In this work we develop an algorithm for signal reconstruction from the magnitude of its Fourier transform in a situation where some (non-zero) parts of the sought signal are known. Although our method does not assume that the known part comprises th
e boundary of the sought signal, this is often the case in microscopy: a specimen is placed inside a known mask, which can be thought of as a known light source that surrounds the unknown signal. Therefore, in the past, several algorithms were suggested that solve the phase retrieval problem assuming known boundary values. Unlike our method, these methods do rely on the fact that the known part is on the boundary. Besides the reconstruction method we give an explanation of the phenomena observed in previous work: the reconstruction is much faster when there is more energy concentrated in the known part. Quite surprisingly, this can be explained using our previous results on phase retrieval with approximately known Fourier phase.
We present a new method for real- and complex-valued image reconstruction from two intensity measurements made in the Fourier plane: the Fourier magnitude of the unknown image, and the intensity of the interference pattern arising from superimpositio
n of the original signal with a reference beam. This approach can provide significant advantages in digital holography since it poses less stringent requirements on the reference beam. In particular, it does not require spatial separation between the sought signal and the reference beam. Moreover, the reference beam need not be known precisely, and in fact, may contain severe errors, without leading to a deterioration in the reconstruction quality. Numerical simulations are presented to demonstrate the speed and quality of reconstruction.
