Fractional Calculus Based Image Simplification Algorithm For Medical applications /Alaa AbdAlrhman Mohammed

Supervisor: Ahmed G. Radwan

Thesis (MS.c)—Nile University, Egypt, 2024.

"Includes bibliographical references"

Contents: Contents Page
V
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XVII
List of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XX
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX
Acknowledgment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII
Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXII
Chapters:
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2. Thesis Motivation and Concepts of Review . . . . . . . . . . . . . . . . 5
2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 Fractional Operator based on GL definition . . . . . . . . . 8
2.2.2 Fractional Operator based on RL Definition . . . . . . . . . 10
2.2.3 Fractional Operator Based on Deformable Definition . . . . 13
2.3 Image Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.1 Fractional Kernels . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Anisotropic Diffusion Filters . . . . . . . . . . . . . . . . . . . . . . 21
2.4.1 Hardware Implementation of Anisotropic Filters on FPGA
Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4.2 Pixel buffer . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.3 Kernel convolution . . . . . . . . . . . . . . . . . . . . . . . 28
2.4.4 G function . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3. New Implementation of GL-based Fractional Operator. 28
3.1 Optimization Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.1.1 Average fixed window optimization method . . . . . . . . . . 28
3.1.2 Optimum α method . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 30
3.1.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 An Improved Approximation of GL Fractional Integral . . . . . . . . 33
3.2.1 Mathematical Analysis . . . . . . . . . . . . . . . . . . . . . . 33
3.2.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.3 Fixed window method . . . . . . . . . . . . . . . . . . . . . . 35
3.2.4 Quadratic approximation method . . . . . . . . . . . . . . . . 35
3.2.5 PWL approximation method . . . . . . . . . . . . . . . . . . . 36
3.2.6 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4. Hardware Accelerator of Fractional-order Derivative/Integral Operator based on Phase Optimized Filters with Applications 41
4.1 Filters Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1.1 Proposed Phase Optimized FIR . . . . . . . . . . . . . . . . . 41
VI
4.1.2 Proposed Phase Optimized IIR . . . . . . . . . . . . . . . . . 44
4.2 FPGA Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.4.1 Heaviside . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.4.2 Edge Detection . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5. Application: Fractional based Image Processing 56
5.1 Performance Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.2 Fractional Image Enhancement . . . . . . . . . . . . . . . . . . . . . 57
5.2.1 Double Fractional Kernel System . . . . . . . . . . . . . . . . 57
5.2.2 Fractional Kernels Optimization . . . . . . . . . . . . . . . . . 58
5.2.3 Single fractional kernels optimization . . . . . . . . . . . . . . 59
5.2.4 Double fractional kernels system optimization . . . . . . . . . 61
5.2.5 Constrained Double Optimization . . . . . . . . . . . . . . . . 62
5.3 Fractional Anisotropic Diffusion Filter . . . . . . . . . . . . . . . . . 66
5.3.1 Fractional Gradient Operator . . . . . . . . . . . . . . . . . . 67
5.3.2 Anisotropic Filter Design . . . . . . . . . . . . . . . . . . . . . 68
5.3.3 Optimizing Fractional Anisotropic Filter . . . . . . . . . . . . 72
5.3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6. Conclusion and Future Work 79
6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
VII
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Appendices:
A. Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Abstract: Fractional calculus is a powerful mathematical tool used to describe complex systems in physics, engineering, and finance. However, implementing fractional order
systems in hardware can be resource-intensive, requiring large amounts of memory
and processing power. Field programmable gate arrays (FPGAs) provide a platform
for implementing fractional order systems that can conduct logic in parallel, resulting
in faster and more efficient processing.
This thesis proposes three new fractional order implementation methods that improve accuracy and reduce resource requirements. The results show that the proposed
methods offer significant improvements in accuracy, power, latency, memory-units usage and DSPs compared to traditional methods. This thesis also proposes a soft IP
library for different fractional operators implementation methods that can be included
in FPGA different applications. In addition to exploring new implementation methods, this thesis investigates the use of fractional calculus in image enhancement and
image simplification. The proposed methods are evaluated using metrics such as peak
signal-to-noise ratio (PSNR) and the average gradient metric (AG) to determine their
effectiveness.
Overall, this thesis makes significant contributions to the field of fractional calculus, that can be of value in many fields. The proposed implementation methods and

X
image processing techniques can be used in a wide range of applications, including
control systems, signal processing, and image processing.
Keywords
Fractional Calculus, FPGA, Matlab, Image Processing, Portable Devices, Anisotropic
Filters, Grünwald–Letnikov, GA, FPA

Text in English, abstracts in English and Arabic