Image Reconstruction from Amplitude-Only and Phase-Only Data in the Fractional Fourier Domain1

Image reconstruction from amplitude-only and phase-only data in the fractional Fourier domain applying the inverse fractional Fourier transform is analyzed on the examples of perfect edges of different contrasts and real-world image. © 2003 MAIK “Nauka/Interperiodica”. 1 The question of the importance of phase and amplitude in the Fourier domain has been studied in many publications [1–4]. There is a widespread belief that the Fourier phase is more important than the Fourier amplitude, based on the fact that the main characteristics of natural images can be recovered from phase-only data by applying the inverse Fourier transform (FT). Indeed, the Fourier amplitude of natural images is mostly concentrated in the low-frequency region. By eliminating the amplitude information, we stress the high frequencies of the Fourier spectrum. This produces an edgeenhancement effect. Nevertheless, it was shown in [4] that the dominance of Fourier phase or amplitude strongly depends on the symmetry and reality of the input signals. Nowadays, the fractional FT, recently introduced in optics [5–6], plays an important role in optical signal and image processing. Convolution and correlation operations in the fractional Fourier domain are applied for pattern recognition, noise reduction, watermarking, encryption, and so on [5, 7]. Often, the application of phase-only filters (POF) in the fractional Fourier domain is proposed for these tasks. Nevertheless, the effect of POFs in fractional Fourier domains has not been discussed in detail at present. In this paper, we consider the possibility of image reconstruction from amplitude-only or phase-only data in fractional Fourier domains on the examples of perfect edges of different contrasts and natural real image. The fractional FT [5] of f(x) is a generalization of the ordinary FT