Novel Deconvolution of Noisy Gaussian Filters with a Modified Hermite Expansion

We have shown (J. Appl. Phys., 1990, 1415-1420) that deconvolving an image which was blurred by a Gaussian filter is equivalent to antidiffusing the image for an appropriate duration of time. However, the antidiffusion algorithm used to show this, based on backward integration of the diffusion equation, is extremely sensitive to noise with numerical errors increasing exponentially with time. Thus, an extremely high signal to noise ratio is required for reconstruction of a blurred image via antidiffusion. In this paper, we introduce a new antidiffusion algorithm which is substantially more robust with respect to noise. This is because each functional component in the series of the reconstructed image is obtained analytically from a corresponding component of the blurred image. We show that the algorithm yields accurate reconstructions of Gaussian-smeared signals and images with extremely low signal to noise ratios.