The fractional calculus (FC) deals with integrals and derivatives of arbitrary (i.e., non-integer) order, and shares its origins with classical integral and differential calculus. The fractional Fourier transform (FRFT), which has been found having many applications in optics and other areas, is a generalization of the usual Fourier transform. The FC and the FRFT are two of the most interesting and useful fractional areas. In recent years, it appears many papers on the FC and FRFT, however, few of them discuss the connection of the two fractional areas. We study their relationship. The relational expression between them is deduced. The expectation of interdisciplinary cross fertilization is our motivation. For example, we can use the properties of the FC (non-locality, etc.) to solve the problem which is difficult to be solved by the FRFT in optical engineering; we can also through the physical meaning of the FRFT optical implementation to explain the physical meaning of the FC. The FC and FRFT approaches can be transposed each other in the two fractional areas. It makes that the success of the fractional methodology is unquestionable with a lot of applications, namely in nonlinear and complex system dynamics and image processing.
Removal of noise is an important step in the image restoration process, and it remains a challenging problem in image processing. Denoising is a process used to remove the noise from the corrupted image, while retaining the edges and other detailed features as much as possible. Recently, denoising in the fractional domain is a hot research topic. The fractional-order anisotropic diffusion method can bring a less blocky effect and preserve edges in image denoising, a method that has received much interest in the literature. Based on this method, we propose a new method for image denoising, in which fractional-varying-order differential, rather than constant-order differential, is used. The theoretical analysis and experimental results show that compared with the state-of-the-art fractional-order anisotropic diffusion method, the proposed fractional-varying-order differential denoising model can preserve structure and texture well, while quickly removing noise, and yields good visual effects and better peak signal-to-noise ratio.
Signal reconstruction, especially for nonstationary signals, occurs in many applications such as optical astronomy, electron microscopy, and x-ray crystallography. As a potent tool to analyze the nonstationary signals, the linear canonical transform (LCT) describes the effect of quadratic phase systems on a wavefield and generalizes many optical transforms. The reconstruction of a finite discrete-time signal from the partial information of its discrete LCT and some known samples under some restrictions is presented. The partial information about its discrete LCT that we have assumed to be available is the discrete LCT phase alone or the discrete LCT magnitude alone. Besides, a reconstruction example is provided to verify the effectiveness of the proposed algorithm.
Newton’s rings pattern always blurs the scanned image when scanning a film using a film scanner. Such phenomenon is a kind of equal thickness interference, which is caused by the air layer between the film and the glass of the scanner. A lot of methods were proposed to prevent the interference, such as film holder, anti-Newton’s rings glass and emulsion direct imaging technology, etc. Those methods are expensive and lack of flexibility. In this paper, Newton’s rings pattern is proved to be a 2-D chirp signal. Then, the fractional Fourier transform, which can be understood as the chirp-based decomposition, is introduced to process Newton’s rings pattern. A digital filtering method in the fractional Fourier domain is proposed to reduce the Newton’s rings pattern. The effectiveness of the proposed method is verified by simulation. Compared with the traditional optical method, the proposed method is more flexible and low cost.
KEYWORDS: Quantization, Fringe analysis, Fourier transforms, Error analysis, Signal to noise ratio, Commercial off the shelf technology, Signal processing, Digital filtering, Image quality, Optical engineering
Newton’s rings fringe pattern is often encountered in optical measurement. The digital processing of the fringe pattern is widely used to enable automatic analysis and improve the accuracy and flexibility. Before digital processing, sampling and quantization are necessary, which introduce quantization errors in the fringe pattern. Quantization errors are always analyzed and suppressed in the Fourier transform (FT) domain. But Newton’s rings fringe pattern is demonstrated to be a two-dimensional chirp signal, and the traditional methods based on the FT domain are not efficient when suppressing quantization errors in such signals with large bandwidth as chirp signals. This paper proposes a method for suppressing quantization errors in the fractional Fourier transform (FRFT) domain, for chirp signals occupies little bandwidth in the FRFT domain. This method has better effect on reduction of quantization errors in the fringe pattern than traditional methods. As an example, a standard Newton’s rings fringe pattern is analyzed in the FRFT domain and then 8.5 dB of improvement in signal-to-quantization-noise ratio and about 1.4 bits of increase in accuracy are obtained compared to the case of the FT domain. Consequently, the image quality of Newton’s rings fringe pattern is improved, which is beneficial to optical metrology.
The continuous fractional Fourier transform (FRFT) can be interpreted as a rotation of a signal in the time-frequency plane and is a powerful tool for analyzing and processing nonstationary signals. Because of the importance of the FRFT, the discrete fractional Fourier transform (DFRFT) has recently become an important issue. We present the computation method for the DFRFT using the adaptive least-mean-square algorithm. First, the DFRFT computation scheme with single angle parameter of the signal block using the adaptive filter system is introduced. Second, considering the transform angles always change in practical applications, the DFRFT computation scheme with adjustable-angle parameter of the signal block using the adaptive filter system is presented. Then we construct two realization structures of the DFRFT computation with simultaneous multiple-angle parameters for each signal block. The proposed computation approaches have the inherent parallel structures, which make them suitable for efficient very large scale integration implementations.