Partial Differential Equations are used in smoothening of images. Under partial differential equations an image is termed as a function; f(x, y), XÎR^{2}. The pixel flux is referred to as an edge stopping function since it ensures that diffusion occurs within the image region but zero at the boundaries; u_{x}(0, y, t) = u_{x}(p, y, t) = u_{y}(x, 0, t) = u_{y}(x, q, t). Nonlinear PDEs tend to adjust the quality of the image, thus giving images desirable outlooks. In the digital world there is need for images to be smoothened for broadcast purposes, medical display of internal organs i.e MRI (Magnetic Resonance Imaging), study of the galaxy, CCTV (Closed Circuit Television) among others. This model inputs optimization in the smoothening of images. The solutions of the diffusion equations were obtained using iterative algorithms i.e. Alternating Direction Implicit (ADI) method, Two-point Explicit Group Successive Over-Relaxation (2-EGSOR) and a successive implementation of these two approaches. These schemes were executed in MATLAB (Matrix Laboratory) subject to an initial condition of a noisy images characterized by pepper noise, Gaussian noise, Brownian noise, Poisson noise etc. As the algorithms were implemented in MATLAB, the smoothing effect reduced at places with possibilities of being boundaries, the parameters C_{v} (pixel flux), C_{f} (coefficient of the forcing term), b (the threshold parameter) alongside time t were estimated through optimization. Parameter b maintained the highest value, while C_{v} exhibited the lowest value implying that diffusion of pixels within the various images i.e. CCTV, MRI & Galaxy was limited to enhance smoothening. On the other hand the threshold parameter (b) took an escalated value across the images translating to a high level of the force responsible for smoothening.
Published in | Science Journal of Applied Mathematics and Statistics (Volume 12, Issue 1) |
DOI | 10.11648/j.sjams.20241201.12 |
Page(s) | 13-19 |
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
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Copyright © The Author(s), 2024. Published by Science Publishing Group |
Partial Differential Equations, Image Processing, Holes, Non-Linear Diffusion Equations, Pixel Flux, Threshold Parameter, Coefficient of the Forcing Term
2.1. Numerical Approaches
2.1.1. Alternating Direction Implicit (ADI) Method
2.1.2. Two-Point Explicit Group Successive Over-Relaxation (2-EGSOR)
2.2. Optimization
3.1. Approximated Parameters Under EGSOR
t | C_{v} | C_{f} | b |
---|---|---|---|
0 | 0.000125 | 0.005 | 4 |
0.111111 | 0.00000006968 | 0.006957 | 4.626973 |
0.222222 | 0.00000001597 | 0.007049 | 4.624980 |
0.333333 | 0.0000003090288 | 0.006933 | 4.624997 |
0.444444 | 0.0000002684966 | 0.006931 | 4.624979 |
0.555555 | 0.000000213459 | 0.007012 | 4.624996 |
0.666666 | 0.00000000845238 | 0.006977 | 4.624980 |
0.777777 | 0.00000022759 | 0.006979 | 4.626915 |
0.888888 | 0.00000000882199 | 0.06988 | 4.624981 |
1 | 0.0000000413959 | 0.07018 | 4.624981 |
t | C_{v} | C_{f} | b |
---|---|---|---|
0 | 0.000008 | 0.005 | 6 |
0.25 | 0.000000068621 | 0.011784 | 5.999747 |
0.5 | 0.000000256962 | 0.011789 | 5.999747 |
0.75 | 0.0000002491727 | 0.011791 | 5.999743 |
1 | 0.000000157290 | 0.011793 | 5.999752 |
3.2. Approximated Parameters Under ADI
t | C_{v} | C_{f} | b |
---|---|---|---|
0 | 0.000125 | 0.005 | 4 |
0.111111 | 0.000003 | 0.012811 | 4.000023 |
0.222222 | 0.00000045924 | 0.012840 | 4.002498 |
0.333333 | 0.00000002024 | 0.012792 | 4.000023 |
0.444444 | 0.0000000731162 | 0.012818 | 4.002586 |
0.555555 | 0.0000001149168 | 0.012883 | 4.000027 |
0.666666 | 0.0000020747 | 0.012809 | 4.002546 |
0.777777 | 0.0000002403366 | 0.012870 | 4.002569 |
0.888888 | 0.001742 | 0.012647 | 4.000024 |
1 | 0.000745 | 0.012642 | 4.011794 |
t | C_{v} | C_{f} | b |
---|---|---|---|
0 | -0.000000068622 | 0.011784 | 5.999747 |
0.25 | 0.00196019959 | 0.01607929220 | 5.999766 |
0.5 | 0.00196019959 | 0.01607929220 | 5.999766 |
0.75 | 0.00196019959102 | 0.01607929220 | 5.999766 |
1 | 0.0196019959102702 | 0.01607929220 | 5.999766 |
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APA Style
Fwamba, R. N., Chepkwony, I., Fwamba, W. S. (2024). Optimization of the Non-Linear Diffussion Equations. Science Journal of Applied Mathematics and Statistics, 12(1), 13-19. https://doi.org/10.11648/j.sjams.20241201.12
ACS Style
Fwamba, R. N.; Chepkwony, I.; Fwamba, W. S. Optimization of the Non-Linear Diffussion Equations. Sci. J. Appl. Math. Stat. 2024, 12(1), 13-19. doi: 10.11648/j.sjams.20241201.12
@article{10.11648/j.sjams.20241201.12, author = {Rukia Nasimiyu Fwamba and Isaac Chepkwony and Wekulo Saidi Fwamba}, title = {Optimization of the Non-Linear Diffussion Equations}, journal = {Science Journal of Applied Mathematics and Statistics}, volume = {12}, number = {1}, pages = {13-19}, doi = {10.11648/j.sjams.20241201.12}, url = {https://doi.org/10.11648/j.sjams.20241201.12}, eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20241201.12}, abstract = {Partial Differential Equations are used in smoothening of images. Under partial differential equations an image is termed as a function; f(x, y), XÎR2. The pixel flux is referred to as an edge stopping function since it ensures that diffusion occurs within the image region but zero at the boundaries; ux(0, y, t) = ux(p, y, t) = uy(x, 0, t) = uy(x, q, t). Nonlinear PDEs tend to adjust the quality of the image, thus giving images desirable outlooks. In the digital world there is need for images to be smoothened for broadcast purposes, medical display of internal organs i.e MRI (Magnetic Resonance Imaging), study of the galaxy, CCTV (Closed Circuit Television) among others. This model inputs optimization in the smoothening of images. The solutions of the diffusion equations were obtained using iterative algorithms i.e. Alternating Direction Implicit (ADI) method, Two-point Explicit Group Successive Over-Relaxation (2-EGSOR) and a successive implementation of these two approaches. These schemes were executed in MATLAB (Matrix Laboratory) subject to an initial condition of a noisy images characterized by pepper noise, Gaussian noise, Brownian noise, Poisson noise etc. As the algorithms were implemented in MATLAB, the smoothing effect reduced at places with possibilities of being boundaries, the parameters Cv (pixel flux), Cf (coefficient of the forcing term), b (the threshold parameter) alongside time t were estimated through optimization. Parameter b maintained the highest value, while Cv exhibited the lowest value implying that diffusion of pixels within the various images i.e. CCTV, MRI & Galaxy was limited to enhance smoothening. On the other hand the threshold parameter (b) took an escalated value across the images translating to a high level of the force responsible for smoothening.}, year = {2024} }
TY - JOUR T1 - Optimization of the Non-Linear Diffussion Equations AU - Rukia Nasimiyu Fwamba AU - Isaac Chepkwony AU - Wekulo Saidi Fwamba Y1 - 2024/04/02 PY - 2024 N1 - https://doi.org/10.11648/j.sjams.20241201.12 DO - 10.11648/j.sjams.20241201.12 T2 - Science Journal of Applied Mathematics and Statistics JF - Science Journal of Applied Mathematics and Statistics JO - Science Journal of Applied Mathematics and Statistics SP - 13 EP - 19 PB - Science Publishing Group SN - 2376-9513 UR - https://doi.org/10.11648/j.sjams.20241201.12 AB - Partial Differential Equations are used in smoothening of images. Under partial differential equations an image is termed as a function; f(x, y), XÎR2. The pixel flux is referred to as an edge stopping function since it ensures that diffusion occurs within the image region but zero at the boundaries; ux(0, y, t) = ux(p, y, t) = uy(x, 0, t) = uy(x, q, t). Nonlinear PDEs tend to adjust the quality of the image, thus giving images desirable outlooks. In the digital world there is need for images to be smoothened for broadcast purposes, medical display of internal organs i.e MRI (Magnetic Resonance Imaging), study of the galaxy, CCTV (Closed Circuit Television) among others. This model inputs optimization in the smoothening of images. The solutions of the diffusion equations were obtained using iterative algorithms i.e. Alternating Direction Implicit (ADI) method, Two-point Explicit Group Successive Over-Relaxation (2-EGSOR) and a successive implementation of these two approaches. These schemes were executed in MATLAB (Matrix Laboratory) subject to an initial condition of a noisy images characterized by pepper noise, Gaussian noise, Brownian noise, Poisson noise etc. As the algorithms were implemented in MATLAB, the smoothing effect reduced at places with possibilities of being boundaries, the parameters Cv (pixel flux), Cf (coefficient of the forcing term), b (the threshold parameter) alongside time t were estimated through optimization. Parameter b maintained the highest value, while Cv exhibited the lowest value implying that diffusion of pixels within the various images i.e. CCTV, MRI & Galaxy was limited to enhance smoothening. On the other hand the threshold parameter (b) took an escalated value across the images translating to a high level of the force responsible for smoothening. VL - 12 IS - 1 ER -