2016/04/23 by Ben Leshem, Leshem, Ben, Oren Raz +5 · 1 citation
Physics and Astronomy · Biochemistry, Genetics and Molecular Biology · Materials Science · #Advanced X-ray Imaging Techniques #Advanced Electron Microscopy Techniques and Applications #Electron and X-Ray Spectroscopy Techniques
paper · pdf · doi:10.48550/arxiv.1604.06933
In this paper we consider the following real-valued and finite dimensional specific instance of the 1-D classical phase retrieval problem. Let \bf F∈ℝN be an N-dimensional vector, whose discrete Fourier transform has a compact support. The sign problem is to recover \bf F from its magnitude |\bf F|. First, in contrast to the classical 1-D phase problem which in general has multiple solutions, we prove that with sufficient over-sampling, the sign problem admits a unique solution. Next, we show that the sign problem can be viewed as a special case of a more general piecewise constant phase problem. Relying on this result, we derive a computationally efficient and robust to noise sign recovery algorithm. In the noise-free case and with a sufficiently high sampling rate, our algorithm is guaranteed to recover the true sign pattern. Finally, we present two phase retrieval applications of the sign problem: (i) vectorial phase retrieval with three measurement vectors; and (ii) recovery of two well separated 1-D objects.