Approximate Fourier phase information in the phase retrieval problem: what it gives and how to use it
Spotlight summary: Very recently, I arranged to meet two people I had never seen before at a local cafe. As an experiment, I decided to not look up their photos on the Internet prior to the meeting. With about 50 or so people at the cafe, I had a problem—that of recognizing 2 people (one of each gender) in a crowd of 50, with the said 2 people having faces that I had never seen before. Interestingly, I had no difficulty recognizing them as they arrived. What helped me to solve an otherwise intractable problem (that is, recognize 2 particular unfamiliar faces from among 50 unfamiliar faces) was the a priori knowledge that we had arranged to meet at a particular time and that none of us had previously met one another.
The point behind this story is that a priori knowledge forms a key ingredient in efficiently solving inverse problems, namely problems in which one seeks to go “backwards” from effects to causes. A classic optical inverse problem is the question of determining a given two-dimensional complex scalar optical field given only the intensity of its corresponding far-field diffraction pattern. This is known as the “phase problem,” and it appears in a rich variety of fields of optics from acoustics and crystallography through to coherent x-ray optics and quantum-mechanical potential scattering. Unfortunately, in general the phase problem is intractable.
But, in both the cafe meeting and the optical phase problem, a priori knowledge comes to the rescue. Over the decades many optics workers have recognized the key importance of such a priori knowledge as the finite extent of the desired two-dimensional field (“finite support”) in rendering tractable the optical inverse problem described above.
One form of a priori knowledge is a rough estimate for the complex field to be reconstructed, or, as a slight restriction of this requirement, a rough estimate of the phase of the field that is to be reconstructed. In this very clever article by Osherovich, Zibulevsky, and Yavneh, the authors have shown that the far-field phase problem can be solved considerably more efficiently when such a rough phase estimate is available. Both mathematical and computational examples are given to demonstrate the authors’ key findings. This important paper adds a powerful new tool to the means for gaining and accelerating the convergence of the inverse problem of phase retrieval from far-field diffraction patterns, and I warmly recommend it to your attention.
-- David M. Paganin
Technical Division: Information Acquisition, Processing, and Display
ToC Category: Image Processing
|OCIS Codes:||(000.3860) General : Mathematical methods in physics|
|(000.4430) General : Numerical approximation and analysis|
|(100.5070) Image processing : Phase retrieval|
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