Abstract

A multiple convolution (e.g., an image formed by convolving several individual components) is automatically deconvolvable, provided that its dimension (i.e., the number of variables of which it is a function) is greater than unity. This follows because the Fourier transform of a K-dimensional function (having compact support) is zero on continuous surfaces (here called zero sheets) of dimension (2K − 2) in a space that effectively has 2K dimensions. A number of important practical applications are transfigured by the concept of the zero sheet. Image restoration can be effected without prior knowledge of the point-spread function, i.e., blind deconvolution is possible even when only a single blurred image is given. It is in principle possible to remove some of the additive noise when the form of the point-spread function is known. Fourier phase can be retrieved directly, and, unlike for readily implementable iterative techniques, complex images can be handled as straightforwardly as real images.

© 1987 Optical Society of America

Some implications of zero sheets for blind deconvolution and phase retrieval

R. H. T. Bates, B. K. Quek, and C. R. Parker
J. Opt. Soc. Am. A 7(3) 468-479 (1990)

Deconvolution and phase retrieval with use of zero sheets

P. J. Bones, C. R. Parker, B. L. Satherley, and R. W. Watson
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Zero tracks for blind deconvolution of blurred ensembles

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