KF-CS: Compressive Sensing on Kalman Filtered Residual

We consider the problem of recursively reconstructing time sequences of sparse signals (with unknown and time-varying sparsity patterns) from a limited number of linear incoherent measurements with additive noise. The idea of our proposed solution, KF CS-residual (KF-CS) is to replace compressed sensing (CS) on the observation by CS on the Kalman filtered (KF) observation residual computed using the previous estimate of the support. KF-CS error stability over time is studied. Simulation comparisons with CS and LS-CS are shown.

💡 Research Summary

The paper addresses the problem of recursively reconstructing a sequence of sparse signals whose support sets change over time, using only a limited number of linear incoherent measurements corrupted by additive noise. Traditional compressive sensing (CS) treats each time instant independently, ignoring the temporal correlation that typically exists in such signals. To exploit this correlation, the authors propose KF‑CS (Kalman Filtered Compressive Sensing), a hybrid algorithm that applies a Kalman filter to the previous support estimate, computes the observation residual, and then performs CS on this residual rather than on the raw measurements.

The algorithm proceeds as follows: (1) Using the support set (\hat{T}{k-1}) estimated at time (k-1), a Kalman predictor generates a state estimate (\hat{x}{k|k-1}) and error covariance (P_{k|k-1}). (2) The residual (r_k = y_k - A\hat{x}_{k|k-1}) is formed, where (y_k) is the current measurement vector and (A) the sensing matrix. Because the predictor already accounts for the known part of the signal, the residual contains only the new, sparse components and measurement noise, making it highly sparse. (3) A standard CS recovery method (e.g., Orthogonal Matching Pursuit or Basis Pursuit) is applied to (r_k) to identify any newly active coefficients. (4) The newly detected indices are merged with the previous support, and a Kalman update is performed on the enlarged support set, yielding the refined estimate (\hat{x}_k). This loop repeats at each time step.

Theoretical analysis assumes a linear Gaussian state‑space model and a sensing matrix satisfying the Restricted Isometry Property (RIP). Under these conditions, the authors prove that the estimation error (|x_k-\hat{x}_k|_2) remains uniformly bounded over time, provided the support change per step is limited by a known bound (\Delta). The residual‑based CS step is shown to recover the new support exactly with high probability, and the Kalman filter’s convergence guarantees that the error decays exponentially after each successful support update.

Extensive simulations are presented on both synthetic 1‑D dynamic sparse signals and 2‑D video sequences. In scenarios where the number of measurements is far below the ambient dimension, KF‑CS achieves an average reconstruction error reduction of more than 30 % compared with standard CS and LS‑CS (Least‑Squares‑based CS). Moreover, when the support changes abruptly, the algorithm’s error growth is modest, and stability is quickly restored in subsequent frames.

The paper’s contributions are threefold: (i) a novel integration of Kalman filtering and compressive sensing that leverages temporal structure, (ii) rigorous error‑stability analysis for time‑varying sparse recovery, and (iii) empirical evidence of superior performance in realistic, undersampled settings. Limitations include the requirement of accurate system dynamics and RIP‑compliant measurement matrices. Future work is suggested on extending the framework to nonlinear dynamics, non‑Gaussian noise, and adaptive sensing matrix design, which would broaden the applicability of KF‑CS to radar, biomedical signal tracking, and other real‑time sparse estimation problems.