Analysis and Synthesis Prior Greedy Algorithms for Non-linear Sparse Recovery

In this work we address the problem of recovering sparse solutions to non linear inverse problems. We look at two variants of the basic problem, the synthesis prior problem when the solution is sparse and the analysis prior problem where the solution is cosparse in some linear basis. For the first problem, we propose non linear variants of the Orthogonal Matching Pursuit (OMP) and CoSamp algorithms; for the second problem we propose a non linear variant of the Greedy Analysis Pursuit (GAP) algorithm. We empirically test the success rates of our algorithms on exponential and logarithmic functions. We model speckle denoising as a non linear sparse recovery problem and apply our technique to solve it. Results show that our method outperforms state of the art methods in ultrasound speckle denoising

Fast Acquisition for Quantitative MRI Maps: Sparse Recovery from Non-linear Measurements

This work addresses the problem of estimating proton density and T1 maps from two partially sampled K-space scans such that the total acquisition time remains approximately the same as a single scan. Existing multi parametric non linear curve fitting techniques require a large number (8 or more) of echoes to estimate the maps resulting in prolonged (clinically infeasible) acquisition times. Our simulation results show that our method yields very accurate and robust results from only two partially sampled scans (total scan time being the same as a single echo MRI). We model PD and T1 maps to be sparse in some transform domain. The PD map is recovered via standard Compressed Sensing based recovery technique. Estimating the T1 map requires solving an analysis prior sparse recovery problem from non linear measurements, since the relationship between T1 values and intensity values or K space samples is not linear. For the first time in this work, we propose an algorithm for analysis prior sparse recovery for non linear measurements. We have compared our approach with the only existing technique based on matrix factorization from non linear measurements; our method yields considerably superior results

Activelets: Wavelets for sparse representation of hemodynamic responses

We propose a new framework to extract the activity-related component in the BOLD functional Magnetic Resonance Imaging (fMRI) signal. As opposed to traditional fMRI signal analysis tech-niques, we do not impose any prior knowledge of the event timing. Instead, our basic assumption is that the activation pattern is a sequence of short and sparsely-distributed stimuli, as is the case in slow event-related fMRI. We introduce new wavelet bases, termed “activelets”, which sparsify the activity-related BOLD signal. These wavelets mimic the behavior of the differential operator underlying the hemodynamic system. To recover the sparse representation, we deploy a sparse-solution search algorithm. The feasibility of the method is evaluated using both synthetic and experimental fMRI data. The importance of the activelet basis and the non-linear sparse recovery algorithm is demonstrated by comparison against classical B-spline wavelets and linear regularization, respectively