  We consider a nonlinear fourth-order diffusion equation that arises in
denoising of image densities. We propose an implicit time-stepping scheme that
employs a primal-dual method for computing the subgradient of the total
variation seminorm. The constraint on the dual variable is relaxed by adding a
\emph{penalty term}, depending on a parameter that determines the weight of the
penalisation. The paper is furnished with some numerical examples showing the
denoising properties of the model considered.
