Kalman Filtered Compressed Sensing

Abstract

We consider the problem of reconstructing time sequences of spatially sparse signals (with unknown and time-varying sparsity patterns) from a limited number of linear "incoherent" measurements, in real-time. The signals are sparse in some transform domain referred to as the sparsity basis. For a single spatial signal, the solution is provided by Compressed Sensing (CS). The question that we address is, for a sequence of sparse signals, can we do better than CS, if (a) the sparsity pattern of the signal's transform coefficients' vector changes slowly over time, and (b) a simple prior model on the temporal dynamics of its current non-zero elements is available. The overall idea of our solution is to use CS to estimate the support set of the initial signal's transform vector. At future times, run a reduced order Kalman filter with the currently estimated support and estimate new additions to the support set by applying CS to the Kalman innovations or filtering error (whenever it is "large").

A slightly shorter version submitted to IEEE Intl. Conf. Image Proc. (ICIP) 2008. (5 pages, 1 figure)

Cited by in corpus (30)

• Sparse Signal Recovery with Temporally Correlated Source Vectors Using Sparse Bayesian Learning
• Modified-CS: Modifying Compressive Sensing for Problems with Partially Known Support
• Dynamic Compressive Sensing of Time-Varying Signals via Approximate Message Passing
• Efficient High-Dimensional Inference in the Multiple Measurement Vector Problem
• LS-CS-residual (LS-CS): Compressive Sensing on Least Squares Residual
• Doubly Robust Smoothing of Dynamical Processes via Outlier Sparsity Constraints
• Recursive Recovery of Sparse Signal Sequences from Compressive Measurements: A Review
• Regularized Modified BPDN for Noisy Sparse Reconstruction with Partial Erroneous Support and Signal Value Knowledge
• Particle Filtering for Large Dimensional State Spaces with Multimodal Observation Likelihoods
• Dynamic Filtering of Time-Varying Sparse Signals via l1 Minimization
• Adaptive-Rate Compressive Sensing Using Side Information
• Sparse Signal Processing with Linear and Nonlinear Observations: A Unified Shannon-Theoretic Approach
• Analyzing Least Squares and Kalman Filtered Compressed Sensing
• Discrete and Continuous-time Soft-Thresholding with Dynamic Inputs
• Statistical Recovery of Simultaneously Sparse Time-Varying Signals from Multiple Measurement Vectors
• Iterated Extended Kalman Smoother-based Variable Splitting for $L_1$-Regularized State Estimation
• Dynamic Iterative Pursuit
• Exploiting Correlation in Sparse Signal Recovery Problems: Multiple Measurement Vectors, Block Sparsity, and Time-Varying Sparsity
• Distributed Representation of Geometrically Correlated Images with Compressed Linear Measurements
• KF-CS: Compressive Sensing on Kalman Filtered Residual
• A Modularized Efficient Framework for Non-Markov Time Series Estimation
• Exact Reconstruction Conditions for Regularized Modified Basis Pursuit
• Stability (over time) of Modified-CS and LS-CS for Recursive Causal Sparse Reconstruction
• A Unified Algorithmic Framework for Dynamic Compressive Sensing
• Norm minimized Scattering Data from Intensity Spectra
• Autonomous Tracking and State Estimation with Generalised Group Lasso
• Nonlinear Compressive Particle Filtering
• Compressed Hypothesis Testing: To Mix or Not to Mix?
• Metalearning: Sparse Variable-Structure Automata
• RL-NCS: Reinforcement learning based data-driven approach for nonuniform compressed sensing