Homotopy Analysis Method

In this section, we briefly introduce the standard homotopy analysis method (HAM) for a general nonlinear problem. This will be the first step to apply the DHAM (as a discrete form of HAM). This method has been known as an analytic method for solving nonlinear problems [97]. The method attracts with a simple way of controlling and adjusting the convergence region and rate of solution series of nonlinear problems. This method has been improved [99, 100, 95, 101, 97, 98] and widely used to solve linear or nonlinear ODE and PDE problems showing great performance. Different fields in science, engineering, finance or technological fields have employed the method to solve or improve many types of problem [184, 1, 72, 79, 50, 2, 139] or even to find some new solutions of a few nonlinear equations which have never been solved by previous analytic methods or even numerical methods [96]. Unlike the perturbation techniques method [116, 117] or the so-called non-perturbation techniques, such as theδ-expansion method [7], Adomian’s decomposition method [5] and so on, HAM does not depend on small/large physical parameters and is valid not only for weakly nonlinear problems, but also for strongly nonlinear problems and at the same time the convergence of the solution has been proved [97]. The method transforms the nonlinear problem to a hight order linear approximation.

Consider a non-linear equation of the following form:

N[u(x, t)] = 0 (2.120) subject to the initial condition

u(x,0) =u0(x) (2.121) where N is a nonlinear operator which represents the whole equation, x ∈ Rn, t

denotes independent variables and u(x) is an unknown function. Based on Liao [99], the following zero-order deformation equation is constructed from the original equation (2.120), as follows:

(1−q)L[φ(x, t;q)−u0(x, t)] =q~H(x, t)N[φ(x, t;q)] (2.122) whereu0(x, t) is an initial guess, ~is an auxiliary parameter,q∈[0,1] is an embedding parameter, φ(x, t;q) is a function of t and q, His a nonzero auxiliary function and L

is an auxiliary linear operator with the following property

L[φ(x, t)] = 0 when φ(x, t) = 0. (2.123) It should be emphasized that we have the freedom to choose the initial approximation, the auxiliary linear operatorL, the auxiliary parameter~and the auxiliary functionH. Obviously, since~̸= 0,H ̸= 0, whenq = 0 andq= 1, it holds thatφ(x, t; 0) =u0(x, t) and φ(x, t; 1) = u(x), t, respectively. Thus, as q increases from 0 to 1, the solution

φ(x, t;q) deforms from the initial guess u0(x, t) to the solution u(x, t). Expanding

φ(x, t;q) in the Taylor series with respect to q, one has

φ(x, t;q) =u0(x, t) + ∞ ∑ m=1 um(x, t)qm, (2.124) where um(x, t) = 1 m! ∂φm(x, t;q) ∂qm q=0 . (2.125)

If the auxiliary linear operator, the initial guess, the auxiliary parameter ~, and the auxiliary function H ̸= 0 are chosen such that the series (2.124) converges at q = 1, one has u(x, t) =u0(x, t) + ∞ ∑ m=1 um(x, t), (2.126)

which must be one of the solutions of the original nonlinear equation, as proved by Liao [94].

Let us define ⃗uk(x) ={u0(x), . . . , uk(x)} as the vector of the composed solutions.

According to the definition for um(x, t) (2.125), the governing equation and the corre-

sponding initial condition ofum(x, t) can be deduced from the zero-order deformation

equation (2.122) and (2.123) as follows. Differentiating the zero-order deformation equation (2.122)m-times with respect toq and dividing them bym! and finally setting

q= 0, we obtain the following mth-order deformation problem:

L[um(x, t)−χmum−1(x, t)] =~HRm[⃗um−1(x, t)] (2.127) where Rm[⃗um−1(x, t)] = 1 (m−1)! ∂m−1N[φ(x, t;q)] ∂qm−1 q=0 (2.128) and χm= { 1, m >1 0, m61

Applying the inverse operatorL−1 on both sides of equation (2.127) we can obtain

um(x, t) =χmum−1(x) +L−1{~HRm[⃗um−1(x, t)]} (2.129) and at the Mth order we haveuM(x, t) =

∑M

may obtain a more accurate approximate solution of the original equation (2.120). Note that when solving themth-order deformation equations (2.127)Rm[⃗um−1(x)] depends only on u0, u1, u2, ..., um−1, which are already known. In our applications we will use two different linear operators, as follows6:

L1(φ(x, t;q)) = ∂φ(x, t;q) ∂t +θφ(x, t;q), (2.130) and L2(φ(x, t;q)) = (1 +t) ∂φ(x, t;q) ∂t +φ(x, t;q), (2.131)

with the properties L1[C1e−t] = 0,L2[₁₊C2_t] = 0 and θ is a positive constant. It can be easy found thatL−₁1 =e−θt∫₀teθτφdτ and L−₂1 = ₁₊1_t∫₀tφdτ.

Below we shall writeuDHAM1 and uDHAM2,notation used to make the distinction between the linear operators L1 and L2 respectively, for finding the solution u(x) of the equation (2.120).

By applying the inverse operators L−₁1 and L−₂1 to both sides of the high-order defor- mation equation (2.127), subject to the initial condition

um(x,0) = 0,

themth terms of the solution are respectively obtained in the following forms:

uDHAM_m 1(x, t) =χmumDHAM−1 1(x, t) +~exp(−θt) ∫ t 0 exp(θt)H1(τ)Rm[⃗uDHAMm−1 1]dτ (2.132) and uDHAM_m 2(x, t) =χmuDHAMm−1 2(x, t) + ~ 1 +t ∫ t 0 H2(τ)Rm[u⃗DHAMm−1 2]dτ (2.133) where Rm[⃗um−1] is as defined by (4.3). Indeed, the solution u(x) of the original non- linear equation (2.120), while using the linear operatorsL1(u) and L2(u) respectively, is expressed in the following form:

uDHAM1(x, t) = +∞ ∑ m=0 amexp(−mt) (2.134) and uDHAM2(x, t) = +∞ ∑ m=0 bm (1 +t)m (2.135)

where theam, bm’s are coefficients depending on x.

According to the rule of solution expression denoted by (2.134) and from equation (2.132), the auxiliary function H(τ) should be in the form H(τ) = e−kτ, where k is an integer. It is found that, when k 6 1, the solution of the high-order deformation

6_{The operatorL}

1(u) in equation (2.130) is a general case of the linear operator applied by Liao et.

equation (2.132) contains the termte−t, which incidentally disobeys the rule of solution expression (2.134). When k > 2, the base e−2t always disappears in the solution expression of the high-order deformation equation (2.132), so that the coefficient of the terme−2t cannot be modified even if the order of approximation tends to infinity. So, according to the so-called rule of coefficient ergodicity by Liao [99], we have to setk= 2, which uniquely determines the corresponding auxiliary function H1(τ) =exp(−2τ).

Following the same discussion above for the rule of solution expression denoted by (2.135) and from equation (2.133), we get H2(τ) = ₍₁₊1_τ)2. Thus, starting from the

initial approximationu0(x, t) =u0(x), we can use the recurrence formulae (2.132) and (2.133) to successively obtainum(x, t) form>1 and at the same time the Mth order

approximation has the formuM(x, t) =

∑M

m=0um(x, t).

The HAM has been applied all these years for continuous functions because of the freedom to use different base functions to approximate a nonlinear problem, its validity for nonlinear problems and a convenient way to adjust the convergence region and the rate of the approximation series. The first attempt at a discrete version of HAM was made recently by Zhu et al. [183] for a diffusion equation. Unfortunately, applying it to a realistic nonlinear PDE models does not suit due to the analytical difficulties in working out the high order approximations demanded by HAM. Sorting this problem out whilst using the HAM’s quality was the motivation of our work presented in Chap- ters 4. In this Chapter you can see how we can slightly modify and apply the method to image denoising and image segmentation tasks.

Chapter 3

Review of Variational Models for

Image Restoration and

Segmentation Techniques

3.1

Introduction

The calculus of variations deals with the theory of finding the maxima and minima of quantities defined as integrals containing unknown functions. The optimisation of ap- propriately chosen functionals with variational models involves the solution of nonlinear partial differential equations (PDEs) derived as necessary optimality conditions.

The history of calculus of variations begins with Newton, then initiated as a sub- ject by the Bernoulli family. The first major contribution was made by the work of Euler, Lagrange and Laplace, with the classical problems of finding the path, curve, surface, etc., for which a given function has a stationary value. Further contributions were made in the nineteenth century by Hamilton, Dirichlet and Hilbert. In modern times, the calculus of variations has a wide-range of applications in classical solutions to minimization problems prescribed by boundary value problems. These problems involve certain types of differential equations, known as the associated Euler–Lagrange equations. Minimization problems that can be analyzed by the calculus of variations such as minimal curve length can be formulated as optimization problems. Such opti- mization problems are fundamental in many areas such as physics, engineering and all branches of mathematics.

In this chapter, we will introduce some basic mathematical analysis of nonlinear minimization principles and their application in image processing techniques. We ex- plore the variational techniques, among the many existing approaches, due to the ex- cellent results obtained from their applications. The main reason behind the success of variational PDE based models in image processing is due to the ease of imposing geometric regularity, such as smoothness. Various existing numerical methods for the realization of these models will be discussed. Mainly for two fundamental problems of image processing this thesis is concerned with: image denoising and digital image

segmentation. Both are important in real life due to their wide application in image restoration and reconstruction. From a mathematical point of view a large amount of research on these topics has been opened and many still remain unsolved.

Some mathematical analysis and properties of the total variation (TV) regular- ization functional are introduced. Some existing models used for solving the partial differential equation such as the Rudin-Osher-Fatemi (ROF) [135] model in denois- ing or Chan-Vese minimization arising in segmentation will be discussed briefly. The pros and cons of different smoothers (local and global) for solving PDEs arising from denoising and segmentation and other existing models are also described.