Enhancing 3-D cell structures in confocal and STED microscopy: a joint model for interpolation, deblurring and anisotropic smoothing

This paper proposes an advanced image enhancement method that is specifically tailored towards 3-D confocal and STED microscopy imagery. Our approach unifies image denoising, deblurring and interpolation in one joint method to handle the typical weaknesses of these advanced microscopy techniques: out-of-focus blur, Poisson noise and low axial resolution. In detail, we propose the combination of (i) Richardson-Lucy deconvolution, (ii) image restoration and (iii) anisotropic inpainting in one single scheme. To this end, we develop a novel PDE-based model that realizes these three ideas. First we consider a basic variational image restoration functional that is turned into a joint interpolation scheme by extending the regularization domain. Next, we integrate the variational representation of Richardson-Lucy deconvolution into our model, and illustrate its relation to Poisson distributed noise. In the following step, we supplement the components of our model with sub-quadratic penalization strategies that increase the robustness of the overall method. Finally, we consider the associated minimality conditions, where we exchange the occurring scalar-valued diffusivity function by a so-called diffusion tensor. This leads to an anisotropic regularization that is aligned with structures in the evolving image. As a further contribution of this paper, we propose a more efficient and faster semi-implicit iteration scheme that also increases the stability. Our experiments on real data sets demonstrate that this joint model achieves a superior reconstruction quality of the recorded cell.