Rayan Saab

Stable sparse approximations via nonconvex optimization

We present theoretical results pertaining to the ability of lscrp minimization to recover sparse and compressible signals from incomplete and noisy measurements. In particular, we extend the results of Candes, Romberg and Tao (2005) to the p < 1 case. Our results indicate that depending on the restricted isometry constants (see, e.g., Candes and Tao (2006; 2005)) and the noise level, lscrp minimization with certain values of p < 1 provides better theoretical guarantees in terms of stability and robustness than lscr1 minimization does. This is especially true when the restricted isometry constants are relatively large.

One-Bit Compressive Sensing With Norm Estimation

Consider the recovery of an unknown signal x from quantized linear measurements. In the one-bit compressive sensing setting, one typically assumes that x is sparse, and that the measurements are of the form sign((a_i, x)) ϵ {±1}. Since such measurements give no information on the norm of x, recovery methods typically assume that ∥x∥₂ = 1. We show that if one allows more generally for quantized affine measurements of the form sign((a_i, x) + b_i), and if the vectors ai are random, an appropriate choice of the affine shifts bi allows norm recovery to be easily incorporated into existing methods for one-bit compressive sensing. In addition, we show that for arbitrary fixed x in the annulus r ∥×∥₂ R, one may estimate the norm ∥×∥₂ up to additive error δ from ≳ R⁴r^-2δ^-2 such binary measurements through a single evaluation of the inverse Gaussian error function. Finally, all of our recovery guarantees can be made universal over sparse vectors in the sense that with high probability, one set of measurements and thresholds can successfully estimate all sparse vectors x in a Euclidean ball of known radius.

Sigma delta quantization for compressed sensing

Recent results make it clear that the compressed sensing paradigm can be used effectively for dimension reduction. On the other hand, the literature on quantization of compressed sensing measurements is relatively sparse, and mainly focuses on pulse-code-modulation (PCM) type schemes where each measurement is quantized independently using a uniform quantizer, say, of step size ¿. The robust recovery result of Cande¿s et al. and Donoho guarantees that in this case, under certain generic conditions on the measurement matrix such as the restricted isometry property, ¿1 recovery yields an approximation of the original sparse signal with an accuracy of O(¿). In this paper, we propose sigma-delta quantization as a more effective alternative to PCM in the compressed sensing setting. We show that if we use an rth order sigma-delta scheme to quantize m compressed sensing measurements of a k-sparse signal in ¿N, the reconstruction accuracy can be improved by a factor of (m/k)(r-1/2)¿ for any 0 < ¿ < 1 if m ¿r k(log N)1/(1-¿) (with high probability on the measurement matrix). This is achieved by employing an alternative recovery method via rth-order Sobolev dual frames.

Bayesian wavefield separation by transform-domain sparsity promotion

Successful removal of coherent-noise sources greatly determines seismic imaging quality. Major advances have been made in this direction, e.g., surface-related multiple elimination (SRME) and interferometric ground-roll removal. Still, moderate phase, timing, amplitude errors, and clutter in predicted signal components can be detrimental. Adopting a Bayesian approach, along with assuming approximate curvelet-domain independence of the to-be-separated signal components, we construct an iterative algorithm that takes predictions produced by, for example, SRME as input and separates these components in a robust manner. In addition, the proposed algorithm controls the energy mismatch between separated and predicted components. Such a control, lacking in earlier curvelet-domain formulations, improves results for primary-multiple separation on synthetic and real data.

Color image desaturation using sparse reconstruction

In this paper, we propose an algorithm to estimate the true values of saturated pixels in color images. Pixel saturation occurs when at least one color channel is clipped at some value below the full dynamic range of the scene, resulting in a loss in image fidelity. The proposed algorithm is based on the assumptions that images are nearly sparse in an appropriate transform domain, and that saturated pixels can be inferred from the structure of non-saturated neighboring pixels. Consequently, we use a hierarchical windowing algorithm which selects image regions containing relatively few saturated pixels for processing. Starting with small sized regions, and progressively increasing the size, we solve a sparsity promoting constrained ℓ1 minimization problem for each selected region to recover the saturated pixels. Moreover, we provide simulation results to show the effectiveness of our algorithm.

A Wavelet Based Approach for the Detection of Coupling in EEG Signals

Electroencephalograms (EEGs) provide a noninvasive way of measuring brainwave activity from sensors placed on the scalp. In this paper we present an approach to measure coupling, or synchrony, between various parts of the brain, critical for motor and cognitive processing, using wavelet coherence of EEG signals. We provide an argument, highlighting the benefits of using this approach as opposed to the regular Fourier based coherence, in the context of localizing short significant bursts of coherence between non-stationary EEG signals, to which regular coherence is insensitive. We further highlight the benefits of the wavelets approach by exploring how a single time-frequency coherence map can be controlled to yield various time and/or frequency resolutions

Algorithm 890: Sparco: A Testing Framework for Sparse Reconstruction

Sparco is a framework for testing and benchmarking algorithms for sparse reconstruction. It includes a large collection of sparse reconstruction problems drawn from the imaging, compressed sensing, and geophysics literature. Sparco is also a framework for implementing new test problems and can be used as a tool for reproducible research. Sparco is implemented entirely in Matlab, and is released as open-source software under the GNU Public License.

Curvelet-based primary-multiple separation from a Bayesian perspective

In this abstract, we present a novel primary-multiple separation scheme which makes use of the sparsity of both primaries and multiples in a transform domain, such as the curvelet transform, to provide estimates of each. The proposed algorithm utilizes seismic data as well as the output of a preliminary step that provides (possibly) erroneous predictions of the multiples. The algorithm separates the signal components, i.e., the primaries and multiples, by solving an optimization problem that assumes noisy input data and can be derived from a Bayesian perspective. More precisely, the optimization problem can be arrived at via an assumption of a weighted Laplacian distribution for the primary and multiple coefficients in the transform domain and of white Gaussian noise contaminating both the seismic data and the preliminary prediction of the multiples, which both serve as input to the algorithm.

A Combined Independent Component Analysis (ICA)/ Empirical Mode Decomposition (EMD) Method to Infer Corticomuscular Coupling

EEG-EMG coherence has been recently used to investigate the motor system in humans. Typically this is performed by calculating the coherence between a single EEG electrode and a rectified EMG channel. However, there are strong biological reasons to expect that the cortical to muscular communication is many-to-many as opposed to one-to-one. Here we describe the use of independent component analysis (ICA) to find linear combinations of EEG channels and EMG channels separately. Empirical mode decomposition (EMD) is then used to determine intrinsic mode functions (IMFs) that estimated the envelope of the EMG ICs. We demonstrate that at least 2 EEG ICs correspond with EMG IC IMFs with much greater significance that the pairwise EEG-EMG comparison. Moreover, the proposed method successfully untangles the ~10 Hz and ~30 Hz effects of the corticomuscular coupling which are thought to underlie different neural processes. We suggest that the ICA/EMD approach is worthy of further exploration

Cortical muscle coupling in Parkinson’s disease (PD) bradykinesia

OBJECTIVES To determine if novel methods establishing patterns in EEG-EMG coupling can infer subcortical influences on the motor cortex, and the relationship between these subcortical rhythms and bradykinesia. BACKGROUND Previous work has suggested that bradykinesia may be a result of inappropriate oscillatory drive to the muscles. Typically, the signal processing method of coherence is used to infer coupling between a single channel of EEG and a single channel of rectified EMG, which demonstrates 2 peaks during sustained contraction: one, approximately 10 Hz, which is pathologically increased in PD, and a approximately 30 Hz peak which is decreased in PD, and influenced by pharmacological manipulation of GABAA receptors in normal subjects. MATERIALS AND METHODS We employed a novel multiperiodic squeezing paradigm which also required simultaneous movements. Seven PD subjects (on and off L-Dopa) and five normal subjects were recruited. Extent of bradykinesia was inferred by reduced relative performance of the higher frequencies of the squeezing paradigm and UPDRS scores. We employed Independent Component Analysis (ICA) and Empirical Mode Decomposition (EMD) to determine EEG/EMG coupling. RESULTS Corticomuscular coupling was detected during the continually changing force levels. Different components included those over the primary motor cortex (ipsilaterally and contralaterally) and over the midline. Subjects with greater bradykinesia had a tendency towards increased approximately 10 Hz coupling and reduced approximately 30 Hz coupling that was erratically reversed with L-dopa. CONCLUSIONS These results suggest that lower approximately 10 Hz peak may represent pathological oscillations within the basal ganglia which may be a contributing factor to bradykinesia in PD.