2D Image Denoising Using Fractional Fourier Transform Algorithm

In the field of digital image processing, image denoising represents a fundamental and critical task. The fractional Fourier transform (FrFT) has emerged as a novel frequency-domain analysis tool that provides fresh perspectives beyond traditional denoising approaches. Compared to conventional Fourier transform, FrFT offers greater flexibility in adjusting the transformation order, enabling optimal balance between time and frequency domains - particularly effective for suppressing noise components in images. The core principle of FrFT-based denoising lies in achieving sparse representation of images in the fractional domain. Noise typically manifests as high-frequency random signals, while meaningful image information distributes within specific frequency ranges. By fine-tuning the fractional order parameter, we can more precisely separate noise energy from image signals. In the fractional frequency domain, noise coefficients tend to be smaller and more dispersed, whereas essential image information concentrates in fewer but larger coefficients. This characteristic allows noise suppression through thresholding techniques or coefficient shrinkage methods, followed by inverse transformation to restore clear images. In implementation, this involves calculating the 2D FrFT using discrete approximation algorithms, applying adaptive thresholding functions like soft-thresholding (sign(x)*max(|x|-λ,0)) or hard-thresholding, and performing inverse FrFT reconstruction. Compared to wavelet transform or traditional Fourier-based denoising, FrFT's advantage stems from its adjustable order parameter that adapts to different noise types and image characteristics. For specific noise patterns, optimized fractional orders can concentrate signal energy more effectively, thereby enhancing denoising performance. However, practitioners must consider trade-offs regarding computational complexity (O(N²logN) for N×N images) and the empirical/optimization-based selection of fractional orders. Implementation typically requires iterative optimization algorithms like gradient descent or genetic algorithms to determine optimal fractional orders. Overall, fractional Fourier transform provides a novel frequency-domain perspective for image denoising, particularly valuable for complex noise scenarios where traditional methods underperform. With ongoing algorithm optimizations (including fast FrFT implementations) and hardware advancements, its potential in image restoration continues to expand. Modern implementations often incorporate machine learning techniques to automate fractional order selection and adaptive threshold determination.