Image processing with the fractional Fourier transform: synthesis, compression and perspective projections

Abstract

In this work, first we give a summary of the fractional Fourier transform including its definition, important properties, generalization to two-dimensions and its discrete counterpart. After that, we repeat the concept of filtering in the fractional Fourier domains and give multi-stage and multi-channel filtering configurations. Due to the nonlinear nature of the problem, the transform orders in fractional Fourier domain filtering configurations have usually not been optimized but chosen uniformly up to date. We discuss the optimization of orders in the multi-channel filtering configuration. In the next part of this thesis, we discuss the application of fractional Fourier transform based filtering configurations to image representation and compression. Next, we introduce the fractional Fourier domain decomposition for continuous signals and systems. In the last part, we analyse perspective projections in the space-frequency plane and show that under certain conditions they can be approximately modeled in terms of the fractional Fourier transform.