This paper studies a difficult and fundamental problem that arises throughout electrical engineering, applied mathematics, and statistics. Suppose that one forms a short linear combination of elementary signals drawn from a large, fixed collection. Given an observation of the linear combination that has been contaminated with additive noise, the goal is to identify which elementary signals participated and to approximate their coefficients. Although many algorithms have been proposed, there is little theory which guarantees that these algorithms can accurately and efficiently solve the problem. This paper studies a method called convex relaxation, which attempts to recover the ideal sparse signal by solving a convex program. This approach is powerful because the optimization can be completed in polynomial time with standard scientific software. The paper provides general conditions which ensure that convex relaxation succeeds. As evidence of the broad impact of these results, the paper describes how convex relaxation can be used for several concrete signal recovery problems. It also describes applications to channel coding, linear regression, and numerical analysis.

A full-rank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combinatorial in nature, are there efficient methods for finding the sparsest solution? These questions have been answered positively and constructively in recent years, exposing a wide variety of surprising phenomena; in particular, the existence of easily-verifiable conditions under which optimally-sparse solutions can be found by concrete, effective computational methods. Such theoretical results inspire a bold perspective on some important practical problems in signal and image processing. Several well-known signal and image processing problems can be cast as demanding solutions of undetermined systems of equations. Such problems have previously seemed, to many, intractable. There is considerable evidence that these problems often have sparse solutions. Hence, advances in finding sparse solutions to underdetermined systems energizes research on such signal and image processing problems – to striking effect. In this paper we review the theoretical results on sparse solutions of linear systems, empirical

Sparse representations of signals have drawn considerable interest in recent years. The assumption that natural signals, such as images, admit a sparse decomposition over a redundant dictionary leads to efficient algorithms for handling such sources of data. In particular, the design of well adapted dictionaries for images has been a major challenge. The K-SVD has been recently proposed for this task [1], and shown to perform very well for various gray-scale image processing tasks. In this paper we address the problem of learning dictionaries for color images and extend the K-SVD-based gray-scale image denoising algorithm that appears in [2]. This work puts forward ways for handling non-homogeneous noise and missing information, paving the way to state-of-the-art results in applications such as color image denoising, demosaicing, and inpainting, as demonstrated in this paper. EDICS Category: COL-COLR (Color processing) I.

In a previous paper, we proposed a general Fourier method that provides an accurate prediction of the approximation error, irrespective of the scaling properties of the approximating functions. Here, we apply our results when these functions satisfy the usual two-scale relation encountered in dyadic multiresolution analysis. As a consequence of this additional constraint, the quantities introduced in our previous paper can be computed explicitly as a function of the refinement filter. This is, in particular, true for the asymptotic expansion of the approximation error for biorthonormal wavelets as the scale tends to zero. One of the contributions of this paper is the computation of sharp, asymptotically optimal upper bounds for the least-squares approximation error. Another contribution is the application of these results to B-splines and Daubechies scaling functions, which yields explicit asymptotic developments and upper bounds. Thanks to these explicit expressions, we can quantify ...

The ridgelet transform [6] was introduced as a sparse expansion for functions on continuous spaces that are smooth away from discontinuities along lines. In this paper, we propose an orthonormal version of the ridgelet transform for discrete and finite -size images. Our construction uses the finite Radon transform (FRAT) [11], [20] as a building block. To overcome the periodization effect of a finite transform, we introduce a novel ordering of the FRAT coefficients. We also analyze the FRAT as a frame operator and derive the exact frame bounds. The resulting finite ridgelet transform (FRIT) is invertible, nonredundant and computed via fast algorithms. Furthermore, this construction leads to a family of directional and orthonormal bases for images. Numerical results show that the FRIT is more effective than the wavelet transform in approximating and denoising images with straight edges.

In 1983, Burt and Adelson introduced the Laplacian pyramid (LP) as a multiresolution representation for images. We study the LP using the frame theory, and this reveals that the usual reconstruction is suboptimal. We show that the LP with orthogonal filters is a tight frame, and thus, the optimal linear reconstruction using the dual frame operator has a simple structure that is symmetric with the forward transform. In more general cases, we propose an efficient filterbank (FB) for the reconstruction of the LP using projection that leads to a proved improvement over the usual method in the presence of noise. Setting up the LP as an oversampled FB, we offer a complete parameterization of all synthesis FBs that provide perfect reconstruction for the LP. Finally, we consider the situation where the LP scheme is iterated and derive the continuous -domain frames associated with the LP.

A new statistical model for characterizing texture images based on wavelet-domain hidden Markov models and steerable pyramids is presented. The new model is shown to capture well both the subband marginal distributions and the dependencies across scales and orientations of the wavelet descriptors. Once it is trained for an input texture image, the model can be easily steered to characterize that texture at any other orientation. After a diagonalization operation, one obtains a rotation-invariant model of the texture image. The effectiveness of the new texture models are demonstrated in retrieval experiments with large image databases, where significant performance gains are shown. Keywords texture characterization, image retrieval, rotation invariance, wavelets, hidden Markov models, steerable pyramids. Corresponding author. Address: see above; Phone: +41 21 693 7663; Fax: +41 21 693 4312. y Also with Department of EECS, UC Berkeley, Berkeley CA 94720, USA. April 23, 2001 DRAFT I.

This paper proposes a rate-distortion optimal a posteriori quantization scheme for Matching Pursuit coefficients. The a posteriori quantization applies to a Matching Pursuit expansion that has been generated off-line, and cannot benefit of any feedback loop to the encoder in order to compensate for the quantization noise. The redundancy of the Matching Pursuit dictionary provides an indicator of the relative importance of coefficients and atom indices, and subsequently on the quantization error. It is used to define a universal upper-bound on the decay of the coefficients, sorted in decreasing order of magnitude. A new quantization scheme is then derived, where this bound is used as an Oracle for the design of an optimal a posteriori quantizer. The latter turns the exponentially distributed coefficient entropy-constrained quantization problem into a simple uniform quantization problem. Using simulations

In this paper we study the nonsubsampled contourlet transform. We address the corresponding filter design problem using the McClellan transformation. We show how zeroes can be imposed in the filters so that the iterated structure produces regular basis functions. The proposed design framework yields filters that can be implemented efficiently through a lifting factorization. We apply the constructed transform in image noise removal where the results obtained are comparable to the state-of-the art, being superior in some cases.

This tutorial paper reviews the theory and design of codes for hiding or embedding information in signals such as images, video, audio, graphics, and text. Such codes have also been called watermarking codes; they can be used in a variety of applications, including copyright protection for digital media, content authentication, media forensics, data binding, and covert communications. Some of these applications imply the presence of an adversary attempting to disrupt the transmission of information to the receiver; other applications involve a noisy, generally unknown, communication channel. Our focus is on the mathematical models, fundamental principles, and code design techniques that are applicable to data hiding. The approach draws from basic concepts in information theory, coding theory, game theory, and signal processing, and is illustrated with applications to the problem of hiding data in images. Keywords—Coding theory, data hiding, game theory, image processing, information theory, security, signal processing, watermarking. I.