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Objective:  The reconstruction of an input based on a sparse combination of signals, known as sparse coding, has found widespread use in signal processing. In this work, the combination of sparse coding with Kalman filtering is explored and its potential is shown on two use-cases.  Methods:  This work extends the Iterative Shrinkage and Thresholding Algorithm with a Kalman filter in the sparse domain. The resulting method may be implemented as a deep unfolded neural network and may be applied to any signal which has a sparse representation and a known or assumed relation between consecutive measurements. This method is evaluated on the use cases of noise reduction in the electrocardiogram (ECG) and the estimation of object motility.  Results:  For ECG denoising, the proposed method achieved an improvement in Signal-to-Noise ratio of 18.6 dB, which is comparable to state-of-the-art. In motility estimation, a correlation of 0.84 with ground truth simulations was found.  Conclusion:  The proposed method was shown to have advantages over sparse coding and Kalman filtering alone. Due to the low complexity and high generalizability of the proposed method, the implementation of context-specific knowledge or an extension to other applications can be readily made.  Significance:  The presented Kalman-ISTA algorithm is a resource-efficient method combining the promise of both sparse coding and Kalman filtering, making it well-suited for various applications.

Deep Learning for Sparse Domain Kalman Filtering With Applications on ECG Denoising and Motility Estimation