A Fourth-Order Compact Finite Difference Scheme for

Vol. 5, No. 4, pp. 361-386 November 2015

East Asian Journal on Applied Mathematics doi: 10.4208/eajam.270415.280915a

A Fourth-Order Compact Finite Difference Scheme for Higher-Order PDE-Based Image Registration Sopida Jewprasert1 , Noppadol Chumchob1,2,∗ and Chantana Chantrapornchai3 1

Department of Mathematics, Faculty of Science, Silpakorn University, Nakorn Pathom 73000, Thailand. 2 Centre of Excellence in Mathematics, CHE, Si Ayutthaya Rd., Bangkok 10400, Thailand. 3 Department of Computer Engineering, Faculty of Engineering, Kasetsart University, Bangkok 10900, Thailand. Received 27 April 2015; Accepted (in revised version) 28 September 2015.

Abstract. Image registration is an ill-posed problem that has been studied widely in recent years. The so-called curvature-based image registration method is one of the most effective and well-known approaches, as it produces smooth solutions and allows an automatic rigid alignment. An important outstanding issue is the accurate and efficient numerical solution of the Euler-Lagrange system of two coupled nonlinear biharmonic equations, addressed in this article. We propose a fourth-order compact (FOC) finite difference scheme using a splitting operator on a 9-point stencil, and discuss how the resulting nonlinear discrete system can be solved efficiently by a nonlinear multi-grid (NMG) method. Thus after measuring the h-ellipticity of the nonlinear discrete operator involved by a local Fourier analysis (LFA), we show that our FOC finite difference method is amenable to multi-grid (MG) methods and an appropriate point-wise smoothing procedure. A high potential point-wise smoother using an outer-inner iteration method is shown to be effective by the LFA and numerical experiments. Real medical images are used to compare the accuracy and efficiency of our approach and the standard secondorder central (SSOC) finite difference scheme in the same NMG framework. As expected for a higher-order finite difference scheme, the images generated by our FOC finite difference scheme prove significantly more accurate than those computed using the SSOC finite difference scheme. Our numerical results are consistent with the LFA analysis, and also demonstrate that the NMG method converges within a few steps. AMS subject classifications: 65F10, 65M55, 68U10 Key words: Curvature image registration, fourth-order compact finite difference scheme, local Fourier analysis, nonlinear multi-grid method, nonlinear biharmonic equation.

∗

Corresponding author. Email addresses: (N. Chumchob),

http://www.global-sci.org/eajam

(S. Jewprasert), (C. Chantrapornchai) 361

⃝2015 c Global-Science Press

362

S. Jewprasert, N. Chumchob and C. Chantrapornchai

1. Introduction Image registration is a fundamental problem in the field of image analysis. The problem arises when two or more images are taken at different times, or from different viewpoints or by different sensors, and then must be compared and integrated into one representation so that the information can be accessed readily and accurately. Over the years, image analysis has been used routinely in medical diagnosis, treatment guidance and monitoring. Medical applications are surveyed in Refs. [31, 43, 44] and references therein. Higher-order partial differential equation (PDE) based image registration methods have proven to be very valuable in a wide range of applications [16–18, 24, 25, 33–35, 37, 38], but the numerical solutions often need to be improved. In deriving the associated PDE in such methods, we may let Ω ⊂ !d denote an image domain and R, T : Ω → ! two images of the same scene, respectively referred to as the reference and template image. The problem is to compute an optimal deformation or displacement field u : x %→u (x) = (u1 (x), u2 (x), · · · , ud (x))⊤

u : !d → !d ,

such that the transformed template T (x + u(x)) = T (u) becomes similar to the reference R. It is also assumed that the given images are smooth, and compactly supported functions map the image domain Ω into V ⊂ [0, ∞), where d ∈ " represents the spatial dimension of the images — usually d = 2 (images) or d = 3 (volume data set) with boundary ∂ Ω. Here we consider the two-dimensional case (d = 2), and that Ω = [0, 1]2 ⊂ !2 and V = [0, 1] for grey intensity images. If the image intensities of R and T are comparable, an obvious method for computing the unknown deformation u would be minimisation of the sum of squared differences: " # ! 1 2 (T (x + u (x)) − R (x)) dx . (1.1) min ) (u) = u 2 Ω Since the minimisation of ) is generally an ill-posed problem in the sense of Hadamard, a regularisation technique is used to impose a constraint on the solution u by a regulariser + that penalises unreasonable and irregular solutions from some a priori knowledge. Thus the image registration problem can be posed as the minimisation problem of a joint energy functional min {,α [u] = ) (u) + α+ (u)} (1.2) u∈"

involving the regularisation parameter α > 0 that compromises similarity and regularity, with u searched over a set - of admissible functions minimising ,α . For two reasons, we select the curvature-based regulariser 2 " 1$ +(u) = (∆ul )2 dx (1.3) 2 l=1 Ω introduced by Fischer & Modersitzki [24]. Firstly, ∆ul in the integrand may be viewed as an approximation to the mean curvature of the surface of ul in !3 since κ M (ul ) = ∇ · /

∇ul

1+|∇ul |2

=

(1+u2l )ul x1

x1 x1

−2ul

u

u

+(1+u2l )ul

x1 l x2 l x1 x2 (1+u2l +u2l )3/2 x1 x2

x2

x2 x2

.

A Fourth-Order Compact Finite Difference Scheme for Higher-Order PDE-Based Image Registration

363

We observe that assuming |∇ul | ≈ 0 yields κ M (ul ) ≈ ∆ul , so this regulariser involving second-order derivatives does penalise oscillations and yields very smooth deformations. Secondly, in contrast to many other PDE-based image registration techniques, such as the elastic image registration method that uses a regulariser based on the linearised elastic potential of the deformation u [5, 10] " 2 & '2 % elas ∂ x l um + ∂ x m ul + (λ/2)(∇ · u)2 )dx (µ > 0 and λ ≥ 0) + (u) = ((µ/4) Ω

l,m=1

or the diffusion image registration method using a regulariser based on the L 2 norm of ∇ul [23] " 2 1% diff + (u) = |∇ul |2 dx , 2 l=1 Ω

our choice of higher-order regulariser (1.3) does not penalise affine linear transformations and hence does not require an additional affine linear pre-registration step to be successful. From the calculus of variations, the minimiser u = (u1 (x 1 , x 2 ), u2 (x 1 , x 2 ))⊤ of the joint energy functional ,α in (1.2) defined by the nonlinear fidelity term ) in (1.1) and the regularisation term + in (1.3) satisfies the Euler-Lagrange (EL) system of two coupled nonlinear biharmonic equations ! f1 (u) + α∆2 u1 = 0 , (1.4) f2 (u) + α∆2 u2 = 0 , subject to the special boundary conditions ∇ul = 0 and ∇∆ul = 0 on ∂ Ω ,

for l = 1, 2 .

(1.5)

Here f l (u) = (T (u) − R) ∂ul T (u) and ∆2 ul are derived from the first variations (or Gâteaux derivatives) of ) and +, respectively. Our main interest here is how to effectively solve the EL system (1.4) subject to the boundary conditions (1.5). We note that various efficient techniques have been proposed for the numerical solution of biharmonic equations in the literature — e.g. see [2,19–21,42,46,47,49,50] and references therein. However, effective numerical solution of the EL system of two coupled nonlinear biharmonic equations (1.4) subject to the boundary conditions (1.5) is rarely encountered. Let us first briefly summarise the existing approaches using standard second-order central (SSOC) finite difference schemes with non-multi-grid or multi-grid methods as follows. (1) SSOC finite difference schemes with non-multi-grid methods. The gradient descent method is the most commonly used non-multi-grid method. The basic idea is to introduce an artificial time variable t and then replace the system of nonlinear elliptic PDE (1.4) with the system of nonlinear parabolic PDE ! ∂ t u1 + α∆2 u1 = − f1 (u) , (1.6) ∂ t u2 + α∆2 u2 = − f2 (u) ,

364


where u = u(x, t) = (u1 (x, t), u2 (x, t))⊤ converges to the solution of (1.4) when t → ∞, assuming an initial solution u(x, 0) — typically, u(x, 0) = 0. This method circumvents the nonlinearity on the right-hand side. For example, applying SSOC finite differences to the first and second terms in (1.4) and using the semi-implicit time marching scheme produces the following discrete system in matrix-vector form: ⎧ , -−1 . / 2 % ⎪ (k+1) (k) (k) (k) ⎪ = I − ατ A u − τ f (u , u ) ⎨ u 1

,

⎪ ⎪ ⎩ u(k+1) = I − ατ 2

l=1 2 %

A

l=1

1

-−1

1

1

2

. / , (k) (k) (k) u2 − τ f2 (u1 , u2 )

k = 1, 2, 3 · · · .

(1.7)

Here I is the identity matrix, f l (u1k , u2k ) is the discretised version of the first term in (1.4), τ > 0 is the time-step determined by a forward difference approximation of the time derivative ∂ t ul , and A is the coefficient matrix from the discretisation of the biharmonic operator ∆2 subject to the boundary conditions (1.5). The additive operator splitting (AOS) scheme [44], the discrete cosine transform (DCT) method [24,25] or the Fourier transform (FT) method [41, 54] can be used to solve the discrete system (1.7) to provide visually pleasing registration results, but they are all very slow to meet the necessary condition for the minimisation represented by (1.2) — specifically, the resulting linear system involved in (1.7) has to be solved many times with a changing right-hand side, so the convergence is very slow (cf. Ref. [16] and references therein). (2) SSOC finite difference schemes with multi-grid methods. The multi-grid (MG) method is one of the most efficient finite difference techniques for solving elliptic PDE. The fundamental idea is to accelerate the convergence of some basic iterative method on the finest grid by relying on the complementary interplay of smoothing and coarse-grid correction principles. The smoothing process, involving the basic iterative method (also known as relaxation or smoother), aims at reducing the high frequency components of the error in the current approximation that cannot be represented on coarser grids — whereas the coarse-grid correction solves for the low frequency components of the error, well represented on the coarser grid [11]. The appropriate combination of these smoothing and coarse-grid correction procedures yields an efficient MG method, which has fast convergence and is independent of the mesh size. The discrete system for the two linear parabolic PDE given by (1.7) at each time step k can be solved efficiently by a linear MG method — e.g. see Ref. [33]. However, as the otherwise convenient MG method is non-optimally convergent for a gradient descent method, previous work by Chumchob & Chen [16] on the SSOC finite difference scheme directly solved the discretised system of non-elliptic PDE used a full approximation scheme nonlinear MG (FAS-NMG) method with robust fixed-point (FP) smoothers. Numerical tests showed that this FAS-NMG method has significantly faster convergence. However, for even more accurate PDE-based image registration with better computational efficiency, in this article we propose a higher-order compact (HOC) finite difference scheme


365

combined with an efficient FAS-NMG method for the numerical solution of the EL system (1.4) as an extension of the numerical method developed in Ref. [16]. We adopt a fourthorder compact (FOC) finite difference scheme for (1.4) on a 9-point compact cell at the interior node, with a higher-order accurate local truncation error leading to more precise numerical results. This FOC finite difference scheme simply incorporates the boundary conditions (1.5), so we do not need to deal with them approximately. Moreover, whereas the SSOC finite difference scheme commonly used to solve (1.4) requires a large number of grid points to be effective, our FOC finite difference scheme provides much more satisfactory and accurate registration results. Finally, we shall demonstrate that our associated FAS-NMG method detailed below efficiently solves the resulting nonlinear discrete system, to deliver fast and accurate registration results for a wide range of real medical images. Section 2 presents the fourth-order compact (FOC) finite difference discretisation strategy for the EL system in (1.4), subject to the boundary conditions (1.5). Our efficient FAS-NMG method to solve the higher-order accurate solution on both the fine and the coarse grids is discussed in Section 3. Local Fourier analysis (LFA) for the associated nonlinear discrete operator and the proposed smoother is also presented in this section. Section 4 describes the numerical experiments that demonstrate the high accuracy of our HOC finite difference scheme, and the computational efficiency of our proposed NMG method in registering real medical images. Our concluding remarks are made in Section 5.

2. Proposed FOC Finite Difference Scheme To solve the EL system (1.4) numerically, each biharmonic equation can be discretised by a higher-order finite difference scheme on a uniform grid — i.e. a FOC finite difference scheme on a 9-point compact cell as envisaged by Stephenson [50]. His approach discretises the biharmonic equation using not only the grid values of the unknown solutions ul but also the grid values of the gradients ul x and ul x at selected grid points, 1 2 and so involves substantial computation to solve the resulting system of six equations in {u1 , u1 x , u1 x , u2 , u2 x , u2 x } to obtain the desired solutions u1 and u2 . However, we prefer 1 2 1 2 to split the EL system (1.4) into a system of four Poisson-type equations and use a 9-point compact cell for a fourth-order approximation of (1.4) as in Ref. [46] and references therein, with the advantage that we only need to solve a system of four algebraic equations rather than six. Thus we convert (1.4) into the following system involving additional unknown functions v1 = −∆u1 and v2 = −∆u2 : ⎧ −∆u1 − v1 = 0 , ⎪ ⎨ −∆u2 − v2 = 0 , (2.1) f (u) − α∆v1 = 0 , ⎪ ⎩ 1 f2 (u) − α∆v2 = 0 , subject to the boundary conditions that become ∇ul = 0 and ∇v l = 0 on ∂ Ω. In order to discretise the EL system (2.1), let . / . / & h' & h' ul i, j = uhl x 1i , x 2 j and v l i, j = v hl x 1i , x 2 j

366


denote the grid function for l = 1, 2 with the grid spacing h = (h1 , h2 ) = (1/n1 , 1/n2 ), where the integers n1 = 1/h1 and n2 = 1/h2 are the number of uniform intervals in the x 1 and x 2 coordinate directions. Each grid point x in the discretised domain Ωh is given by x = (x 1i , x 2 j )⊤ = ((2i − 1)h1 /2, (2 j − 1)h2 /2)⊤ for 1 ≤ i ≤ n1 and 1 ≤ j ≤ n2 .

Let (z0h )i, j = z0h (x 1i , x 2 j ) denote the grid function over the discretised domain Ωh for l

l

0l = 1, · · · , 4 , where z h = uh or v h for l = 1, 2. We can write the SSOC finite difference l l 0l operators as (δ2x z0h )i, j = 1

l

(z0h )i+1, j − 2(z0h )i, j + (z0h )i−1, j l

l h21

l

,

(δ2x z0h )i, j = l

2

(z0h )i, j+1 − 2(z0h )i, j + (z0h )i, j−1 l

l

l

h22

.

Using Taylor series expansions at the grid point (x 1i , x 2 j )⊤, we obtain (z0l 2 )i, j = x

β

(δ2x z0h )i, j β l

−

h2β 12

(z0l 4 )i, j + 3 (h4β ) , x

β = 1, 2 ,

(2.2)

β

where z0l m = ∂ m z0l /∂ x βm is the m-th partial derivative with respect to x β of the function z0l . x

β

Our FOC finite difference scheme for (2.1) is based on the finite difference operators 1

(δ¯2x z0h )i, j β l

h2β

δ2x β

2

(z0h )i, j ,

β = 1, 2 .

(2.3)

(δ2x z0h )i, j = (δ¯2x z0h )i, j + 3 (h4β ) ,

β = 1, 2 ,

(2.4)

2 h h 4 (δ¯−2 x δ x z0 )i, j = (z0 )i, j + 3 (hβ ) ,

β = 1, 2 ,

(2.5)

= 1+

12

l

According to the Numerov formula [1], β

l

l x2

β

l

or symbolically

β

β

l

l x2 l

¯−2 = (δ¯2 )−1 denotes the inverse of δ ¯2 . where δ x x x β

β

β

We now apply the above FOC approximations to the second-order derivatives involved in (2.1), yielding symbolically the discrete EL system ⎧ 2 h h 4 ¯−2 2 −(δ¯−2 ⎪ x 1 δ x 1 + δ x 2 δ x 2 )(u1 )i, j − (v1 )i, j = 3 (h ) , ⎪ ⎪ ⎪ ⎨ −(δ¯−2 δ2 + δ¯−2 δ2 )(uh ) − (v h ) = 3 (h4 ) , x x x x 2 i, j 2 i, j 1

1

2

2

2 h h h h 4 ¯−2 2 ⎪ −α(δ¯−2 ⎪ x 1 δ x 1 + δ x 2 δ x 2 )(v1 )i, j + f1 (u1 , u2 )i, j = 3 (h ) , ⎪ ⎪ ⎩ 2 h h h h 4 ¯−2 2 −α(δ¯−2 x δ x + δ x δ x )(v2 )i, j + f2 (u1 , u2 )i, j = 3 (h ) , 1

1

2

(2.6)

2

where 3 (h4 ) means the truncated terms 3 (h41 + h42 ). By multiplying the symbolic operators


367

¯2 δ ¯2 = δ¯2 δ¯2 and dropping the 3 (h4 ) terms, (2.6) can be rewritten δ x x x x 1

2

2

1

⎧ −(δ¯2x δ2x + δ¯2x δ2x )(uh1 )i, j − δ¯2x δ¯2x (v1h )i, j = 0 , ⎪ ⎪ 2 1 1 2 1 2 ⎪ ⎪ ⎨ −(δ¯2 δ2 + δ¯2 δ2 )(uh ) − δ¯2 δ¯2 (v h ) = 0 , x x x x 2 i, j x x 2 i, j 2

1

2

or

1

2

1

2

¯2 δ¯2 f h (uh , uh )i, j = 0 , ⎪ −α(δ¯2x δ2x + δ¯2x δ2x )(v1h )i, j + δ ⎪ x1 x2 1 1 2 2 1 1 2 ⎪ ⎪ ⎩ 2 2 2 2 h 2 ¯2 h h h ¯ ¯ ¯ −α(δ x δ x + δ x δ x )(v2 )i, j + δ x δ x f2 (u1 , u2 )i, j = 0 , ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

1

1

2

1

(2.7)

2

−4 h (uh1 )i, j − 5 h (v1h )i, j = 0 , 45 6 3 61h ( z h )i, j −4 h (uh2 )i, j − 5 h (v2h )i, j = 0 , 3 45 6 62h ( z h )i, j −α4 h (v1h )i, j + 5 h f1h (uh1 , uh2 )i, j = 0 , 3 45 6 63h ( z h )i, j −α4 h (v2h )i, j + 5 h f2h (uh1 , uh2 )i, j = 0 , 3 45 6 64h ( z h )i, j

(2.8)

where (z h )i, j = ((z1h )i, j , (z2h )i, j , (z3h )i, j , (z4h )i, j )⊤ = ((uh1 )i, j , (uh2 )i, j , (v1h )i, j , (v2h )i, j )⊤ and the symbolic operators are ¯2 δ2 4 h = δ¯2x δ2x + δ x x 2

1

1

(2.9)

2

and 5 h =1+

h21 12

δ2x + 1

h22 12

δ2x , 2

(2.10)

respectively. In passing, we note that the FOC approximations in (2.8) need to be adjusted at the image boundary ∂ Ωh using the homogeneous Neumann boundary conditions.

3. An Efficient FAS-NMG Method As previously indicated, several MG methods have been developed to solve linear or nonlinear discrete systems. Recently, two specialised MG methods have been proposed to solve sparse linear systems arising from a three-dimensional Poisson equation on a cubic domain involving a FOC finite difference scheme over unequal mesh sizes [27], and also an efficient MG method to solve the stream function formulation of the incompressible Navier-Stokes equations [52]. For the application of MG computational methods to image registration, reference may be made to Refs. [6, 14–18, 22, 26, 29, 33, 36, 39, 40, 51, 53, 57]. We now proceed to discuss our proposed MG method for the discretised system (2.8) of two nonlinear second-order PDEs. Due to its nonlinearity, the use of a FAS-NMG method

368


seems natural, but a nonlinear relaxation method that exhibits good smoothing properties is crucial. Let us denote our nonlinear discrete system in (2.8) by ⎧ & h' h 6 = g1h , ⎪ 1 z ⎪ ⎪ & ' ⎪ ⎨ 6 h zh = g h , 2 2 (3.1) & h' h ⎪ 63 z = g3h , ⎪ ⎪ ⎪ & ' ⎩ 64h z h = g4h ,

but drop the symbol (·)i, j for simplicity and set g0l = 0 on the finest grid for 0l = 1, · · · , 4.

3.1. The FAS-NMG algorithm

& '⊤ Let z h = z h1 , z h2 , z h3 , z h4 be the approximation of z h after a few smoothing iterations in a pre-smoothing step on a fine-grid problem. The algebraic error e h of the solution is then h given by e h = z h − z , where z h denotes the exact solution of (3.1). The residual equation system for the 0lth equation is 60h (z h + e h ) − 60h (z h ) = g0h − 60h (z h ) = r0h . l

l

l

l

l

In order to correct the approximate solution z h on the fine grid, we need the error e h , but its computation is prohibitively expensive on the fine grid. However, since high frequency components of the error in the pre-smoothing step have already been removed by the smoother, we can transfer the nonlinear system to the coarse grid as follows: h

h

60h (z + e h ) = r0h + 60h (z ) 6 3l 45l 6 3 l 45 60h (z h ) l

g0h l

H

−→

H

60H (z + e H ) = r0H + 60H (z ) 6 3l 45l 6 , 3 l 45 60H (z H ) l

(3.2)

g0H l

where H is the index for the new cell size H1 × H2 with H1 ≥ h1 and H2 ≥ h2 , and the right-hand side g0H ̸= 0 on the coarse grid. After the nonlinear residual equation (3.2) l on the coarse grid has been solved with a method of choice, the coarse-grid correction H e H = z H − z is interpolated back to that on the fine grid e h , which can now be used to h h h update the approximate solution z of the original system on the fine grid z new = z + e h (coarse-grid correction step). In the last step of the FAS-NMG method, the smoother is again applied to remove high frequency parts of the interpolated error (post-smoothing step). The MG components for solving (3.1) are as follows. 1. The standard coarsening method is used in the coarse-grid domain ΩH by doubling the grid size in each space direction — i.e. hl → 2hl = H l for l = 1, 2. 2. The intergrid transfer operators are determined by averaging and bilinear interpolation techniques, for the restriction and interpolation operators denoted respectively by IhH and I Hh — cf. [6, 30, 53, 55, 56] for details.


369

3. The discretisation coarse grid approximation (DCA) method is applied to compute the coarse-grid operator of 60h (z h ), where the EL system is re-discretised directly. For l

4 h (z0l h ) and 5 h (z0l h ), the corresponding coarse-grid parts 4 H (z0l H ) and 5 H (z0l H ) are obtained by restricting z0l h and a simple adaptation of the grid size to the discretised operators; and for f lH (z1H , z2H ), we first use the restriction operator for the first and second components of z h (i.e. z1h and z2h ) and the given images Rh and T h , and then compute the corresponding coarse-grid part of f lh (z1h , z2h ).

In addition, our FAS-NMG method requires a suitable smoother based on some iterative relaxation method, which is often the decisive factor in determining whether or not any MG method converges as previously mentioned. This issue will be discussed after we introduce the local Fourier analysis (LFA), to analyse the smoothing property of the smoother. The pseudo-code implementation of our proposed FAS-NMG Algorithm 3.1 is: Algorithm 3.1 FAS-NMG Algorithm → z h ← FAS-NMG(z h , α, − ϵ)

1) Select α, ϵ = (ϵ1 , ϵ2 , ϵ3 , ϵ4 )⊤ and initial guess solutions z ini t ial = (z h1 , z h2 , z h3 , z h4 )⊤ on the finest grid h

h

2) Set K = 0, [z h ]K = z ini t ial , ϵ72 = ϵ2 + 1, ϵ73 = ϵ3 + 1, and ϵ74 = ϵ4 + 1 & ' 3) While K < ϵ1 AND ϵ72 > ϵ2 AND ϵ73 > ϵ3 AND ϵ74 > ϵ4

3.1) [z h ]K+1 ← FASCYC(z h , g1h , g2h , g3h , g4h , Rh , T h , ν1 , ν2 , α, Siter) ! # ||g0h −60h (z h )||2 l l 3.2) Compute ϵ72 = mean | 0l = 1, · · · , 4 h ||g0h −60h (z ini tial )||2 l

3.3) Compute ϵ73 =

) h (Rh ,T h (u h )) ) h (Rh ,T h )

l

at step K + 1,

& ' [Recall that ) h Rh , T h (·) ∼

h2 h h 2 2 ||R , T (·) ||2 ]

8 8 8 h h h h K+1 h K 8 h h h )) − ) (R , T ((u ) ))8 3.4) Compute the difference ϵ74 = 8) (R , T ((u )

3.5) Set K = K + 1 end h

h

z ← FASCYC(z , g1h , g2h , g3h , g4h , Rh , T h , ν1 , ν2 , α, Siter) 3.1.1) If Ωh = coarsest g r id (|Ωh | = 16 × 16), solve (3.1) using a method of our choice and then stop. Else continue with following step. 3.1.2) Pre-smoothing: 9 : h h For k = 1 to ν1 , z ← Smoother(z , g1h , g2h , g3h , g4h , Rh , T h , α, Siter)

370


3.1.3) Restriction to the coarse grid: H h H h H h H h z 1 ← IhH z 1 , z 2 ← IhH z 2 , z 3 ← IhH z 3 , z 4 ← IhH z 4 , RH ← IhH Rh , T H ← IhH T h 3.1.4) Set initial solution the coarse-grid problem: 9 H the : 9 H for : H H H 7 z1 , 7 z2H , 7 z3H , 7 z4H ← z 1 , z 2 , z 3 , z 4

3.1.5) Compute the new right-hand side & H 'for the coarse-grid problem: & h' h H g1H ← IhH (g1h − 61h (z )) + 61H z , g2H ← IhH (g2h − 62h z ) + 62H (z ) & H' h h H g3H ← IhH (g3h − 63h (z )) + 63H z , g4H ← IhH (g4h − 64h (z )) + 64H (z )

3.1.6) Implement the FAS-NMG method on the coarse-grid problem: H H z ← FASCYC(z , g1H , g2H , g3H , g4H , RH , T H , ν1 , ν2 , α, Siter) 3.1.7) Add the coarse-grid corrections: & H ' h & H ' h h h z 1 ← z 1 + I Hh z 1 − 7 z1H , z 2 ← z 2 + I Hh z 2 − 7 z2H & H & H ' h ' h h h z 3 ← z 3 + I Hh z 3 − 7 z3H , z 4 ← z 4 + I Hh z 4 − 7 z4H

3.1.8) Post-smoothing: 9 : h h For k = 1 to ν2 , z ← Smoother(z , g1h , g2h , g3h , g4h , Rh , T h , α, Siter)

In practice, the FAS-NMG method is stopped if the maximum number ϵ1 of V− or W−cycles is reached (usually ϵ1 = 20), or the mean of the relative residuals obtained from the discrete EL system (3.1) is smaller than a small prescribed number ϵ2 > 0 (typically ϵ2 = 10−4 ), or the relative reduction of the dissimilarity ϵ73 is smaller than some ϵ3 > 0 (we usually assign ϵ3 = 0.3, so that the relative reduction of the dissimilarity decreases about 70%), or the change in two consecutive steps of the data fitting term ) is smaller than a small number ϵ4 > 0 (typically ϵ4 = 10−6 ). There can also be some improvement by embedding our FAS-NMG method in the soh called full multi-grid (FMG) method, to obtain a good initial guess z ini t ial allowing excellent error reduction properties as follows: h

h

z ← FMG(z , g1h , g2h , g3h , g4h , Rh , T h , ν1 , ν2 , α, Siter) 1) If Ωh = coarsest g r id (|Ωh | = 16 × 16), solve (3.1) using a method of choice and then stop. Else continue with following step. 2) Restriction to the coarse grid: H h H h H h H h z 1 ← IhH z 1 , z 2 ← IhH z 2 , z 3 ← IhH z 3 , z 4 ← IhH z 4 , RH ← IhH Rh , T H ← IhH T h 3) Implement the FMG step on the next coarser grid: H H z ← FMG(z , g1H , g2H , g3H , g4H , RH , T H , ν1 , ν2 , α, Siter) 4) Interpolation to the next finer grid: h H h H h H h H z 1 ← I Hh z 1 , z 2 ← I Hh z 2 , z 3 ← I Hh z 3 , z 4 ← I Hh z 4 5) Implement the FAS-NMG method on the next finer grid: h h z ← FASCYC(z , g1h , g2h , g3h , g4h , Rh , T h , ν1 , ν2 , α, Siter)


371

3.2. Local Fourier analysis (LFA) and the measure of h-ellipticity Local Fourier analysis (LFA) is a powerful tool to develop efficient MG methods. Although LFA was originally developed for linear discrete operators with constant coefficients on infinite grids, it can also be applied to more general nonlinear discrete systems. An infinite grid is first assumed to eliminate the effect of boundary conditions, and it is then also assumed that the discrete nonlinear operator can be linearised and replaced locally by a new operator with constant coefficients [53]. This approach has proven to be very useful in understanding MG methods applied to nonlinear problems — cf. [3, 4, 8, 9, 12, 13, 16–18, 28, 29, 32, 40, 48] for interesting examples and discussions. The h-ellipticity measure of a discrete operator is an important quantity in the LFA framework. Some h-ellipticity can be used as a sufficiency condition for the existence of an efficient point-wise smoother for the discrete operator associated with (2.1). Thus let N[z]z = G[z]

(3.3)

denote our linearised PDE system of (2.1) arising from the fixed-point (FP) iteration scheme (a global linearisation method), adapted from the so-called robust FP iteration method [16] by freezing coefficients at some FP step. Recalling that z = (z1 , z2 , z3 , z4 )⊤ = (u1 , u2 , v1 , v2 )⊤ and z = (z 1 , z 2 , z 3 , z 4 )⊤ = (u1 , u2 , v 1 , v 2 )⊤ represent the exact solution and current approximation, we have the operators N[¯z ] and G[¯z ] resulting from the global linearistion at ¯z given by ⎡

and

⎤ −∆ 0 −I 0 0 −∆ 0 −I ⎥ ⎢ N[¯z ] = ⎣ ⎦ σ11 (ξ) σ12 (ξ) −α∆ 0 σ21 (ξ) σ22 (ξ) 0 −α∆ ⎡

⎤ g1 ⎢ g ⎥ G[¯z ] = ⎣ 2 ⎦, g3 + σ11 (ξ)¯ u1 + σ12 (ξ)¯ u2 − f1 (¯ u1 , u¯2 ) g4 + σ21 (ξ)¯ u1 + σ22 (ξ)¯ u2 − f2 (¯ u1 , u¯2 )

where σl1 (ξ) = ∂u1 f l (¯ u1 , u¯2 ) = (∂ul T (u ¯ ))(∂u1 T (u ¯ )) + (T (u ¯ ) − R)(∂u1 ul T (u ¯ )) ,

σl2 (ξ) = ∂u2 f l (¯ u1 , u¯2 ) = (∂ul T (u ¯ ))(∂u2 T (u ¯ )) + (T (u ¯ ) − R)(∂u2 ul T (u ¯ )) ,

ξ = (x 1 , x 2 )⊤ and g0l = 0 for 0l = 1, · · · , 4. We note that σ21 (ξ) = σ12 (ξ), and that σl m (ξ) can be approximated by σl m(ξ) = (∂ul T (u ¯ ))(∂um T (u ¯ )) for m = 1, 2 — e.g. see [40] and [44, pages 56, 79]. Applying the FOC finite difference scheme proposed in Section 2 in (3.3) yields the discrete problem Nh [z h ]z h = Gh [z h ]

(3.4)

372


where the symbol ‘(·)i, j ’ is dropped, ⎡

−4 h 0 ⎢ 0 −4 h Nh [¯z h ] = ⎢ h ⎣ σ11 (ζ)5 σ12 (ζ)5 h σ21 (ζ)5 h σ22 (ζ)5 h

⎤ −5 h 0 0 −5 h ⎥ ⎥, h ⎦ −α4 0 h 0 −α4

⎡

⎤ 5 h G1 ⎢ 5 h G2 ⎥ ⎥ Gh [¯z ] = ⎢ ⎣ 5 h G3 ⎦ , 5 h G4

5 h G1 = g1 , 5 h G2 = g2 , 5 h G3 = g3 + σ11 (ζ)5 h u¯1 + σ12 (ζ)5 h u¯2 − 5 h f1 (¯ u1 , u¯2 ), and h h h h 5 G4 = g4 + σ21 (ζ)5 u¯1 + σ22 (ζ)5 u¯2 − 5 f2 (¯ u1 , u¯2 ). We now proceed to show that the linearised discrete operator Nh [¯z h ] in (3.4) provides sufficient h-ellipticity, similar to applications discussed elsewhere [16–18, 29, 40, 53, 56]. For simplicity, our analysis is carried out over the infinite grid A B Ωh∞ = x ∈ Ω|x = (x 1i , x 2 j )⊤ = ((2i − 1) h1 /2, (2 j − 1) h2 /2)⊤ , i, j ∈ #2 , (3.5)

where h1 = h2 = h = 1/n. We note that the stencils corresponding to −4 h and −5 h are ⎤ ⎡ ⎡ ⎤ 0 1 0 −1 −4 −1 2 h −4 h $ ⎣ −4 20 −4 ⎦ and − 5 h $ − ⎣ 1 8 1 ⎦ . 2 0 1 0 −1 −4 −1

Let ϕ h (θ , x) = exp(iθ x/h) · 0I be/the grid functions, where 0I = (1, 1, 1, 1)⊤ , θ = (θ1 , θ2 )⊤ ∈ Θ = (−π, π]2 , x ∈ Ω∞ and i = −1. Due to the locality aspect of LFA, our analysis applies h to each grid point separately — i.e. we consider the local discrete system Nh (ξ)z h = Gh (ξ) centred and defined only within a small neighbourhood of each grid point ξ = (x 1i , x 2 j )⊤

and u h (ξ) = (uh1 (ξ), uh2 (ξ))⊤ . Applying the discrete operator Nh (ξ) to the grid functions 0 h (ξ, θ )ϕ h (θ , x) — we use the Fourier ϕ h (θ , x) —- i.e. on considering Nh (ξ)ϕ h (θ , x) = N symbol ⎡ ⎤ Ch (θ ) Ch (θ ) −4 0 −5 0 ⎢ Ch (θ ) Ch (θ ) ⎥ 0 −4 0 −5 ⎥. 0 h (ξ, θ ) = ⎢ N (3.6) ⎣ σ11 (ζ)5 ⎦ Ch (θ ) σ12 (ζ)5 Ch (θ ) −α4 Ch (θ ) 0 Ch (θ ) σ22 (ζ)5 Ch (θ ) Ch (θ ) σ21 (ζ)5 0 −α4

Ch (θ ) Details of Fourier symbols of systems of PDE are given in Refs. [53, 56], and here 4 h h h C (θ ) denote the Fourier symbols of the symbolic operators 4 C and 5 C , respectively. and 5 0 h (ξ, θ ) as [53, 56] The measure of h-ellipticity is defined via N Eh (Nh (ξ)) =

0 h (ξ, θ ))| : θ ∈ Θhigh } min{| det(N 0 h (ξ, θ ))| : θ ∈ Θ} max{| det(N

,

(3.7)

where Θhigh = Θ\( − π/2, π/2]2 denotes the range of high frequencies and 0 h (ξ, θ )) = α2 (4 Ch (θ ))4 + αc1 (4 Ch (θ ))2 (5 Ch (θ ))2 + c2 (5 Ch (θ ))4 det(N

(3.8)

373


0 h (ξ, θ ) with represents the determinant of N

c1 = σ11 (ξ) + σ22 (ξ) and c2 = σ11 (ξ)σ22 (ξ) − σ12 (ξ)σ21 (ξ) .

Ch (θ ) and −5 Ch (θ ), we have For −4

Ch (θ ) = 20 − 8 (cos(θ1 ) + cos(θ2 )) − 4 cos(θ1 ) cos(θ2 ) −4

and

such that

and

Ch (θ ) = −h2 (4 + cos θ1 + cos θ2 ) , −5

8 8 8 8 π 0 h (ξ, θ ))8 = 88det(N 0 h (ξ, − , 0))88 = α2 (12)4 + 3600αc1h4 + 625c2 h8 min 8det(N θ ∈Θhigh 2

8 8 8 8 0 h (ξ, θ ))8 = 8det(N 0 h (ξ, π, π))8 = α2 (32)4 + 4096αc1 h4 + 16c2 h8 . max 8det(N θ ∈Θ

Consequently, the measure of h-ellipticity for the discrete operator Nh (ξ) is given by Eh (Nh (ξ)) =

α2 (12)4 + 3600αc1 h4 + 625c2h8 α2 (32)4 + 4096αc1 h4 + 16c2 h8

(3.9)

such that lim Eh (Nh (ξ)) =

h→0

3 >0, 8

(3.10)

bounded away from 0 for all possible choices of α, h > 0 and for all possible values of σ11 (ξ), σ12 (ξ), σ21 (ξ) and σ22 (ξ) (i.e. the results do not depend on the given images) over the whole discrete domain Ωh . Thus it can be expected that the discrete system Nh [z h ]z h = Gh [z h ] is appropriate for our MG treatment, and also indicates that a point-wise errorsmoothing procedure can be constructed.

3.3. The proposed smoother We have used the LFA to inform theoretically the choice of error-smoothing procedures for our FAS-NMG Algorithm 3.1. In order to obtain a high potential point-wise smoother, we propose using an outer-inner iteration method as follows. Starting from an initial guess z [0] (typically z [0] = 0) in the outer iteration, we compute a sequence of approximate solutions z [1] , z [2] , z [3] , · · · , z [ν] , z [ν+1] , · · · by solving the discrete PDE system obtained from the global linearistion method by the FP iteration scheme discretised by the FOC finite difference scheme given in (3.4) — i.e. N[z [ν] ]z [ν+1] = G[z [ν] ] ,

(3.11)

374


where ν denotes the index for the FP or outer iteration step and the symbols ‘h’ and ‘(·)i, j ’ are dropped for simplicity. We then solve the resulting linear system for the inner iteration by the successive over relaxation (SOR) method, also known as the ω-PCGS (point-wise collective Gauss-Seidel) relaxation method. Thus each grid point (x 1i , x 2 j ) can be determined by this outer-inner iteration method as [k+1]

(z [ν+1] )i, j

= (1 − ω)(z [ν+1] )i, j + ω(N[z [ν] ]i, j )−1 (G[z [ν] ])i, j [k]

[k+1/2]

(3.12)

where ω ∈ (0, 2) is the relaxation parameter (typically ω = 3/4 = 0.75), and ⎡ ⎤ 20 0 −4h2 0 0 20 0 −4h2 ⎥ ⎢ N[z [ν] ] = ⎣ ⎦, 2 2 4h (σ11 )i, j 4h (σ12 )i, j 20α 0 4h2 (σ21 )i, j 4h2 (σ22 )i, j 0 20α

[k+1/2]

(G[z [ν] ])i, j

⎛

[k+1/2]

(G1 [z [ν] ])i, j

⎜ (G [z [ν] ])[k+1/2] ⎜ 2 i, j =⎜ ⎝ (G3 [z [ν] ])[k+1/2] i, j [k+1/2]

⎞

⎟ ⎟ ⎟, ⎠

(G4 [z [ν] ])i, j . / . / [k+1/2] [ν] [ν+1] [k] [ν+1] [k+1] [ν+1] [k] [ν+1] [k+1] (G1 [z [ν] ])i, j =(PG1 )i, j +4 (u1 )i+1, j +(u1 )i−1, j +4 (u1 )i, j+1 +(u1 )i, j−1 [ν+1] [k] [ν+1] [k+1] [ν+1] [k] [ν+1] [k+1] )i+1, j+1 + (u1 )i+1, j−1 + (u1 )i−1, j+1 + (u1 )i−1, j−1 / h2 . [ν+1] [k] [ν+1] [k+1] [ν+1] [k] [ν+1] [k+1] (v1 )i+1, j + (v1 )i−1, j + (v1 )i, j+1 + (v1 )i, j−1 ,

+ (u1 +

2

[k+1/2]

(G2 [z [ν] ])i, j

[ν+1] [k] [ν+1] [k+1] [ν+1] [k] [ν+1] [k+1] )i+1, j+1 + (u2 )i+1, j−1 + (u2 )i−1, j+1 + (u2 )i−1, j−1 2 . / h [ν+1] [k] [ν+1] [k+1] [ν+1] [k] [ν+1] [k+1] (v2 )i+1, j + (v2 )i−1, j + (v2 )i, j+1 + (v2 )i, j−1 ,

+ (u2 +

. / . / [ν] [ν+1] [k] [ν+1] [k+1] [ν+1] [k] [ν+1] [k+1] =(PG2 )i, j +4 (u2 )i+1, j +(u2 )i−1, j +4 (u2 )i, j+1 +(u2 )i, j−1

2

. . / . / [ν] [ν+1] [k] [ν+1] [k+1] [ν+1] [k] [ν+1] [k+1] =(PG3 )i, j +α 4 (v1 )i+1, j +(v1 )i−1, j +4 (v1 )i, j+1 +(v1 )i, j−1 / [ν+1] [k] [ν+1] [k+1] [ν+1] [k] [ν+1] [k+1] + (v1 )i+1, j+1 + (v1 )i+1, j−1 + (v1 )i−1, j+1 + (v1 )i−1, j−1 [k+1/2]

(G3 [z [ν] ])i, j

−

−

h2 (σ11 )i, j . 2 h2 (σ12 )i, j . 2

[ν+1] [k] )i+1, j

+ (u1

[ν+1] [k] )i+1, j

+ (u2

(u1 (u2

[ν+1] [k+1] )i−1, j

+ (u1

[ν+1] [k] )i, j+1

+ (u1

[ν+1] [k+1] )i−1, j

+ (u2

[ν+1] [k+1] )i, j−1

[ν+1] [k] )i, j+1

+ (u2

[ν+1] [k+1] )i, j−1

/ /

,

. . / . / [ν] [ν+1] [k] [ν+1] [k+1] [ν+1] [k] [ν+1] [k+1] =(PG4 )i, j +α 4 (v2 )i+1, j +(v2 )i−1, j +4 (v2 )i, j+1 +(v2 )i, j−1 / [ν+1] [k] [ν+1] [k+1] [ν+1] [k] [ν+1] [k+1] + (v2 )i+1, j+1 + (v2 )i+1, j−1 + (v2 )i−1, j+1 + (v2 )i−1, j−1 [k+1/2]

(G4 [z [ν] ])i, j

−

h2 (σ21 )i, j . 2

[ν+1] [k] )i+1, j

(u1

[ν+1] [k+1] )i−1, j

+ (u1

[ν+1] [k] )i, j+1

+ (u1

[ν+1] [k+1] )i, j−1

+ (u1

/


375

Algorithm 3.2 The proposed outer-inner iteration method for our NMG-FAS Algorithm 3.1 Denote by α the regularisation parameter PCGSiter the maximum number of ω−PCGS iterations h

h

z ← FP-PCGS(z , g1h , g2h , g3h , g4h , Rh , T h , α, PCGSiter) h

1) Use input parameters to compute σlm (¯z h )i, j , (PG0h )i, j , and (Nh [z ]i, j )−1 l for l, m = 1, 2, 1 ≤ i, j ≤ n and 0l = 1, 2, 3, 4. 2) Perform ω−PCGS steps h [k+1]

2.1) Update (z )i, j by the ω−PCGS relaxation method as given by (3.12) for all 1 ≤ i, j ≤ n for k = 0, 1, 2, · · · , PCGSiter.

− [ν]

h2 (σ22 )i, j . 2

(PG1 )i, j = (g1 )i, j , [ν]

[ν+1] [k] )i+1, j

(u2

[ν]

[ν+1] [k+1] )i−1, j

+ (u2

[ν+1] [k] )i, j+1

+ (u2

[ν+1] [k+1] )i, j−1

+ (u2

/

,

(PG2 )i, j = (g2 )i, j ,

' h2 & 8 f1 (u [ν] )i, j +f1 (u [ν] )i+1, j +f1 (u [ν] )i−1, j +f1 (u [ν] )i, j+1 +f1 (u [ν] )i, j−1 2 / h2 (σ11 )i, j . [ν] [ν] [ν] [ν] [ν] 8(u1 )i, j + (u1 )i+1, j + (u1 )i−1, j + (u1 )i, j+1 + (u1 )i, j−1 + 2 2 / h (σ12 )i, j . [ν] [ν] [ν] [ν] [ν] + 8(u2 )i, j + (u2 )i+1, j + (u2 )i−1, j + (u2 )i, j+1 + (u2 )i, j−1 , 2

(PG3 )i, j =(g3 )i, j −

' h2 & 8 f2 (u [ν] )i, j +f2 (u [ν] )i+1, j +f2 (u [ν] )i−1, j +f2 (u [ν] )i, j+1 +f2 (u [ν] )i, j−1 2 2 / h (σ21 )i, j . [ν] [ν] [ν] [ν] [ν] 8(u1 )i, j + (u1 )i+1, j + (u1 )i−1, j + (u1 )i, j+1 + (u1 )i, j−1 + 2 / h2 (σ22 )i, j . [ν] [ν] [ν] [ν] [ν] 8(u2 )i, j + (u2 )i+1, j + (u2 )i−1, j + (u2 )i, j+1 + (u2 )i, j−1 , + 2 ' & '& f1 (u [ν] )i, j = T (u [ν] )i, j − (R)i, j T (u [ν] )i+1, j − T (u [ν] )i−1, j /(2h) , & '& ' f2 (u [ν] )i, j = T (u [ν] )i, j − (R)i, j T (u [ν] )i, j+1 − T (u [ν] i, j−1 )/(2h) , [ν]

(PG4 )i, j =(g4 )i, j −

and the superscripts k, k+1/2, and k+1 denote the current, intermediate and new approximations computed by the PCGS method, respectively. The implementation of our proposed smoother (3.11) based on the FP (outer) and ω−PCGS (inner) iteration methods (3.12) on a fine grid is summarised in Algorithm 3.2.

376


3.4. Smoothing analysis for the proposed smoother A quantitative measure of the smoothing efficiency for a given relaxation method is the smoothing factor, denoted by µ in the LFA and numerically computed for test problems, defining the worst asymptotic error reduction in performing one smoothing step from all high-frequency components of the error between the exact solution and the current approximation. More details on the LFA smoothing analysis are given in Refs. [53, 56]. We proceed to analyse the smoothing properties of our proposed smoother via (3.11) and (3.12). As pointed out in many cases of nonlinear operators with varying coefficients [3, 4, 7, 12, 13, 16, 18, 28, 29, 32, 40, 48], the smoothing factor is x-dependent. Thus it is customary to look for the maximum over the local smoothing factors of the linearised operator Nh (ξ) — i.e. (3.13) µ∗loc = maxµloc . ζ∈Ωh

In order to determine µloc for the case ω = 1, let us again consider the local discrete [+] [0] [−] system Nh (ξ)z h (ξ) = Gh (ξ). By using the splitting Nh (ξ) = Nh (ξ) + Nh (ξ) + Nh (ξ), it is possible to write the local inner iterations of (3.11) as [+]

[0]

h

[−]

h

h

Nh (ξ)z new (ξ) + Nh (ξ)z new(ξ) + Nh (ξ)z ol d (ξ) = Gh (ξ) ,

(3.14)

where z hol d (ξ) and z hnew(ξ) denote the approximations to the exact solution z h (ξ) before and after the inner smoothing step, respectively. Here ⎡ ⎤ h h −4[+/0/−] 0 −5[+/0/−] 0 ⎢ ⎥ h h 0 −4[+/0/−] 0 −5[+/0/−] ⎢ ⎥ [+/0/−] Nh (ξ) = ⎢ ⎥, h h h σ12 (ζ)5[+/0/−] −α4[+/0/−] 0 ⎣ σ11 (ζ)5[+/0/−] ⎦ h h h σ21 (ζ)5[+/0/−] σ22 (ζ)5[+/0/−] 0 −α4[+/0/−] ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 0 0 0 0 0 −1 −4 −1 h h h 0 ⎦, −4[0] 0 −4 ⎦, − 4[+] $ ⎣ −4 0 $ ⎣ 0 20 0 ⎦, −4[−] $⎣ 0 −1 −4 −1 0 0 0 0 0 0 ⎤ ⎤ ⎤ ⎡ ⎡ ⎡ 0 0 0 0 0 0 0 1 0 h2 h2 h2 h h h $ − ⎣ 0 8 0 ⎦, −5[−] $ − ⎣ 0 0 1 ⎦. − 5[+] $ − ⎣ 1 0 0 ⎦, −5[0] 2 2 2 0 1 0 0 0 0 0 0 0

By subtracting (3.14) from Nh (ξ)z h (ξ) = Gh (ξ), and defining e hnew(ξ) = z h (ξ) − z hnew(ξ) and e hol d (ξ) = z h (ξ) − z hol d (ξ), we obtain the system of local error equations [+]

[0]

[−]

Nh (ξ)e hnew(ξ) + Nh (ξ)e hnew(ξ) + Nh (ξ)e hol d (ξ) = 0 , or e hnew(ξ) = Sh (ξ)e hol d (ξ) , [+]

[0]

(3.15) [−]

where Sh (ξ) = −[Nh (ξ) + Nh (ξ)]−1 [Nh (ξ)] is the amplification factor. The effect of Sh (ξ) on the grid functions ϕ h (θ , x) within Θhigh = Θ\(−π/2, π/2]2 determines the

377


smoothing properties of the proposed smoother. Thus by considering the grid function h h h h h ϕ h (θ , x) we can represent the Fourier symbols of −4[+] , −4[0] , −4[−] , −5[+] , −5[0] and h −5[−] by

h h −4[+] (θ ) = −e−i(θ1 +θ2 ) − 4e−iθ1 − e−i(θ1 −θ2 ) − 4e−iθ2 , −4[0] (θ ) = 20 ,

h −4[−] (θ ) = −4e iθ2 − e i(θ1 −θ2 ) − 4e iθ1 − e i(θ1 +θ2 ) ,

h h h −5[+] (θ ) = −(h2 /2)(e−iθ1 + e−iθ2 ) , −5[0] (θ ) = −4h2 , −5[−] (θ ) = −(h2 /2)(e iθ2 + e iθ1 ) ,

respectively. The local smoothing factor is therefore given by J K µloc = sup |ρ(0 Sh (ξ, θ ))| : θ ∈ Θhigh ,

(3.16)

where ρ indicates the spectral radius of 0 Sh (ξ, θ ). Here 0 Sh (ξ, θ )) is the Fourier symbol of Sh (ξ) defined by

where

M−1 L M L 0 0 [−] (ξ, θ ) ∈ %4 × %4 , 0 [0] (ξ, θ ) + N 0 [+] (ξ, θ ) N Sh (ξ, θ )) = − N h h h

0 [+/0/−] (ξ, θ ) N ⎡h h h −4[+/0/−] (θ ) 0 −5[+/0/−] (θ ) 0 ⎢ h h 0 −4[+/0/−] (θ ) 0 −5[+/0/−] (θ ) ⎢ =⎢ h h h 0 ⎣ σ11 (ζ)5[+/0/−] (θ ) σ12 (ζ)5[+/0/−] (θ ) −α4[+/0/−] (θ ) h h h σ21 (ζ)5[+/0/−] (θ ) σ22 (ζ)5[+/0/−] (θ ) 0 −α4[+/0/−] (θ )

(3.17)

⎤ ⎥ ⎥ ⎥ ⎦

can be used to compute (3.17). For the case ω ̸= 1, the local smoothing factor can be defined similarly to (3.16) — viz. J K 0h (ξ, θ , ω))| : θ ∈ Θhigh , µloc = sup |ρ(S

(3.18)

where the Fourier symbol of the amplification factor 0 Sh (ξ, θ , ω) is given by

L M−1 0 [0] (ξ, θ ) + ωN 0 [+] (ξ, θ ) 0 Sh (ξ, θ , ω)) = N h h L M [0] 0 [−] (ξ, θ ) ∈ %4 × %4 . 0 × (1 − ω)Nh (ξ, θ ) − ωN h

(3.19)

The effectiveness of the proposed smoother is now tested by computing the smoothing rate for three data sets of real medical images, as shown in Fig. 1 on a 64 × 64 grid. On average, it emerges below that the optimal value of µ∗loc is 0.464 at ω = 0.75 and a fixed value of α = 10−4 , after performing 10 outer iterations with PCGSiter = 5.

378


Rh

Th

256 × 256

4. Numerical Experiments and Results We now present numerical results from several test cases, to assess the accuracy and efficiency of our proposed numerical techniques. In all of these numerical experiments, n1 = n2 = n (i.e. we assume the mesh size h1 = h2 = h = 1/n), and bilinear interpolation was used to compute the transformed template image T h (u). All numerical algorithms were implemented in MATLAB (version R2011a) and run on a MacBook Pro under OS X 10.10, at a 2.5 GHz clock speed and equipped with an Intel Core i5 and 8 GB of RAM. We compare the proposed FOC finite difference scheme with the central fourth-order (CFO) and SSOC finite difference schemes, using the FAS-NMG framework represented in Algorithms 3.1 to solve the EL system (2.1). The CFO finite difference operators were given

379

A Fourth-Order Compact Finite Difference Scheme for Higher-Order PDE-Based Image Registration Example 1

0

10

FOC−MG CFO−MG SSOC−MG

−2

10 Mean of relative residuals

Mean of relative residuals


−2

10

−4

10

−6

10

−8

−4

10

−6

10

−8

10

10

−10

10

Example 2

0

10

0

−10

2

4

6 8 10 MG cycles (steps)

12

10

14

0

2

4


12

14

Example 3

0

10


−2

Mean of relative residuals

10

−4

10

−6

10

−8

10

−10

10

0

2

4


12

14

−

256 × 256

by (δ2x z0h )i, j 1 l (δ2x z0h )i, j 2 l

=

−(z0h )i+2, j + 16(z0h )i+1, j − 30(z0h )i, j + 16(z0h )i−1, j − (z0h )i+2, j

,

=

−(z0h )i, j+2 + 16(z0h )i, j+1 − 30(z0h )i, j + 16(z0h )i, j−1 − (z0h )i, j+2

.

l

l

l

l

l

12h21

l

l

l 12h22

l

l

The three MG methods (FOC-MG, CFO-MG and SSOC-MG) primarily differ in the discretisation method used. The other MG components are essentially identical for the three methods. In particular, all of these methods share the following basic components: a uniform mesh overlapping grids Ωh , Ω2h , Ω4h , · · · , Ω1/16 ; intergrid transfer operators that use the averaging technique for the fine-to-coarse grid transfer of residuals; and the bi-linear interpolation technique for the coarse-to-fine grid transfer of corrections; the coarse-grid oper-

380

h

T (u)


FOC-MG T h (u)

SSOC-MG T h (u)

ϵ73 = 0.0703

ϵ73 = 0.0895

ϵ73 = 0.0003

ϵ73 = 0.0006

ϵ73 = 0.0299

ϵ73 = 0.0410

ϵ73

256 × 256 Rh

ator using the DCA method; the MG smoothers using the FP iteration scheme in Algorithm 3.2 for the FOC-MG method and the FP iteration schemes, which were slightly adapted from those of previous work [16] for the CFO-MG and SSOC-MG methods with the same parameter PCGSiter = 5. The numbers of pre- and post-smoothing steps were ν1 = ν2 = 10 for all MG methods; and a W-cycle control algorithm was used. The iteration stopped when ϵ72 < 10−8 (the mean of the relative residuals is less than 10−8 ). For each test, α was well selected to deliver the good qualities of the registered images. In order to estimate a rea-


381

ϵ73

Problem Example 1 h = 1/32 h = 1/64 h = 1/128 h = 1/256 Example 2 h = 1/32 h = 1/64 h = 1/128 h = 1/256 Example 3 h = 1/32 h = 1/64 h = 1/128 h = 1/256

FOC-MG M/7 ϵ3 /CPUs

CFO-MG M/7 ϵ3 /CPUs

SSOC-MG M/7 ϵ3 /CPUs

6/0.0280/1.3140 6/0.0376/3.4964 6/0.0554/9.7224 6/0.0703/30.1004

16/0.0287/3.7621 ∗/ ∗ /∗ ∗/ ∗ /∗ ∗/ ∗ /∗

8/0.0390/0.5440 8/0.0485/2.5766 8/0.0670/8.2978 7/0.0845/26.4317

6/0.0183/1.2189 6/0.0048/3.2681 6/0.0010/9.1025 6/0.0003/29.8091

18/0.0192/4.2348 ∗/ ∗ /∗ ∗/ ∗ /∗ ∗/ ∗ /∗

8/0.0262/1.1451 8/0.0067/2.9146 8/0.0021/8.2242 7/0.0006/27.5968

6/0.0076/1.3542 6/0.0092/3.7829 6/0.0128/9.5382 6/0.0299/30.4103

18/0.0087/4.9026 ∗/ ∗ /∗ ∗/ ∗ /∗ ∗/ ∗ /∗

8/0.0106/0.9171 8/0.0163/3.4860 8/0.0270/8.8553 7/0.0410/28.1158

ϵ73 = 0.0710, CPUs = 3.3488 ϵ73 = 0.0005, CPUs = 3.4498 ϵ73 = 0.0301, CPUs = 3.7868 256 × 256

sonable α automatically, we adapted our FAS-NMG method to follow the ‘cooling’ process suggested in Refs. [16,29], which resembles the L-curve method in other inverse problems. In Table 1, one can see three quantities: the MG cycles ‘M’; the relative reduction of dissimilarity ‘7 ϵ3 ’ between Rh and T h (u) defined in Algorithm 3.1; and run times ‘CPUs’ (in minutes) used by the MG methods in reducing the mean of the relative residuals below 10−8. For all registration problems shown in Fig. 1† . the CFO-MG method either converges very slowly or diverges when the mesh size h tends to zero, whereas the FOC-MG and SSOC-MG methods both converge in a few MG steps and their convergence behaviour does not depend on the number of grid points. The registration results indicate not only that the CFO finite difference scheme leads to poor MG efficiency, but also that the proposed FOC finite difference scheme is a better choice for computing fast and accurate solutions of the †

The test images are from Ref. [45] — see also http://www.siam.org/books/fa06/.

382

S. Jewprasert, N. Chumchob and C. Chantrapornchai L2

Problem Example 4 h = 1/32 h = 1/64 h = 1/128 h = 1/256 Example 5 h = 1/32 h = 1/64 h = 1/128 h = 1/256

L∞

FOC-MG e h2 /Order2

SSOC-MG e h2 /Order2

FOC-MG e h∞ /Order∞

SSOC-MG e h∞ /Order∞

5.05e − 5/− 3.10e − 6/4.02 1.93e − 7/4.00 1.20e − 8/4.00

1.02e − 2/− 2.51e − 3/2.02 6.20e − 4/2.01 1.54e − 4/2.00

2.51e − 5/− 1.48e − 6/4.08 8.80e − 8/4.07 5.45e − 9/4.01

5.08e − 3/− 1.24e − 3/2.03 3.07e − 4/2.01 7.65e − 5/2.00

5.21e − 5/− 3.17e − 6/4.03 1.98e − 7/4.00 1.23e − 8/4.00

1.15e − 2/− 2.75e − 3/2.06 6.71e − 4/2.03 1.67e − 4/2.00

2.15e − 5/− 1.29e − 6/4.05 7.95e − 8/4.00 1.23e − 8/4.00

5.01e − 3/− 1.21e − 3/2.04 3.01e − 4/2.01 7.51e − 4/2.00

Curvature-based image registration. Moreover, Fig. 2 confirms that the rate of convergence for the FOC-MG method is better than for the CFO-MG and SSOC-MG methods. However, the FOC-MG and CFO-MG methods are computationally more expensive than the SSOCMG method, for the same discretisation parameter h. This is no surprise, as the numbers of arithmetical operations in the FOC and CFO finite difference schemes are greater than that for the SSOC finite difference scheme. Nevertheless, as expected of a HOC finite difference scheme, the test results on the uncontrolled experiments where the exact solutions are not known (shown in Table 1 and Fig. 3) indicate that the registered images computed by the FOC finite difference scheme is much more accurate than those computed from the SSOC finite difference scheme when the mesh size h tends to zero — cf. the bold numbers in Table 1. As shown in Fig. 4 and Table 1, one single FMG cycle is sufficient to reach the same level of registration qualities obtained from those of the FOC-MG and SSOC-MG methods. This is remarkable, demonstrating that our proposed FMG method is evidently fast and accurate for solving Curvature-based image registration in real applications. To further demonstrate the accuracy of our FOC-MG method, we designed a new test — viz. a two image registration as shown in Fig. 5 (Examples 4 and 5). Separate template images T1h and T2h were obtained by transforming the reference image‡ Rh with u h1 (x) = (0.5 cos(−pi/6)x 1−sin(−pi/6)x 2−(10/256), sin(−pi/6)x 1+0.5 cos(−pi/6)x 2+(10/256))⊤ as a linear transformation and u h2 (x) = (0.1 cos(3θ ))x 1 , (0.1 cos(3θ )x 2 )⊤ as a nonlinear transformation, where θ = tan−1 (x 2 /x 1 ). As can be seen, the true solutions for Examples 4 and 5 can be obtained under these deformations u h1 and u h2 . The Lγ -error and the order of Lγ -error of the numerical solution vector U h are defined by e hγ = ∥U h − U hexact ∥γ h and Orderγ = log2 (e 2h γ /e γ ) for γ = 2, ∞, respectively. It is clear from Table 2 that the FOC-MG and SSOC-MG methods provide the numerical solutions to the expected orders of accuracy in both the L2 -norm and the L∞ -norm. Moreover, the FOC-MG method computes much more accurate solutions than the SSOC-MG method when the number of grid points increases, as expected with the FOC finite difference scheme.

‡

The test image by http://www.liv.ac.uk/ cmchenke/softw.htm.


383

Rh

T1h

T2h

(a)

(b) Rh

R

h

T2h

T1h

5. Conclusion We have developed a FOC finite difference scheme for Curvature-based image registration. To further improve the method for solving the resulting nonlinear discrete system, we modified the unified approach proposed in Ref. [16] for designing the robust FAS-NMG method with a high potential smoother based on an outer-inner iteration scheme. The LFA and numerical experiments confirm the effectiveness in smoothing the errors of the proposed smoother. The numerical experiments not only support our LFA prediction but also demonstrate that the registered images computed by the FOC-MG method are significantly more accurate than those computed from the SSOC-MG method, and that the FOC-MG method converges in a few MG steps. Indeed, our fast and accurate FAS-NMG method can be recommended for a wide range of real applications.

Acknowledgments The authors express their thanks to the referees for a number of very useful suggestions. The second author’s work was partially supported by the Thailand Research Fund Grant ♯TRG5680037 and the Faculty of Science Research Fund, Silpakorn University, Thailand.

384


References [1] R.P. Agarwal and Y.M. Wang, Some recent developments of the Numerov’s method, Comput. Math. Appl. 42, 561-592 (2001). [2] I. Altas, J. Dym, M.M. Gupta and R.P. Manohar, Multigrid solution of automatically generated high-order discretizations for the biharmonic equation, SIAM J. Sc. Comput. 19, 1575-1585 (1998). [3] N. Badshah and K. Chen, Multigrid method for the Chan-Vese model in variational segmentation, Comm. Comput. Phys. 4, 294-316 (2008). [4] N. Badshah and K. Chen, On two multigrid algorithms for modelling variational multiphase image segmentation, IEEE Trans. Image Proc. 18, 1097-1106 (2009). [5] R. Bajcsy and S. Kovaˇciˇc, Multiresolution elastic matching, Comput. Vision Graph. Image Proc. 46, 1-21 (1989). [6] W.L. Briggs, V.E. Henson and S.F. McCormick, A Multigrid Tutorial (2nd Edition). SIAM Publications, Philadelphia (2000). [7] C. Brito-Loeza and K. Chen, Multigrid method for a modified curvature driven diffusion model for image inpainting, JCM 26, 856-875 (2008). [8] C. Brito-Loeza and K. Chen, Fast numerical algorithms for Euler’s Elastica digital inpainting model, Int. J. Mod. Math. 5, 157-182 (2010). 2010. [9] C. Brito-Loeza and K. Chen, Multigrid algorithm for high order denoising, SIAM J. Imaging Sci. 3, 363-389 (2010). [10] C. Broit, Optimal registration of deformed images. PhD thesis, University of Pennsylvania (1981). [11] O. Bröker, M. J. Grote, C. Mayer and A. Reusken, Robust parallel smoothing for multigrid via sparse approximate inverses, SIAM J. Sci. Comput. 23, 1396-1417 (2001). [12] T.F. Chan and K. Chen, On a nonlinear multigrid algorithm with primal relaxation for the image total variation minimisation, Num. Algorithms 41, 387-411 (2006). [13] T.F. Chan and K. Chen, An optimization-based multilevel algorithm for total variation image denoising, Multiscale Mod. Sim. 5, 615-645 (2006). [14] N. Chumchob, Vectorial total variation-based regularization for variational image registration, IEEE Trans. Image Proc. 22, 4551-4559 (2013). [15] N. Chumchob and K. Chen, A variational approach for discontinuity-preserving image registration, East-West J. Math. Special volume, 266-282 (2010). [16] N. Chumchob and K. Chen, A robust multigrid approach for variational image registration models, J. Comput. Appl. Math. 236, 653-674 (2011). [17] N. Chumchob and K. Chen, Improved variational image registration model and a fast algorithm for its numerical approximation, Num. Meth. Partial Diff. Eq. 28, 1966-1995 (2012). [18] N. Chumchob, K. Chen and C. Brito, A fourth order variational image registration model and its fast multigrid algorithm, SIAM J. Multiscale Mod. Sim. 9, 89-128 (2011). [19] M. Dehghan and A. Mohebbi, Multigrid solution of high order discretisation for threedimensional biharmonic equation with Dirichlet boundary conditions of second kind, Appl. Math. Comput. 180, 575-593 (2006). [20] L.W. Ehrlich, Solving the biharmonic equation as coupled finite difference equations, SIAM J. Num. Anal. 8, 278-287 (1971). [21] D.J. Evans and R.K. Mohanty, Block iterative methods for the numerical solution of twodimensional non-linear biharmonic equations, Int. J. Comput. Math. 69, 371-390 (1998). [22] C.F. Schauf, S. Henn and K. Witsch, Multigrid based total variation image registration, Comput. Visual Sci. 11, 101-113 (2008).


385

[23] B. Fischer and J. Modersitzki, Fast diffusion registration, Contemporary Math. 313, 117-129 (2002). [24] B. Fischer and J. Modersitzki, Curvature-based image registration, J. Math. Imaging Vision 18, 81-85 (2003). [25] B. Fischer and J. Modersitzki, A unified approach to fast image registration and a new curvature based registration technique, Linear Alg. Appl. 380, 107-124 (2004). [26] S. Gao, L. Zhang, H. Wang, R. Crevoisier de, D.D. Kuban, R. Mohan and L. Dong, A deformable image registration method to handle distended rectums in prostate cancer radiotherapy, Med. Phys. 33, 3304-3312 (2006). [27] Y. Ge, Multigrid method and fourth-order compact difference discretization scheme with unequal mesh sizes for 3D Poisson equation, J. Comput. Phys. 229, 6381-6391 (2010). [28] E. Haber, R. Horesh and J. Modersitzki, Numerical optimization for constrained image registration, Num. Linear Alg. Appl. 17, 343–359 (2010). [29] E. Haber and J. Modersitzki, A multilevel method for image registration, SIAM J. Sci. Comput. 27, 1594-1607 (2006). [30] W. Hackbusch, Multi-Grid Methods and Applications. Springer-Verlag Berlin Heidelberg New York (1985). [31] J.V. Hajnal, D.L.G. Hill and D.J. Hawkes, Medical Image Registration, The Biomedical Engineering Series, CRC Press (2001). [32] S. Hamilton, M. Benzi and E. Haber, New multigrid smoothers for the Oseen problem Num. Linear Algebra Appl. 17, 557-576 (2010). [33] S. Henn, A multigrid method for a fourth-order diffusion equation with application to image processing, SIAM J. Sci. Comput. 27, 831-849 (2005). [34] S. Henn, A full Curvature-based algorithm for image registration, J. Math. Imaging Vision 24, 195-208 (2006). [35] S. Henn, A translation and rotation invariant Gauss-Newton like scheme for image registration, BIT Num. Math. 46, 325-344 (2006). [36] S. Henn and K. Witsch, Iterative multigrid regularization techniques for image matching, SIAM J. Sci. Comput. 23, 1077-1093 (2001). [37] S. Henn and K. Witsch, Image registration based on multiscale energy information, Multiscale Mod. Sim. 4, 584-609 (2005). [38] S. Henn and K. Witsch, A variational image registration approach based on curvature scale space, LNCS 3459, 143-154 (2005). [39] L. Hömke, A multigrid method for anisotropic PDE in elastic image registration, Num. Linear Algebra Appl. 13, 215-229 (2006). [40] H. Köstler, K. Ruhnau and R. Wienands, Multigrid solution of the optical flow system using a combined diffusion- and curvature-based regularizer, Num. Linear Algebra Appl. 15, 201-218 (2008). [41] J. Larrey-Ruiz, R. Verdú-Monedero and J. Morales-Sánchez, A Fourier domain framework for variational image registration, J. Math. Imaging Vision 32, 57-72 (2008). [42] L. Bauer and E.L Riessl, Block five diagonal matrices and the fast numerical solution of the biharmonic equation, Math. Comput. 26, 311-326 (1972). [43] J.B.A. Maintz and M.A. Viergever, A survey of medical image registration, Med. Image. Anal. 2, 1-36 (1998). [44] J. Modersitzki, Numerical Methods for Image Registration, Oxford (2004). [45] J. Modersitzki, FAIR: Flexible Algorithms for Image Registration, SIAM Publications, Philadelphia (2009). [46] R.K. Mohanty, A new high accuracy finite difference discretization for the solution of 2D nonlinear

386


biharmonic equations using coupled approach, Num. Meth. Partial Diff. Eq. 26, 931-944 (2010). [47] R.K. Mohanty and P. K. Pandey, Difference methods of order two and four for systems of mildly nonlinear biharmonic problems of second kind in two space dimensions, Num. Meth. Partial Diff. Eq. 12, 707-717 (1996). [48] B. Seynaeve, E. Rosseel, B. Nicolaï and S. Vandewalle. Fourier mode analysis of multigrid methods for partial differential equations with random coefficients, J. Comput. Phys. 224, 132149 (2007). [49] J. Smith, The coupled equation approach to the numerical solution of the biharmonic equation by finite differences, SIAM J. Num. Anal. 7, 104-111 (1970). [50] J.W. Stephenson, Single cell discretization of order two and four for biharmonic problems, J. Comput. Phys. 55, 65–80 (1984). [51] M. Stürmer, H. Köstler and U. Rüde, A fast full multigrid solver for applications in image processing, Num. Linear Algebra Appl. 15, 187-200 (2008). [52] Z.F. Tian and P.X. Yu, An efficient compact difference scheme for solving the streamfunction formulation of the incompressible Navier-Stokes equations, J. Comput. Phys. 230, 6404-6419 (2011). [53] U. Trottenberg, C. Oosterlee and A. Schüller, Multigrid, Academic Press (2001). [54] R. Verdú-Monedero, J. Larrey-Ruiz and J. Morales-Sánchez, Frequency implementation of the Euler-Lagrange equations for variational image registration, IEEE Signal Proc. Lett. 15, 321– 324 (2008). [55] P. Wesseling, An Introduction to Multigrid Methods, Edwards, Philadelphia (2004). [56] R. Wienands and W. Joppich, Practical Fourier Analysis for Multigrid Method, Chapman & Hall/CRC (2005). [57] D. Zikic, W. Wein, A. Khamene, D.A. Clevert and N. Navab, Fast deformable registration of 3D−ultrasound data using a variational approach, LNCS 4190, 915-923 (2006).