## The fourier phase problem

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##### Author

##### Date

1984##### Permanent Link

http://hdl.handle.net/10092/7708##### Thesis Discipline

Electrical Engineering##### Degree Grantor

University of Canterbury##### Degree Level

Doctoral##### Degree Name

Doctor of PhilosophyTheoretical and practical aspects of solving the Fourier phase problem are presented. Reconstructing an image, or the phase of its Fourier transform (visibility), when only the magnitude, or intensity, of its visibility is given, presents what is called a phase problem. If it is to be a Fourier phase problem, then samples of the visibility magnitude must be available at at least twice the Nyquist rate in each direction in Fourier space. The phase reconstruction algorithms which are discussed in this thesis can then be employed. The fields of technical science in which Fourier phase problems arise are listed and commented on. Particular attention is paid to high-resolution optical astronomy, and to a lesser extent to radio astronomy. Recent advances in astronomical imaging techniques are discussed and illustrated with results obtained from optical laboratory simulations. The phase problem associated with X-ray crystallography is also discussed and is shown not to be a Fourier phase problem. The question of uniqueness of the solution to the phase problem is considered. Practical constraints that can be placed on real-world images are shown to reduce the ambiguity of the solution, especially in more than one dimension. The constraint of non-negative realness, or positivity, of the image, is particularly restrictive. Existing theory on the uniqueness question is reviewed. One-dimensional phase problems usually have ambiguous solutions, whereas in two (or more) dimensions it appears that solutions are almost always unique. Some phase reconstruction algorithms that have proved effective in practice are explained. Algorithms for solving one-dimensional phase problems are nearly all direct, but additional information is usually necessary to resolve the inherent ambiguity of the solution. Two- (or more) dimensional algorithms, on the other hand, are nearly all iterative. The practical implementation of a discrete two-dimensional phase reconstruction algorithm is described. The algorithm, called crude phase estimation (CPE), is shown to produce useful estimates of the visibility phase. A method of optimising the fideility of the estimated phase is described which requires no a priori information about the image. Further improvement of the estimated phase is obtained by iterative processing of the kind pioneered by Fienup. Certain kinds of image are identified as being difficult to reconstruct unless CPE and the iterative algorithms are supplemented by ancillary procedures. Two such procedures are described. The first, called pre-filtering, is designed to reduce noise. Although CPE and the iterative algorithms are not unstable with noisy data, the quality of the reconstruction is shown to be enhanced by pre-filtering. The second ancillary procedure has two stages called defogging and refogging. They are shown to assist in reconstructing images which are comprised of faint detail superimposed on large amplitude backgrounds. Defogging modifies the visibility magnitude before the actual reconstruction algorithms are invoked. Fine detail in the image is usually recovered more effectively in this way. In a range of situations of potential practical importance, the detail is often impossible to recover without the aid of the ancillary procedures. CPE, the iterative algorithms and the ancillary procedures are combined into a composite phase reconstruction scheme. Examples are presented using both computer-generated data and data measured in the optical laboratory. The composite scheme is shown to be particularly useful for reconstructing images of the kind occuring in high-resolution astronomy.