Discrete combined fractional Fourier transform and its application to image enhancement

Many generalizations of the Fourier transform (FT) have been introduced in the literature, including fractional Fourier transform (FrFT) and linear canonical transform (LCT). This paper aims to extend the discrete combined Fourier transform (DCFT) to the FrFT domains and investigate its applications in image enhancement. The DCFT of a signal is a linear combination of the forward and inverse DFTs of the signal. The discrete combined fractional Fourier transform (DCFrFT) introduced in this work can be seen as a generalization of the of DCFT to the FrFT domains. Simulation results of the proposed DCFrFT and the original signal recovery are given to validate the proposed approach. To present an example of application of the proposed combined transform, we propose a methodology for image enhancement in which the root filtering (RF) technique is extended to the DCFrFT domains. Here the proper selection of the angle parameter of the proposed DCFrFT and the RF parameter are crucial in the enhancement process. For this purpose, simulation results for different values of the RF parameter are carried out and its appropriate value is chosen based on the subjective evaluation. It is seen here that the enhancement approach in the proposed transform domain can improve degraded images when an appropriate RF parameter is used.