<title>Blind deconvolution of 2-D and 3-D fluorescent micrographs</title>

Abstract

This paper presents recent results of our reconstructions of 3-D data from Drosophila chromosomes as well as our simulations with a refined version of the algorithm used in the former. It is well known that the calibration of the point spread function (PSF) of a fluorescence microscope is a tedious process and involves esoteric techniques in most cases. This problem is further compounded in the case of confocal microscopy where the measured intensities are usually low. A number of techniques have been developed to solve this problem, all of which are methods in blind deconvolution. These are so called because the measured PSF is not required in the deconvolution of degraded images from any optical system. Our own efforts in this area involved the maximum likelihood (ML) method, the numerical solution to which is obtained by the expectation maximization (EM) algorithm. Based on the reasonable early results obtained during our simulations with 2-D phantoms, we carried out experiments with real 3-D data. We found that the blind deconvolution method using the ML approach gave reasonable reconstructions. Next we tried to perform the reconstructions using some 2-D data, but we found that the results were not encouraging. We surmised that the poor reconstructions were primarily due to the large values of dark current in the input data. This, coupled with the fact that we are likely to have similar data with considerable dark current from a confocal microscope prompted us to look into ways of constraining the solution of the PSF. We observed that in the 2-D case, the reconstructed PSF has a tendency to retain values larger than those of the theoretical PSF in regions away from the center (outside of those we considered to be its region of support). This observation motivated us to apply an upper bound constraint on the PSF in these regions. Furthermore, we constrain the solution of the PSF to be a bandlimited function, as in the case in the true situation. We have derived two separate approaches for implementing the constraint. One approach involves the mathematical rigors of Lagrange multipliers. This approach is discussed in another paper. The second approach involves an adaptation of the Gershberg Saxton algorithm, which ensures bandlimitedness and non-negativity of the PSF. Although the latter approach is mathematically less rigorous than the former, we currently favor it because it has a simpler implementation on a computer and has smaller memory requirements. The next section describes briefly the theory and derivation of these constraint equations using Lagrange multipliers.