Fractional Fourier Transform for Two-Dimensional Image Processing

In the field of image processing, denoising serves as a crucial preprocessing step for numerous applications. While Fourier transform is a common approach, it assumes signal stationarity. For non-stationary signals, the fractional Fourier transform (FrFT) provides superior handling capabilities. As a novel mathematical tool, FrFT introduces fractional-order derivatives to characterize non-stationary signals effectively. When implementing 2D fractional Fourier transform for image denoising, the process typically involves: 1. Separable transformation applying 1D FrFT sequentially along rows and columns 2. Using discrete FrFT algorithms based on eigen decomposition of DFT matrix 3. Adjusting fractional order parameter (α) to optimize noise separation in hybrid time-frequency domain 4. Applying thresholding in fractional Fourier domain before inverse transformation This approach enables better noise removal by leveraging the fractional transform's ability to represent signals between pure time and frequency domains, ultimately enhancing image quality through optimized time-frequency localization.