Most of compressed sensing (CS) theory to date is focused on incoherent sensing, that is, columns from the sensing matrix are highly uncorrelated. However, sensing systems with naturally occurring correlations arise in many applications, such as signal detection, motion detection and radar. Moreover, in these applications it is often not necessary to know the support of the signal exactly, but instead small errors in the support and signal are tolerable. Despite the abundance of work utilizing incoherent sensing matrices, for this type of tolerant recovery we suggest that coherence is actually beneficial . We promote the use of coherent sampling when tolerant support recovery is acceptable, and demonstrate its advantages empirically. In addition, we provide a first step towards theoretical analysis by considering a specific reconstruction method for selected signal classes.
Low-rank matrix recovery addresses the problem of recovering an unknown low-rank matrix from few linear
measurements. Nuclear-norm minimization is a tractable approach with a recent surge of strong theoretical
backing. Analagous to the theory of compressed sensing, these results have required random measurements.
For example, m ≥ Cnr Gaussian measurements are sufficient to recover any rank-r n x n matrix with high
probability. In this paper we address the theoretical question of how many measurements are needed via any
method whatsoever - tractable or not. We show that for a family of random measurement ensembles, m ≥ 4nr-4r2 measurements are sufficient to guarantee that no rank-2r matrix lies in the null space of the measurement
operator with probability one. This is a necessary and sufficient condition to ensure uniform recovery of all rank-r
matrices by rank minimization. Furthermore, this value of m precisely matches the dimension of the manifold
of all rank-2r matrices. We also prove that for a fixed rank-r matrix, m ≥ 2nr - r2 + 1 random measurements
are enough to guarantee recovery using rank minimization. These results give a benchmark to which we may
compare the efficacy of nuclear-norm minimization.
Recent developments of new medical treatment techniques put challenging demands on ultrasound imaging
systems in terms of both image quality and raw data size. Traditional sampling methods result in very large
amounts of data, thus, increasing demands on processing hardware and limiting the flexibility in the postprocessing
In this paper, we apply Compressed Sensing (CS) techniques to analog ultrasound signals, following the recently
developed Xampling framework. The result is a system with significantly reduced sampling rates which, in turn,
means significantly reduced data size while maintaining the quality of the resulting images.
We address the problem of motion blur removal from an image sequence that was acquired by a sensor with
nonlinear response. Motion blur removal in purely linear settings has been studied extensively in the past. In
practice however, sensors exhibit nonlinearities, which also need to be compensated for. In this paper we study
the problem of joint motion blur removal and nonlinearity compensation. Two naive approaches for treating this
problem are to apply the inverse of the nonlinearity prior to a deblurring stage or following it. These strategies
require a preliminary motion estimation stage, which may be inaccurate for complex motion fields. Moreover,
even if the motion parameters are known, we provide theoretical arguments and also show through simulations
that theses methods yield unsatisfactory results. In this work, we propose an efficient iterative algorithm for
joint nonlinearity compensation and motion blur removal. Our approach relies on a recently developed theory for
nonlinear and nonideal sampling setups. Our method does not require knowledge of the motion responsible for
the blur. We show through experiments the effectiveness of our method compared with alternative approaches.