Non-causal Fractinal Filter And Its Application To Image Processing

Fractional calculus is three centuries old as the conventional calculus, but not very popular among science and engineering community. Fractional calculus, which means to generalize the differentia-tion and integration into fractional and complex order, is a generalization of the integer-order calculus. Fractional calculus with the continuous order differentiation and integration extends the power of the integer-order calculus. With the high speed development of computer science and the increasing ability of calculation in recent years, realization of fractional calculus becomes feasible. As a new tool, it has been applied in many research fields, such as materials, rheology, seismic data processing, electromag-netic theory, control theory, signal processing, and etc. At present, fractional calculus has been adopted in image processing and made some remarkable achievements.In signal processing, a causal filter is a linear and time-invariant causal system. The word causal indicates that the filter output depends only on past and present inputs. A filter whose output depends only on future inputs is anti-causal. A filter whose output also depends on future inputs is noncausal. Systems (including filters) that are realizable (i.e., that operate in real time) must be causal because such systems cannot act on a future inputs, such as signal processing, control and etc. Replacing the time dimension by the spacial dimension, such as in 2D image processing, noncausal filters are realizable. Noncausal filter has the freedom of letting its output depend on the past, present, and future inputs, hence, it makes full use of the input information.This paper focus on the design and implementation of non-causal fractional filters, and their ap-plication to image edge detection and the estimation of the blur parameters of motion-blurred image. One analyse and studies the amplitude-frequency characteristics and the phase-frequency characteristics of forward and backward fractional calculus. Non-causal signal processing is introduced in fractional calculus, two non-causal fractional 0-degree phase filters (Adding backward integral to forward inte-gral, a non-causal fractional 0-degree phase low-pass filters is obtained. Adding backward derivative to forward derivative, a non-causal fractional 0-degree phase high-pass filters is obtained), two non-causal fractional 90-degree phase filters (Subtracting backward integral from forward integral, a non-causal fractional 90-degree phase low-pass filters is obtained. Subtracting backward derivative from forward derivative, a non-causal fractional 90-degree phase high-pass filters is obtained), a cascade non-causal fractional filters of fractional integral and derivative, and a non-causal fractional directional differentia-tor are derived. According to characteristics of 2D image signal, these non-causal filters are deduced in space domain.For traditional integer-order edge detector, selectivity and immunity to noise is conflicting, i.e., the increment of one causes the other to decrease. In this paper, a novel cascade noncausal fractional gradient operator is derived based on the proposed cascade noncausal fractional filter of fractional-order integral and derivative. The phase characteristic of the cascade noncausal fractional gradient operator, whose integral and derivative both contribute phase, performs 90° phase shift of the traditional first derivative in the small order of derivative. Hence the novel fractional gradient operator can effective-ly improve the selectivity/immunity-to-noise compromise. The new edge detector, formed from the novel fractional gradient, is applied to edge detection and the results are analyzed, emphasizing on the compromise between selectivity and noise suppression. Both objective and subjective comparisons with other edge detectors, are carried out, including the evaluations through the use of the Benchmark Berkeley Segmentation Dataset (BSDS500).The blur direction and extent of motion-blurred image, which are introduced by relative motion between a camera and its object scene, are needed in the methods of image restoration, such as blind deconvolution. As an extension to the fractional-order derivative, a noncausal fractional-order direc-tional derivative operator is devised, which is robust to noise. Based on this new operator, a novel method identifying blur parameters is developed in this work. The performance comparisons between the proposed method and the state-of-the-art method are also presented, demonstrating that the former provides better immunity to noise and capacity to identify motion blur extent, especially for large blur length.