| In many branches of Physics and Engineering one comes across the problem of reconstructing a function f using the Fourier transform F, when only partial information about the transform and the function is available. One of the most common examples is to reconstruct f when only the magnitude |f| of the function and the magnitude |F| of the Fourier transform are known. This problem occurs in electron microscopy and wavefront sensing. Another problem which occurs in astronomy and crystallography is to reconstruct f when only |F| and some constraints on f, e.g., f ≥ 0, are available. In this paper we study the latter problem in a context where f is univariate and discrete. We make use of Fienup's analysis and adapt the Gerchberg-Saxton algorithm to our problem. We devise ways to eliminate indeterminacy and we suggest ways to improve the rate of convergence of this algorithm. |
