This paper addresses the problem of stably recovering sparse or compressible signals from compressed sensing measurements that have undergone optimal nonuniform scalar quantization, i.e., minimizing the common$\ell _{2}$-norm distortion. Generally, this quantized compressed sensing (QCS) problem is solved by minimizing the$\ell _{1}$-norm constrained by the$\ell _{2}$-norm distortion. In such cases, remeasurement and quantization of the reconstructed signal do not necessarily match the initial observations, showing that the whole QCS model is not consistent. Our approach considers instead that quantization distortion more closely resembles heteroscedastic uniform noise, with variance depending on the observed quantization bin. Generalizing our previous work on uniform quantization, we show that for nonuniform quantizers described by the “compander” formalism, quantization distortion may be better characterized as having bounded weighted$\ell _{p}$-norm ($p \geqslant 2$), for a particular weighting. We develop a new reconstruction approach, termed Generalized Basis Pursuit DeNoise (GBPDN), which minimizes the$\ell _{1}$-norm of the signal to reconstruct constrained by this weighted$\ell _{p}$-norm fidelity. We prove that, for standard Gaussian sensing matrices and$K$sparse or compressible signals in$ \BBR ^{N}$with at least$\Omega ((K \log N/K)^{p/2})$measurements, i.e., under strongly oversampled QCS scenario, GBPDN is$\ell _{2}-\ell _{1}$instance optimal and stable recovers all such sparse or compressible signals. The reconstruction error decreases as$O(2^{-B}/\sqrt {p+1})$given a budget of$B$bits per measurement. This yields a reduction by a factor$\sqrt {p+1}$of the reconstruction error compared to the one produced by$\ell _{2}$-norm constrained decoders. We also propose an primal-dual proximal splitting scheme to solve the GBPDN program which is efficient for large-scale problems. Interestingly, extensive simulations testing the GBPDN effectiveness confirm the trend predicted by the theory, that the reconstruction error can indeed be reduced by increasing$p$, but this is achieved at a much less stringent oversampling regime than the one expected by the theoretical bounds. Besides the QCS scenario, we also show that GBPDN applies straightforwardly to the related case of CS measurements corrupted by heteroscedastic generalized Gaussian noise with provable reconstruction error reduction.

The electroencephalography source estimation problem consists of inferring cortical activation from measurements of electrical potential taken on the scalp surface. This inverse problem is intrinsically ill-posed. In particular the dimensionality of cortical sources greatly exceeds the number of electrode measurements, and source estimation requires regularization to obtain a unique solution. In this work, we introduce a novel regularization function called cortical graph smoothing, which exploits knowledge of anatomical connectivity available from diffusion-weighted imaging. Given a weighted graph description of the anatomical connectivity of the brain, cortical graph smoothing penalizes the weighted sum of squares of differences of cortical activity across the graph edges, thus encouraging solutions with consistent activation across anatomically connected regions. We explore the performance of the cortical graph smoothing source estimates for analysis of the event related potential for simple motor tasks, and compare against the commonly used minimum norm, weighted minimum norm, LORETA and sLORETA source estimation methods. Evaluated over a series of 18 subjects, the proposed cortical graph smoothing method shows superior localization accuracy compared to the minimum norm method, and greater relative peak intensity than the other comparison methods.

The source estimation problem for EEG consists of estimating cortical activity from measurements of electrical potential on the scalp surface. This is a underconstrained inverse problem as the dimensionality of cortical source currents far exceeds the number of sensors. We develop a novel regularization for this inverse problem which incorporates knowledge of the anatomical connectivity of the brain, measured by diffusion tensor imaging. We construct an overcomplete wavelet frame, termed cortical graph wavelets, by applying the recently developed spectral graph wavelet transform to this anatomical connectivity graph. Our signal model is formed by assuming that the desired cortical currents have a sparse representation in these cortical graph wavelets, which leads to a convex ℓ1-regularized least squares problem for the coefficients. On data from a simple motor potential experiment, the proposed method shows improvement over the standard minimum-norm regularization.

Following the Compressed Sensing (CS) paradigm, this paper studies the problem of recovering sparse or compressible signals from (scalar) non-uniformly quantized measurements. We show that a simple adaptation of the Basis Pursuit De-Quantizer introduced earlier, that is, a sign sensitive weighting of their ℓp-norm fidelity constraint, yields good SNR improvements in the signal reconstruction. As a good indication of this improvement origin, we prove theoretically that a similar decoder, using a particular side-position-to-level oracle, displays a reduction of the reconstruction error when both the number of measurements and the moment p of the constraint increase. This follows the oversampling principle underlined in our previous study for uniformly quantized CS, with an additional gain provided by the non-uniform quantization. We conclude this paper by showing the efficiency of the approach on 1-D and 2-D signal examples.

In this paper, we study the problem of recovering sparse or compressible signals from uniformly quantized measurements. We present a new class of convex optimization programs, or decoders, coined Basis Pursuit DeQuantizer of moment p (BPDQp), that model the quantization distortion more faithfully than the commonly used Basis Pursuit DeNoise (BPDN) program. Our decoders proceed by minimizing the sparsity of the signal to be reconstructed subject to a data-fidelity constraint expressed in the ℓp-norm of the residual error for 2 ≤ p ≤ ∞. We show theoretically that, (i) the reconstruction error of these new decoders is bounded if the sensing matrix satisfies an extended Restricted Isometry Property involving the Iρ norm, and (ii), for Gaussian random matrices and uniformly quantized measurements, BPDQpperformance exceeds that of BPDN by dividing the reconstruction error due to quantization by √(p + 1). This last effect happens with high probability when the number of measurements exceeds a value growing with p, i.e., in an oversampled situation compared to what is commonly required by BPDN = BPDQ2. To demonstrate the theoretical power of BPDQp, we report numerical simulations on signal and image reconstruction problems.

In this paper, following the Compressed Sensing (CS) paradigm, we study the problem of recovering sparse or compressible signals from uniformly quantized measurements. We present a new class of convex optimization programs, or decoders, coined Basis Pursuit DeQuantizer of moment p (BPDQp), that model the quantization distortion more faithfully than the commonly used Basis Pursuit DeNoise (BPDN) program. Our decoders proceed by minimizing the sparsity of the signal to be reconstructed while enforcing a data fidelity term of bounded ¿p-norm, for 2pdecoders outperforms that of BPDN, with reconstruction error due to quantization divided by. This reduction relies on a modified Restricted Isometry Property of the sensing matrix expressed in the ¿p-norm (RIPp); a property satisfied by Gaussian random matrices with high probability. We conclude with numerical experiments comparing BPDQpand BPDN for signal and image reconstruction problems.

We develop a statistical model to describe the spatially varying behavior of local neighborhoods of coefficients in a multiscale image representation. Neighborhoods are modeled as samples of a multivariate Gaussian density that are modulated and rotated according to the values of two hidden random variables, thus allowing the model to adapt to the local amplitude and orientation of the signal. A third hidden variable selects between this oriented process and a nonoriented scale mixture of Gaussians process, thus providing adaptability to the local orientedness of the signal. Based on this model, we develop an optimal Bayesian least squares estimator for denoising images and show through simulations that the resulting method exhibits significant improvement over previously published results obtained with Gaussian scale mixtures.

We present a general framework for combination of two distinct local denoising methods. Interpolation between the two methods is controlled by a spatially varying decision function. Assuming the availability of clean training data, we formulate a learning problem for determining the decision function. As an example application we use Weighted Kernel Ridge Regression to solve this learning problem for a pair of wavelet-based image denoising algorithms, yielding a "hybrid" denoising algorithm whose performance surpasses that of either initial method.

We develop a statistical model for images that explicitly captures variations in local orientation and contrast. Patches of wavelet coefficients are described as samples of a fixed Gaussian process that are rotated and scaled according to a set of hidden variables representing the local image contrast and orientation. An optimal Bayesian least squares estimator is developed by conditioning upon and integrating over the hidden orientation and scale variables. The resulting denoising procedure gives results that are visually superior to those obtained with a Gaussian scale mixture model that does not explicitly incorporate local image orientation.