Existing generative model-based approaches to linear inverse problems typically use the acquired noisy measurement to condition or guide sampling in the image domain. In this work, we introduce Dual Flow, a novel joint flow model for solving linear inverse problems by jointly estimating the clean image and its corresponding noise-free measurement from the observation. Specifically, Dual Flow formulates image reconstruction and measurement denoising as a coupled estimation problem, using coupled flows in the image and measurement domains. The acquired noisy measurement y0 is used only to initialize the measurement state; thereafter, the image and measurement states are jointly updated based on their current values. By construction, the final image and measurement estimates are guaranteed to satisfy the underlying forward model. We derive the sampling updates for the joint flow and show that they are governed by the conditional mean of the clean image given the current joint state. We further derive a closed-form expression for a sufficient statistic of the joint state and show that the required conditional mean can be obtained by solving an image-domain Gaussian denoising problem involving operator-dependent, correlated noise. We train a velocity-prediction network using this formulation to estimate the conditional mean from the derived sufficient statistic. Dual Flow requires no hyperparameter tuning beyond selecting the number of inference steps and achieves substantially shorter reconstruction times than competing methods. Extensive experiments across various linear inverse problems and image datasets demonstrate state-of-the-art performance in both distortion metrics and perceptual quality. Additional analyses assess robustness to model mismatch, as well as the ability to generate diverse plausible reconstructions and characterize reconstruction uncertainty.
Flow matching methods for imaging inverse problems typically incorporate measurements through network conditioning or guidance during sampling. Neither approach explicitly applies the forward operator within the learned conditional velocity field. We develop a principled measurement-conditional velocity parameterization that does. For a linear interpolation path, we express the optimal velocity through the posterior mean E[x1∣xt,y] and show that this mean is the unique minimizer of a variational objective with an explicit data-consistency term. The velocity defined by this minimizer provably transports the source distribution to the measurement-conditioned posterior. This result leads to a forward operator-aware velocity field that is trained end-to-end and requires no separate guidance during sampling. Across five imaging tasks, our method achieves leading reconstruction quality with up to 50× fewer network evaluations than competitive flow-based methods. Varying the number of sampling steps also controls the distortion-perception trade-off without retraining.
Shirin Shoushtari, Edward P. Chandler, Xiao Shi +1
Flow-based generative models have emerged as powerful priors for solving inverse problems. One option is to directly optimize the initial latent code (noise), such that the flow output solves the inverse problem. However, this requires backpropagating through the entire generative trajectory, incurring high memory costs and numerical instability. We propose MS-Flow, which represents the trajectory as a sequence of intermediate latent states rather than a single initial code. By enforcing the flow dynamics locally and coupling segments through trajectory-matching penalties, MS-Flow alternates between updating intermediate latent states and enforcing consistency with observed data. This reduces memory consumption while improving reconstruction quality. We demonstrate the effectiveness of MS-Flow over existing methods on image recovery and inverse problems, including inpainting, super-resolution, and computed tomography.
Alexander Denker, Zeljko Kereta, Carola-Bibiane Schönlieb +1
Department of Computer Science, University College London · Department of Applied Mathematics and Theoretical Physics, University of Cambridge.
Flow models have recently emerged as powerful generative priors for solving inverse problems. Despite their success, most existing flow-based inverse problem solvers initialize the reconstruction process with random Gaussian noise. However, recent studies in image generation have shown that the choice of initial noise can significantly influence generation outcomes. The role of initialization as a distinct design choice in flow-based inverse problem solving remains comparatively underexplored. In this work, we systematically evaluate noise initialization strategies while holding the downstream flow-based inverse solver fixed. We propose Retrieval-Augmented Initialization with Dual-Guided Inversion (RA-DGI), a method that constructs improved initial noise by leveraging retrieved proxy clean images and dual-guided inversion. Extensive experiments across multiple inverse problems demonstrate that the quality of the initial noise substantially affects reconstruction fidelity and perceptual performance. RA-DGI outperforms standard random initialization and achieves superior results compared to inference-time scaling strategies, while requiring less computational overhead. These findings highlight initialization as a critical yet previously overlooked factor in flow-based inverse problem solvers.
Hansam Cho, Seoung Bum Kim
Department of Industrial and Management Engineering, Korea University