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.

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Audio inpainting and audio declipping are important problems in audio signal processing, which are encountered in various practical applications. A number of approaches has been proposed in the literature to address these problems, most successful of which are based on sparsity of the audio signals in certain dictionary representations. Non-negative matrix factorization (NMF) is another powerful tool that has been successfully used in applications such as audio source separation. In this paper we propose a new algorithm that makes use of a low rank NMF model to perform audio inpainting and declipping. In addition to utilizing for the first time the NMF model to perform audio inpainting in presence of arbitrary losses in time domain, the proposed approach also introduces a novel way to enforce additional constraints on the signal magnitude in order to improve the performance in declipping applications. The proposed approach is shown to have a comparable performance with the state of the art dictionary based methods while providing a number of advantages.

Imaging of cardiac perfusion with MR is a challenging area of research especially due to the motion of the heart and limited time of data acquisition. Compressed sensing is a popular signal estimation method recently adopted by researchers in MRI which can improve the spatial and/or temporal resolution of the acquired images by reducing the number of necessary samples for image reconstruction. This paper focuses on performance of temporal regularization with total variation and wavelets in compressed sensing. The impact of the choice of regularization parameters on the image quality and the temporal variation of intensity in region of interests (ROIs) are discussed. It is found that selecting the regularization parameter so as to optimize the quality of the reconstructed image sequence as a whole, leads to erroneous reconstruction of certain regions due to over regularization.