Summary

We have designed fast algorithms for wavelet-regularized deconvolution of 3-D fluorescence micrographs. The key idea is to use a multilevel optimization strategy together with subband-adapted steps to speed up the convergence. We are also investigating ways to automatically select the hyper parameters of the algorithm (regularization factor and number of iterations).

Introduction

Modern experimental biology makes extensive use of fluorescent probes for selectively labeling structures of interest. These are then visualized in 3-D using optical fluorescence microscopy. The optical sectioning, which is the key to accessing the third physical dimension (depth), is either achieved optically using confocal microscopy, or numerically by applying image enhancement techniques. When applied to widefield systems, the latter approach is commonly dubbed "deconvolution microscopy." One of its main difficulties lies in the size of the sets of data that are routinely produced in 3-D fluorescence microscopy. This puts a strong limitation on the computational complexity of potential deconvolution procedures, and also explains why the algorithms used in current commercial systems are still rudimentary in comparison to the current state of research on inverse problems.

Main Contribution

Wavelet-domain ℓ ₁ -regularization is a promising approach to solving inverse problems. In their 2004 landmark paper, Daubechies et al. proved that one could solve such linear inverse problems by means of a "thresholded Landweber" (TL) algorithm. While this iterative procedure is simple to implement, it is known to converge slowly.

We have applied the technique to 3-D deconvolution and have accelerated it by optimizing the step sizes in each subband separately. We have also proposed a multilevel version of the algorithm that is inspired from the multigrid techniques used for solving PDEs, but with one important difference: instead of cycling through coarser versions of the problem (REDUCE part of multigrid), the multilevel algorithm cycles through the successive wavelet subspaces. The method works with arbitrary wavelet representations; it typically yields a 10-fold speed increase over the standard TL algorithm, while providing the same restoration quality. We have demonstrated its applicability to real-world 3-D deconvolution microscopy.