Inversion of Fourier Transforms by Means of Scale-Frequency Series

We report on inversion of the Fourier transform when the frequency variable can be scaled in a variety of different ways that improve the resolution of certain parts of the frequency domain. The corresponding inverse Fourier transform is shown to exist in the form of two dual scale-frequency series. Upon discretization of the continuous scale factor, this Fourier transform series inverse becomes a certain nonharmonic double series, a discretized scale-frequency (DSF) series. The DSF series is also demonstrated, theoretically and practically, to be rate-optimizable with respect to its two free parameters, when it satisfies, as an entropy maximizer, a pertaining recursive nonlinear programming problem incorporating the entropy-based uncertainty principle. 1. Introduction We revisit the classical problem of recovering a Fourier transformable signal in , , from its Fourier transform; see, for example, [1, 2], image . The space is the set of all real functions for which the integral exists. Obviously, the function to be integrated in (1) is the product and when the variable , the function will oscillate extremely rapidly. Then in order to follow the variations of the product meaningfully in a -quadrature formula, see, for example, [3], for (1), there is a need for a large number of -points, even for slowly varying . Consequently, the faster the decreases as , the more the tractable computationally the integral (1) will be, and the more accurate are its numerical evaluations. Bandlimiting of signals has been a practical way for easing the previous problem. A fact that has so far been motivating the wide engineering interest in bandlimited signals, for which where is the band width of with an image having the compact support of . Inversion of bandlimited signals is currently performed by projections onto convex sets (POCS) algorithms [4], which encompass the Gerchberg-Papoulis algorithm [5]. In this work we report on a novel robust semianalytical method for integration of (1) to recover a signal that is not necessarily bandlimited. In this method, the frequency variable can be scaled in a variety of different ways that improve the resolution of certain parts of the frequency domain. The corresponding inverse Fourier transform is shown to exist in the form of two dual scale-frequency series. Upon discretization of the continuous scale factor, this Fourier transform series inverse becomes a certain nonharmonic [6] double series, a discretized scale-frequency (DSF) series. This DSF series is demonstrated, for the first time in this work, to be the proper