Diffusion-Based Inverse Problems

Stochastic differential equation (SDE)-based generative models have achieved substantial progress in conditional generation via training-free differentiable loss-guided approaches. However, existing methodologies utilizing posterior sam- pling typically confront a substantial estimation error, which results in inaccu- rate gradients for guidance and leading to inconsistent generation results. To mitigate this issue, we propose that performing an additional backward denois- ing step and Monte-Carlo sampling (ABMS) can achieve better guided diffu- sion, which is a plug-and-play adjustment strategy. To verify the effectiveness of our method, we provide theoretical analysis and propose the adoption of a dual-focus evaluation framework, which further serves to highlight the critical problem of cross-condition interference prevalent in existing approaches. We conduct experiments across various task settings and data types, mainly includ- ing conditional online handwritten trajectory generation, image inverse problems (inpainting, super resolution and gaussian deblurring) molecular inverse design and so on. Experimental results demonstrate that our approach can be effec- tively used with higher order samplers and consistently improves the quality of generation samples across all the different scenarios.

Cold and generalized diffusion models have recently shown strong potential for sparse-view CT reconstruction by explicitly modeling deterministic degradation processes. However, existing sampling strategies often rely on ad hoc sampling controls or fixed schedules, which remain sensitive to error accumulation and sampling instability. We propose ReCo-Diff, a residual-conditioned diffusion framework that leverages observation residuals through residual-conditioned self-guided sampling. At each sampling step, ReCo-Diff first produces a null (unconditioned) baseline reconstruction and then conditions subsequent predictions on the observation residual between the predicted image and the measured sparse-view input. This residual-driven guidance provides continuous, measurement-aware correction while preserving a deterministic sampling schedule, without requiring heuristic interventions. Experimental results demonstrate that ReCo-Diff consistently outperforms existing cold diffusion sampling baselines, achieving higher reconstruction accuracy, improved stability, and enhanced robustness under severe sparsity.

Low-field (LF) magnetic resonance imaging (MRI) democratizes access to diagnostic imaging but is fundamentally limited by low signal-to-noise ratio and significant tissue contrast distortion due to field-dependent relaxation dynamics. Reconstructing high-field (HF) quality images from LF data is a blind inverse problem, severely challenged by the scarcity of paired training data and the unknown, non-linear contrast transformation operator. Existing zero-shot methods, which assume simplified linear degradation, often fail to recover authentic tissue contrast. In this paper, we propose DACT(Diffusion-Based Adaptive Contrast Transport), a novel zero-shot framework that restores HF-quality images without paired supervision. DACT synergizes a pre-trained HF diffusion prior to ensure anatomical fidelity with a physically-informed adaptive forward model. Specifically, we introduce a differentiable Sinkhorn optimal transport module that explicitly models and corrects the intensity distribution shift between LF and HF domains during the reverse diffusion process. This allows the framework to dynamically learn the intractable contrast mapping while preserving topological consistency. Extensive experiments on simulated and real clinical LF datasets demonstrate that DACT achieves state-of-the-art performance, yielding reconstructions with superior structural detail and correct tissue contrast.

Recovering radiometric fidelity from unknown dynamic range compression (UDRC), such as low-light enhancement and HDR reconstruction, is a challenging blind inverse problem, due to the unknown forward model and irreversible information loss introduced by compression. To address this challenge, we first identify monotonicity as the fundamental physical invariant shared across UDRC tasks. Leveraging this insight, we introduce the cascaded monotonic Bernstein (CaMB) operator to parameterize the unknown forward model. CaMB enforces monotonicity as a hard architectural inductive bias, constraining optimization to physically consistent mappings and enabling robust and stable operator estimation. We further integrate CaMB with a plug-and-play diffusion framework, proposing CaMB-Diff. Within this framework, the diffusion model serves as a powerful geometric prior for structural and semantic recovery, while CaMB explicitly models and corrects radiometric distortions through a physically grounded forward operator. Extensive experiments on a variety of zero-shot UDRC tasks, including low-light enhancement, low-field MRI enhancement, and HDR reconstruction, demonstrate that CaMB-Diff significantly outperforms state-of-the-art zero-shot baselines in terms of both signal fidelity and physical consistency. Moreover, we empirically validate the effectiveness of the proposed CaMB parameterization in accurately modeling the unknown forward operator.

Diffusion models have gained prominence as powerful generative tools for solving inverse problems due to their ability to model complex data distributions. However, existing methods typically rely on complete knowledge of the forward observation process to compute gradients for guided sampling, limiting their applicability in scenarios where such information is unavailable. In this work, we introduce Constrained Particle Seeking (CPS), a novel gradient-free approach that leverages all candidate particle information to actively search for the optimal particle while incorporating constraints aligned with high-density regions of the unconditional prior. Unlike previous methods that passively select promising candidates, CPS reformulates the inverse problem as a constrained optimization task, enabling more flexible and efficient particle seeking. We demonstrate that CPS can effectively solve both image and scientific inverse problems, achieving results comparable to gradient-based methods while significantly outperforming gradient-free alternatives. Code is available at https://github.com/deng-ai-lab/CPS.

Training-free diffusion priors enable inverse-problem solvers without retraining, but for nonlinear forward operators data consistency often relies on repeated derivatives or inner optimization/MCMC loops with conservative step sizes, incurring many iterations and denoiser/score evaluations. We propose a training-free solver that replaces these inner loops with a hard measurement-space feasibility constraint (closed-form projection) and an analytic, model-optimal step size, enabling a small, fixed compute budget per noise level. Anchored at the denoiser prediction, the correction is approximated via an adjoint-free, ADMM-style splitting with projection and a few steepest-descent updates, using one VJP and either one JVP or a forward-difference probe, followed by backtracking and decoupled re-annealing. We prove local model optimality and descent under backtracking for the step-size rule, and derive an explicit KL bound for mode-substitution re-annealing under a local Gaussian conditional surrogate. We also develop a latent variant and a one-parameter pixel$\rightarrow$latent hybrid schedule. Experiments achieve competitive PSNR/SSIM/LPIPS with up to 19.5$\times$ speedup, without hand-coded adjoints or inner MCMC.

Diffusion models have emerged as powerful priors for solving inverse problems in computed tomography (CT). In certain applications, such as neutron CT, it can be expensive to collect large amounts of measurements even for a single scan, leading to sparse data sets from which it is challenging to obtain high quality reconstructions even with diffusion models. One strategy to mitigate this challenge is to leverage a complementary, easily available imaging modality; however, such approaches typically require retraining the diffusion model with large datasets. In this work, we propose incorporating an additional modality without retraining the diffusion prior, enabling accelerated imaging of costly modalities. We further examine the impact of imperfect side modalities on cross-modal guidance. Our method is evaluated on sparse-view neutron computed tomography, where reconstruction quality is substantially improved by incorporating X-ray computed tomography of the same samples.

Accurate Speed-of-Sound (SoS) reconstruction from acoustic waveforms is a cornerstone of ultrasound computed tomography (USCT), enabling quantitative velocity mapping that reveals subtle anatomical details and pathological variations often invisible in conventional imaging. However, practical utility is hindered by the limitations of existing algorithms; traditional Full Waveform Inversion (FWI) is computationally intensive, while current deep learning approaches tend to produce oversmoothed results lacking fine details. We propose DiffSOS, a conditional diffusion model that directly maps acoustic waveforms to SoS maps. Our framework employs a specialized acoustic ControlNet to strictly ground the denoising process in physical wave measurements. To ensure structural consistency, we optimize a hybrid loss function that integrates noise prediction, spatial reconstruction, and noise frequency content. To accelerate inference, we employ stochastic Denoising Diffusion Implicit Model (DDIM) sampling, achieving near real-time reconstruction with only 10 steps. Crucially, we exploit the stochastic generative nature of our framework to estimate pixel-wise uncertainty, providing a measure of reliability that is often absent in deterministic approaches. Evaluated on the OpenPros USCT benchmark, DiffSOS significantly outperforms state-of-the-art networks, achieving an average Multi-scale Structural Similarity of 0.957. Our approach provides high-fidelity SoS maps with a principled measure of confidence, facilitating safer and faster clinical interpretation.

Color polarization demosaicking (CPDM) aims to reconstruct full-resolution polarization images of four directions from the color-polarization filter array (CPFA) raw image. Due to the challenge of predicting numerous missing pixels and the scarcity of high-quality training data, existing network-based methods, despite effectively recovering scene intensity information, still exhibit significant errors in reconstructing polarization characteristics (degree of polarization, DOP, and angle of polarization, AOP). To address this problem, we introduce the image diffusion prior from text-to-image (T2I) models to overcome the performance bottleneck of network-based methods, with the additional diffusion prior compensating for limited representational capacity caused by restricted data distribution. To effectively leverage the diffusion prior, we explicitly model the polarization uncertainty during reconstruction and use uncertainty to guide the diffusion model in recovering high error regions. Extensive experiments demonstrate that the proposed method accurately recovers scene polarization characteristics with both high fidelity and strong visual perception.

Plug-and-Play diffusion prior (PnPDP) frameworks have emerged as a powerful paradigm for solving imaging inverse problems by treating pretrained generative models as modular priors. However, we identify a critical flaw in prevailing PnP solvers (e.g., based on HQS or Proximal Gradient): they function as memoryless operators, updating estimates solely based on instantaneous gradients. This lack of historical tracking inevitably leads to non-vanishing steady-state bias, where the reconstruction fails to strictly satisfy physical measurements under heavy corruption. To resolve this, we propose Dual-Coupled PnP Diffusion (DC-PnPDP), which restores the classical dual variable to provide integral feedback, progressively enforce agreement between the data-consistency and prior. However, this rigorous geometric coupling introduces a secondary challenge: the accumulated dual residuals exhibit spectrally colored, structured artifacts that violate the Additive White Gaussian Noise (AWGN) assumption of diffusion priors, causing severe hallucinations. To bridge this gap, we introduce Spectral Homogenization (SH), a frequency-domain adaptation mechanism that modulates these structured residuals into statistically compliant pseudo-AWGN inputs. This effectively aligns the solver's rigorous optimization trajectory with the denoiser's valid statistical manifold. Extensive experiments on CT and MRI reconstruction demonstrate that our approach resolves the bias-hallucination trade-off, achieving state-of-the-art fidelity with significantly accelerated convergence. The code is available at https://github.com/duchenhe/DC-PnPDP

The choice of prior is central to solving ill-posed imaging inverse problems, making it essential to select one consistent with the measurements $y$ to avoid severe bias. In Bayesian inverse problems, this could be achieved by evaluating the model evidence $p(y \mid M)$ under different models $M$ that specify the prior and then selecting the one with the highest value. Diffusion models are the state-of-the-art approach to solving inverse problems with a data-driven prior; however, directly computing the model evidence with respect to a diffusion prior is intractable. Furthermore, most existing model evidence estimators require either many pointwise evaluations of the unnormalized prior density or an accurate clean prior score. We propose DiME, an estimator of the model evidence of a diffusion prior by integrating over the time-marginals of posterior sampling methods. Our method leverages the large amount of intermediate samples naturally obtained during the reverse diffusion sampling process to obtain an accurate estimation of the model evidence using only a handful of posterior samples (e.g., 20). We also demonstrate how to implement our estimator in tandem with recent diffusion posterior sampling methods. Empirically, our estimator matches the model evidence when it can be computed analytically, and it is able to both select the correct diffusion model prior and diagnose prior misfit under different highly ill-conditioned, non-linear inverse problems, including a real-world black hole imaging problem.

Diffusion models have recently emerged as powerful priors for solving inverse problems. While computed tomography (CT) is theoretically a linear inverse problem, it poses many practical challenges. These include correlated noise, artifact structures, reliance on system geometry, and misaligned value ranges, which make the direct application of diffusion models more difficult than in domains like natural image generation. To systematically evaluate how diffusion models perform in this context and compare them with established reconstruction methods, we introduce DM4CT, a comprehensive benchmark for CT reconstruction. DM4CT includes datasets from both medical and industrial domains with sparse-view and noisy configurations. To explore the challenges of deploying diffusion models in practice, we additionally acquire a high-resolution CT dataset at a high-energy synchrotron facility and evaluate all methods under real experimental conditions. We benchmark ten recent diffusion-based methods alongside seven strong baselines, including model-based, unsupervised, and supervised approaches. Our analysis provides detailed insights into the behavior, strengths, and limitations of diffusion models for CT reconstruction. The real-world dataset is publicly available at zenodo.org/records/15420527, and the codebase is open-sourced at github.com/DM4CT/DM4CT.

Physical fields are typically observed only at sparse, time-varying sensor locations, making forecasting and reconstruction ill-posed and uncertainty-critical. We present SOLID, a mask-conditioned diffusion framework that learns spatiotemporal dynamics from sparse observations alone: training and evaluation use only observed target locations, requiring no dense fields and no pre-imputation. Unlike prior work that trains on dense reanalysis or simulations and only tests under sparsity, SOLID is trained end-to-end with sparse supervision only. SOLID conditions each denoising step on the measured values and their locations, and introduces a dual-masking objective that (i) emphasizes learning in unobserved void regions while (ii) upweights overlap pixels where inputs and targets provide the most reliable anchors. This strict sparse-conditioning pathway enables posterior sampling of full fields consistent with the measurements, achieving up to an order-of-magnitude improvement in probabilistic error and yielding calibrated uncertainty maps (\r{ho} > 0.7) under severe sparsity.

This article addresses the issue of estimating observation parameters (response and error parameters) in inverse problems. The focus is on cases where regularization is introduced in a Bayesian framework and the prior is modeled by a diffusion process. In this context, the issue of posterior sampling is known to be thorny, and a recent paper proposes a notably simple and effective solution. Additionally, it opens an remarkable flexibility when it comes to estimating observation parameters. The proposed strategy enables to define an optimal estimator for both observation parameters and image of interest. Furthermore, the strategy provides a means for uncertainty quantification. In addition, MCMC algorithms allow for the computation of estimates and properties of posteriors, while offering some guarantees. The paper presents several numerical experiments that clearly confirm the computational efficiency and the quality of both estimates and uncertainty quantification.

This paper addresses the issue of inversion in cases where (1) the observation system is modeled by a linear transformation and additive error, (2) the problem is ill-posed and regularization relies on a Bayesian strategy, (3)~the prior is modeled by a diffusion process adjusted on an available large set of examples. In this context, it is known that the issue of posterior sampling is a thorny one and the paper introduces a Gibbs algorithm. It appears that this avenue has not been explored, and we show that it is particularly effective and remarkably simple. In addition, it provides clear elements regarding convergence guarantees in a specific case and arguments supporting such guarantees in practical cases. The results are clearly confirmed by numerical simulations based on a toy example.

Score-based diffusion models are a recently developed framework for posterior sampling in Bayesian inverse problems, enabling high-quality reconstructions in inverse problems by leveraging expressive prior distributions learned from empirical data. Despite their strong empirical performance and growing interest within the machine learning community, their behaviour in realistic, severely ill-posed inverse problems with experimental measurement data remains under-explored. Diffuse optical tomography (DOT) is an inverse boundary value problem that uses boundary measurements of near-infrared light to recover spatially varying absorption and scattering parameters in biological tissue. The problem is highly ill-posed and particularly sensitive to both measurement noise and modelling errors. We introduce a regularization strategy by constructing a mixed score consisting of a learned component and a model-based component. We show that the resulting mixed score approximates the score of a corresponding mixture distribution locally and in the small diffusion-time regime, providing a theoretical justification for the approach. We compare four approaches for difference imaging in DOT: a classical model-based method, an approximate score-based diffusion method (DPS), an exact posterior sampling method (UCoS) and a novel, regularized version of UCoS. We show that both the model-based approach and approximate diffusion-based sampling degrade significantly in the presence of limited-view geometry and real experimental data, whereas UCoS yields more accurate reconstructions.

We propose a novel training-free method for inpainting with off-the-shelf text-to-image models. While guidance-based methods in theory allow generic models to be used for inverse problems such as inpainting -- in practice their effectiveness is limited, leading to the necessity of specialized inpainting-specific models. In this work, we argue the missing ingredient for training-free generic model usage is proper optimization of the initial noise sample. We optimize the initial noise to approximately reproduce the unmasked image, in as few as tens of optimization steps, then use it with a conventional training-free inpainting method. Critically, we propose two core ideas that make this possible: (i) we perform linear approximation that avoids the costly and often impractical unrolling required to relate the initial noise sample to model output -- which potentially is why this relationship was previously overlooked; and (ii) perform spectral preconditioning by optimizing the initial noise sample in the spectral domain with Adam, which stabilizes the optimization. We demonstrate our method on various inpainting tasks, outperforming the state of the art. Project website: https://ubc-vision.github.io/sonic/

Diffusion models provide powerful generative priors for solving inverse problems by sampling from a posterior distribution conditioned on corrupted measurements. Existing methods primarily follow two paradigms: direct methods, which approximate the likelihood term, and proximal methods, which incorporate intermediate solutions satisfying measurement constraints into the sampling process. Under standard Gaussian approximations and locally-linear measurements, we demonstrate that these approaches differ fundamentally in their treatment of the diffusion denoiser's Jacobian within the likelihood term. While this Jacobian encodes critical prior knowledge of the data distribution, training-induced non-idealities can degrade performance in zero-shot settings. In this work, we bridge direct and proximal approaches by proposing a principled Jacobian-Aware Posterior Sampler (JAPS). JAPS leverages the Jacobian's prior knowledge while mitigating its detrimental effects through a corresponding proximal solution, requiring no additional computational cost. Additionally, we integrate our guidance into DDIM sampling, with a corrected conditional factor that has been missing in previous works. Our method enhances reconstruction quality across diverse linear and nonlinear noisy imaging tasks, outperforming existing diffusion-based baselines in perceptual quality while maintaining or improving distortion metrics.

From a Bayesian perspective, score-based diffusion solves inverse problems through joint inference, embedding the likelihood with the prior to guide the sampling process. However, this formulation fails to explain its practical behavior: the prior offers limited guidance, while reconstruction is largely driven by the measurement-consistency term, leading to an inference process that is effectively decoupled from the diffusion dynamics. We show that the diffusion prior in these solvers functions primarily as a warm initializer that places estimates near the data manifold, while reconstruction is driven almost entirely by measurement consistency. Based on this observation, we introduce DAPS++, which fully decouples diffusion-based initialization from likelihood-driven refinement, allowing the likelihood term to guide inference more directly while maintaining numerical stability and providing insight into why unified diffusion trajectories remain effective in practice. By requiring fewer function evaluations (NFEs) and measurement-optimization steps, DAPS++ achieves high computational efficiency and robust reconstruction performance across diverse image restoration tasks.

Video inverse problems are fundamental to streaming, telepresence, and AR/VR, where high perceptual quality must coexist with tight latency constraints. Diffusion-based priors currently deliver state-of-the-art reconstructions, but existing approaches either adapt image diffusion models with ad hoc temporal regularizers - leading to temporal artifacts - or rely on native video diffusion models whose iterative posterior sampling is far too slow for real-time use. We introduce InstantViR, an amortized inference framework for ultra-fast video reconstruction powered by a pre-trained video diffusion prior. We distill a powerful bidirectional video diffusion model (teacher) into a causal autoregressive student that maps a degraded video directly to its restored version in a single forward pass, inheriting the teacher's strong temporal modeling while completely removing iterative test-time optimization. The distillation is prior-driven: it only requires the teacher diffusion model and known degradation operators, and does not rely on externally paired clean/noisy video data. To further boost throughput, we replace the video-diffusion backbone VAE with a high-efficiency LeanVAE via an innovative teacher-space regularized distillation scheme, enabling low-latency latent-space processing. Across streaming random inpainting, Gaussian deblurring and super-resolution, InstantViR matches or surpasses the reconstruction quality of diffusion-based baselines while running at over 35 FPS on NVIDIA A100 GPUs, achieving up to 100 times speedups over iterative video diffusion solvers. These results show that diffusion-based video reconstruction is compatible with real-time, interactive, editable, streaming scenarios, turning high-quality video restoration into a practical component of modern vision systems.

This work introduces MedForj, a suite of 3D foundational generative priors based on diffusion models. The MedForj models were trained on $72{,}659$ 1~mm isotropic 3D $T_1$-weighted MRI human brain image volumes from $38{,}174$ subjects, drawn from a curated corpus of $80{,}675$ volumes from $42{,}506$ subjects spanning $38$ publicly available datasets. These training images were manually inspected to exclude those with poor quality and excessive pathology, and otherwise were minimally processed. The models include six different diffusion training strategies: rectified flow, latent diffusion rectified flow, flow matching, velocity prediction, clean prediction, and noise prediction. Image samples produced by each of these models were compared to each other and against real, ground truth data under downstream segmentation distributions, FID, five inverse problems, and blind human inspection in an observer study. Flow matching was the strongest strategy overall, achieving the best inverse problem solving results at $28.80$~dB PSNR and $0.874$ SSIM averaged over the five forward problems, the highest rate of reconstructions judged real by blind human raters at $72.6\%$, and the closest per-structure match to real segmented anatomy in a permutation test. It was not best everywhere: rectified flow produced the most convincing unconditional samples in the observer study and the best FID, and the latent rectified-flow model achieved the smallest joint distributional distance to real anatomy. No other strategy, however, performed consistently well across all four evaluations. We therefore recommend flow matching as the default MedForj prior, while releasing every strategy so that the choice can be revisited per application. All model weights and corresponding code are publicly available at https://github.com/piksl-research/medforj.

Solving ill-posed inverse problems requires powerful and flexible priors. We propose leveraging pretrained latent diffusion models for this task through a new training-free approach, termed Diffusion-regularized Wasserstein Gradient Flow (DWGF). Specifically, we formulate the posterior sampling problem as a Wasserstein gradient flow in the latent space of an expected negative log posterior objective, regularized by a Kullback-Leibler divergence to the diffusion prior. We demonstrate the performance of our method on standard benchmarks using StableDiffusion (Rombach et al., 2022) as the prior.

Diffusion models (DMs) have proven to be effective in modeling high-dimensional distributions, leading to their widespread adoption for representing complex priors in Bayesian inverse problems (BIPs). However, current DM-based posterior sampling methods proposed for solving common BIPs rely on heuristic approximations to the generative process. To exploit the generative capability of DMs and avoid the usage of such approximations, we propose an ensemble-based algorithm that performs posterior sampling without the use of heuristic approximations. Our algorithm is motivated by existing work that combines DM-based methods with the sequential Monte Carlo (SMC) method. By examining how the prior evolves through the diffusion process encoded by the pre-trained score function, we derive a modified partial differential equation (PDE) governing the evolution of the corresponding posterior distribution. This PDE includes a modified diffusion term and a reweighting term, which can be simulated via stochastic weighted particle methods. Theoretically, we prove that the error between the true posterior and the empirical distribution of the generated samples can be bounded in terms of the training error of the pre-trained score function and the number of particles in the ensemble. Empirically, we validate our algorithm on several inverse problems in imaging to show that our method gives more accurate reconstructions compared to existing DM-based methods. Our code is available at the following Github repository~https://github.com/HaoxuanSteveC00/AFDPS-TMLR.

Rapid urbanization demands efficient monitoring of turbulent wind and pollutant dispersion, yet existing reconstruction and sensor placement strategies fail under realistic sparsity constraints. Here, we introduce Diff--SPORT, a diffusion-based framework that combines a generative diffusion prior with maximum a posteriori inference and Shapley-value attribution for high-fidelity flow reconstruction and optimal sensor placement. By training a diffusion prior model once over a domain, Diff--SPORT enables non-linear optimal sensor placement and near-real-time flow reconstruction from sparse measurements orders of magnitude faster than RANS or LES simulations, consistently outperforming state-of-the-art methods. The framework also extends, without algorithmic modification, to experimental passive scalar concentration dataset, a direct proxy for pollutant dispersion, measured in a 1:2400 scale water-flume model of the Beijing Haidian neighbourhood under realistic urban flow conditions. Shapley-guided sensor placement achieves up to 57% lower reconstruction error than randomly placed sensors at extreme sparsity, identifying compact and physically interpretable configurations. These results establish Diff--SPORT as a modular foundation offering a zero-shot alternative to retraining-intensive downstream strategies, supporting scalable urban flow monitoring for air quality management and resilient city design.

Diffusion models represent the state-of-the-art for solving inverse problems such as image restoration tasks. Diffusion-based inverse solvers incorporate a likelihood term to guide prior sampling, generating data consistent with the posterior distribution. However, due to the intractability of the likelihood, most methods rely on isotropic Gaussian approximations, which can push estimates off the data manifold and produce inconsistent, poor reconstructions. We propose Equivariance Regularized (EquiReg) diffusion, a general plug-in framework that improves posterior sampling by penalizing trajectories that deviate from the data manifold. EquiReg formalizes manifold-preferential equivariant functions that exhibit low equivariance error for on-manifold samples and high error for off-manifold ones, thereby guiding sampling toward symmetry-preserving regions of the solution space. We highlight that such functions naturally emerge when training non-equivariant models with augmentation or on data with symmetries. EquiReg's largest gains are under reduced sampling and measurement consistency steps, where many methods suffer severe quality degradation. By regularizing trajectories toward the manifold, EquiReg implicitly accelerates convergence and enables high-quality reconstructions. EquiReg consistently improves performance in linear and nonlinear image restoration tasks and solving partial differential equations. Our code is available at https://github.com/Anima-Lab/EquiReg

We consider the problem of reconstructing a HxWx31 hyperspectral image from a $H\times W$ grayscale snapshot measurement that is captured using only a single diffractive lens and a filterless panchromatic photosensor. This problem is severely ill-posed, but we present a model that produces high-quality results in simulation and experiment. We make efficient use of limited training data by creating a conditional denoising diffusion model that operates on small patches in a shift-invariant manner. During inference, we synchronize per-patch hyperspectral predictions using guidance by physical consistency with the system's optical point spread function. Our experiments reveal that the patch size can be as small as the point spread function, with local optical cues being the main source of information about complete spectra. Also, by drawing multiple samples, our model provides per-pixel uncertainty estimates that strongly correlate with reconstruction error.

Diffusion models excel in solving imaging inverse problems due to their ability to model complex image priors. However, their reliance on large, clean datasets for training limits their practical use where clean data is scarce. In this paper, we propose EMDiffusion, an expectation-maximization (EM) approach to train diffusion models from corrupted observations. Our method alternates between reconstructing clean images from corrupted data using a known diffusion model (E-step) and refining diffusion model weights based on these reconstructions (M-step). This iterative process leads the learned diffusion model to gradually converge to the true clean data distribution. We validate our method through extensive experiments on diverse computational imaging tasks, including random inpainting, denoising, and deblurring, achieving new state-of-the-art performance.

Diffusion models have recently gained traction as a powerful class of deep generative priors, excelling in a wide range of image restoration tasks due to their exceptional ability to model data distributions. To solve image restoration problems, many existing techniques achieve data consistency by incorporating additional likelihood gradient steps into the reverse sampling process of diffusion models. However, the additional gradient steps pose a challenge for real-world practical applications as they incur a large computational overhead, thereby increasing inference time. They also present additional difficulties when using accelerated diffusion model samplers, as the number of data consistency steps is limited by the number of reverse sampling steps. In this work, we propose a novel diffusion-based image restoration solver that addresses these issues by decoupling the reverse process from the data consistency steps. Our method involves alternating between a reconstruction phase to maintain data consistency and a refinement phase that enforces the prior via diffusion purification. Our approach demonstrates versatility, making it highly adaptable for efficient problem-solving in latent space. Additionally, it reduces the necessity for numerous sampling steps through the integration of consistency models. The efficacy of our approach is validated through comprehensive experiments across various image restoration tasks, including image denoising, deblurring, inpainting, and super-resolution.