Unrolled Networks are Conditional Probability Flows in MRI Reconstruction
Kehan Qi, Saumya Gupta, Xiaoling Hu, Qingqiao Hu, Weimin Lyu, Chao Chen
TL;DR
This paper reveals that unrolled networks in MRI reconstruction are discretizations of conditional probability flows and introduces FLow-Aligned Training (FLAT) to enhance stability and quality of the reconstructions.
Contribution
It theoretically connects unrolled networks to Flow Matching and proposes FLAT to improve cascade stability and reconstruction quality.
Findings
FLAT achieves more stable intermediate reconstructions.
Improved final image quality across MRI datasets.
Theoretical proof linking unrolled networks to probability flows.
Abstract
Unrolled networks have been widely used for Magnetic Resonance Imaging (MRI) reconstruction due to their efficiency. However, they typically exhibit unstable output quality across cascades, resulting in sub-optimal final reconstruction results. In this work, we address this inherent limitation of unrolled networks, drawing inspiration from recent Flow Matching paradigm. We first theoretically prove that unrolled networks are discretizations of conditional probability flows. This connection shows that unrolled networks and Flow Matching are analogous in MRI reconstruction. Building upon this insight, we propose FLow-Aligned Training (FLAT), which (1) derives important cascade parameters from the Flow Matching discretization; and (2) aligns intermediate reconstructions with the ideal Flow Matching trajectory to improve cascade iteration stability and convergence. Experiments on three MRI…
Peer Reviews
Decision·Submitted to ICLR 2026
1. This paper is well-written and has a comprehensive literature review. 2. This paper includes experiments that compares the proposed method to methods that are from different categories.
1. The authors claim that the core innovation of FLAT defines the velocity alignment between unrolling iterations and flow matching. It is formulated by first defining two different velocities at the k-th timestep: a. **Ideal Discretized Velocity ($v_{t_k}$):** This is the target velocity, defined as the discrete temporal derivative using the ground truth. $v_{t_{k}} = (x_{t_{k}}^{\*} - x^{(k+1)}) / (t_{k} - t_{k+1})$, $x_{t_{k}}^{*}$ is the linearly interpolated ground truth at time $t_
- I think the overall idea of using a flow ODE characterization to describe unrolled networks is great. Hence I really wanted to like this paper. Unfortunately both the theory and execution has substantial flaws.
1) The proof of the main result is fundamentally flawed: - The argument hinges on writing out p(y|x_t). Unfortunately Eq. 1 does not apply to intermediate points on the trajectory, which is well-known in the literature. p(y|x_t) would need to be calculated as (in the authors' notation): \int p(y|x_0) p(x_0| x_t) dx_0, since we only know the relationship between y and x_0 (i.e. Eq. 1). This breaks down the whole proof. There are many works on approximating this integral in the diffusion inverse p
Overall, this paper provides a novel and useful insight into existing unrolled techniques which I think would be of interest to the broader MRI recon community since unrolled techniques are very popular. The connection between flows and unrolls is interesting and clearly improves performance which is great. The authors did a good job comparing to other SOTA methods.
I do believe that the paper would benefit from testing their method on various acceleration levels of MRI data. They show results for R=8 but I would also like to see what their performance gains are at higher (and) lower acceleration levels like R=4 and 12. Additionally I would like to know what the wall clock time is for running inference of their method vs. the other methods presented. They present the number of iterations compared to other techniques, but it would be nice to see the actual
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Taxonomy
TopicsAdvanced MRI Techniques and Applications · Functional Brain Connectivity Studies · Medical Imaging Techniques and Applications