Properties and Implications of Phase Problems for Multidimensional Images

Phase retrieval is concerned with recovery of an image from measurements of the amplitude of its Fourier transform [1]. It is of considerable practical importance in areas such as microscopy, ultrasonic imaging, astronomy, and crystallography. There are a number of applications, such as in x-ray crystallography, electron microscopy, ultrasonic imaging, and geophysical imaging, where the image to be reconstructed is three-dimensional. Uniqueness properties of phase problems in higher dimensions are examined, and implications in a number of applications of x-ray crystallography are described.