Reconstruction of complex signals from intensities of Fourier-transform pairs

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Abstract
The problem of retrieving a complex function when both its square modulus and the square modulus of its Fourier transform are known is considered. When these intensities are directly assumed to be data, it amounts to performing the inversion of a quadratic operator. The solution is found to be the global minimum of an appropriate functional. Moreover, inasmuch as the unknown function is modeled within a finite-dimensional set, the data are also consistently represented within finite-dimensional subspaces, and a coherent discretization of the problem results. Because the assumed formulation involves nonquadratic functionals, the crucial problem of the existence of local minima in the course of the minimization procedure is discussed. The main factors affecting these minima can be identified, such as the amount of available independent data. Furthermore, quadraticity makes it possible to define an efficient conjugate-gradient-based minimization procedure. The numerical results confirm the distinguishing feature of the proposed approach—its ability to obtain the solution starting from a completely random guess.

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