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Title: Performance Analysis of Fractional Derivative in Image Processing Applications
Authors: Kaur, Kanwarpreet
Supervisor: Singh, Kulbir
Jindal, Neeru
Keywords: Fractional Fourier Transform;Fractional Order Calculus;Image Edge Detection;Image Enhancement;Riesz Fractional Order Derivative
Issue Date: 24-Sep-2021
Abstract: The primary objective of the reported work is to thoroughly explore the aspects of Fractional Order Derivatives (FODs) in the prospective applications of image processing. The majority of integer-order based algorithms in the prevailing literature are not able to accurately model the systems owing to the problem of loss of information details, noise sensitivity, and deterioration in the smooth regions of an image. Moreover, in the case of color images, additional issues such as loss of correlation and information details exist when the color images are converted into grayscale images before processing or individual color channels are processed. Several techniques based on optimization, learning, and fractional exist in literature to deal with these issues but the fractional based methods are more efficient in terms of computational cost for smaller datasets. Despite being efficient, FODs such as Riemann-Liouville (RL) and Grünwald-Letnikov (GL) possess the demerit in terms of phase shifting that can cause image blurred distortion. In order to resolve these issues, Riesz FOD (RFOD) is considered in this work which possesses the inherent property of zero phase shifting. However, instead of utilizing the RFOD solely on the basis of theoretical concepts and findings, its mathematical and experimental analysis is conducted in spatial, FrFT, and quaternion domains. The imperative objective of the presented work is to incorporate the concept of RFOD in Fractional Fourier Transform (FrFT) and quaternion domain for utilizing the benefit of the fractional parameter in achieving design flexibility. To validate the proficiency of the proposed concept, the applications of edge detection and image enhancement are considered as they can further assist in the high-level image processing applications. The robustness of RFOD in signal and image processing applications is confirmed judiciously by evaluating it in spatial, FrFT, and quaternion domain orderly. In the spatial domain, RFOD is incorporated in the Unsharp Masking (UM) for image enhancement. Upon subjecting to evaluation based on several standard images of various datasets as well as Fundus images, RFOD yielded better image visual quality as compared to RL, GL, and other image enhancement techniques. Thus, the promising results obtained by RFOD in the spatial domain for the image enhancement further stimulated to explore RFOD in the frequency domain. Hence, mathematical analysis of RFOD is carried out in the FrFT domain, thus, deriving a novel closed-form analytical expression for RFOD in the FrFT domain which is further utilized for the filtering application of signal processing. The efficacy of the low pass Finite Impulse Response (FIR) differentiator designed on the basis of RFOD in the FrFT domain is validated by considering a design example of signal corrupted with high-frequency chirp noise. The simulation results exhibit that RFOD outperforms the RL and Caputo FODs in the FrFT domain in terms of minimum Root Mean Square Error (RMSE) of 0.115136 for the fractional order ranging from 0.2-0.6. Moreover, the proposed concept of RFOD in FrFT domain is extended for the two-dimensional applications of image sharpening and Homomorphic Filtering (HF) for the improvement in image visual quality. The substantial performance of RFOD in the FrFT domain further encouraged to utilize this exceptional combination for applications of image processing. Henceforth, RFOD is used to obtain the masks by employing various interpolation methods for the applications of image enhancement and edge detection. Consequently, a unified approach based on RFOD in the FrFT domain is developed for edge detection and enhancement of grayscale images which provided superior results than the existing FOD based approaches for edge detection and image enhancement. The concept of edge detection is further extended to color images. A quaternion based RFOD edge detection approach is developed that processes all the channels of a color image simultaneously to avoid the problem of loss of information details and correlation among color channels. Extensive experimentation is conducted on the standard datasets to demonstrate its effectiveness in comparison to the existing techniques. The proposed technique is further utilized for enhancing the color images. Another important point that is considered in both these approaches is that selection of mask size is done on the basis of both visual comparison and performance metrics. To further evaluate the adequacy of the proposed approaches, uncontrolled conditions are taken into consideration. Although some of the existing approaches are tested against the uncontrolled features of noise and illumination, the JPEG compression artifacts are not considered in the existing edge detection techniques. Therefore, the robustness of the developed approaches is validated by subjecting them to not only noise and variation in illumination but also JPEG compression. Thus, a unique concept of integration of RFOD with FrFT as well as quaternion domain is devised in the presented work to provide effective and reliable techniques for the applications of signal and image processing. The future work can be dedicated to extend the present work for the applications of computer vision and pattern recognition.
Description: Ph.D Thesis
Appears in Collections:Doctoral Theses@ECED

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