# Stable Backward Diffusion Models that Minimise Convex Energies

## TL;DR

This paper introduces a stable backward diffusion model based on convex energies with range constraints, suitable for image enhancement tasks, and demonstrates its effectiveness through theoretical analysis and experiments.

## Contribution

It develops a new class of convex energy-based backward diffusion models with stability guarantees using simple explicit discretisation.

## Key findings

- The model is stable under explicit time discretisation.
- Convex energies enable effective backward diffusion stabilization.
- Experimental results show improved image contrast enhancement.

## Abstract

The inverse problem of backward diffusion is known to be ill-posed and highly unstable. Backward diffusion processes appear naturally in image enhancement and deblurring applications. It is therefore greatly desirable to establish a backward diffusion model which implements a smart stabilisation approach that can be used in combination with an easy to handle numerical scheme. So far, existing stabilisation strategies in literature require sophisticated numerics to solve the underlying initial value problem. We derive a class of space-discrete one-dimensional backward diffusion as gradient descent of energies where we gain stability by imposing range constraints. Interestingly, these energies are even convex. Furthermore, we establish a comprehensive theory for the time-continuous evolution and we show that stability carries over to a simple explicit time discretisation of our model. Finally, we confirm the stability and usefulness of our technique in experiments in which we enhance the contrast of digital greyscale and colour images.