Faculty Bibliography

We propose a novel formulation for relaxed analysis-based sparsity in multiple dictionaries as a general type of prior for images, and apply it for Bayesian estimation in image restoration problems. Our formulation of a l2-relaxed l0 pseudo-norm prior allows for an especially simple maximum a posteriori estimation iterative marginal optimization algorithm, whose convergence we prove. We achieve a significant speedup over the direct (static) solution by using dynamically evolving parameters through the estimation loop. As an added heuristic twist, we fix in advance the number of iterations, and then empirically optimize the involved parameters according to two performance benchmarks. The resulting constrained dynamic method is not just fast and effective, it is also highly robust and flexible. First, it is able to provide an outstanding tradeoff between computational load and performance, in visual and objective, mean square error and structural similarity terms, for a large variety of degradation tests, using the same set of parameter values for all tests. Second, the performance benchmark can be easily adapted to specific types of degradation, image classes, and even performance criteria. Third, it allows for using simultaneously several dictionaries with complementary features. This unique combination makes ours a highly practical deconvolution method.

BACKGROUND: This paper addresses the problem of detecting sleep spindles and K-complexes in human sleep EEG. Sleep spindles and K-complexes aid in classifying stage 2 NREM human sleep. NEW METHOD: We propose a non-linear model for the EEG, consisting of a transient, low-frequency, and an oscillatory component. The transient component captures the non-oscillatory transients in the EEG. The oscillatory component admits a sparse time-frequency representation. Using a convex objective function, this paper presents a fast non-linear optimization algorithm to estimate the components in the proposed signal model. The low-frequency and oscillatory components are used to detect K-complexes and sleep spindles respectively. RESULTS AND COMPARISON WITH OTHER METHODS: The performance of the proposed method is evaluated using an online EEG database. The F1 scores for the spindle detection averaged 0.70 +/- 0.03 and the F1 scores for the K-complex detection averaged 0.57 +/- 0.02. The Matthews Correlation Coefficient and Cohen's Kappa values were in a range similar to the F1 scores for both the sleep spindle and K-complex detection. The F1 scores for the proposed method are higher than existing detection algorithms. CONCLUSIONS: Comparable run-times and better detection results than traditional detection algorithms suggests that the proposed method is promising for the practical detection of sleep spindles and K-complexes.

The fused lasso problem involves the minimization of the sum of a quadratic, a TV term and an l(1) term. The solution can be obtained by applying a TV denoising filter followed by soft-thresholding. However, soft-thresholding introduces a certain bias to the non-zero coefficients. In order to prevent this bias, we propose to replace the l(1) penalty with a non-convex penalty. We show that the solution can similarly be obtained by applying a modified thresholding function to the result of the TV-denoising filter.

This paper addresses the problem of sparsity penalized least squares for applications in sparse signal processing, e. g., sparse deconvolution. This paper aims to induce sparsity more strongly than L1 norm regularization, while avoiding non-convex optimization. For this purpose, this paper describes the design and use of non-convex penalty functions (regularizers) constrained so as to ensure the convexity of the total cost function to be minimized. The method is based on parametric penalty functions, the parameters of which are constrained to ensure convexity of F. It is shown that optimal parameters can be obtained by semidefinite programming (SDP). This maximally sparse convex (MSC) approach yields maximally non-convex sparsity-inducing penalty functions constrained such that the total cost function is convex. It is demonstrated that iterative MSC (IMSC) can yield solutions substantially more sparse than the standard convex sparsity-inducing approach, i.e., L1 norm minimization.

This paper seeks to combine linear time-invariant (LTI) filtering and sparsity-based denoising in a principled way in order to effectively filter (denoise) a wider class of signals. LTI filtering is most suitable for signals restricted to a known frequency band, while sparsity-based denoising is suitable for signals admitting a sparse representation with respect to a known transform. However, some signals cannot be accurately categorized as either band-limited or sparse. This paper addresses the problem of filtering noisy data for the particular case where the underlying signal comprises a low-frequency component and a sparse or sparse-derivative component. A convex optimization approach is presented and two algorithms derived: one based on majorization-minimization (MM), and the other based on the alternating direction method of multipliers (ADMM). It is shown that a particular choice of discrete-time filter, namely zero-phase noncausal recursive filters for finite-length data formulated in terms of banded matrices, makes the algorithms effective and computationally efficient. The efficiency stems from the use of fast algorithms for solving banded systems of linear equations. The method is illustrated using data from a physiological-measurement technique (i.e., near infrared spectroscopic time series imaging) that in many cases yields data that is well-approximated as the sum of low-frequency, sparse or sparse-derivative, and noise components.

This paper addresses the suppression of transient artifacts in signals, e.g., biomedical time series. To that end, we distinguish two types of artifact signals. We define "Type 1" artifacts as spikes and sharp, brief waves that adhere to a baseline value of zero. We define "Type 2" artifacts as comprising approximate step discontinuities. We model a Type 1 artifact as being sparse and having a sparse time-derivative, and a Type 2 artifact as having a sparse time-derivative. We model the observed time series as the sum of a low-pass signal (e.g., a background trend), an artifact signal of each type, and a white Gaussian stochastic process. To jointly estimate the components of the signal model, we formulate a sparse optimization problem and develop a rapidly converging, computationally efficient iterative algorithm denoted TARA ("transient artifact reduction algorithm"). The effectiveness of the approach is illustrated using near infrared spectroscopic time-series data.