  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.
