Geodesic Diffusion Models for Efficient Medical Image Enhancement

Diffusion models generate data by learning to reverse a forward process, where samples are progressively perturbed with Gaussian noise according to a predefined noise schedule. From a geometric perspective, each noise schedule corresponds to a unique trajectory in probability space from the data distribution to a Gaussian prior. However, prior diffusion models rely on empirically chosen schedules that may not be optimal. This inefficiency necessitates many intermediate time steps, resulting in high computational costs during both training and sampling. To address this, we derive a family of geodesic noise schedules corresponding to the shortest paths in probability space under the Fisher-Rao metric. Based on these schedules, we propose Geodesic Diffusion Models (GDMs), which significantly improve training and sampling efficiency by minimizing the energy required to transform between probability distributions. This efficiency further enables sampling to start from an intermediate distribution in conditional image generation, achieving state-of-the-art results with as few as 6 steps. We evaluated GDM on two medical image enhancement tasks: CT image denoising and MRI image super-resolution. Experimental results show that GDM achieved state-of-the-art performance while reducing training time by 20- to 30-fold compared to Denoising Diffusion Probabilistic Models (DDPMs) and 4- to 6-fold compared to Fast-DDPM, and accelerating sampling by 160- to 170-fold and 1.6-fold, respectively. These gains support the use of GDM for efficient model development and real-time clinical applications. Our code is publicly available at: https://github.com/mirthAI/GDM-VE.