Designing Gabor windows using convex optimization

Redundant Gabor frames admit an infinite number of dual frames, yet only the canonical dual Gabor system, constructed from the minimal l2-norm dual window, is widely used. This window function however, might lack desirable properties, e.g. good time-frequency concentration, small support or smoothness. We employ convex optimization methods to design dual windows satisfying the Wexler-Raz equations and optimizing various constraints. Numerical experiments suggest that alternate dual windows with considerably improved features can be found

A proximal iteration for deconvolving Poisson noisy images using sparse representations

We propose an image deconvolution algorithm when the data is contaminated by Poisson noise. The image to restore is assumed to be sparsely represented in a dictionary of waveforms such as the wavelet or curvelet transforms. Our key contributions are: First, we handle the Poisson noise properly by using the Anscombe variance stabilizing transform leading to a {\it non-linear} degradation equation with additive Gaussian noise. Second, the deconvolution problem is formulated as the minimization of a convex functional with a data-fidelity term reflecting the noise properties, and a non-smooth sparsity-promoting penalties over the image representation coefficients (e.g. $\ell_1$-norm). Third, a fast iterative backward-forward splitting algorithm is proposed to solve the minimization problem. We derive existence and uniqueness conditions of the solution, and establish convergence of the iterative algorithm. Finally, a GCV-based model selection procedure is proposed to objectively select the regularization parameter. Experimental results are carried out to show the striking benefits gained from taking into account the Poisson statistics of the noise. These results also suggest that using sparse-domain regularization may be tractable in many deconvolution applications with Poisson noise such as astronomy and microscopy

Deconvolution under Poisson noise using exact data fidelity and synthesis or analysis sparsity priors

In this paper, we propose a Bayesian MAP estimator for solving the deconvolution problems when the observations are corrupted by Poisson noise. Towards this goal, a proper data fidelity term (log-likelihood) is introduced to reflect the Poisson statistics of the noise. On the other hand, as a prior, the images to restore are assumed to be positive and sparsely represented in a dictionary of waveforms such as wavelets or curvelets. Both analysis and synthesis-type sparsity priors are considered. Piecing together the data fidelity and the prior terms, the deconvolution problem boils down to the minimization of non-smooth convex functionals (for each prior). We establish the well-posedness of each optimization problem, characterize the corresponding minimizers, and solve them by means of proximal splitting algorithms originating from the realm of non-smooth convex optimization theory. Experimental results are conducted to demonstrate the potential applicability of the proposed algorithms to astronomical imaging datasets

Learning to Transform Time Series with a Few Examples

We describe a semi-supervised regression algorithm that learns to transform one time series into another time series given examples of the transformation. This algorithm is applied to tracking, where a time series of observations from sensors is transformed to a time series describing the pose of a target. Instead of defining and implementing such transformations for each tracking task separately, our algorithm learns a memoryless transformation of time series from a few example input-output mappings. The algorithm searches for a smooth function that fits the training examples and, when applied to the input time series, produces a time series that evolves according to assumed dynamics. The learning procedure is fast and lends itself to a closed-form solution. It is closely related to nonlinear system identification and manifold learning techniques. We demonstrate our algorithm on the tasks of tracking RFID tags from signal strength measurements, recovering the pose of rigid objects, deformable bodies, and articulated bodies from video sequences. For these tasks, this algorithm requires significantly fewer examples compared to fully-supervised regression algorithms or semi-supervised learning algorithms that do not take the dynamics of the output time series into account