Balancing Sparsity and Rank Constraints in Quadratic Basis Pursuit

We investigate the methods that simultaneously enforce sparsity and low-rank structure in a matrix as often employed for sparse phase retrieval problems or phase calibration problems in compressive sensing. We propose a new approach for analyzing the trade off between the sparsity and low rank constraints in these approaches which not only helps to provide guidelines to adjust the weights between the aforementioned constraints, but also enables new simulation strategies for evaluating performance. We then provide simulation results for phase retrieval and phase calibration cases both to demonstrate the consistency of the proposed method with other approaches and to evaluate the change of performance with different weights for the sparsity and low rank structure constraints

Convex Optimization Approaches for Blind Sensor Calibration using Sparsity

We investigate a compressive sensing framework in which the sensors introduce a distortion to the measurements in the form of unknown gains. We focus on blind calibration, using measures performed on multiple unknown (but sparse) signals and formulate the joint recovery of the gains and the sparse signals as a convex optimization problem. We divide this problem in 3 subproblems with different conditions on the gains, specifially (i) gains with different amplitude and the same phase, (ii) gains with the same amplitude and different phase and (iii) gains with different amplitude and phase. In order to solve the first case, we propose an extension to the basis pursuit optimization which can estimate the unknown gains along with the unknown sparse signals. For the second case, we formulate a quadratic approach that eliminates the unknown phase shifts and retrieves the unknown sparse signals. An alternative form of this approach is also formulated to reduce complexity and memory requirements and provide scalability with respect to the number of input signals. Finally for the third case, we propose a formulation that combines the earlier two approaches to solve the problem. The performance of the proposed algorithms is investigated extensively through numerical simulations, which demonstrates that simultaneous signal recovery and calibration is possible with convex methods when sufficiently many (unknown, but sparse) calibrating signals are provided

Blind Phase Calibration in Sparse Recovery

International audienceWe consider a {\em blind} calibration problem in a compressed sensing measurement system in which each sensor introduces an unknown phase shift to be determined. We show that this problem can be approached similarly to the problem of phase retrieval from quadratic measurements. Furthermore, when dealing with measurements generated from multiple unknown (but sparse) signals, we extend the approach for phase retrieval to solve the calibration problem in order to recover the signals jointly along with the phase shift parameters. Additionally, we propose an alternative optimization method with less computation complexity and memory requirements. The proposed methods are shown to have significantly better recovery performance than individual recovery of the input signals when the number of input signals are sufficiently large

CBC4CS (Convex Blind Calibration for Compressive Sensing) Toolbox

CBC4CS is a Matlab package to reproduce the convex blind calibration experiments of the following papers:[1] Cagdas Bilen, Gilles Puy, Rémi Gribonval, Laurent Daudet, "Convex Optimization Approaches for Blind Sensor Calibration using Sparsity", IEEE Transationc on Signal Processing 2014 .[2] ,Cagdas Bilen, Gilles Puy, Rémi Gribonval, Laurent Daudet, "Blind Phase Calibration in Sparse Recovery", EUSIPCO 2013