Analytic properties of the Hartley transform and their implications

The Hartley transform is an integral transform closely related to the Fourier transform. It has some advantages over the Fourier transform in the analysis of real signals as it avoids the use of complex arithmetic. However, the Hartley transform has other applications in signal and image reconstruction related to traditional phase retrieval problems. These can he understood by examining the analytic properties of the Hartley transform in the complex plane. In this paper, the analytic continuation of the Hartley transform into the complex plane is derived and its properties discussed. It is shown that for signals or images of finite extent, the Hartley transform is analytic in the entire finite complex plane, and this is used to derive properties of its complex zeros. Hilbert transform-type relationships for the Hartley transform, related to causal and analytic-signals, are also derived. The analytic properties derived are used to study the problem of image reconstruction from the Hartley transform intensity. Uniqueness and reconstruction algorithms for one- and two-dimensional problems are discussed, and examples are presented. Generation of image moments from the Hartley transform intensity is also described.>