Reconstruction Of A Univariate Discrete Function From The Magnitude Of Its Fourier Transform

Author

Alia Khurram, Southern Illinois University CarbondaleFollow

Date of Award

1-1-2009

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Kammler,David

Abstract

In many branches of Physics and Engineering one comes across the problem of reconstructing a function $f$ using the Fourier transform $F$, when only partial information about the transform and the function is available. One of the most common examples is to reconstruct $f$ when only the magnitude $|f|$ of the function and the magnitude $|F|$ of the Fourier transform are known. This problem occurs in electron microscopy and wavefront sensing. Another problem which occurs in astronomy and crystallography is to reconstruct $f$ when only $|F|$ and some constraints on $f$, e.g., $f \geq 0$, are available. In this paper we study the latter problem in a context where $f$ is univariate and discrete. We make use of Fienup's analysis and adapt the Gerchberg-Saxton algorithm to our problem. We devise ways to eliminate indeterminacy and we suggest ways to improve the rate of convergence of this algorithm.