An Investigation of Different Regularisation Terms Used in Variational Imaging Problems
Accelerated First Order Methods for Variational Imaging
Variational imaging problems are used in a wide range of imaging applications such as magnetic resonance imaging (mri)reconstruction and optical flow.We begin by studying smooth problems and partially non-smooth problems in the form of tikhonov denoising and total variation (tv)denoising, respectively.For denoising, we study an accelerated gradient method with adaptiverestart, which shows a very rapid convergence rate.However, it is not straightforward to apply this fast algorithm to tv denoising, due to the non-smoothness of its built-in regularisation.To tackle this issue, we propose to utilise duality to convert such a non-smooth problem into a smooth one so that the accelerated gradient method with restart applies naturally.However, we notice that both tikhonov and tv regularisations have drawbacks, in the form of blurred image edges and staircase artefacts, respectively.To overcome these drawbacks, we propose a novel adaption to total generalisedvariation (tgv) regularisation called total smooth variation (tsv), which retains edges and meanwhile does not produce results which contain staircaseartefacts.Compared to existing state-of-the-art regularisations (e.g.Tv), tsv is shown to obtain more effective results on denoising problems as well as advanced imaging applications such as magnetic resonance imaging (mri)reconstruction and optical flow.
