Application of the split-gradient method to 3D image deconvolution in fluorescence microscopy

The methods of image deconvolution are important for improving the quality of the detected images in the different modalities of fluorescence microscopy such as wide‐field, confocal, two‐photon excitation and 4Pi. Because deconvolution is an ill‐posed problem, it is, in general, reformulated in a statistical framework such as maximum likelihood or Bayes and reduced to the minimization of a suitable functional, more precisely, to a constrained minimization, because non‐negativity of the solution is an important requirement. Next, iterative methods are designed for approximating such a solution. In this paper, we consider the Bayesian approach based on the assumption that the noise is dominated by photon counting, so the likelihood is of the Poisson‐type, and that the prior is edge‐preserving, as derived from a simple Markov random field model. By considering the negative logarithm of the a posteriori probability distribution, the computation of the maximum a posteriori (MAP) estimate is reduced to the constrained minimization of a functional that is the sum of the Csiszár I‐divergence and a regularization term. For the solution of this problem, we propose an iterative algorithm derived from a general approach known as split‐gradient method (SGM) and based on a suitable decomposition of the gradient of the functional into a negative and positive part. The result is a simple modification of the standard Richardson–Lucy algorithm, very easily implementable and assuring automatically the non‐negativity of the iterates. Next, we apply this method to the particular case of confocal microscopy for investigating the effect of several edge‐preserving priors proposed in the literature using both synthetic and real confocal images. The quality of the restoration is estimated both by computation of the Kullback–Leibler divergence of the restored image from the detected one and by visual inspection. It is observed that the noise artefacts are considerably reduced and desired characteristics (edges and minute features as islets) are retained in the restored images. The algorithm is stable, robust and tolerant at various noise (Poisson) levels. Finally, by remarking that the proposed method is essentially a scaled gradient method, a possible modification of the algorithm is briefly discussed in view of obtaining fast convergence and reduction in computational time.