A method of restoring the discrete-Fourier-transform (DFT) spectrum of a diffraction-limited (DL) image from a narrow-observation segment of the DL image is presented. The DL spectral-restoration process is the dual of the more common DL image-restoration process with the roles of the frequency and space reversed. Applications of a spectrum restoration include increasing the field of view of existing imaging systems and extracting precise frequency components of a large DL image by using only a small segment of the entire image. This method could also be employed for image data compression, which is of interest in digital video applications. Several differences between the implementations of the image- and the spectrum-restoration processes are described. The estimate is constrained to have an upper bound on the number of frequency components contained in the Fourier spectrum. The bound is the number of samples acquired at the Nyquist rate for the length of the image. The magnitude of the DFT spectrum is also bounded. These constraints define a large number of possible solutions. The desired solution is then selected such that the distance, defined in a function-theoretic sense, between the measured and the estimated images is an optimum. A number of such measures are investigated. Numerical experiments show that this approach yields results that are highly immune to measurement noise.
© 1982 Optical Society of America
R. Mammone and G. Eichmann, "Restoration of discrete Fourier spectra using linear programming," J. Opt. Soc. Am. 72, 987-992 (1982)