Geometry of the Phase Retrieval Problem: Graveyard of Algorithms

Recovering the phase of the Fourier transform is a ubiquitous problem in  imaging applications from astronomy to nanoscale X-ray diffraction  imaging. Despite the efforts of a multitude of scientists, from  astronomers to mathematicians, there is, as yet, no satisfactory  theoretical or algorithmic solution to this class of problems. Written  for mathematicians, physicists and engineers working in image analysis  and reconstruction, this book introduces a conceptual, geometric  framework for the analysis of these problems, leading to a deeper  understanding of the essential, algorithmically independent, difficulty  of their solutions. Using this framework, the book studies standard  algorithms and a range of theoretical issues in phase retrieval and  provides several new algorithms and approaches to this problem with the  potential to improve the reconstructed images. The book is lavishly  illustrated with the results of numerous numerical experiments that  motivate the theoretical development and place it in the context of  practical applications.