Optimization of the Non-Linear Diffussion Equations

Rukia Nasimiyu Fwamba; Isaac Chepkwony; Wekulo Saidi Fwamba

doi:doi:10.11648/j.sjams.20241201.12

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Optimization of the Non-Linear Diffussion Equations

Rukia Nasimiyu Fwamba^*

, Isaac Chepkwony

, Wekulo Saidi Fwamba

Published in Science Journal of Applied Mathematics and Statistics (Volume 12, Issue 1)

Received: 27 February 2024 Accepted: 12 March 2024 Published: 2 April 2024

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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; 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
Creative Commons	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.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Partial Differential Equations, Image Processing, Holes, Non-Linear Diffusion Equations, Pixel Flux, Threshold Parameter, Coefficient of the Forcing Term

1. Introduction

Partial differential equations (PDE’s)

[22]

describe fundamental laws of physics. Examples: Hydrodynamics - Navier Stokes equations. Electrodynamics - Maxwell’s equations. Thermodynamics - Diffusion equation. Quantum Mechanics - Schr¨odinger equation. Typically, these equations determine the evolution of a scalar or vector field that depends on space and time

[31]

. Steady state problems in two or three spatial dimensions are also described by PDE’s.

PDEs got some basic properties including;

a) Linearity: Express the PDE in terms of a differential operator,

(1)

, is a function of the independent variable.

Linearity means that

is a linear operator: For any function

and

is a constant

b) Homogeneity, a PDE is homogeneous if it can be expressed as

. The PDE is inhomogeneous if it has a so called source term:

, where

is some function of the independent variables (e.g. Poisson’s equation in electrostatics). Note that one can add solutions of the homogeneous PDE to the inhomogeneous solution. If a PDE is both linear and homogeneous then it obeys the principle of superposition: If

and

are two solutions of a linear, homogeneous PDE, then

is also a solution, where

are some constants.

In order to have a well defined problem we not only need the partial differential equation that governs the physics, but also a set of boundary conditions (BC) and initial conditions (IC) to specify the problem. e.g. Heat equation for an insulated bar with its two ends immersed in heat baths

[21]

Typical boundary conditions:

Dirichlet boundary conditions: The value of the function is specified at the boundary. e.g. temperature of heat bath.

von Neumann boundary conditions: The normal gradient of the function is specified at the boundary. e.g. the electric field from a given potential at the boundary.

Cauchy boundary conditions: Both Dirichlet and von von Neumann boundary conditions are specified.

There are usually two different problems that arise in physics:

Initial value problem: This deals with time evolution of

. At a certain time

, the value of

(and possibly its derivative) is identified

[29]

Boundary value problem: Steady state, time independent problems. The value of

and/or its derivative is determined along a boundary that encloses the region of interest. e.g. electrostatic problems where the potential at the boundaries is specified. In some cases can also have open boundaries.

The applications of partial differential equations (PDEs) to computer vision and image processing started as early as the 1960s

[11]

. However, this technique did not take root until the introduction of the concept of scale space by Koenderink

[19]

and Witkin

[30]

in the 1980s. Perona and Malik’s work on anisotropic diffusion

[26]

further drew great interest from researchers towards PDE-based methods. Nowadays, PDEs have been successfully used to model image processing and computer vision

[10, 8]

. There are three methods used to design PDEs. For the first method, PDEs are written down directly, based on some mathematical understandings on the properties of the PDEs (e.g., anisotropic diffusion

[26]

, shock filter

[23]

and curve evolution based equations

[28]

. The second methods basically define an energy functional first, which collects the wish list of the desired properties of the output image, and then derives the evolution equations by computing the Euler-Lagrange equation of the energy functional

[15]

). The third kind of methods are axiomatic approaches, which first prescribe the properties, i.e., axioms, that the PDEs should hold and then deduce the form of PDEs from these axioms (e.g., Alvarez et al.’s axiomatic formulation of the scale-space theory

[1]

). All of these methods require both good insight to what properties to hold and high mathematical skills, in order to acquire the desired output.

Consider a diffusion equation coupled to chemical reactions modeled by a nonlinear term

;

(2)

This is a physical process composed of two individual processes:

is the concentration of a substance that is locally generated by a chemical reaction

, while

is spreading in space because of diffusion

[25]

. There are obviously two time scales: one for the chemical reaction and one for diffusion. Typically, fast chemical reactions require much finer time stepping than slower diffusion processes

[5]

. It could therefore be advantageous to split the two physical effects in separate models

[12]

and use different numerical methods for the two. A natural spitting in the present case is;

(3)

(4)

Non-linear Diffusion equations can be applied to initially given images. Let the image to be processed be,

is a domain in space.

is a surface and

. Introducing time

this image deforms in a partial differential evolution equation (3) according to;

(5)

where:

is the evolving image

[20]

is an operator that characterizes the given algorithm. The solution of this differential equation gives the processed image at scale

[27]

Perona et al

[26]

transformed the diffusion equation from linear to non- linear restoring borderlines. Currently the following nonlinear PDE is used,

(6)

is an unknown function defined in

This equation is accompanied by Neumann boundary conditions and initial conditions;

, the unit normal vector to the boundary of

. Assuming that

, a bounded rectangular domain

, a scaling interval,

, a non-increasing function,

is smooth,

, a smoothing kernel (e.g. the Gauss function),

(7)

is the Dirac measure at the point,

This research modeled an equation of nonlinear diffusion for image processing, optimized the nonlinear heat equation. The objectives of this research was to; optimize the nonlinear heat equation, establish viable values for

that enhance the image smoothening process.

2. Materials and Methods

Digital images are given on discrete (regular) grids

[13]

. This lends itself for discretizing the diffusion equation to obtain numerical schemes

[24]

that can be solved using MATLAB. The considered numerical schemes are stable and efficient.

As the numerical algorithms are executed in MATLAB, Optimization is done to establish the most appropriate values of

. These values are responsible for smoothening of images that are initially noisy

[14]

A model below was used in this research;

(8)

2.1. Numerical Approaches

There is a wide body of literature on the numerical solution of PDE’s

[7]

. Many algorithms are available including:

Finite difference method which is the most common approach. Based on approximating derivatives by finite differences. e.g.

where

Finite Element method: Basic idea is to divide volume of selected region into discrete pieces (polygons) and approximate the solution by simple functions on these pieces. Complex geometries whose boundaries are curved are easily implemented too. It is therefore common among engineers to model solids and structures.

Spectral methods: Approximate the solution and the boundary condition with a Fourier series and substitute into the PDE. Use the inverse Fast Fourier transform to compute the Fourier coefficients of the solution.

Monte Carlo: Use stochastic methods to find solutions of a PDE.

The fact that linear PDEs fall into three canonical forms

[17]

is not of importance from computational point of view. What is relevant whether we are dealing with;

Time evolution - Initial value problem: Wave equation, diffusion equations. Compute the time evolution for a given initial condition subject to boundary conditions.

Static solution - Boundary value problem: Laplace equation, Poisson equation: Find solutions that satisfy the boundary conditions around the region of interest.

Because of the way the boundary conditions need to be enforced, the numerical methods will be different for these two types of problems. In boundary value problems one cannot “integrate in from the boundary”

[18]

in the same sense one can “integrate forward in time” for initial value problems. For static problems the goal of the numerical method is to converge on the correct solution everywhere at once.

To apply numerical methods of solution to a domain it must be divided into equal spaced points. An example of a rectangular 2D domain can be an image or a photograph.

2.1.1. Alternating Direction Implicit (ADI) Method

The numerical discretization in time was conducted using the Alternate Direction Implicit method The merit of using the ADI method is good stability even when there is alarge convection to diffusion ratio (i.e. large Peclet number, P e > 2), which is typically the case for pultrusion.

Alternating direction implicit (ADI) methods modify Crank-Nicolson to maintain convergence but introduce the tridiagonal matrices which are more efficient

[4]

The ADI methods are based on a factored form. The factors are used to split each full timestep into two partial timesteps

[9]

. The equation for ADI below was used to solve the non-linear diffusion equation in MATLAB;

(9)

2.1.2. Two-Point Explicit Group Successive Over-Relaxation (2-EGSOR)

2-EGSOR iterative method can be displayed in a grid as below:

Download: Download full-size image

Figure 1. Grid points in 2-EGSOR.

The following equation was used in MATLAB to smoothen sampled noisy images of CCTV, MRI & Galaxy nature

(10)

2.2. Optimization

Let

denote a subset of the plane and

be a family of gray scale images, and then anisotropic diffusion is defined as;

(11)

, Laplacian

, gradient

, divergence operator

, Diffusion coefficient controls the rate of diffusion

[16]

and is usually chosen as a function of the image gradient so as to preserve edges in the image

[3]

Perona et al

[26]

pioneered the idea of anisotropic diffusion and proposed two functions for the diffusion coefficient:

(12)

And

(13)

Controls the sensitivity to edges and is chosen experimentally or as a function of the noise in the image.

Let

denote the manifold of smooth images, then the diffusion equations presented above can be interpreted as the gradient descent equations for the minimization of the energy

[6]

function

defined by,

(14)

Where

is a real-valued function which is intimately related to the diffusion coefficient. Then for any compactly supported infinitely differentiable test function

(15)

The last line of

[15]

follows from multidimensional integration by parts

[2]

. Letting

denote the gradient of

with respect to

the inner product evaluated at

, this gives;

(16)

Thus, the gradient descent equations on the function

are given by;

(17)

Thus by letting

the anisotropic diffusion equations are obtained.

Optimization was done for (8) using the objective function;

(18)

, pixel flux

, Coefficient of theForcing term

, the threshold parameter

The minimized values of

were applied in the numerical schemes of ADI and 2-EGSOR in MATLAB generating smoothened images.

3. Results

Parameter estimation gives the approximate values of time and the constants C_v, C_f and b in (8).

3.1. Approximated Parameters Under EGSOR

Table 1. Parameter estimation for MRI image 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

Download: Download full-size image

Figure 2. Time against C_v,C_fand b for MRI image under EGSOR.

Download: Download full-size image

Figure 3. Time against Cv, Cf and b for CCTV image under EGSOR.

Table 2. Parameter estimation for CCTV image under EGSOR.

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

Table 3. Parameter estimation for MRI image 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

Download: Download full-size image

Figure 4. Time against C_v,C_fand b for MRI image under ADI.

Table 4. Parameter estimation for CCTV image under ADI.

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

Download: Download full-size image

Figure 5. Time against C_v, C_f, b for CCTV image under ADI.

4. Discussion

From the graphs the parameter

which is the threshold parameter maintains the highest value as opposed to

which is the pixel flux, it registers the lowest value implying pixel diffusion is limited in the smoothening process. On the other hand the forcing term coefficient

is of moderate value, inputing minimum force into the image in the course of smoothening hence limiting blurring.

5. Conclusion

The paper sort to optimize parameters

. Parameter

works best when optimized to the highest value as compared to

. Smoothening of images can be done well when parameters are minimized. I recommend use of other computer programs other than MATLAB in future optimization.

Abbreviations

MRI: Magnetic Resonance Imaging

CCTV: Closed Circuit Television

ADI: Alternating Direction Implicit

2-EGSOR: Two-point Explicit Group Successive Over-Relaxation

MATLAB: Matrix Laboratory

PDEs: Partial differential equations

Conflicts of Interest

The authors declare no conflicts of interest.

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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

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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

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Fwamba RN, Chepkwony I, Fwamba WS. Optimization of the Non-Linear Diffussion Equations. Sci J Appl Math Stat. 2024;12(1):13-19. doi: 10.11648/j.sjams.20241201.12

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@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}
}

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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 -

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Author Information

Rukia Nasimiyu Fwamba

Department of Mathematics & Actuarial Science, Kenyatta University, Nairobi, Kenya

Contact Email

http://orcid.org/0009-0008-1651-3800
Isaac Chepkwony

Department of Mathematics & Actuarial Science, Kenyatta University, Nairobi, Kenya

Contact Email

http://orcid.org/0000-0001-5528-9487
Wekulo Saidi Fwamba

Department of Disaster Preparedness and Engineering Management, Masinde Muliro University of Science and Technology, Kakamega, Kenya

Contact Email

http://orcid.org/0009-0007-0931-4347

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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

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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

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Fwamba RN, Chepkwony I, Fwamba WS. Optimization of the Non-Linear Diffussion Equations. Sci J Appl Math Stat. 2024;12(1):13-19. doi: 10.11648/j.sjams.20241201.12

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@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}
}

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