Denosing Using Wavelets and Projections onto the L1-Ball

Both wavelet denoising and denosing methods using the concept of sparsity are based on soft-thresholding. In sparsity based denoising methods, it is assumed that the original signal is sparse in some transform domains such as the wavelet domain and the wavelet subsignals of the noisy signal are projected onto L1-balls to reduce noise. In this lecture note, it is shown that the size of the L1-ball or equivalently the soft threshold value can be determined using linear algebra. The key step is an orthogonal projection onto the epigraph set of the L1-norm cost function.Comment: Submitted to Signal Processing Magazin

Range-doppler radar target detection using denoising within the compressive sensing framework

Compressive sensing (CS) idea enables the reconstruction of a sparse signal from a small set of measurements. CS approach has applications in many practical areas. One of the areas is radar systems. In this article, the radar ambiguity function is denoised within the CS framework. A new denoising method on the projection onto the epigraph set of the convex function is also developed for this purpose. This approach is compared to the other CS reconstruction algorithms. Experimental results are presented1. © 2014 EURASIP

Walking Behavior Change Detector for a “Smart” Walker

AbstractThis study investigates the design of a novel real-time system to detect walking behavior changes using an accelerometer on a rollator. No sensor is required on the user. We propose a new non-invasive approach to detect walking behavior based on the motion transfer by the user on the walker. Our method has two main steps; the first is to extract a gait feature vector by analyzing the three-axis accelerometer data in terms of magnitude, gait cycle and frequency. The second is to classify gait with the use of a decision tree of multilayer perceptrons. To assess the performance of our technique, we evaluated different sampling window lengths of 1, 3 an 5seconds and four different Neural Network architectures. The results revealed that the algorithm can distinguish walking behavior such as normal, slow and fast with an accuracy of about 86%. This research study is part of a project aiming at providing a simple and non-invasive walking behavior detector for elderly who use rollators

Walking Behavior Change Detector for a “Smart” Walker

AbstractThis study investigates the design of a novel real-time system to detect walking behavior changes using an accelerometer on a rollator. No sensor is required on the user. We propose a new non-invasive approach to detect walking behavior based on the motion transfer by the user on the walker. Our method has two main steps; the first is to extract a gait feature vector by analyzing the three-axis accelerometer data in terms of magnitude, gait cycle and frequency. The second is to classify gait with the use of a decision tree of multilayer perceptrons. To assess the performance of our technique, we evaluated different sampling window lengths of 1, 3 an 5seconds and four different Neural Network architectures. The results revealed that the algorithm can distinguish walking behavior such as normal, slow and fast with an accuracy of about 86%. This research study is part of a project aiming at providing a simple and non-invasive walking behavior detector for elderly who use rollators

Diffusion-adapted spatial filtering of fMRI data for improved activation mapping in white matter

Brain activation mapping using fMRI data has been mostly focused on finding detections in gray matter. Activations in white matter are harder to detect due to anatomical differences between both tissue types, which are rarely acknowledged in experimental design. However, recent publications have started to show evidence for the possibility of detecting meaningful activations in white matter. The shape of the activations arising from the BOLD signal is fundamentally different between white matter and gray matter, a fact which is not taken into account when applying isotropic Gaussian filtering in the preprocessing of fMRI data. We explore a graph-based description of the white matter developed from diffusion MRI data, which is capable of encoding the anisotropic domain. Based on this representation, two approaches to white matter filtering are tested, and their performance is evaluated on both semi-synthetic phantoms and real fMRI data. The first approach relies on heat kernel filtering in the graph spectral domain, and produced a clear increase in both sensitivity and specificity over isotropic Gaussian filtering. The second approach is based on spectral decomposition for the denosing of the signal, and showed increased specificity at the cost of a lower sensitivity.Novel approach to white matter filtering We introduced new advanced methods for filtering brain scans. Using them, we managed to improve the detection of activity in the white matter of the brain

Sparse modelling of natural images and compressive sensing

This thesis concerns the study of the statistics of natural images and compressive sensing for two main objectives: 1) to extend our understanding of the regularities exhibited by natural images of the visual world we regularly view around us, and 2) to incorporate this knowledge into image processing applications. Previous work on image statistics has uncovered remarkable behavior of the dis tributions obtained from filtering natural images. Typically we observe high kurtosis, non-Gaussian distributions with sharp central cusps, which are called sparse in the literature. These results have become an accepted fact through empirical findings us ing zero mean filters on many different databases of natural scenes. The observations have played an important role in computational and biological applications, where re searchers have sought to understand visual processes through studying the statistical properties of the objects that are being observed. Interestingly, such results on sparse distributions also share elements with the emerging field of compressive sensing. This is a novel sampling protocol where one seeks to measure a signal in already com pressed format through randomised projections, while the recovery algorithm consists of searching for a constrained solution with the sparsest transformed coefficients. In view of prior art, we extend our knowledge of image statistics from the monochrome domain into the colour domain. We study sparse response distributions of filters constructed on colour channels and observe the regularity of the distributions across diverse datasets of natural images. Several solutions to image processing problems emerge from the incorporation of colour statistics as prior information. We give a Bayesian treatment to the problem of colorizing natural gray images, and formulate image compression schemes using elements of compressive sensing and sparsity. We also propose a denoising algorithm that utilises the sparse filter responses as a regular- isation function for the effective attenuation of Gaussian and impulse noise in images. The results emanating from this body of work illustrate how the statistics of natural images, when incorporated with Bayesian inference and sparse recovery, can have deep implications for image processing applications

Composite Minimization: Proximity Algorithms and Their Applications

ABSTRACT Image and signal processing problems of practical importance, such as incomplete data recovery and compressed sensing, are often modeled as nonsmooth optimization problems whose objective functions are the sum of two terms, each of which is the composition of a prox-friendly function with a matrix. Therefore, there is a practical need to solve such optimization problems. Besides the nondifferentiability of the objective functions of the associated optimization problems and the larger dimension of the underlying images and signals, the sum of the objective functions is not, in general, prox-friendly, which makes solving the problems challenging. Many algorithms have been proposed in literature to attack these problems by making use of the prox-friendly functions in the problems. However, the efficiency of these algorithms relies heavily on the underlying structures of the matrices, particularly for large scale optimization problems. In this dissertation, we propose a novel algorithmic framework that exploits the availability of the prox-friendly functions, without requiring any structural information of the matrices. This makes our algorithms suitable for large scale optimization problems of interest. We also prove the convergence of the developed algorithms. This dissertation has three main parts. In part 1, we consider the minimization of functions that are the sum of the compositions of prox-friendly functions with matrices. We characterize the solutions to the associated optimization problems as the solutions of fixed point equations that are formulated in terms of the proximity operators of the dual of the prox-friendly functions. By making use of the flexibility provided by this characterization, we develop a block Gauss-Seidel iterative scheme for finding a solution to the optimization problem and prove its convergence. We discuss the connection of our developed algorithms with some existing ones and point out the advantages of our proposed scheme. In part 2, we give a comprehensive study on the computation of the proximity operator of the ℓp-norm with 0 ≤ p \u3c 1. Nonconvexity and non-smoothness have been recognized as important features of many optimization problems in image and signal processing. The nonconvex, nonsmooth ℓp-regularization has been recognized as an efficient tool to identify the sparsity of wavelet coefficients of an image or signal under investigation. To solve an ℓp-regularized optimization problem, the proximity operator of the ℓp-norm needs to be computed in an accurate and computationally efficient way. We first study the general properties of the proximity operator of the ℓp-norm. Then, we derive the explicit form of the proximity operators of the ℓp-norm for p ∈ {0, 1/2, 2/3, 1}. Using these explicit forms and the properties of the proximity operator of the ℓp-norm, we develop an efficient algorithm to compute the proximity operator of the ℓp-norm for any p between 0 and 1. In part 3, the usefulness of the research results developed in the previous two parts is demonstrated in two types of applications, namely, image restoration and compressed sensing. A comparison with the results from some existing algorithms is also presented. For image restoration, the results developed in part 1 are applied to solve the ℓ2-TV and ℓ1-TV models. The resulting restored images have higher peak signal-to-noise ratios and the developed algorithms require less CPU time than state-of-the-art algorithms. In addition, for compressed sensing applications, our algorithm has smaller ℓ2- and ℓ∞-errors and shorter computation times than state-ofthe- art algorithms. For compressed sensing with the ℓp-regularization, our numerical simulations show smaller ℓ2- and ℓ∞-errors than that from the ℓ0-regularization and ℓ1-regularization. In summary, our numerical simulations indicate that not only can our developed algorithms be applied to a wide variety of important optimization problems, but also they are more accurate and computationally efficient than stateof- the-art algorithms

Optical flow estimation via steered-L1 norm

Global variational methods for estimating optical flow are among the best performing methods due to the subpixel accuracy and the ‘fill-in’ effect they provide. The fill-in effect allows optical flow displacements to be estimated even in low and untextured areas of the image. The estimation of such displacements are induced by the smoothness term. The L1 norm provides a robust regularisation term for the optical flow energy function with a very good performance for edge-preserving. However this norm suffers from several issues, among these is the isotropic nature of this norm which reduces the fill-in effect and eventually the accuracy of estimation in areas near motion boundaries. In this paper we propose an enhancement to the L1 norm that improves the fill-in effect for this smoothness term. In order to do this we analyse the structure tensor matrix and use its eigenvectors to steer the smoothness term into components that are ‘orthogonal to’ and ‘aligned with’ image structures. This is done in primal-dual formulation. Results show a reduced end-point error and improved accuracy compared to the conventional L1 norm