Jean-Luc Starck

Image decomposition via the combination of sparse representations and a variational approach

The separation of image content into semantic parts plays a vital role in applications such as compression, enhancement, restoration, and more. In recent years, several pioneering works suggested such a separation be based on variational formulation and others using independent component analysis and sparsity. This paper presents a novel method for separating images into texture and piecewise smooth (cartoon) parts, exploiting both the variational and the sparsity mechanisms. The method combines the basis pursuit denoising (BPDN) algorithm and the total-variation (TV) regularization scheme. The basic idea presented in this paper is the use of two appropriate dictionaries, one for the representation of textures and the other for the natural scene parts assumed to be piecewise smooth. Both dictionaries are chosen such that they lead to sparse representations over one type of image-content (either texture or piecewise smooth). The use of the BPDN with the two amalgamed dictionaries leads to the desired separation, along with noise removal as a by-product. As the need to choose proper dictionaries is generally hard, a TV regularization is employed to better direct the separation process and reduce ringing artifacts. We present a highly efficient numerical scheme to solve the combined optimization problem posed by our model and to show several experimental results that validate the algorithms performance.

Sparse Solution of Underdetermined Systems of Linear Equations by Stagewise Orthogonal Matching Pursuit

Finding the sparsest solution to underdetermined systems of linear equations y = Φx is NP-hard in general. We show here that for systems with “typical”/“random” Φ, a good approximation to the sparsest solution is obtained by applying a fixed number of standard operations from linear algebra. Our proposal, Stagewise Orthogonal Matching Pursuit (StOMP), successively transforms the signal into a negligible residual. Starting with initial residual r0 = y, at the s -th stage it forms the “matched filter” ΦTrs-1, identifies all coordinates with amplitudes exceeding a specially chosen threshold, solves a least-squares problem using the selected coordinates, and subtracts the least-squares fit, producing a new residual. After a fixed number of stages (e.g., 10), it stops. In contrast to Orthogonal Matching Pursuit (OMP), many coefficients can enter the model at each stage in StOMP while only one enters per stage in OMP; and StOMP takes a fixed number of stages (e.g., 10), while OMP can take many (e.g., n). We give both theoretical and empirical support for the large-system effectiveness of StOMP. We give numerical examples showing that StOMP rapidly and reliably finds sparse solutions in compressed sensing, decoding of error-correcting codes, and overcomplete representation.

Gray and color image contrast enhancement by the curvelet transform

We present in this paper a new method for contrast enhancement based on the curvelet transform. The curvelet transform represents edges better than wavelets, and is therefore well-suited for multiscale edge enhancement. We compare this approach with enhancement based on the wavelet transform, and the Multiscale Retinex. In a range of examples, we use edge detection and segmentation, among other processing applications, to provide for quantitative comparative evaluation. Our findings are that curvelet based enhancement out-performs other enhancement methods on noisy images, but on noiseless or near noiseless images curvelet based enhancement is not remarkably better than wavelet based enhancement.

Redundant Multiscale Transforms and Their Application for Morphological Component Separation

Publisher Summary This chapter presents an alternative deterministic methodology, based on sparsity, toward the problem of morphological component analysis (MCA) and anchors this method with some conclusive theoretical results, essentially guaranteeing successful separation under some conditions. The chapter also demonstrates the use of MCA in several applications for images. A major role in the application of the MCA method is played by the dictionaries chosen for decomposition. A wide survey of possible fast-implementation dictionaries taken from the wavelet theory is presented in the chapter, along with ways to use these dictionaries in linear and nonlinear settings. The combination of multi-scale transforms leads to a powerful method in the MCA framework. For some applications such de-noising or de-convolution, MCA is, however, not the best way to combine the different transforms and to benefit from the advantages of each of them.

The Undecimated Wavelet Decomposition and its Reconstruction

This paper describes the undecimated wavelet transform and its reconstruction. In the first part, we show the relation between two well known undecimated wavelet transforms, the standard undecimated wavelet transform and the isotropic undecimated wavelet transform. Then we present new filter banks specially designed for undecimated wavelet decompositions which have some useful properties such as being robust to ringing artifacts which appear generally in wavelet-based denoising methods. A range of examples illustrates the results

Inpainting and Zooming Using Sparse Representations

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 sparse-representation-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.

Deconvolution in Astronomy: A Review

This article reviews different deconvolution methods. The all-pervasive presence of noise is what makes deconvolution particularly difficult. The diversity of resulting algorithms reflects different ways of estimating the true signal under various idealizations of its properties. Different ways of approaching signal recovery are based on different instrumental noise models, whether the astronomical objects are pointlike or extended, and indeed on the computational resources available to the analyst. We present a number of recent results in this survey of signal restoration, including in the areas of superresolution and dithering. In particular, we show that most recent published work has consisted of incorporating some form of multiresolution in the deconvolution process.

Wavelets, Ridgelets, and Curvelets for Poisson Noise Removal

In order to denoise Poisson count data, we introduce a variance stabilizing transform (VST) applied on a filtered discrete Poisson process, yielding a near Gaussian process with asymptotic constant variance. This new transform, which can be deemed as an extension of the Anscombe transform to filtered data, is simple, fast, and efficient in (very) low-count situations. We combine this VST with the filter banks of wavelets, ridgelets and curvelets, leading to multiscale VSTs (MS-VSTs) and nonlinear decomposition schemes. By doing so, the noise-contaminated coefficients of these MS-VST-modified transforms are asymptotically normally distributed with known variances. A classical hypothesis-testing framework is adopted to detect the significant coefficients, and a sparsity-driven iterative scheme reconstructs properly the final estimate. A range of examples show the power of this MS-VST approach for recovering important structures of various morphologies in (very) low-count images. These results also demonstrate that the MS-VST approach is competitive relative to many existing denoising methods.

Morphological Component Analysis: An Adaptive Thresholding Strategy

In a recent paper, a method called morphological component analysis (MCA) has been proposed to separate the texture from the natural part in images. MCA relies on an iterative thresholding algorithm, using a threshold which decreases linearly towards zero along the iterations. This paper shows how the MCA convergence can be drastically improved using the mutual incoherence of the dictionaries associated to the different components. This modified MCA algorithm is then compared to basis pursuit, and experiments show that MCA and BP solutions are similar in terms of sparsity, as measured by the lscr1 norm, but MCA is much faster and gives us the possibility of handling large scale data sets.

Astronomical image representation by the curvelet transform

We outline digital implementations of two newly developed multiscale representation systems, namely, the ridgelet and curvelet transforms. We apply these digital transforms to the problem of restoring an image from noisy data and compare our results with those obtained via well established methods based on the thresholding of wavelet coefficients. We show that the curvelet transform allows us also to well enhance elongated features contained in the data. Finally, we describe the Morphological Component Analysis, which consists in separating features in an image which do not present the same morphological characteristics. A range of examples illustrates the results.