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Efficient Plug-and-Play method for Dynamic Imaging Via Kalman Smoothing

Benjamin Hawkes, Mike Davies, Victor Elvira, Audrey Repetti

TL;DR

The paper tackles dynamic imaging under a linear Gaussian state-space model, where measurements follow $y_t = H_t x_t + r_t$ and the state evolves as $x_t = A_t x_{t-1} + q_t$. It introduces a Plug-and-Play KS-ADMM method that integrates a deep denoiser into the KS-ADMM loop, replacing the regularization proximal step with a learning-based denoiser $\mathcal{D}$ and performing state updates via Kalman smoothing. The approach yields improved expressivity and substantial speedups for long sequences by keeping KS for the state-space fidelity and using a denoiser for priors, demonstrated on a synthetic 2D+t deblurring task where it outperforms batch PnP-ADMM in speed while preserving PSNR. This work enables scalable, high-quality dynamic imaging with learned priors and can be extended to non-linear dynamics in future work.

Abstract

State-space models (SSM) are common in signal processing, where Kalman smoothing (KS) methods are state-of-the-art. However, traditional KS techniques lack expressivity as they do not incorporate spatial prior information. Recently, [1] proposed an ADMM algorithm that handles the state-space fidelity term with KS while regularizing the object via a sparsity-based prior with proximity operators. Plug-and-Play (PnP) methods are a popular type of iterative algorithms that replace proximal operators encoding prior knowledge with powerful denoisers such as deep neural networks. These methods are widely used in image processing, achieving state-of-the-art results. In this work, we build on the KS-ADMM method, combining it with deep learning to achieve higher expressivity. We propose a PnP algorithm based on KS-ADMM iterations, efficiently handling the SSM through KS, while enabling the use of powerful denoising networks. Simulations on a 2D+t imaging problem show that the proposed PnP-KS-ADMM algorithm improves the computational efficiency over standard PnP-ADMM for large numbers of timesteps.

Efficient Plug-and-Play method for Dynamic Imaging Via Kalman Smoothing

TL;DR

The paper tackles dynamic imaging under a linear Gaussian state-space model, where measurements follow and the state evolves as . It introduces a Plug-and-Play KS-ADMM method that integrates a deep denoiser into the KS-ADMM loop, replacing the regularization proximal step with a learning-based denoiser and performing state updates via Kalman smoothing. The approach yields improved expressivity and substantial speedups for long sequences by keeping KS for the state-space fidelity and using a denoiser for priors, demonstrated on a synthetic 2D+t deblurring task where it outperforms batch PnP-ADMM in speed while preserving PSNR. This work enables scalable, high-quality dynamic imaging with learned priors and can be extended to non-linear dynamics in future work.

Abstract

State-space models (SSM) are common in signal processing, where Kalman smoothing (KS) methods are state-of-the-art. However, traditional KS techniques lack expressivity as they do not incorporate spatial prior information. Recently, [1] proposed an ADMM algorithm that handles the state-space fidelity term with KS while regularizing the object via a sparsity-based prior with proximity operators. Plug-and-Play (PnP) methods are a popular type of iterative algorithms that replace proximal operators encoding prior knowledge with powerful denoisers such as deep neural networks. These methods are widely used in image processing, achieving state-of-the-art results. In this work, we build on the KS-ADMM method, combining it with deep learning to achieve higher expressivity. We propose a PnP algorithm based on KS-ADMM iterations, efficiently handling the SSM through KS, while enabling the use of powerful denoising networks. Simulations on a 2D+t imaging problem show that the proposed PnP-KS-ADMM algorithm improves the computational efficiency over standard PnP-ADMM for large numbers of timesteps.
Paper Structure (9 sections, 13 equations, 3 figures, 1 algorithm)

This paper contains 9 sections, 13 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Maximum a Posteriori framework
  4. ADMM and KS-ADMM iterations
  5. ADMM computation of ${\mathb{x}}\xspace^{(k+1)}$
  6. KS-ADMM computation of ${\mathb{x}}\xspace^{(k+1)}$
  7. Proposed PnP KS-ADMM approach
  8. Simulation results
  9. Conclusion

Figures (3)

  • Figure 1: Ground truth images for frames $t\in \{1, 4, 7, 10, 13\}$ (top row), with associated measurements (middle row, average PSNR $24.05$dB), and reconstruction obtained with PnP KS-ADMM (bottom row, average PSNR $28.55$dB).
  • Figure 2: Reconstruction time for the considered PnP methods, with respect to (top) frame numbers for fixed image size $N=64^2$, and (bottom) image sizes for fixed frame number $T=5$.
  • Figure 3: Average PSNR profile with respect to iterations for the different considered PnP methods.