DIPA: Distilled Preconditioned Algorithms for Solving Imaging Inverse Problems
Authors: Romario Gualdrón-Hurtado, Roman Jacome, Leon Suarez, Henry Arguello
Organizations: Department of Computer Science, Universidad Industrial de Santander, Bucaramanga, 680002, Colombia · Department of Electrical Engineering, Universidad Industrial de Santander, Bucaramanga, 680002, Colombia
Solving imaging inverse problems has usually been addressed by designing proper prior models of the underlying signal. However, minimizing the data fidelity term poses significant challenges due to the ill-conditioned sensing matrix caused by physical constraints in the acquisition system. Thus, preconditioning techniques have been adopted in classical optimization theory to address ill-conditioned data-fidelity minimization by transforming the algorithm gradient step to achieve faster convergence and better numerical stability. We extend the preconditioning concept beyond convergence acceleration and use it to improve reconstruction quality. We introduce DIPA: Distilled Preconditioned Algorithms, where a preconditioning operator (PO) is optimized using teacher-guided distillation criteria. Unlike standard model-compression KD, the teacher and student differ by the sensing operators available during reconstruction: the teacher uses a simulated, better-conditioned, and more informative sensing matrix, whereas the student uses the physically feasible sensing matrix. We design different distillation loss functions to transfer different properties of the teacher algorithm to the preconditioned student. The PO can be linear (L-DIPA), allowing interpretability, or non-linear (N-DIPA), parametrized by a neural network, offering better scalability. We validate the proposed PO design across several imaging modalities, including magnetic resonance imaging, compressed sensing, and super-resolution imaging.
Pretrained diffusion models represent image distributions through a continuum of progressively smoothed distributions. This multiscale structure organizes generation from global structure to fine detail and supports high-quality, diverse samples. We exploit the same multiscale diffusion prior for linear imaging inverse problems. Rather than using the pretrained model only as a denoiser in an outer iteration, we define a surrogate likelihood whose center is aligned with the clean-image coordinate and whose covariance accounts for residual diffusion uncertainty. This construction defines an explicit surrogate posterior path, from which we derive continuous posterior dynamics. A tunable Langevin component supports target tracking and allows the amount of posterior exploration to be adapted to the application. We prove endpoint consistency and a finite-horizon tracking bound and, in the exact-score setting, first-order weak accuracy. For computation, we derive the Posterior-Dynamics Implicit--Explicit sampler (PD-IMEX), a stable method using one score evaluation per diffusion scale and an implicit data-consistency update. Experiments on deblurring, super-resolution, and inpainting show strong reconstruction quality at 100 score evaluations, coarse-grid stability, and controllable fidelity--diversity behavior.
Recently, Deep Image Prior (DIP) has demonstrated strong capabilities for solving inverse imaging problems (IIPs) by optimizing a randomly initialized convolutional neural network in a training-data-free regime. However, DIP suffers from overfitting to noisy measurements due to network over-parameterization, making early stopping (ES) essential. The most successful ES method tracks fluctuations in the running variance of the network output to detect overfitting. However, in many applications, these fluctuations may appear prematurely, leading to unstable reconstructions. In this paper, we first show that nearly optimal DIP early stopping can be achieved when two independent noisy copies of the degraded image are available. Motivated by this observation, and since obtaining two fully independent copies is infeasible, we propose an overfitting detection framework based on constructing pseudo self-referenced images, resulting in three IIP-specific algorithms. Our approach is further supported by theoretical results on single-reference validation, pseudo-validation estimation, and the impact of shared noise. Across different IIPs, ranging from natural image restoration to medical image reconstruction, and under varying noise levels and noise types, our methods consistently outperform existing DIP early stopping approaches, all without requiring an accurate estimate of the noise level.
Chaoyan Huang, Cheng-Han Huang, Ismail R. Alkhouri +1
Diffusion-based methods have achieved remarkable empirical success in solving inverse problems. However, many existing posterior samplers either lack rigorous theoretical guarantees or incur substantial computational overhead. We propose a simple and efficient algorithm, called \pddim, for solving linear inverse problems with diffusion priors via a DDIM-type sampler. Our method requires only lightweight, coordinate-wise modifications to the standard DDIM update, while explicitly incorporating the measurement model. The key idea is to perform posterior sampling separately along each singular direction of the measurement operator: for each direction, the sampler follows the learned diffusion prior when the observation signal-to-noise ratio (SNR) is below the corresponding diffusion SNR, and switches to a calibrated measurement-based predictor otherwise. We prove that the proposed sampler converges to the Bayesian posterior conditioned on the measurements. Empirical results show that the proposed sampler performs favorably against existing diffusion-based posterior samplers across a range of image restoration tasks, achieving the best performance on the majority of evaluation metrics considered. Overall, our results convert posterior sampling for noisy linear inverse problems to simple coordinate-wise DDIM updates, yielding an efficient, easy-to-implement algorithm with provable posterior consistency.