Diffusion Posterior Sampling for General Noisy Inverse Problems

Real-world scientific and industrial imaging tasks frequently require reconstructing high-quality images from partial, corrupted, or indirect sensor measurements. While modern diffusion-based generative artificial intelligence models have emerged as powerful tools for solving these inverse imaging problems, existing techniques primarily operate under the assumption of ideal, noiseless conditions or simple linear transformations. When confronted with real-world complexities such as sensor noise and nonlinear measurement processes, traditional diffusion methods often fail dramatically, either amplifying noise artifacts or becoming computationally intractable due to expensive matrix decompositions.

The article evaluates and demonstrates a new framework called Diffusion Posterior Sampling, which extends generative diffusion models to efficiently handle general noisy and nonlinear inverse problems without requiring problem-specific retraining or complex matrix decompositions.

The researchers developed an analytical approximation that estimates measurement likelihood during the intermediate steps of the generative diffusion process. By utilizing classical statistical denoising theory (Tweedie's formula), the method computes an empirical posterior mean at each step and applies a gradient-based consistency update directly in the standard image domain, omitting the strict geometric projection steps that previously caused noise amplification. The authors validated this approach on 1,000 validation images each from two widely used standard image datasets (FFHQ human faces and ImageNet) across both Gaussian and Poisson noise settings. Evaluated tasks included super-resolution, inpainting, linear and motion deblurring, and two challenging nonlinear tasks: non-uniform deblurring and Fourier phase retrieval.

The experimental findings demonstrate that the proposed method consistently outperforms existing state-of-the-art restoration methods in perceptual image quality across all tested configurations. On linear inverse problems under Gaussian noise, the framework achieved the lowest perceptual error scores, significantly outperforming projection-based methods and prior diffusion baselines. For example, in random inpainting on the FFHQ dataset, the method achieved a Fréchet Inception Distance of 21.19, compared to 29.26 for manifold constrained gradients and 69.71 for spectral-based diffusion restoration models. In nonlinear Fourier phase retrieval, the proposed approach achieved a perceptual score of 55.61, outperforming traditional standard algorithms such as Hybrid Input-Output, which scored 96.40. Furthermore, because the algorithm operates entirely in the image domain via standard automatic differentiation, it seamlessly handles complex operators such as motion blur and neural-network-approximated nonlinear blur that break alternative spectral-decomposition methods.

These results demonstrate that generative diffusion models can be deployed in noisy, complex, and nonlinear imaging pipelines without costly architectural modifications or custom hardware-prohibitive computations. Eliminating matrix decompositions reduces computational overhead when dealing with complex forward measurement models, while avoiding strict projection constraints substantially reduces the operational risk of generating severe noise-induced artifacts. This makes diffusion-based image reconstruction substantially more viable for practical, safety-critical domains such as medical scanning, optical imaging, and remote sensing.

Organizations evaluating computational imaging pipelines should consider adopting this sampling methodology for general noisy image restoration, particularly when dealing with nonlinear operators or non-separable blur kernels. To optimize computational resources, practitioners should maintain step sizes in the recommended stable range (between 0.1 and 1.0) and use multiple random initializations when solving non-unique nonlinear problems such as phase retrieval. Further engineering development is recommended to integrate fast accelerated solvers, which can bridge the gap between high-quality iterative diffusion sampling and the latency demands of real-time applications.

The primary limitation of the method is its computational runtime: completing a full iterative reconstruction requires around 78 to 95 seconds on a standard graphics processing unit for 256×256 images, making it slower than specialized low-iteration methods in compute-constrained environments. Additionally, highly ambiguous problems such as Fourier phase retrieval remain susceptible to non-uniqueness and initialization sensitivity. Nevertheless, high confidence is supported by rigorous mathematical error bounds and broad empirical validation across diverse imaging modalities.