accuracy of this generalized model with respect to the previous model assuming zero antenna mispointing.

This paper presents an unsupervised Bayesian algorithm for hyperspectral image unmixing, accounting for endmember variability. The pixels are modeled by a linear combination of endmembers weighted by their corresponding abundances. However, the endmembers are assumed random to consider their variability in the image. An additive noise is also considered in the proposed model, generalizing the normal compositional model. The proposed algorithm exploits the whole image to benefit from both spectral and spatial information. It estimates both the mean and the covariance matrix of each endmember in the image. This allows the behavior of each material to be analyzed and its variability to be quantified in the scene. A spatial segmentation is also obtained based on the estimated abundances. In order to estimate the parameters associated with the proposed Bayesian model, we propose to use a Hamiltonian Monte Carlo algorithm. The performance of the resulting unmixing strategy is evaluated through simulations conducted on both synthetic and real data.

This paper introduces a new model for delay/Doppler altimetry, taking into account the effect of antenna mispointing. After defining the proposed model, the effect of the antenna mispointing on the altimetric waveform is analyzed as a function of along-track and across-track angles. Two least squares approaches are investigated for estimating the parameters associated with the proposed model. The first algorithm estimates four parameters including the across-track mispointing (which affects the echo's shape). The second algorithm uses the mispointing angles provided by the star-trackers and estimates the three remaining parameters. The proposed model and algorithms are validated via simulations conducted on both synthetic and real data.

The concept of delay/Doppler (DD) altimetry (DDA) has been under study since the mid-1990s, aiming at reducing the measurement noise and increasing the along-track resolution in comparison with the conventional pulse-limited altimetry. This paper introduces a new model for the mean backscattered power waveform acquired by a radar altimeter operating in synthetic aperture radar mode, as well as an associated least squares (LS) estimation algorithm. As in conventional altimetry (CA), the mean power can be expressed as the convolution of three terms: the flat surface impulse response (FSIR), the probability density function of the heights of the specular scatterers, and the time/frequency point target response of the radar. An important contribution of this paper is to derive an analytical formula for the FSIR associated with DDA. This analytical formula is obtained for a circular antenna pattern, no mispointing, no vertical speed effect, and a uniform scattering. The double convolution defining the mean echo power can then be computed numerically, resulting in a 2-D semi-analytical model called the DD map (DDM). This DDM depends on three altimetric parameters: the epoch, the sea surface wave height, and the amplitude. A multi-look model is obtained by summing all the reflected echoes from the same along-track surface location of interest after applying appropriate delay compensation (range migration) to align the DDM on the same reference. The second contribution of this paper concerns the estimation of the parameters associated with the multi-look semi-analytical model. An LS approach is investigated by means of the Levenberg-Marquardt algorithm. Simulations conducted on simulated altimetric waveforms allow the performance of the proposed estimation algorithm to be appreciated. The analysis of Cryosat-2 waveforms shows an improvement in parameter estimation when compared to the CA.

Much attention has been recently devoted to the analysis of coastal altimetric waveforms. When approaching the coast, altimetric waveforms are sometimes corrupted by peaks caused by high reflective areas inside the illuminated land surfaces or by the modification of the sea state close to the shoreline. This paper introduces a new parametric model for these peaky altimetric waveforms. This model assumes that the received altimetric waveform is the sum of a Brown echo and an asymmetric Gaussian peak. The asymmetric Gaussian peak is parameterized by a location, an amplitude, a width, and an asymmetry coefficient. A maximum-likelihood estimator is studied to estimate the Brown plus peak model parameters. The Cramér-Rao lower bounds of the model parameters are then derived providing minimum variances for any unbiased estimator, i.e., a reference in terms of estimation error. The performance of the proposed model and the resulting estimation strategy are evaluated via many simulations conducted on synthetic and real data. Results obtained in this paper show that the proposed model can be used to retrack efficiently standard oceanic Brown echoes as well as coastal echoes corrupted by symmetric or asymmetric Gaussian peaks. Thus, the Brown with Gaussian peak model is useful for analyzing altimetric measurements closer to the coast.

This paper presents a nonlinear mixing model for hyperspectral image unmixing. The proposed model assumes that the pixel reflectances are nonlinear functions of pure spectral components contaminated by an additive white Gaussian noise. These nonlinear functions are approximated using polynomial functions leading to a polynomial postnonlinear mixing model. A Bayesian algorithm and optimization methods are proposed to estimate the parameters involved in the model. The performance of the unmixing strategies is evaluated by simulations conducted on synthetic and real data.

This paper studies a hierarchical Bayesian model for nonlinear hyperspectral image unmixing. The proposed model assumes that the pixel reflectances are polynomial functions of linear mixtures of pure spectral components contaminated by an additive white Gaussian noise. The parameters involved in this model satisfy constraints that are naturally expressed within a Bayesian framework. A Gibbs sampler allows one to sample the unknown abundances and nonlinearity parameters according to the joint posterior of interest. The performance of the resulting unmixing strategy is evaluated thanks to simulations conducted on synthetic and real data.

This paper studies estimation algorithms for nonlinear hyperspectral image unmixing. The proposed unmixing model assumes that the pixel reflectances are polynomial functions of linear mixtures of pure spectral components contaminated by an additive white Gaussian noise. A hierarchical Bayesian algorithm and an optimization method are proposed for solving the resulting unmixing problem. The parameters involved in the proposed model satisfy constraints that are naturally included in the estimation procedure. The performance of the unmixing strategies is evaluated thanks to simulations conducted on synthetic and real data.

Nonlinear models have recently shown interesting properties for spectral unmixing. This paper considers a generalized bilinear model recently introduced for unmixing hyperspectral images. Different algorithms are studied to estimate the parameters of this bilinear model. The positivity and sum-to-one constraints for the abundances are ensured by the proposed algorithms. The performance of the resulting unmixing strategy is evaluated via simulations conducted on synthetic and real data.

Coastal altimetric waveforms may be corrupted by peaks. A simple parametric model was recently introduced to model peaky altimetric waveforms. This model assumes that the received altimetric waveform is the sum of a Brown echo and a Gaussian peak. This model has provided interesting results for symmetric peaks affecting altimetric signals. However, it is not appropriate for altimetric signals corrupted by asymmetric peaks. This paper introduces a Brown with asymmetric Gaussian peak model for altimetric waveforms. The parameters of this model are estimated by a maximum likelihood estimator. The performance of the proposed model and the resulting estimation strategy is evaluated via simulations con ducted on synthetic and real data.

Nonlinear models have recently shown interesting properties for spectral unmixing. This paper studies a generalized bilinear model and a hierarchical Bayesian algorithm for unmixing hyperspectral images. The proposed model is a generalization not only of the accepted linear mixing model but also of a bilinear model that has been recently introduced in the literature. Appropriate priors are chosen for its parameters to satisfy the positivity and sum-to-one constraints for the abundances. The joint posterior distribution of the unknown parameter vector is then derived. Unfortunately, this posterior is too complex to obtain analytical expressions of the standard Bayesian estimators. As a consequence, a Metropolis-within-Gibbs algorithm is proposed, which allows samples distributed according to this posterior to be generated and to estimate the unknown model parameters. The performance of the resulting unmixing strategy is evaluated via simulations conducted on synthetic and real data.