A Short Note on Compressed Sensing with Partially Known Signal Support

This short note studies a variation of the Compressed Sensing paradigm introduced recently by Vaswani et al., i.e. the recovery of sparse signals from a certain number of linear measurements when the signal support is partially known. The reconstruction method is based on a convex minimization program coined "innovative Basis Pursuit DeNoise" (or iBPDN). Under the common $\ell_2$-fidelity constraint made on the available measurements, this optimization promotes the ($\ell_1$) sparsity of the candidate signal over the complement of this known part. In particular, this paper extends the results of Vaswani et al. to the cases of compressible signals and noisy measurements. Our proof relies on a small adaption of the results of Candes in 2008 for characterizing the stability of the Basis Pursuit DeNoise (BPDN) program. We emphasize also an interesting link between our method and the recent work of Davenport et al. on the $\delta$-stable embeddings and the "cancel-then-recover" strategy applied to our problem. For both approaches, reconstructions are indeed stabilized when the sensing matrix respects the Restricted Isometry Property for the same sparsity order. We conclude by sketching an easy numerical method relying on monotone operator splitting and proximal methods that iteratively solves iBPDN

On Modified l_1-Minimization Problems in Compressed Sensing

Sparse signal modeling has received much attention recently because of its application in medical imaging, group testing and radar technology, among others. Compressed sensing, a recently coined term, has showed us, both in theory and practice, that various signals of interest which are sparse or approximately sparse can be efficiently recovered by using far fewer samples than suggested by Shannon sampling theorem. Sparsity is the only prior information about an unknown signal assumed in traditional compressed sensing techniques. But in many applications, other kinds of prior information are also available, such as partial knowledge of the support, tree structure of signal and clustering of large coefficients around a small set of coefficients. In this thesis, we consider compressed sensing problems with prior information on the support of the signal, together with sparsity. We modify regular l_1 -minimization problems considered in compressed sensing, using this extra information. We call these modified l_1 -minimization problems. We show that partial knowledge of the support helps us to weaken sufficient conditions for the recovery of sparse signals using modified ` 1 minimization problems. In case of deterministic compressed sensing, we show that a sharp condition for sparse recovery can be improved using modified ` 1 minimization problems. We also derive algebraic necessary and sufficient condition for modified basis pursuit problem and use an open source algorithm known as l_1 -homotopy algorithm to perform some numerical experiments and compare the performance of modified Basis Pursuit Denoising with the regular Basis Pursuit Denoising

Generalized-KFCS: Motion estimation enhanced Kalman filtered compressive sensing for video

In this paper, we propose a Generalized Kalman Filtered Compressive Sensing (Generalized-KFCS) framework to reconstruct a video sequence, which relaxes the assumption of a slowly changing sparsity pattern in Kalman Filtered Compressive Sensing [1, 2, 3, 4]. In the proposed framework, we employ motion estimation to achieve the estimation of the state transition matrix for the Kalman filter, and then reconstruct the video sequence via the Kalman filter in conjunction with compressive sensing. In addition, we propose a novel method to directly apply motion estimation to compressively sensed samples without reconstructing the video sequence. Simulation results demonstrate the superiority of our algorithm for practical video reconstruction.This work was partially supported by EPSRC Research Grant EP/K033700/1, the Fundamental Research Funds for the Central Universities (No. 2014JBM149), and the Scientific Research Foundation for the Returned Overseas Chinese Scholars (of State Education Ministry).This is the author accepted manuscript. The final version is available from IEEE via http://dx.doi.org/10.1109/ICIP.2014.702525

Generalized-KFCS: Motion estimation enhanced Kalman filtered compressive sensing for video

In this paper, we propose a Generalized Kalman Filtered Compressive Sensing (Generalized-KFCS) framework to reconstruct a video sequence, which relaxes the assumption of a slowly changing sparsity pattern in Kalman Filtered Compressive Sensing [1, 2, 3, 4]. In the proposed framework, we employ motion estimation to achieve the estimation of the state transition matrix for the Kalman filter, and then reconstruct the video sequence via the Kalman filter in conjunction with compressive sensing. In addition, we propose a novel method to directly apply motion estimation to compressively sensed samples without reconstructing the video sequence. Simulation results demonstrate the superiority of our algorithm for practical video reconstruction.This work was partially supported by EPSRC Research Grant EP/K033700/1, the Fundamental Research Funds for the Central Universities (No. 2014JBM149), and the Scientific Research Foundation for the Returned Overseas Chinese Scholars (of State Education Ministry).This is the author accepted manuscript. The final version is available from IEEE via http://dx.doi.org/10.1109/ICIP.2014.702525

TV-min and Greedy Pursuit for Constrained Joint Sparsity and Application to Inverse Scattering

This paper proposes a general framework for compressed sensing of constrained joint sparsity (CJS) which includes total variation minimization (TV-min) as an example. TV- and 2-norm error bounds, independent of the ambient dimension, are derived for the CJS version of Basis Pursuit and Orthogonal Matching Pursuit. As an application the results extend Cand`es, Romberg and Tao's proof of exact recovery of piecewise constant objects with noiseless incomplete Fourier data to the case of noisy data.Comment: Mathematics and Mechanics of Complex Systems (2013