Spatial Affine transformations of images by using fractional shift fourier transform

Affine transformations of images are common used in modern signal processing, image processing, and computer graphics areas. Take rotation of images for example, we have to calculate the positions of pixels after rotation and use methods like cubic, bilinear, and nearest neighbor interpolation to interpolate some pixels in between. These interpolation and spatial-domain-based methods are slow, cumbersome and the PSNR (Peak signal to noise ratio) is also not satisfactory. In this paper, we propose frequency domain based method for Affine transformations by applying proposed fractional shift Fourier transform and modified inverse Fourier transform. For rotation of gray-scale images, three shearing matrices and fractional shift Fourier transform are applied to complete the task. Fast Fourier transform (FFT) algorithm can be used to implement Fourier transform and therefore our method is very fast and precise. For the general case, i.e. Affine transformations, we combine shearing matrices, fractional shift Fourier transform and Modified inverse Fourier transform method to achieve arbitrary Affine transformations with small distortion (above 50dB PSNR).