Space-variant image reconstruction via Cauchy regularisation: Application to Optical Coherence Tomography

Abstract

We propose a smooth, non-convex and content-adaptive regularisation model for single-image super-resolution of murine Optical Coherence Tomography (OCT) data. We follow a sparse-representation approach where sparsity is modelled with respect to a suitable dictionary generated from high-resolution OCT data. To do so, we employ a pre-learned dictionary tailored to model α-stable statistics in the non-Gaussian case, i.e. α<2. The image reconstruction problem renders here particularly challenging due to the high level of noise degradation and to the heterogeneity of the data at hand. As a regulariser, we employ a separable Cauchy-type penalty. To favour adaptivity to image contents, we propose a space-variant modelling by which the local degree of non-convexity given by the local Cauchy shape parameter is estimated via maximum likelihood. For the solution of the reconstruction problem, we consider an extension of the cautious Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm where the descent direction is suitably updated depending on the local convexity of the functional. Our numerical results show that the combination of a space-variant modelling with a tailored optimisation strategy improves reconstruction results and allows for an effective segmentation with standard approaches.