Phase Retrieval
In phase retrieval a signal or image is reconstructed from the modulus (absolute value, magnitude) of its discrete Fourier transform. For example, the source of the modulus data may be the Fraunhofer diffraction pattern formed when an object is illuminated with coherent light.
The projection to the Fourier modulus constraint, say PA, is accomplished by first computing the discrete Fourier transform of the signal or image, rescaling the moduli to agree with the data, and then inverse transforming the result. This is a projection, in the sense that the Euclidean distance to the constraint is minimized, because (i) the discrete Fourier transform, as a unitary transformation, preserves distance, and (ii) rescaling the modulus (without modifying the phase) is the smallest change that realizes the modulus constraint.
To recover the unknown phases of the Fourier transform the difference map relies on the projection to another constraint, PB. This may take several forms, as the object being reconstructed may be known to be positive, have a bounded support, etc. In the reconstruction of the surface image, for example, the effect of the projection PB was to nullify all values outside a rectangular support, and also to nullify all negative values within the support.
Read more about this topic: Difference-map Algorithm
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