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.