Upper Error Bounds for Score-Based Inverse Problem Solving in Imaging

Diffusion models have gained tremendous popularity for generating diverse and high quality images. We adapt diffusion models for solving inverse problems in imaging. Moreover, we quantify the uncertainties in the reconstructed images by deriving pixel-wise upper error bounds dependent on the determined variance without relying on the ground-truth. Especially in high-stake applications such as healthcare, well-calibrated uncertainties are vital for reliable decision making. For example in magnetic resonance imaging, undersampling the k-space plays a crucial role in clinical applications and it is highly important to be aware of the uncertainties as the diagnosis and further treatments depend on the reconstructed images. In this work, we focus on the score-based generative models through stochastic differential equations and show that an unconditional diffusion model trained on a specific dataset (BSDS and fastMRI) can be utilized for solving various inverse problems, e.g. denoising, inpainting and zero-filling and the uncertainty quantification yields a strong correlation between the squared error and the variance.