The problem of reconstructing a signal or image from a finite number of spectral data is the problem of finding an optimal approximation to one function by another. We describe here methods for signal recovery and phase retrieval based on the theory of best approximation in weighted Hilbert spaces 1. These methods have been under development for some time for use in a variety of 1-D and 2-D spectral estimation problems. A new interpretation of these methods will be discussed based on the close analogy between the reconstruction of a non-negative function from finitely many values of its Fourier transform, and the design of approximate Wiener filters2. We also describe the relationship between these methods and other spectral estimation procedures. The Wiener filter produces as an output an estimate of a signal when presented with an input of signal plus noise. Its design, under restrictions of limited data, gives a clear picture of the role of the prior estimate for both linear and non-linear signal recovery procedures, and the significance of this is discussed.
C L Byrne,
M. A. Fiddy,
"Approximation Of Wiener Filters And Solving Inverse Problems", Proc. SPIE 0976, Statistical Optics, (2 December 1988); doi: 10.1117/12.948532; https://doi.org/10.1117/12.948532