The Phase Retrieval Algorithm as a Dynamical System

Can you reconstruct the face of a man from only his shadow? This is the type of question that needs to be answered every day by researchers involved in a number of scientific areas from x-ray crystallography to astronomy. In all these fields an image or an object has to be reconstructed from very limited data, usually the scatter or diffraction pattern. In certain cases, the scatter pattern produced by an object can be simulated by taking the modules of its Fourier transform, however to reconstruct the object from its Fourier transform, we need to know both its modulus and its phase. The problem of reconstructing an object from only its Fourier modulus, or equivalently, the retrieving of an object’s Fourier phase, not surprisingly, is known as the phase retrieval problem. This problem belongs to a branch of mathematics known as inverse problems. In principle, a discrete object in two or more dimensions with fixed support can almost always be reconstructed from its power spectrum [14], however the ‘error reduction algorithm’, commonly used for t his task, often converges to the wrong solution. It appears to be clear that the theory of phase retrieval is inconsistent from the practice of the phase retrieval algorithm and this dissertation is primarily concerned with addressing this difference. By considering the error reduction algorithm as a dynamical system, we can consider the behaviour of the algorithm as it moves over the entire solution space. This allows us to consider the occurrence of discontinuities in the solution space which partition the space into many separate regions, and hence describe a reason for the multiple solutions that exist. Finally, we discus the use of multi-resolution algorithms, a technique from image processing, in reducing the number of discontinuities in the solution space. This reduces the number of separate regions and so increases our chances of converging to the true solution.

Name dc:type.qualificationname

PhD

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

De Montfort University

Year dc:date.issued

2004

Author dc:creator

• Topiwala, Diven

Licence

• https://dora.dmu.ac.uk/bitstreams/fd052377-5872-4734-a29b-5d01c05f3819/download