Faculty Bibliography

In this paper, we discuss about applications of different methods for decomposing a signal over elementary waveforms chosen in a family called a dictionary (atomic representations) in optical coherence tomography (OCT). If the representation is learned from the data, a nonparametric dictionary is defined with three fundamental properties of being data-driven, applicability on 3D, and working in multi-scale, which make it appropriate for processing of OCT images. We discuss about application of such representations including complex wavelet based K-SVD, and diffusion wavelets on OCT data. We introduce complex wavelet based K-SVD to take advantage of adaptability in dictionary learning methods to improve the performance of simple dual tree complex wavelets in speckle reduction of OCT datasets in 2D and 3D. The algorithm is evaluated on 144 randomly selected slices from twelve 3D OCTs taken by Topcon 3D OCT-1000 and Cirrus Zeiss Meditec. Improvement of contrast to noise ratio (CNR) (from 0.9 to 11.91 and from 3.09 to 88.9, respectively) is achieved. Furthermore, two approaches are proposed for image segmentation using diffusion. The first method is designing a competition between extended basis functions at each level and the second approach is defining a new distance for each level and clustering based on such distances. A combined algorithm, based on these two methods is then proposed for segmentation of retinal OCTs, which is able to localize 12 boundaries with unsigned border positioning error of 9.22 +/-3.05 mum, on a test set of 20 slices selected from 13 3D OCTs.

BACKGROUND: Optical coherence tomography offers a potential biomarker tool in Parkinson's disease (PD). A mathematical model quantifying symmetry, breadth, and depth of the fovea was applied. METHODS: Nintey-six subjects (72 PD and 24 healthy controls) were included in the study. Macular scans of each eye were obtained on two different optical coherence tomography devices: Cirrus and RTVue. RESULTS: The variables corresponding to the cardinal gradients of the fovea were the most sensitive indicators of PD for both devices. Principal component analysis distinguished 65% of PD patients from controls on Cirrus, 57% on RTVue. CONCLUSION: Parkinson's disease shallows the superior/inferior and to a lesser degree nasal-temporal foveal slope. The symmetry, breadth, and depth model fits optical coherence tomography data derived from two different devices, and it is proposed as a diagnostic tool in PD.

Adaptive transforms are required for better signal analysis and processing. Key issue in finding the optimal expansion basis for a given signal is the representation of signal information with very few elements of the basis. In this context a key role is played by the multiscale transforms that allow signal representation at different resolutions. This paper presents a method for building a multiscale transform with adaptive scale dilation factors. The aim is to promote sparsity and adaptiveness both in time and scale. To this aim interscale relationships of wavelet coefficients are used for the selection of those scales that measure significant changes in signal information. Then, a wavelet transform with variable dilation factor is defined accounting for the selected scales and the properties of coprime numbers. Preliminary experimental results in image denoising by Wiener filtering show that the adaptive multiscale transform is able to provide better reconstruction quality with a minimum number of scales and comparable computational effort with the classical dyadic transform.

Image restoration is one of the most fundamental issues in imaging science. Total variation regularization is widely used in image restoration problems for its capability to preserve edges. In the literature, however, it is also well known for producing staircase artifacts. In this work we extend the total variation with overlapping group sparsity, which we previously developed for one dimension signal processing, to image restoration. A convex cost function is given and an efficient algorithm is proposed for solving the corresponding minimization problem. In the experiments, we compare our method with several state-of-the-art methods. The results illustrate the efficiency and effectiveness of the proposed method in terms of PSNR and computing time. (C) 2014 Elsevier Inc. All rights reserved.

This letter considers the problem of signal denoising using a sparse tight-frame analysis prior. The l(1) norm has been extensively used as a regularizer to promote sparsity; however, it tends to under-estimate non-zero values of the underlying signal. To more accurately estimate non-zero values, we propose the use of a non-convex regularizer, chosen so as to ensure convexity of the objective function. The convexity of the objective function is ensured by constraining the parameter of the non-convex penalty. We use ADMM to obtain a solution and show how to guarantee that ADMM converges to the global optimum of the objective function. We illustrate the proposed method for 1D and 2D signal denoising.

This letter formulates a convex generalized total variation functional for the estimation of discontinuous piecewise linear signals from corrupted data. The method is based on (1) promoting pairwise group sparsity of the second derivative signal and (2) decoupling the principle knot parameters so they can be separately weighted. The proposed method refines the recent approach by Ongie and Jacob.

Algorithms for signal denoising that combine wavelet-domain sparsity and total variation (TV) regularization are relatively free of artifacts, such as pseudo-Gibbs oscillations, normally introduced by pure wavelet thresholding. This paper formulates wavelet-TV (WATV) denoising as a unified problem. To strongly induce wavelet sparsity, the proposed approach uses non-convex penalty functions. At the same time, in order to draw on the advantages of convex optimization (unique minimum, reliable algorithms, simplified regularization parameter selection), the non-convex penalties are chosen so as to ensure the convexity of the total objective function. A computationally efficient, fast converging algorithm is derived.

Total variation (TV) denoising is an effective noise suppression method when the derivative of the underlying signal is known to be sparse. TV denoising is defined in terms of a convex optimization problem involving a quadratic data fidelity term and a convex regularization term. A non-convex regularizer can promote sparsity more strongly, but generally leads to a non-convex optimization problem with non-optimal local minima. This letter proposes the use of a non-convex regularizer constrained so that the total objective function to be minimized maintains its convexity. Conditions for a non-convex regularizer are given that ensure the total TV denoising objective function is convex. An efficient algorithm is given for the resulting problem.

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.