Representing the image to be inpainted in an appropriate sparse representation dictionary, and combining elements from Bayesian statistics and modern harmonic analysis, we introduce an expectation maximization (EM) algorithm for image inpainting and interpolation. From a statistical point of view, the inpainting/interpolation can be viewed as an estimation problem with missing data. Toward this goal, we propose the idea of using the EM mechanism in a Bayesian framework, where a sparsity promoting prior penalty is imposed on the reconstructed coefficients. The EM framework gives a principled way to establish formally the idea that missing samples can be recovered/ interpolated based on sparse representations. We first introduce an easy and efficient sparserepresentation-based iterative algorithm for image inpainting. Additionally, we derive its theoretical convergence properties. Compared to its competitors, this algorithm allows a high degree of flexibility to recover different structural components in the image (piecewise smooth, curvilinear, texture, etc.). We also suggest some guidelines to automatically tune the regularization parameter.

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Long-memory noise is common to many areas of signal processing and can seriously confound estimation of linear regression model parameters and their standard errors. Classical autoregressive moving average (ARMA) methods can adequately address the problem of linear time invariant, short-memory errors but may be inefficient and/or insufficient to secure type 1 error control in the context of fractal or scale invariant noise with a more slowly decaying autocorrelation function. Here we introduce a novel method, called wavelet-generalized least squares (WLS), which is (to a good approximation) the best linear unbiased (BLU) estimator of regression model parameters in the context of long-memory errors. The method also provides maximum likelihood (ML) estimates of the Hurst exponent (which can be readily translated to the fractal dimension or spectral exponent) characterizing the correlational structure of the errors, and the error variance. The algorithm exploits the whitening or Karhunen-Loé ve-type property of the discrete wavelet transform to diagonalize the covariance matrix of the errors generated by an iterative fitting procedure after both data and design matrix have been transformed to the wavelet domain. Properties of this estimator, including its Cramè r-Rao bounds, are derived theoretically and compared to its empirical performance on a range of simulated data. Compared to ordinary least squares and ARMA-based estimators, WLS is shown to be more efficient and to give excellent type 1 error control. The method is also applied to some real (neurophysiological) data acquired by functional magnetic resonance imaging (fMRI) of the human brain. We conclude that wavelet-generalized least squares may be a generally useful estimator of regression models in data complicated by long-memory or fractal noise. © 2002 Elsevier Science

| This paper gives essential insights into the use of sparsity and morphological diversity in image decomposition and source separation by reviewing our recent work in this field.The idea to morphologically decompose a signal into its building blocks is an important problem in signal processing and has farreaching applications in science and technology. Starck et al. is an iterative thresholding algorithm that is capable of decoupling the signal content. Sparsity and morphological diversity have also been used as a novel and effective source of diversity for blind source separation (BSS), hence extending the MCA to multichannel data. Building on these ingredients, we will provide an overview the generalized MCA introduced by the authors in [3] and [4] as a fast and efficient BSS method. We will illustrate the application of these algorithms on several real examples. We conclude our tour by briefly describing our software toolboxes made available for download on the Internet for sparse signal and image decomposition and separation.

A paradigm independent multistage strategy based on the Unsupervised Fuzzy Clustering Analysis (UFCA) and its potential for fMRI data analysis are presented. The influence of the fuzziness index is studied using Receiver Operating Characteristics (ROC) methodology and an interval of choice, around the widely used value 2, is shown to yield the best performance. The ill‐balanced data problem is also overcome using a pre‐processing step to reduce the number of voxels presented to the method. Statistical and anatomical criteria are proposed to exclude some voxels and enhance the UFCA sensitivity. An original postprocessing step aiming at statistically characterizing the obtained clusters is also developed. Two similarity criteria are used: the correlation coefficient on temporal profiles and a novel fuzzy overlap coefficient on membership degree maps. This final step provides a useful analysis tool to study intra‐individual reproducibility of the classes across series (stimulation vs. stimulation, noise vs. noise or stimulation vs. noise). Finally, a comparison between this technique and some existing or locally developed postprocessing algorithms is presented using ROC methods. Its sensitivity and robustness is compared to the classical FCA or other techniques as a function of several parameters such as Contrast‐to‐Noise Ratio (CNR) and noise amplitude. Even without knowledge about the paradigm, the hemodynamic response function and the number of clusters, the performances of the proposed strategy are comparable to those of the classical approaches where extensive prior knowledge has to be added. Hum. Brain Mapping 10:160–178, 2000. © 2000 Wiley‐Liss, Inc.

. On the number of clusters and the fuzziness index for unsupervised FCA application to BOLD fMRI time series.

Context. One of the main challenges of modern cosmology is to understand the nature of the mysterious dark energy that causes the cosmic acceleration. The integrated Sachs-Wolfe (ISW) effect is sensitive to dark energy, and if detected in a universe where modified gravity and curvature are excluded, presents an independent signature of dark energy. The ISW effect occurs on large scales where cosmic variance is high and where owing to the Galactic confusion we lack large amounts of data in the CMB as well as large-scale structure maps. Moreover, existing methods in the literature often make strong assumptions about the statistics of the underlying fields or estimators. Together these effects can severely limit signal extraction. Aims. We aim to define an optimal statistical method for detecting the ISW effect that can handle large areas of missing data and minimise the number of underlying assumptions made about the data and estimators. Methods. We first review current detections (and non-detections) of the ISW effect, comparing statistical subtleties between existing methods, and identifying several limitations. We propose a novel method to detect and measure the ISW signal. This method assumes only that the primordial CMB field is Gaussian. It is based on a sparse inpainting method to reconstruct missing data and uses a bootstrap technique to avoid assumptions about the statistics of the estimator. It is a complete method, which uses three complementary statistical methods. Results. We apply our method to Euclid-like simulations and show we can expect a ∼7σ model-independent detection of the ISW signal with WMAP7-like data, even when considering missing data. Other tests return ∼5σ detection levels for a Euclid-like survey. We find that detection levels are independent from whether the galaxy field is normally or lognormally distributed. We apply our method to the 2 Micron All Sky Survey (2MASS) and WMAP7 CMB data and find detections in the 1.0−1.2σ range, as expected from our simulations. As a by-product, we have also reconstructed the full-sky temperature ISW field from the 2MASS data.Conclusions. We present a novel technique based on sparse inpainting and bootstrapping, which accurately detects and reconstructs the ISW effect.

Reconstructing the cosmic microwave background (CMB) in the Galactic plane is extremely difficult due to the dominant foreground emissions such as dust, free-free or synchrotron. For cosmological studies, the standard approach consists in masking this area where the reconstruction is insufficient. This leads to difficulties for the statistical analysis of the CMB map, especially at very large scales (to study for instance the low quadrupole, integrated Sachs Wolfe effect, axis of evil, etc.). We investigate how well some inpainting techniques can recover the low-spherical harmonic coefficients. We introduce three new inpainting techniques based on three different kinds of priors: sparsity, energy, and isotropy, which we compare. We show that sparsity and energy priors can lead to extremely high-quality reconstruction, within 1% of the cosmic variance for a mask with a sky coverage larger than 80%.