Robust, noniterative, and computationally efficient modification of van Cittert deconvolution optical figuring

A modification of van Cittert deconvolution (VCD) is introduced and shown to yield a robust, noniterative (or closed-form), and numerically efficient method of deconvolution. This modification removes the restrictions limiting the applicability of conventional VCD only to shapes and relative positions of the convolved functions for which it converges, while also avoiding the ill effects of zeros in these functions. The resulting method is computationally efficient because it is noniterative and uses the fast Fourier transform. In contrast to the convergences obtained with VCD, those obtained with this modified method are ensured by their expansion in terms of an introduced auxiliary function rather than the convolved functions. This permits both the general removal of the above limitations and arbitrarily accurate deconvolution of infinitely sampled input data even in the presence of input data noise. For discretely sampled input data the accuracy of this modification is shown to be limited only by the implicit bandwidth of the input data density. To exemplify its numerical and analytical advantages, I apply the method to computer control of optical surface figuring. I also demonstrate an intrinsic means of optimal frequency filtering of raw input data made available by this modification. The advantages of this procedure are also applicable to image restoration.