Convex Optimization Approaches for Blind Sensor Calibration Using Sparsity

Abstract

Abstract-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.

Robust blind calibration via total least squares

“…Recent results on blind calibration employ a variety of methods [2,3,4,5]. In [2], the authors provide a method of detecting sensor faults and correcting for sensor drift using spatial kriging and Kalman filtering.…”

“…In this paper, we wish to move away from the uniform phenomenon model. Most similar to our formulation is work in calibration for compressed sensing measurements [5], where the true signal is only assumed to be sparse in a known basis. This work also uses an optimization framework for calibration, but has no theoretical guarantees and is only for measurements that are a compressed linear combination of true signal values.…”

Outlier-robust calibration method for sensor networks

“…The outlier-free vectors y(t j ) are thus inaccessible and one must deal with corrupted sensor readings z(t j ) instead. If 4 Please note that other strategies-e.g., assuming to know the sum of the values of the unknown calibration parameters [18]-have been proposed in the literature and can be also applied with these methods instead. In the remainder of the paper, we follow the same strategy as in [15], [16].…”

“…However, in the case of fixed sensors, other assumptions are needed. For fixed sensors, state-of-the-art calibration approaches assume to known the low-rank subspace where the sensed phenomenon lies [15], [16] or consider a compressed sensing framework [17], [18]. Other approaches use statistical properties 3 of the sensed phenomenon to derive estimates of the calibration parameters [19].…”

Blind mobile sensor calibration using an informed nonnegative matrix factorization with a relaxed rendezvous model

“…On the contrary, in our previous work [12], we revisited BMSC as an informed matrix This work was funded by the "OSCAR" project within the Région Nord Pas de Calais "Chercheurs Citoyens" Program. 1 It should be noticed that calibration may refer to several different-while sometimes linked-problems and have been tackled, e.g., for fixed sensor gain calibration [2,3], gain/offset calibration [4] or gain/phase calibration [5][6][7][8]. factorization problem which provided the calibration of all the sensors at the same time.…”