A FOURIER THEORY OF STATISTICAL IMAGE FORMATION

Abstract

The statistical approach must be used to describe the image when object brightness, optical pupil characteristics, and image detection are all subject to random fluctuations. Use of the statistical characteristic W (the Fourier transform of probability density) is found to clarify the phenomenon and to result in a linear theory. There is a characteristic function W, corresponding to the joint statistics of the log modulus and the phase, for: the object spectrum, optical transfer function, detection transfer function and image spectrum. These four W-functions are the statistical analogy to the four Fourier spectra themselves, if the object radiation is either perfectly coherent or perfectly incoherent. Thus, in analogy to the ordinary Fourier theory of image formation, there is (1) a statistical transfer theorem linking object and image fluctuations, (2) a statistical transfer function for the optics, which may be computed from the optical pupil statistics, (3) sampling theorems, and other analogous results. The W-functions are found to determine all moments of each spectral distribution, and to imply that the moments themselves obey a transfer theorem. Also, the optical characteristic W-function seems to be useful as a quality criterion of optical system stability. In the deterministic limit, the statistical theory goes over into the ordinary Fourier theory of image formation. Random detection noise is a natural parameter of the theory, so that application to practical problems seems eminently possible.