Distribution of FRFT Coefficients of Natural Images

Abstract

Abstract. For the convenience of providing temporal and spectral information by a single variable, fractional Fourier transformation (FRFT) is more and more applied to image processing recently. This paper focuses on the statistical regularity of FRFT coefficients of natural images and proposes that the real and imaginary parts of FRFT coefficients of natural images take on the generalized Gaussian distribution, the coefficient modulus follow the gamma distribution and the coefficient phase angles tend to the uniform distribution, moreover, the real and imaginary parts of coefficient phases similar to the extended beta distribution. These underlying statistics can provide theoretical basis for image processing in FRFT, such as dimensionality reduction, feature extraction, smooth denoising, digital forensics, watermarking, etc.Fractional Fourier transformation (FRFT), as a kind of generalized Fourier transformation (FT), can be interpreted as a rotation of the signal in the time-frequency plane. Different from wavelet transformation, short-time Fourier transformation, Gabor transformation and other common two-parameters time frequency distributions, FRFT can provide the related local information of the signal in the time domain and the frequency domain simultaneously by a single variable, thus it has a wide application prospect in the field of signal processing [1]. FRFT is also introduced into the image processing, which is a significant branch of signal processing, such as the data compression [2], the image registration [3], the facial expression recognition [4], the image encryption [5], the digital watermarking [6] and so on. For the image processing based on FRFT, it is necessary to learn the related prior knowledge of FRFT coefficients of natural images. However, to the author's knowledge, there has not been relevant research report currently. Thus, this paper will focus on the statistical probability distribution of FRFT coefficients of natural images. As the FRFT phase gaining more and more attention for carrying the important image texture information [4], this paper will explore the statistical distribution of the amplitude and phase parts also with the real and imaginary parts of FRFT coefficients of natural images.