# Geodesic Diffusion Models for Efficient Medical Image Enhancement

arXiv: https://arxiv.org/abs/2503.00745 · HTML: https://mlsanity.com/paper/2503.00745/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Medical imaging and segmentation, one of 14 directions in Diffusion models.
- Builds on 27 of its 32 references that are in the map.
- First posted on arXiv 2 Mar 2025.

## Placed in

- Diffusion models / [Medical imaging and segmentation](https://mlsanity.com/diffusion/medical-imaging-and-segmentation/)

## Builds on (most cited first, 27 of 27)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [Image Super-Resolution Using Deep Convolutional Networks](https://mlsanity.com/paper/1501.00092/) (2014)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [DPM-Solver++: Fast Solver for Guided Sampling of Diffusion Probabilistic Models](https://mlsanity.com/paper/2211.01095/) (2022)
- [UniPC: A Unified Predictor-Corrector Framework for Fast Sampling of Diffusion Models](https://mlsanity.com/paper/2302.04867/) (2023)
- [Simple diffusion: End-to-end diffusion for high resolution images](https://mlsanity.com/paper/2301.11093/) (2023)
- [Knowledge Distillation in Iterative Generative Models for Improved Sampling Speed](https://mlsanity.com/paper/2101.02388/) (2021)
- [Brain Imaging Generation with Latent Diffusion Models](https://mlsanity.com/paper/2209.07162/) (2022)
- [Efficient Diffusion Training via Min-SNR Weighting Strategy](https://mlsanity.com/paper/2303.09556/) (2023)
- [Low-Dose CT with a Residual Encoder-Decoder Convolutional Neural Network (RED-CNN)](https://mlsanity.com/paper/1702.00288/) (2017)
- [Scalable Adaptive Computation for Iterative Generation](https://mlsanity.com/paper/2212.11972/) (2022)
- [On the Importance of Noise Scheduling for Diffusion Models](https://mlsanity.com/paper/2301.10972/) (2023)
- [Unsupervised Medical Image Translation with Adversarial Diffusion Models](https://mlsanity.com/paper/2207.08208/) (2022)
- [Diffusion Normalizing Flow](https://mlsanity.com/paper/2110.07579/) (2021)
- [DU-GAN: Generative Adversarial Networks with Dual-Domain U-Net Based Discriminators for Low-Dose CT Denoising](https://mlsanity.com/paper/2108.10772/) (2021)
- [Improved Noise Schedule for Diffusion Training](https://mlsanity.com/paper/2407.03297/) (2024)
- [SEEDS: Exponential SDE Solvers for Fast High-Quality Sampling from Diffusion Models](https://mlsanity.com/paper/2305.14267/) (2023)
- [Fast-DDPM: Fast Denoising Diffusion Probabilistic Models for Medical Image-to-Image Generation](https://mlsanity.com/paper/2405.14802/) (2024)
- [Conditional Diffusion Models for Weakly Supervised Medical Image Segmentation](https://mlsanity.com/paper/2306.03878/) (2023)

## Abstract (arXiv)

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.