Recovering Missing Data in Coherent Diffraction Imaging

In coherent diffraction imaging (CDI) experiments, the intensity of the scattered wave impinging on an object is measured on an array of detectors. These measurements can be interpreted as samples of the square of the modulus of the Fourier transform of the unknown scattering density. A beamstop obstructs the forward scattered wave and, hence, the modulus Fourier data from a neighborhood of $k=0$ cannot be measured. In this note, we describe a linear method for recovering this unmeasured modulus Fourier data from the measured values and an estimate of the support of the image's autocorrelation function without consideration of phase retrieval. We analyze the effects of noise, and the conditioning of this problem, which grows exponentially with the modulus of the maximum spatial frequency not measured.