Recently, diffusion models havebeen used to solvevarious inverse problems in anunsupervised manner withappropriate modifications tothesampling process.

Denoising and score estimation have long been known to be linked via the classical Tweedie's formula. In this work, we first extend the latter to a wider range of distributions often called "energy models" and denoted elliptical distributions in this work. Next, we examine an alternative view: we consider the denoising posterior $P(X|Y)$ as the optimizer of the energy score (a scoring rule) and derive a fundamental identity that connects the (path-) derivative of a (possibly) non-Euclidean energy score to the score of the noisy marginal. This identity can be seen as an analog of Tweedie's identity for the energy score, and allows for several interesting applications; for example, score estimation, noise distribution parameter estimation, as well as using energy score models in the context of "traditional" diffusion model samplers with a wider array of noising distributions.

The first order derivative of a data density can be estimated efficiently by denoising score matching, and has become an important component in many applications, such as image generation and audio synthesis. Higher order derivatives provide additional local information about the data distribution and enable new applications. Although they can be estimated via automatic differentiation of a learned density model, this can amplify estimation errors and is expensive in high dimensional settings. To overcome these limitations, we propose a method to directly estimate high order derivatives (scores) of a data density from samples. We first show that denoising score matching can be interpreted as a particular case of Tweedie's formula. By leveraging Tweedie's formula on higher order moments, we generalize denoising score matching to estimate higher order derivatives. We demonstrate empirically that models trained with the proposed method can approximate second order derivatives more efficiently and accurately than via automatic differentiation. We show that our models can be used to quantify uncertainty in denoising and to improve the mixing speed of Langevin dynamics via Ozaki discretization for sampling synthetic data and natural images.

Recently, there has been extensive research interest in training deep networks to denoise images without clean reference.However, the representative approaches such as Noise2Noise, Noise2Void, Stein's unbiased risk estimator (SURE), etc. seem to differ from one another and it is difficult to find the coherent mathematical structure. To address this, here we present a novel approach, called Noise2Score, which reveals a missing link in order to unite these seemingly different approaches.Specifically, we show that image denoising problems without clean images can be addressed by finding the mode of the posterior distribution and that the Tweedie's formula offers an explicit solution through the score function (i.e. the gradient of loglikelihood). Our method then uses the recent finding that the score function can be stably estimated from the noisy images using the amortized residual denoising autoencoder, the method of which is closely related to Noise2Noise or Nose2Void. Our Noise2Score approach is so universal that the same network training can be used to remove noises from images that are corrupted by any exponential family distributions and noise parameters. Using extensive experiments with Gaussian, Poisson, and Gamma noises, we show that Noise2Score significantly outperforms the state-of-the-art self-supervised denoising methods in the benchmark data set such as (C)BSD68, Set12, and Kodak, etc.

Diffusion models have recently enabled state-of-the-art reconstruction of positron emission tomography (PET) images while requiring only image training data. However, domain shift remains a key concern for clinical adoption: priors trained on images from one anatomy, acquisition protocol or pathology may produce artefacts on out-of-distribution data. We propose integrating steerable conditional diffusion (SCD) with our previously-introduced likelihood-scheduled diffusion (PET-LiSch) framework to improve the alignment of the diffusion model's prior to the target subject. At reconstruction time, for each diffusion step, we use low-rank adaptation (LoRA) to align the diffusion model prior with the target domain on the fly. Experiments on realistic synthetic 2D brain phantoms demonstrate that our approach suppresses hallucinated artefacts under domain shift, i.e. when our diffusion model is trained on perturbed images and tested on normal anatomy, our approach suppresses the hallucinated structure, outperforming both OSEM and diffusion model baselines qualitatively and quantitatively. These results provide a proof-of-concept that steerable priors can mitigate domain shift in diffusion-based PET reconstruction and motivate future evaluation on real data.

Diffusion models have been firmly established as principled zero-shot solvers for linear and nonlinear inverse problems, owing to their powerful image prior and iterative sampling algorithm. These approaches often rely on Tweedie's formula, which relates the diffusion variate $\mathbf{x}_t$ to the posterior mean $\mathbb{E} [\mathbf{x}_0 | \mathbf{x}_t]$, in order to guide the diffusion trajectory with an estimate of the final denoised sample $\mathbf{x}_0$. However, this does not consider information from the measurement $\mathbf{y}$, which must then be integrated downstream. In this work, we propose to estimate the conditional posterior mean $\mathbb{E} [\mathbf{x}_0 | \mathbf{x}_t, \mathbf{y}]$, which can be formulated as the solution to a lightweight, single-parameter maximum likelihood estimation problem. The resulting prediction can be integrated into any standard sampler, resulting in a fast and memory-efficient inverse solver. Our optimizer is amenable to a noise-aware likelihood-based stopping criteria that is robust to measurement noise in $\mathbf{y}$. We demonstrate comparable or improved performance against a wide selection of contemporary inverse solvers across multiple datasets and tasks.