RED-PSM: Regularization by Denoising of Factorized Low Rank Models for Dynamic Imaging

TL;DR

This work tackles the challenge of dynamic tomography under extreme undersampling by marrying a non-parametric partially separable (PSM) spatio-temporal model with Regularization by Denoising (RED). The RED-PSM framework formulates a bilinear, hard-constrained optimization $f=\Lambda\Psi^T$ and leverages a scalable bi-convex ADMM, enhanced by a learned spatial prior via a pre-trained CNN denoiser. The authors provide a convergence analysis proving that, under mild conditions (including a gradient rule for RED and Lipschitz denoisers), the algorithm converges to a stationary point of the objective; they also demonstrate empirical improvements over state-of-the-art TD-DIP methods in dynamic CT and cardiac dMRI, with substantial speedups and a patch-based variant enabling high-resolution scalability. The approach yields strong reconstruction quality with reduced data requirements and offers practical convergence guarantees, making it suitable for real-time or near-real-time dynamic imaging scenarios. Overall, RED-PSM represents a principled, data-efficient, and scalable framework for dynamic imaging in tomography and MRI.

Abstract

Dynamic imaging addresses the recovery of a time-varying 2D or 3D object at each time instant using its undersampled measurements. In particular, in the case of dynamic tomography, only a single projection at a single view angle may be available at a time, making the problem severely ill-posed. We propose an approach, RED-PSM, which combines for the first time two powerful techniques to address this challenging imaging problem. The first, are non-parametric factorized low rank models, also known as partially separable models (PSMs), which have been used to efficiently introduce a low-rank prior for the spatio-temporal object. The second is the recent Regularization by Denoising (RED), which provides a flexible framework to exploit the impressive performance of state-of-the-art image denoising algorithms, for various inverse problems. We propose a partially separable objective with RED and a computationally efficient and scalable optimization scheme with variable splitting and ADMM. Theoretical analysis proves the convergence of our objective to a value corresponding to a stationary point satisfying the first-order optimality conditions. Convergence is accelerated by a particular projection-domain-based initialization. We demonstrate the performance and computational improvements of our proposed RED-PSM with a learned image denoiser by comparing it to a recent deep-prior-based method known as TD-DIP. Although the main focus is on dynamic tomography, we also show performance advantages of RED-PSM in a cardiac dynamic MRI setting.

Figures (13)

• Figure 1: Imaging geometry for time-varying tomography of the object $f_t$ with single measurement $g_t$ at each time instant for $t \in \{0,1,2\}$.
• Figure 2: The RED-PSM framework. The deep denoiser $D_\phi$ is trained on slices of static objects similar to the object of interest, and the learned spatial prior is used at inference time.
• Figure 3: Ground-truth frames uniformly sampled in time for $P=4$, for the time-varying walnut (top) and compressed object (bottom).
• Figure 4: Reconstruction metrics for the time-varying walnut and compressed material vs. $P$ using different methods. For TD-DIP, the metrics reported are assuming a "stopping oracle" that stops the iterations at the best PSNR reconstruction. The minimum and maximum values over three different runs with different random initial values are shown with bars, and the mean values are connected by dashed lines
• Figure 5: Comparison of reconstructed object frames at two time instants using different methods for $P$=256, and the corresponding normalized absolute reconstruction errors for (A) the time-varying walnut, and (B) compressed object.

Theorems & Definitions (12)

• Theorem 1
• Lemma 2
• proof
• Lemma 3
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof