Performance Evaluation of Fractional-Order Digital Filter for Signal and Image Processing Applications

Abstract

Fractional-order calculus (FOC), which is often contemplated as a branch of pure mathematics, has encapsulated the attention of eminent researchers from several backgrounds due to its profound applications in various fields of science and engineering. The mathematical tool of FOC, which is, in fact, the generalization of integer-order calculus, is concerned with differentiation and integration of non-integer orders. FOC concept emerged from the generalization of integer-order calculus in two most prominent ways: One based upon the direct generalization of the limit of difference quotients defining integer-order derivatives and another one based upon the inversion of the generalization of n-fold integer-order successive integration rule (a.k.a. Cauchy’s formula for repeated integration). These generalizations provided two different ways to define fractional-order derivatives and hence, referred to as Gr¨unwald-Letnikov (GL) and Riemann-Liouville (RL) fractional-order derivatives, respectively.From signal and image processing viewpoint, the definitions of fractional-order derivatives can be viewed in terms of convolution, where the filter or kernel function is of power-law type. A natural query that arises is: Why the kernel is of power-law type? This is because of the association of power-law in Cauchy’s integral formula. Fractional-order operators exhibiting power-law kernel have found tremendous applications in natural and man-made phenomena. However, it has been pointed out recently that there exists an artificial singularity at the initial point of the power-law kernel, due to which many real-life phenomena cannot be implemented exactly. Therefore, several mathematicians made an attempt to replace the power-law kernel with other appropriate kernels (such as, exponential kernel, Mittag-Leffler kernel) and hence, led to the development of various fractional-order operators based upon different kernel functions.On the other hand, a significant attention has been paid by the research community on the study of fractional-order digital differentiators (FODDs), i.e., digital discrete-time systems that can perform non-integer order differentiation. The fundamental advantage of FODDs is that they provide an extra degree of freedom which helps in optimizing the performance of methods based upon traditional integer-order calculus. Therefore, research work carried out in this thesis emphasis on the design of power-law kernel and Mittag-Leffler kernel based fractional-order digital filters for signal and image processing applications.Out of power-law kernel based GL and RL fractional-order derivatives, GL fractional-order derivative has been intensively-studied and extensively-utilized for the design of FODD due to its discrete nature. However, there seems to be an improvement window for designing filters based upon RL fractional-order differential operator. Hence, initial research work in this thesis aims at establishing a closed-form analytical formulation for the design of Riemann- Liouville based fractional-order digital differentiator and further authenticates the efficacy of the proposed design through simulation results as compared to GL-based methods.Furthermore, to explore the applicability of recently introduced fractional-order operators, non-local and non-singular Mittag-Leffler kernel based Atangana-Baleanu-Caputo fractional-order digital filter (herein, referred to as ABC-FODF) is proposed. Atangana- Baleanu fractional-order differential operator has been extensively investigated in various fields of science and engineering. However, applications of AB fractional-order operator in signal and image processing applications are limited, and hence, the research work carried out in this thesis aims to add to its applications by establishing the design of ABC-FODF. Furthermore, the applications of the proposed ABC-FODF are discussed for R-peak detection in electrocardiogram (ECG) signals and digital image sharpening.After establishing closed-form analytical formulations for the design of fractional-order digital filters, further research work aims at studying various fractional-order integral operators for image denoising applications. For this, a generalized fractional integral (GFI) mask which incorporates both RL and AB fractional-order integral operators is proposed. The proposed GFI mask provide a more flexible and appropriate tool for image denoising applications and hence, avoids the construction of different masks for each definition separately. Furthermore, a GFI-based adaptive mask (GFI-AM) is put forward to select the fractional order based upon local features of an image. The performance of various fractional-order operators i.e., GL, RL, and AB is tested for images affected by Gaussian and salt and pepper noise. From the simulation studies conducted, it is seen that the proposed GFI-AM incorporating AB fractional integral operator outperforms the conventional operators in removal of noise from images.Furthermore, as a major contribution of this research work, a generalized fractional integral and fractional differential based adaptive algorithm (GFIFD-AA) is proposed for the detection and removal of salt and pepper noise in images. The proposed algorithm is segregated into two parts: In the first part, a novel noise detection method for the detection of salt and pepper noise (SP) is introduced. In the second part, the detected noisy pixels are processed by utilizing the proposed GFI-AM. In order to preserve the texture and edge details of the images, a generalized fractional differential based adaptive mask (GFD-AM) is proposed, and hence, the non-noisy pixels are processed by utilizing the proposed GFD-AM. Therefore, the proposed GFI-AM and GFD-AM are utilized together in the framework of proposed GFIFD-AA for the effective removal of salt and pepper noise and preservation of details in images. The proposed method exhibits a profound advantage in avoiding misclassification of original pixels with intensity values 0 or 255 as noisy pixels and hence, is effective in detecting SP noise in black and white images as well as synthetic images (where every pixel of an image is of value 0 or 255). The simulation studies conducted on several standard as well as synthetic images indicated the superiority of the proposed method against well-established state-of-the-art methods.Hence, the study carried out in this thesis has unfolded significant advantages of fractionalorder calculus for signal and image processing applications and hence, opens up a new paradigm of research based on recently introduced fractional-order operators for various signal filtering, image enhancement and adaptive image denoising applications.