Analyzing Least Squares and Kalman Filtered Compressed Sensing

In recent work, we studied the problem of causally reconstructing time sequences of spatially sparse signals, with unknown and slow time-varying sparsity patterns, from a limited number of linear "incoherent" measurements. We proposed a solution called Kalman Filtered Compressed Sensing (KF-CS). The key idea is to run a reduced order KF only for the current signal's estimated nonzero coefficients' set, while performing CS on the Kalman filtering error to estimate new additions, if any, to the set. KF may be replaced by Least Squares (LS) estimation and we call the resulting algorithm LS-CS. In this work, (a) we bound the error in performing CS on the LS error and (b) we obtain the conditions under which the KF-CS (or LS-CS) estimate converges to that of a genie-aided KF (or LS), i.e. the KF (or LS) which knows the true nonzero sets.Comment: Proc. IEEE Intl. Conf. Acous. Speech Sig. Proc. (ICASSP), 200

LS-CS-residual (LS-CS): Compressive Sensing on Least Squares Residual

We consider the problem of recursively and causally reconstructing time sequences of sparse signals (with unknown and time-varying sparsity patterns) from a limited number of noisy linear measurements. The sparsity pattern is assumed to change slowly with time. The idea of our proposed solution, LS-CS-residual (LS-CS), is to replace compressed sensing (CS) on the observation by CS on the least squares (LS) residual computed using the previous estimate of the support. We bound CS-residual error and show that when the number of available measurements is small, the bound is much smaller than that on CS error if the sparsity pattern changes slowly enough. We also obtain conditions for "stability" of LS-CS over time for a signal model that allows support additions and removals, and that allows coefficients to gradually increase (decrease) until they reach a constant value (become zero). By "stability", we mean that the number of misses and extras in the support estimate remain bounded by time-invariant values (in turn implying a time-invariant bound on LS-CS error). The concept is meaningful only if the bounds are small compared to the support size. Numerical experiments backing our claims are shown.Comment: Accepted (with mandatory minor revisions) to IEEE Trans. Signal Processing. 12 pages, 5 figure

Linear Dynamic Sparse Modelling for functional MR imaging

The reconstruction quality of a functional MRI sequence is determined by reconstruction algorithms as well as the information obtained from measurements. In this paper, we propose a Linear Dynamic Sparse Modelling method which is composed of measurement design and reconstruction processes to improve the image quality from both aspects. This method models an fMRI sequence as a linear dynamic sparse model which is based on a key assumption that variations of functional MR images are sparse over time in the wavelet domain. The Hierarchical Bayesian Kalman filter which follows the model is employed to implement the reconstruction process. To accomplish the measurement design process, we propose an Informative Measurement Design (IMD) method. The IMD method addresses the measurement design problem of selecting k feasible measurements such that the mutual information between the unknown image and measurements is maximised, where k is a given budget and the mutual information is extracted from the linear dynamic sparse model. The experimental results demonstrated that our proposed method succeeded in boosting the quality of functional MR images

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