Trajectory Constraints for Imaging Inverse Problems

Diffusion models have achieved strong performance in imaging inverse problems, but typically rely on domain-matched pretrained models and external training data. Moreover, errors introduced at intermediate reverse-sampling steps can accumulate and degrade the final reconstruction, with their propagation strongly influenced by how successive states are related. We propose TRACE, a training-free TRAjectory-Constrained rEconstruction framework for imaging inverse problems. TRACE replaces the pretrained diffusion model with an untrained overparameterized network optimized directly from the observed measurements and organizes reconstruction as a sequence of intermediate states. At each step, TRACE enforces stochastic proximal consistency (SPC) by inferring each reconstruction from a stochastically perturbed predecessor under measurement consistency, while proximally constraining the resulting estimate to its corresponding unperturbed state. This formulation admits a proximal interpretation, from which we derive stability bounds characterizing the variation between consecutive states under approximate network updates. A local Jacobian analysis further characterizes how SPC regularizes the network-parameterized reconstruction. Experiments on natural-image restoration and CT reconstruction demonstrate competitive performance against both untrained and pretrained neural-prior methods. We further show that the proposed trajectory constraint can be incorporated into existing diffusion-based inverse solvers to improve reconstruction performance.