Geodesic Diffusion Models for Efficient Medical Image Enhancement

TL;DR

Geodesic Diffusion Models (GDM) address the computational bottlenecks of diffusion models by constraining diffusion trajectories to geodesics in probability space under the Fisher–Rao metric. This yields energy-efficient forward and reverse processes, enabling conditional medical image enhancement with significantly fewer denoising steps via Geodesic Truncated Sampling (GTS). Empirical results on CT denoising and MRI super-resolution show state-of-the-art performance while reducing training time by 20–30× over DDPM and 4–6× over Fast-DDPM, and accelerating sampling by 160–170× relative to DDPM. The approach promises practical impact for real-time clinical applications and sets the stage for broader 3D and multi-modal extensions.

Abstract

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.

Figures (5)

• Figure 1: Overview of Geodesic Diffusion Models (GDM). By constraining the diffusion process to follow geodesic paths with minimal energy in probability space, GDM improves both efficiency and fidelity. The radar plot shows performance on the CT denoising task, where GDM achieves higher accuracy and substantially faster training and sampling compared to prior diffusion-based methods.
• Figure 2: Training and sampling pipeline of GDM. During training, the model learns an efficient continuous geodesic diffusion process. During sampling, noise is first added into condition image to simulate the intermediate image, and then only a few discrete steps are required to generate a high-quality image.
• Figure 3: A family of geodesic noise schedulers for different $\alpha_{1}$ and $\sigma_{1}$
• Figure 4: Qualitative results for CT image denoising. Compared with CNN- (RED-CNN) and GAN-based (DU-GAN) methods, which blur or lose fine anatomical structures, diffusion-based models better preserve lung fissures. Our proposed GDM produces the sharpest and most continuous fissure reconstruction, closely matching the normal-dose reference.
• Figure 5: Qualitative results of MRI image super-resolution. While CNN- (miSRCNN) and GAN-based (miSRGAN) methods produce blurry outputs and other diffusion models lose fine details, our proposed GDM reconstructs sharper anatomical structures and closely matches the ground-truth center slice.