Approach for Calculating an Analytical PSF

Since image restoration algorithms are highly sensitive to the accuracy of the calculated PSF, a significant portion of our research effort is devoted to accurately modeling the PSF which is a highly non-linear problem. Our approach is based on analytical modeling of the wavefront phase using Zernike polynomials. The wave front phase is reconstructed using a phase retrieval algorithm. First, we must make reasonable guesses of significant Zernike polynomial coefficients, and use these as input to a phase retrieval algorithm (i.e., Gerchberg and Saxton 1972, Fienup 1982) to produce an estimate of the FOC phase map, which is an array of complex numbers. In the Fourier domain, we scale the phase map by an observed PSF to more accurately account for the HST/FOC combination. After scaling, the inverse transform provides the required information in the pupil domain. We mask and scale the inverse transform data by the well-known geometry of the components contributing to the pupil (e.g., spider supports, mirror handling pads, central obscuration, as well as primary and secondary mirror surface errors). We then repeat this process several times to converge with minimal differences between the result and the observed PSF. The final result is an improved estimate of the phase map. The resulting map has to be smoothed and unwrapped to obtain a continuous phase map. We then use this result as input to a total least squares estimate algorithm (QR factorization) to provide improved Zernike coefficients which ultimately permit a final calculated PSF whose point-to-point differences are minimized with respect to the observed PSF. Of course, at this point, the observed and calculated PSFs are compared by a variety of other means including inspection of the core and wings to insure that morphologically they are as close as possible - if not, some important Zernike term may have been left out. Moreover, one has to contend with achieving a local rather than a global minimum in this process, a problem that has yet to be solved for the phase retrieval stage. We present the mathematical methodology for modeling the PSF in the next section.