Stable Backward Diffusion Models that Minimise Convex Energies
This work addresses the instability in backward diffusion for image processing applications, offering a stabilised model with simpler numerics, though it appears incremental as it builds on existing stabilisation strategies.
The authors tackled the ill-posed and unstable inverse problem of backward diffusion in image enhancement and deblurring by deriving a class of space-discrete one-dimensional backward diffusion models as gradient descent of convex energies with range constraints, achieving stability that carries over to a simple explicit time discretisation and confirming it in experiments on contrast enhancement of digital greyscale and colour images.
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.