PHASE RETRIEVAL FOR CHARACTERISTIC FUNCTIONS OF CONVEX BODIES AND RECONSTRUCTION FROM COVARIOGRAMS

摘要

The Phase Retrieval Problem of Fourier analysis involves^ determining a function f on R~n from the modulus |f| of its Fourier transform f . This problem arises naturally and frequently in various areas of science, such as X-ray crystallogra-phy, electron microscopy, optics, astronomy, and remote sensing, in which only the magnitude of the Fourier transform can be measured and the phase is lost. (Sometimes, as when reconstructing an object from its far-field diffraction pattern, it is the squared modulus If 12 that is directly measured.) In 1984, Rosenblatt [42] wrote that the Phase Retrieval Problem "arises in all experimental uses of diffracted electromagnetic radiation for determining the intrinsic detailed structure of a diffracting object." Today, the word "all" is perhaps too strong in view of recent advances in coherent diffraction imaging. In any case, the literature is vast; see the surveys [32], [34], [36], and [42], as well as the articles [9] and [18] and the references given there.