IMAGE REGISTRATION BASED ON MULTISCALE ... - Semantic Scholar

MULTISCALE MODEL. SIMUL. Vol. 4, No. 2, pp. 584–609

c 2005 Society for Industrial and Applied Mathematics

IMAGE REGISTRATION BASED ON MULTISCALE ENERGY INFORMATION∗ STEFAN HENN† AND KRISTIAN WITSCH† Abstract. We propose a novel multiscale approach for the image registration problem, i.e., to find a deformation that maps one image onto another. The image registration problem is confirmed to be mathematical ill-posed due to the fact that determining the unknown components of the displacements merely from the images is an underdetermined problem. The approach presented here utilizes an auxiliary regularization term based on the energy of a plate with free edges, which incorporates smoothness constraints into the deformation field. One of the important aspects of this approach is that the energy does not penalize affine-linear functions. Consequently, the kernel of the Euler–Lagrange equation is spanned by all rigid motions. Hence, the presented approach is invariant under planar rotation and translation. In order to find an optimal deformation, we solve a sequence of subproblems with decreasing regularization parameter. In this framework the regularization parameter can be regarded as a scale parameter, which captures information at multiple spatial scales. We analyze the multiscale nature of a solution. Key words. multiscale, image processing, image registration, variational methods, regularization, biharmonic differential equation, functional minimization AMS subject classifications. 35J35, 49K20, 65J20, 68U10 DOI. 10.1137/040604194

1. Introduction. Many applications in the modern computer era are based on images. In the past two decades many methods in image processing have raised a strong interest in the mathematical community. As the field requires higher levels of reliability and efficiency, classical mathematical scopes including powerful tools to solve PDEs, variational methods, and multiscale approaches become relevant to answer fundamental questions in image processing. The mathematical point of view has opened new ways to handle classical image processing issues (restoration, segmentation, optical flow computation, registration, etc.). 1.1. The image registration problem. In this paper we consider the so-called image registration problem. Image registration, also known as image matching or image mapping, is one of the most challenging problems in image processing. The importance of the problem can be seen at a glance in the following probably incomplete list of applications: medical imaging, geophysics, virtual reality, robotics, and meteorology; see, e.g., [3, 4, 9, 12, 13, 17, 20, 21, 22, 37]. A good survey of some practical applications is given in [5, 28] and the references therein. Mainly two different registration techniques for medical applications have been discussed in the literature. On the one hand, Grenander and his coworkers have introduced the generation of diffeomorphisms as flows (solutions of an ODE) in a framework which guarantees smoothness and consistency; see [5, 18]. Here the introduction of a time-dependent vector field (restricted to the case of homogeneous norms) has been proposed in [14]. In [30], it has been shown that this also provides a ∗ Received by the editors February 17, 2004; accepted for publication (in revised form) February 4, 2005; published electronically July 18, 2005. http://www.siam.org/journals/mms/4-2/60419.html † Mathematisches Institut, Heinrich-Heine-Universit¨ at D¨ usseldorf, Universit¨ atsstraße 1, D-40225 D¨ usseldorf, Germany ([email protected], [email protected]).

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way to define a group, which shares many of the properties of finite-dimensional Lie groups. On the other hand, so-called variational registration techniques deal with a direct regularization of the problem, typically adding a gradient-based convex regularization functional to a similarity functional of the images; see, e.g., [3, 11, 13, 16, 17, 20, 21, 15]. The regularization energy is regarded as a penalty for displacements resulting from the deformation of the images. This approach is related to the well-known classical Tikhonov regularization of the originally ill-posed problem; see [38, 39]. Taking into account a time-step discretization this methodology is closely related to iterative Tikhonov regularization methods [21, 35]. 1.2. Multiscale nature of image registration. Image registration problems are often multiscale problems in nature; namely, the reason for a displacement is governed by effects on different scales. This phenomenon is given, e.g., in human brain mapping. Here the displacements often come from global transformations (translation and rotation) as well as from the different morphology of complex neuroanatomical shapes of the underlying brains. The main aim of this paper is to propose a novel variational multiscale approach for the image registration problem, which is attractive both from a theoretical and a practical point of view. Here, from an abstract point of view, we encounter a solution which illustrates more precisely the link between scale space and image registration. This approach provides a multiscale description of the displacement fields, one in which the notion of scale is based on curvature scale space, rather than on conventional multiscale approaches which are widely known and used concepts in image analysis; cf. [1, 2, 4, 11, 15, 24, 25, 32, 33, 36, 40, 42]. We emphasize that standard multiscale techniques, such as the multigrid method [11, 21], the wavelet method [3], multilevel methods [4, 21], and scale space approaches for the underlying images [2, 13, 15] have been used in order to speed up the computations and to find large nonlinear displacements. From a practical point of view, the proposed multiscale framework unifies existing variational approaches, which are normally classified due to the underlying transformations into either affine-linear or nonlinear approaches. Therefore the proposed approach is a flexible image registration scheme, which treats the deformations on different spatial scales. This is an attractive option in the situation where no a priori knowledge of the displacements is available. 1.3. A variational formulation. A general formulation of the image registration problem can be posed as follows: Given are two images (possibly from a sequence) R (the so-called reference image) and T (called the template image). We assume that in continuous variables the images can be represented by compactly supported functions T, R : R2 → R and that the template is distorted by a deformation x − u(x) with a displacement field u : R2 → R2 , whose components u1 and u2 are functions of the variables x = (x1 , x2 )t . The goal of image registration is to determine a displacement field u (out of an underlying set of displacements X ) in such a way that the transformed template T (x−u(x)) matches the reference R. In contrast to the approaches in [18, 29] we do not explicitly require the deformation x − u(x) to be a diffeomorphism. Usually this will the case. In rare cases the transformation may become singular. For such images the biological relevance of the requirement of invertibility is questionable. A further discussion of this point can be found in [8]. One of the most popular approaches is to define an energy function whose minimization determines the optimal transformation. Several such functions have been proposed, e.g., in [13, 22, 23, 26, 41], each characterized by a different set of properties.

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STEFAN HENN AND KRISTIAN WITSCH

For instance, in the situation where the intensities of the given images are comparable, a proper choice is the L2 (Ω)-distance of the two images: 2 T (x − u(x)) − R(x) dx. (1.1) DLSQ [R, T, u(x)] = Ω

This is a common criterion (see, e.g., [3, 7, 9, 17, 20, 21, 29]) in the case where the images are recorded with the same imaging machinery. In general, if the images are recorded with different imaging machinery, the so-called multimodal registration, the DLSQ [R, T, u(x)] functional is not an appropriate measure. To cope with this difficulty, statistical (see, e.g., [22, 23, 15, 27, 26, 41]) and morphological methods [13] have been proposed. For a so-called registration energy D[R, T, u], which measures the disparity between T (x − u(x)) and R, the image registration problem can be identified, in that way, with a minimization problem: (1.2)

Find u ∈ X , such that D[R, T, u] is minimal.

Unfortunately, this problem is not well-posed: Solutions, if they exist, are in general neither unique nor stable. Different solutions can give very similar outputs, and small data errors can yield very different solutions. Therefore, the approximations u of (1.2) may be useless. One has to define better approximate solutions. Since the problem is ill-posed, we have to apply a regularizing technique to solve the problem in a stable way. This leads us to solve the following nonlinear minimization problem: min D[R, T, u] + R[u] (1.3) u∈X

with a regularization parameter > 0, which controls the quality of the fit of the data, as measured by D[R, T, u], and the variability of the approximate solution, as measured by the regularization term R[u]. 1.4. Review of common regularization techniques. The choice of R depends crucially on the underlying application. Many regularization methods are discussed in the literature. They incorporate desired features of the displacements into the minimization problem and determine what part of X is preserved and what part is eliminated. For instance, so-called elastic registration techniques deal with a regularization term given by the energy of an elastic deformation (see, e.g., [9, 21, 23, 15]). Here the minimization over H01 (Ω) × H01 (Ω) ⊂ X may be interpreted physically as the deformation of a clamped elastic membrane. This model, of course, allows only elastic deformations and penalizes others, in particular affine-linear ones. The minimization of the same problem over H 1 (Ω) × H 1 (Ω) admits constant displacements. Solutions of the approach presented in [17] may be interpreted physically as the displacement of a plate subject to a load. Here the authors propose a regularization 2 term R[u] = l=1 Ω Δ2 ul dΩ, which involves higher-order derivatives of u. Although the regularization term is neutral with respect to affine-linear displacements, the functional is not H 2 (Ω)×H 2 (Ω)-coercive (see, e.g., [19]), the kernel is spanned by infinitely many elements, and the question of existence of minima is unsolved. To overcome this problem the approach is restricted to the space ∂Δui ∂ui 2 ui ∈ H (Ω), = = 0, i = 1, 2 ⊂ H 2 (Ω) × H 2 (Ω) (1.4) ∂n ∂n


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of displacements. As a consequence the affine-linear displacements are penalized by the underlying function space. Other fourth-order approaches are proposed in [3, 31]. They add a Helmholtz term, yielding a positive definite operator penalizing affinelinear transformations. However, explicitly preserving affine-linear transformations requires an energy term which uses only second-order derivatives on the boundary. This paper mainly addresses the issue of developing a novel regularization functional and a multiscale image registration approach. In order to regularize (1.2), we use for each component of the displacement field u an approximation for the sum of the squared principal curvatures. It turns out that this approach is equivalent to using the energy of a thin plate with free edges. This particular choice contains various important aspects: (a) the proposed energy is H 2 (Ω) × H 2 (Ω)-coercive, yielding the existence of minima by standard Hilbert space existence theory; (b) the proposed energy is described by a symmetric positive semidefinite bilinear form; (c) the kernel of the proposed energy consists only of the affine-linear displacements, and consequently the energy is invariant under planar rotation and translation; (d) the underlying regularization parameter is related to a scale parameter, since it specifies the curvature of the resulting displacements; (e) this multiscale nature of the approach provides a description that can be made robust to noise, because it is based on solving the registration problem at multiple scales; (f) the proposed multiscale framework unifies existing image registration approaches, which are normally classified with respect to the underlying transformations into either affine-linear or nonlinear approaches. In the next section we describe the proposed regularization method more precisely. 2. A translation and rotation invariant regularization energy. In this section we propose a novel regularization energy for the image registration problem based on an approximation of the sum of squared principal curvature for each component of a displacement field u. 2.1. Minimal curvature approach. The principal curvatures κ1 (ul (x1 , x2 ), x1 , x2 )

and κ2 (ul (x1 , x2 ), x1 , x2 )

(denoted for simplicity as κ1 (ul ) and κ2 (ul )) completely define the local curvature structure of the lth component ul (l = 1, 2) of the displacement field u. Consider the sum of squared principal curvature of ul κ21 (ul ) + κ22 (ul ) = (κ1 (ul ) + κ2 (ul ))2 − 2κ1 (ul )κ2 (ul ) = H 2 (ul ) − 2K(ul ) with mean curvature H(ul ) = κ1 (ul ) + κ2 (ul ) =

(1 + ul 2y )ulxx − 2ulx uly ulxy + (1 + ul 2x )ulyy (1 + ulx + uly )3/2

and Gaussian curvature K(ul ) = κ1 (ul )κ2 (ul ) =

ulxx ulyy − ul 2xy . (1 + ulx + uly )2

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For the case where the nonlinear deflection is small, i.e., ∇ul ≈ 0, the sum of squared principal curvature can be approximated by κ21 (ul ) + κ22 (ul ) ≈ Δ2 ul − 2(ulxx ulyy − ul 2xy ). Based on this approximation, we propose the regularization energy G[u] =

(2.1)

2

B[ul , ul ],

l=1

which measures the curvature on a domain Ω for a displacement field u, where the bilinear form B is given by B[ul , vl ] =

1 2

2 ∂ 2 ul ∂ vl ∂ 2 ul ∂ 2 vl + + ∂x2 ∂y 2 ∂x2 ∂y 2 Ω

2

∂ ul ∂ 2 vl ∂ 2 ul ∂ 2 vl ∂ 2 vl ∂ 2 ul + 2 − dΩ. + ∂x∂y ∂x∂y ∂x2 ∂y 2 ∂x2 ∂y 2

For each component of a displacement field u this energy has its physical origin in the variational formulation of the plate problem with free edges [10] P[ul ] = B[ul , ul ] − f, ul ,

(2.2)

which models the equilibrium position of a plate of constant thickness under the load of a transverse force f and flexural rigidity . The use of the proposed energy term G has important computational aspects, which are investigated in the following. 2.2. The Euler–Lagrange equation of the approach. The bilinear form B is symmetric and positive semidefinite over H 2 (Ω) and positive definite over V =

v ∈ H (Ω), 2

vdx = Ω

x1 vdx =

Ω

x2 vdx = 0 ⊂ H 2 (Ω).

Ω

Consequently, the Lax–Milgram theorem can be used to prove the existence and uniqueness of a weak solution of (2.2). A weak solution v ∗ ∈ V is characterized by !

the necessary condition P [v] = 0, which is given by the variational equation (2.3)

B[v, φ] = f, φ for every ϕ ∈ V.

Referring to the Riesz theorem of the representation of a linear functional, one can write (2.4)

B[v, φ] = Lv, φ = f, φ for every φ ∈ V.

Here the linear operator L is given by ⎧ ⎪ ⎨ ΔΔv(x1 , x2 ) for (x1 , x2 ) ∈ Ω, (Lv)(x1 , x2 ) = B1 [v(x1 , x2 )] for (x1 , x2 ) ∈ ∂Ω, (2.5) ⎪ ⎩ B2 [v(x1 , x2 )] for (x1 , x2 ) ∈ ∂Ω,


589

with ∂ Δv(x1 , x2 ) − K[v(x1 , x2 )], ∂n ∂ 2 v(x1 , x2 ) , B2 [v(x1 , x2 )] = 2 ∂n2

2 ∂ ∂ v(x1 , x2 ) 2 ∂ v(x1 , x2 ) ∂ 2 v(x1 , x2 ) 2 nx1 nx2 , (nx1 − nx2 ) + − K[v(x1 , x2 )] = ∂s ∂x1 ∂x2 ∂x22 ∂x21

B1 [v(x1 , x2 )] = −

where n = (nx1 , nx2 ) stands for the normal component in outward direction, and s stands for the tangential component vertical to n. The classical solution u∗ ∈ C 4 (Ω) of (2.3) is characterized by the Euler–Lagrange equation ⎧ ⎪ ⎨ ΔΔv(x1 , x2 ) = f (x1 , x2 ) for each (x1 , x2 ) ∈ Ω, for each (x1 , x2 ) ∈ ∂Ω, B1 [v(x1 , x2 )] = 0 (Lv)(x1 , x2 ) = (2.6) ⎪ ⎩ B2 [v(x1 , x2 )] = 0 for each (x1 , x2 ) ∈ ∂Ω. The set of solutions of (2.3) over H 2 (Ω) is given by v(x1 , x2 ) = v ∗ (x1 , x2 ) + p(x1 , x2 ) (see, e.g., [34]), with an affine-linear function p(x1 , x2 ) = ax1 + bx2 + c and a, b, c ∈ R. This has the following consequence for the proposed energy G. Remark 2.1. The kernel of the proposed regularization energy G has a nontrivial kernel containing only affine-linear transformations, i.e., G[M · (x1 , x2 )t + b] = 0

M ∈ R2×2 and b ∈ R2 .

for all

Consequently, G is invariant under planar rotation and translation. 3. A multiscale minimization approach. As mentioned above, we have to minimize the functional (3.1)

D[u(x)] + G[u(x)] = D[u(x)] +

2

B[ul (x), ul (x)] ; min.!

l=1

=:J [u(x)]

over H 2 (Ω) × H 2 (Ω). We now present an iterative scheme for minimizing (3.1). 3.1. Minimization strategy. A minimizer u(x) = (u1 (x), u2 (x))t of (3.1) is characterized by the necessary condition B[ul (x), ϕl (x)] + Du [ϕl (x)] = Lul (x), ϕl (x) + fl (u(x)), ϕl (x) !

= Lul (x) + fl (u(x)), ϕl (x) = 0 for all ϕl (x) ∈ H 2 (Ω), l = 1, 2, with the Gˆ ateaux derivative D[u(x) + sv(x)] − D[u(x)] Du [v(x)] = lim s→0 s t =: f1 (u(x)), v1 (x), f2 (u(x)), v2 (x) of D and v(x) = (v1 (x), v2 (x))t ∈ L2 (Ω) × L2 (Ω). Classical solutions fulfill (3.2)

Lul (x) − fl (u(x)) = 0

for l = 1, 2,

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which is equivalent to (L + αI)ul (x) = −fl (u(x)) + αul (x)

(3.3)

for l = 1, 2,

with α ∈ R. If we require α > 0, the operator on the left-hand side is invertible, and we can use a semi-implicit iteration for (3.3). This results in a sequence of linear subproblems given by (k+1)

(αI + L)ul

(k)

(x) = −fl (u(k) (x)) + αul (x)

for l = 1, 2

or, equivalently, (3.4)

(k+1) 1 (k) I + L ul (x) = − fl (u(k) (x)) + ul (x) . α α (k)

=:C(α,)

=:bl,α (x)

This equation can be seen as the semi-implicit time stepping, with α1 > 0 the length of the time step, of the fourth-order diffusion equations given by the following parabolic PDE: (3.5)

∂ui (x,t) ∂t

+ Δ2 ui (x, t) = −f (u(x, t)) on Ω × (0, T ), ui (x, 0) = u∗i (x) on Ω

supplemented by the boundary conditions B1 [ui (x, t)] = B2 [ui (x, t)] = 0 for i = 1, 2. Here the diffusion time t is an artificial evolution parameter and u∗ (x) represents the initial displacement field. 3.2. Multiscale properties of C(α, ). In order to express the sensitivity of a displacement field u at a point x = (x1 , x2 )t to perturbations in the images (resp., f (u)) at other locations ξ = (ξ1 , ξ2 )t , we consider the explicit solution of (3.4) (k+1) (k) (3.6) (x) = G (x, ξ) ∗ bl,α (ξ)dξ ul Ω

for a domain with infinite extent. The associated Green’s function (see [6]) is given by the sum of modified Bessel functions of the second kind. Here the regularization parameter can be used to adjust the extent of the in(k) fluence of bα (x). On the one hand, we have a rapid decay of G for → 0 (see Figure 1). This means that the transformations are highly localized and therefore sensitive to discontinuities or noise in the images. On the other hand, for → ∞ the width of the influence becomes large. Consequently, a huge parameter penalizes higher oscillations in the solution and prefers global transformations. This behavior points out the role of as a scale-selection parameter. This is analyzed in the next section. Of course, the explicit description of a solution (3.6) could only be of use for theoretical considerations. For the purpose of the numerical solution of (3.4) we have to approximate the displacements on a finite-dimensional space. 4. Iterative multiscale image registration. In general the Euler–Lagrange equation (3.2) will have multiple solutions. Consequently, the asymptotic state of (3.4) will depend on the initial data. To avoid convergence to irrelevant local minima, we present a scale space-based minimization approach in this section. Here the main idea is to decompose the solution into components with different spatial scales.


591

Fig. 1. The extent of influence for G .

4.1. Energy-based scale space decomposition. In this section, we assume that Sh = (Δh Δh , B1,h , B2,h ) is a symmetric discretization of the differential operator L (for details see Appendix A) and (λi , Vi ) the eigensystem of Sh , with 0 = λ1 = λ2 = λ3 < λ4 ≤ · · · ≤ λn−1 ≤ λn . Let Ch (α, ) = (Ih + α Sh ) be the discretization of C(α, ); then it follows directly that (λ∗ , Vi ), with λ∗ = 1 + λαi ≥ 1, is an eigensystem of Ch (α, ). The next result relates the solution of the kth iterate of (k+1)

Ch (α, )ul

(4.1)

(k)

(x) = bl,α (x)

to the eigensystem of Sh . (k+1) (x) of the kth step of (4.1) is given by Theorem 4.1. The solution ul (k+1)

(4.2)

ul

(x) =

n i=1

(k+1)

α (k) Vi , bl,α (x)Vi α + λi

(k)

and ||ul (x)|| ≤ ||bl,α (x)|| < ∞ when α > 0. Proof. Consider the eigendecomposition of the matrix Ch (α, ) = Ih + Sh = V ΛV T , α with a diagonal matrix Λ = diag(λ∗ (α, )), with the positive eigenvalues (4.3)

λ∗i (α, ) = 1 +

λi ≥ 1, α

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and the eigenvalues λi of Sh . Consequently, the inverse of Ch (α, ) is given by 1 Ch−1 (α, ) = V diag(1/λ∗i (α, ))V T = V diag V T. 1 + λαi Therefore, the solution of (4.1) is given by (k+1) ul (x)

=

1 = V diag i 1 + λ α

(k) C −1 (α, )bl,α (x)

(k)

V T bl,α (x) =

(k)

n i=1

α (k) Vi , bl,α (x)Vi . α + λi

(k+1)

Since ||bl,α (x)|| < ∞ for α > 0, from (4.2) it follows that ||ul (x)|| < ∞ directly. The multiscale nature of the proposed approach can be seen by the following result. Theorem 4.2. By using (2.1) the energy of (a) us = (Vs , Vs )t , where Vs is the eigenvector belonging to λs ≥ 0, ( 1 ≤ s ≤ n) is given by G[us ] = 2λs and, consequently, G[un ] ≥ · · · ≥ G[u4 ] > G[u3 ] = G[u2 ] = G[u1 ] = 0; (b) u(k+1) (x) is given by G[u

(k+1)

(x)] =

n

2 l=1

i=4

α (k) Vi , bl,α (x) α + λi

2 λi .

Proof. (a) Recall from (2.1) that G[(Vi , Vi )t ] =

2

B[Vi , Vi ] = 2B[Vi , Vi ] = 2LVi , Vi = 2λi .

l=1

(b) By Theorem 4.1, we get n 2 n α α (k) (k) (k+1) G[u Vi , bl,α (x) Vi , Vj , bl,α (x) Vj ]= B α + λi α + λj i=1 j=1 l=1 n n 2 α α (k) (k) L = Vi , bl,α (x) Vi , Vj , bl,α (x) Vj α + λi α + λj i=1 j=1 l=1 n 2 n α α (k) (k) = Vi , bl,α (x) Vj , bl,α (x) LVi , Vj α + λi α + λj j=1 l=1 i=1 n

2 2 α (k) Vi , bl,α (x) λi . = α + λi i=1 l=1

Since λ1 = λ2 = λ3 = 0, we get the assertion.


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Fig. 2. Three-dimensional plot of the first four eigenvectors (with nonzero eigenvalue) of the biharmonic equation with free edges for 18 × 18 grid points.

Fig. 3. Contour plot of the last two eigenvectors of the biharmonic equation with free edges for 18 × 18 grid points.

From the last theorem it can be seen directly that the eigenvectors Vi of the operator Sh can be regarded as a sequence of spatial scales. Here the coarsest scale is given by the affine-linear functions corresponding to the eigenvalues λ1 = λ2 = λ3 = 0. The eigenvectors corresponding to the smallest positive eigenvalues are spatially smooth with small curvature; see Figure 2. Eigenvectors corresponding to large eigenvalues are oscillatory in space; see Figure 3.

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The major difficulty in the proposed image registration approach is to choose a suitable value for that gives an essential reduction of the similarity functional D[R, T, u]. The role of the parameter as a scale-selection parameter (see section 3.2) leads us to the central idea of the multiscale approach presented in the next section. Here we determine sequential optimal displacements, so that an essential reduction of the similarity functional D[R, T, u] is performed. 4.2. Iterative multiscale minimization. The set of eigenvectors of Ch (α, ) (resp., Sh ) can be regarded as a scale space. The representation (4.2) explicitly shows the role of as a scale constant. For large values of we have the following result. Theorem 4.3. Let 0; then the solution u(k+1) (x) of the kth step of (4.1) is given by (k+1)

ul

(4.4)

(x) ≈

3

(k)

Vi , bl,α (x)Vi .

i=1

Proof. Since λ1 = λ2 = λ3 = 0, from (4.2) it follows that (k+1)

ul

(x) =

3

(k)

Vi , bl,α (x)Vi +

i=1

n i=4

α (k) Vi , bl,α (x)Vi α + λi

and, consequently, (k+1)

lim ul

→∞

(x) =

3

(k)

Vi , bl,α (x)Vi .

i=1

First, it follows directly from representation (4.2) that decreasing incorporates functions with higher curvature. Second, the parameter allows us to balance the influence of both terms in the functional J [u(x)]. Theorem 4.4. Assume it is possible to calculate solutions uk (x) of the energy Jk [u(x)] = D[u(x)] + k G[u(x)] for a strictly monotone decreasing sequence {k }k∈N . Then for all n > m it yields (4.5)

G[un (x)] ≤ G[um (x)]

and

D[um (x)] ≤ D[un (x)].

Proof. Consider (4.6)

Jm [um (x)] = D[um (x)] + m G[um (x)] ≤ D[un (x)] + m G[un (x)] = Jm [un (x)]

and (4.7)

Jn [un (x)] = D[un (x)] + n G[un (x)] ≤ D[um (x)] + n G[um (x)] = Jn [um (x)].

With (4.7) and (4.6), (n − m )G[un (x)] ≤ (n − m )G[um (x)] follows, and we get G[un (x)] ≤ G[um (x)]

and D[um (x)] ≤ D[un (x)].


595

As a consequence of the last theorem, we embed the minimization of (3.1) into a scale space framework, which treats different scales efficiently. We start the minimization with initial displacement u(0) = 0 and a large initial scale parameter, i.e., 0 0. As a consequence of Theorem 4.3 the minimization is aimed at affine-linear transformations and is reduced to (4.4). We carry out iteration (4.1) until D[u(k+1) (x)] > D[u(k) (x)].

(4.8)

Then we change to a finer scale; i.e., we choose s := τ s−1 with some decay rate τ ∈ (0, 1). This is done for a number of scale parameters s < s−1 < · · · < 1 < 0 , where the coarse-scale solution serves as initial data for solving the problem at finer scales. Here for smaller the iteration incorporates more and more finer-scaled functions. Theorem 4.4 illustrates that decreasing should give a better match of the images. When is decreased, (3.5) shows that the underlying parabolic equation becomes stiffer, and therefore the time step α1 has to be decreased; i.e., α has to be increased. This can be done using a step-size control. Another option is to choose the parameter α so as to impose D[u(k+1) (x)] < D[u(k) (x)], but in all our examples an increase of α by a factor of 2 was sufficient when was decreased by a factor of 10. This stabilizes the iteration sufficiently. In practice, decreasing too much leads to strong artifacts due to the influence of high-frequency structures in the image data, but this is indicated by an increase of the registration energy. In that case the iteration is stopped as described in (4.8). 5. Numerical results. In order to illustrate the advantages of the proposed approach, we present numerical results for four experiments. In all examples, we set 1 0 = 106 and α0 = 10 and apply iteration (4.1). In order to measure the distance between the images, we use the L2 (Ω)-distance of the images; see (1.1). The Gâteaux derivative of DLSQ [R, T, u(x)] at u(k) (x) is given by f LSQ (u(k) (x)) = −2 · ∇T (x − u(k) (x)) · (T (x − u(k) (x)) − R(x)). We use the second-order approximation for ∇T at a point (xi , xj ) ∈ Ω. This leads to f

LSQ

(u

(k)

1 (xi , xj )) = h

Ti+1,j − Ti−1,j Ti,j+1 − Ti,j−1

(Ti,j − Ri,j ),

where (k) (k) Ti,j = T xi − u1,h (xi , xj ), xj − u2,h (xi , xj ) , Ri,j = R(xi , xj ). The focus of the present paper is not on fast numerical solvers of (4.1); therefore we ignore complexity issues and simply use a Cholesky factorization of Ch (α, ). This makes sense when the image size is moderate (here (128 × 128)) and the linear system has to be solved many times with the changing right-hand side. Then the Cholesky factorization can be computed once and used for all further iteration steps (with the same parameter setting). To illustrate the advantages of the new multiscale approach, we present the following synthetic example, which illustrates the properties of the proposed regularization.

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5.1. A synthetic example. In order to study the characteristics of the presented multiscale approach we start with a synthetic example. Figure 4(a) displays the reference image R given by overlying three squares with decreasing size and gray values positioned in the middle of the image domain Ω. In contrast the template image T is a rotated and translated version of the reference. This means that an appropriate affine-linear transformation would produce a perfect registration result in this example. We apply iteration (4.1) with the operator as proposed in (2.5). As a consequence of Theorem 4.3 the multiscale minimization process is performed on the coarsest scale, i.e., on the affine-linear functions (see Figure 5(b)). Hence, the iteration succeeds in completely matching these two images (except for interpolation artifacts resulting from the bilinear image transformation), without changing the scale parameter ; see Figure 5(a). In contrast, iteration (4.1) with the same parameter setting and the biharmonic operator with double Neumann boundary conditions (as introduced in [17]) finds a constant displacement (see Figure 5(d)). The result presented in Figure 5(c) is optimal, since the underlying boundary conditions (1.4) penalize a rotation. These findings are stressed by the solid line in Figure 6. Here the history of the decreasing difference DLSQ [u] between the images after each iteration step is given. The dashed line in Figure 6 reflects the behavior of the iteration when using the curvature regularization. Here by the 15th iteration the image difference is reduced down to 60 and cannot be decreased by the following steps. The impact of the proposed approach can be seen in the solid line. It also reduces the image difference by a translation down to 60 within the first 15 iterations. Then the translation is recognized as being optimal, and hence the image difference is decreased by rotating the template in the following iterations. Although the proposed multiscale minimization approach (see section 4.2) can improve the registration result for the operator proposed in [17] by using finer scales (see Figure 5(e)), this is not done by determining the underlying rotation. This is obvious from the displacement field displayed in Figure 5(f), which is the transformation from the image depicted in Figure 5(c) to the image (after applying multiscale minimization) in Figure 5(e).


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(f) Displacement field from (c) to (e)

Fig. 5. Results of the synthetic example.

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Fig. 7. (a) Reference image R(x). (b) Template image T δ (x) with the superimposed reference contour.

5.2. Registration of magnetic resonance (MR) images of a human brain 1. As a second example, we used two MR images of a human brain. Figure 7 shows a typical alignment of a brain data set with a standard brain of a human brain atlas (different individuals). Here the MR image displayed on the right-hand side is assumed to be the deformed template and should be matched onto the corresponding reference MR image (Figure 7(a)). Notice that the template image includes a nonlinear deformation, a rotation, and a translation. In this example, the template image T (x) contains artificial white Gaussian noise, so that ||T − T δ ||2 = ||δ(x)||2 equals 36% of ||T ||2 . This is a serious problem in practical applications, where the deformations must be estimated using noisy data. The impact of the proposed approach can be seen in Figure 8. To give an idea of the effect from the displacements, we display the transformed template image during the iteration with the superimposed contour of the reference image. Since the iteration is started with a large 0 , only affine-linear transformations are taken into account. By decreasing the iteration incorporates more and more functions with smaller


(a) k = 1

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Fig. 8. (a) Template image T (x − u(k) (x))); (a)–(f) deformed template after k = 1, 25, 50, 100, 125, and 175 iteration steps of the proposed approach. All images are presented with the superimposed reference contour.

spatial scale. The graph in Figure 9 illustrates the behavior of the iteration. Here one can observe the decreasing least squares functionals for this experiment. The template image is transformed by a large rigid motion. Consequently, the iteration

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needs 145 steps to determine the underlying affine-linear transformation. Then the parameter is recognized as being too large and is reduced down to 103 (first arrow in Figure 9). In the following 10 iteration steps the value of the functional decreases drastically. Then the parameter is reduced again down to 100 . In this example the value of ||R(x) − T δ (x − u(k) (x))||22 converges down to the squared noise level ||δ(x)||22 . 5.3. Registration of MR images of a human brain 2. The previous examples were presented to demonstrate the principle and reliability of the proposed approach. Most work is done on the coarsest scales, i.e., by the affine-linear functions. To give an idea of the full multiscale potential of the proposed approach, we present a very delicate artificial example in this section. The template image T (x) (Figure 10(a)) is given by a rotated and translated version of the reference image (Figure 10(b)) with preserved rows and columns flipped in the left/right direction. For this example, we have used a decay rate of τ = 10−1 . In the first iteration steps (with = 106 ) the template is transformed only by coarse-scale basis functions (rigid motion). During the iteration the parameter is decreased successively by ← τ , as described in section 4.2. Consequently, the resulting displacements incorporates more and more finer-scaled functions. In order to illustrate the effect of calculated displacements, we show the transformed template T (x − u(k) (x)) after k = 27, 33, 104, 113, 202, 215, 235, and 242 iteration steps in Figure 11(a)–(h). In this example, we demonstrate how the presented multiscale minimization approach separates between different features of the displacement field. Figure 12 gives a multiscale representation of the first component u1 (abs) of the displacement field. The larger values of the scale parameter correspond to coarse scales of the displacement field, while the finer scales are detected with decreasing . As can be seen in Figure 12(a), for large the displacements are given by an affine-linear function corresponding to the coarsest scale. If we continue the iteration into finer scales (smaller ), mainly highly located displacements, due to the high-frequency structures of the images, are detected. These findings are stressed by Figure 13. Here we observe a strong decay of D as the scale parameter decreases. The arrows give the link to the corresponding result in Figures 11 and 12.


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5.4. Registration of X-rays of a human hand. For the last example we consider Figure 14. Figure 14(a) displays the reference R, whereas the template T is depicted in Figure 14(b). Notice that the transformation from the template to the reference image includes a rotation, a translation, and nonlinear deformations. To underline the role of the proposed approach, we superimpose in Figure 15 the transformed template image by the contour of the reference during the iteration. In each step the template was altered by the computed displacements. One can easily see how the deformation fields produces an image becoming more and more similar to the reference image. As a consequence of the large regularization parameter the iteration first determines the affine-linear transformations. Then the value of is recognized as being too large, and hence decreased in the following iterations down to = 1. Figure 16 shows the graph of the decreasing difference between the images after each step of the iteration. After 12 iteration steps the parameter is reduced to 1 = 0 · 10−3 = 103 (first arrow in Figure 16). Using this parameter setting, the value of DLSQ decreases drastically during the following iteration steps, until the parameter is again reduced after the 35th iteration step.


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Fig. 12. Level set representation (abs) for different levels of the first component of the displacement field u1 for the example presented in Figure 11.

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6. Conclusion. In this paper we have proposed a novel image registration approach. Image registration strategies currently used are normally classified due to the underlying transformations into either affine-linear (the whole image undergoes the same type of motion) or nonlinear approaches; e.g., the transformation reflects physical properties of an elastic deformation. The presented multiscale framework unifies these existing approaches. The presented approach may be interpreted as the deformation of a plate with free edges and is consequently neutral with respect to translations and rotations. This is an important aspect, since the presented approach allows a rigid alignment of the underlying images. Other approaches penalize affine-linear transformations by using Dirichlet or Neumann boundary conditions (see, e.g., [4, 9, 21, 17]) or by adding a Helmholtz term [3, 31]. In a theoretical analysis of the proposed model it turns out that the regularization parameter can be used as a scale parameter. Experimental results indicate that the


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Fig. 15. From left to right and from top to bottom: Template image T (x − u(k) (x))) (top left); deformed template after k = 1, 5 10, 15, 30, and 60 iteration steps. All images are presented with the superimposed reference contour.

approach also works well when the images are contaminated by small errors (noise). Although we have only shown two-dimensional results, the extension of the approach for three-dimensional problems is currently under investigation. In this case, the developed multiscale approach has to be combined with fast multigrid solvers for (3.4).

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Appendix A. Discretization. In order to solve (2.6) on the unit square Ω = {(x, y)| 0 < x, y < 1}, we discretize at the grid points, which are at (xi , yj ) with xi = ih, yj = jh, and h = N1 . We abbreviate ui,j and fi,j . We identify the vectors u and f as either a two-dimensional (nx , ny ) vector or a (nx ny , 1) column vector u = (u1,1 , . . . , unx ,ny )t

and f = (f1,1 , . . . , fnx ,ny )t

(in a row-wise ordering) corresponding to u and f in the continuous equation (2.6). Various approaches for the discretization of the biharmonic equation with Dirichlet or Neumann boundary conditions have been considered in the literature. Applying these approaches on (2.6) lead to an unsymmetric discretization matrix. Another approach to determine a discretization of the operator L is to use (2.4). Here for a given discretization B h of B the discretization matrix Lh of L is characterized by B h [uh , uh ] = Lh u, u =

n n

Lk,l uk ul

k=0 l=0

and therefore Lhk,l =

∂2 B h [uh , uh ] ∂uk ∂ul

for all

1 ≤ k, l ≤ n.

For example, the well-known 13-point approximation with truncation error of order h2 1 20ui,j − 8(ui+1,j + ui−1,j + ui,j+1 + ui,j−1 ) + 2(ui+1,j+1 + ui−1,j−1 h2 + ui−1,j+1 + ui+1,j−1 ) + ui+2,j + ui−2,j + ui,j+2 + ui,j−2 = fi,j for i, j = 2, . . . , n − 2 is obtained by choosing


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• one line of ghost points at the boundaries (without the corners), 2 2 • standard second-order approximations for ∂∂xu2 and ∂∂yu2 of B[u, u], • the 4-point approximation with truncation error of order h2 to approximate ∂2u ∂x∂y of B[u, u], • the trapezoidal rule for the first integral and the midpoint rule for the second integral in (2.2), 2 • the 4-point approximation to approximate ∂u∂i ∂uj B[u, u], which is exact in this case. The entries Lk,l can most comfortably be written as two-dimensional point dependent difference stars with at most 5 × 5 entries. Exploiting the obvious symmetries in the problem (x ↔ 1 − x, y ↔ 1 − y, and x ↔ y) approximately 18 of the stencils are sufficient to determine all. These are, for instance, ⎛ ⎜ ⎜ ⎜ ⎜ ⎝

0 0 1 0 0

0 1 0 0 2 −8 2 0 −8 20 −8 1 2 −8 2 0 0 1 0 0

⎛

0 0 1 0 3 ⎜ 0 −8 2 2 ⎜ 1 ⎜ 19 12 −8 ⎜ 2 −6 3 ⎝ 0 −8 2 2 0 0 1 0

⎞ ⎟ ⎟ ⎟ for all ⎟ ⎠

0 0 1 0 0

⎛

0 0 1 0 0 3 ⎜ 0 −8 2 0 2 ⎜ 1 ⎜ −6 19 −8 1 2 ⎜ 3 ⎝ 0 1 −6 0 2 1 0 0 0 0 2 ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ ⎝

1 0 0 0 2 1 0 −3 1 2 0 −1 5 −3 1 0 0 −1 2 0 0 0 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

1 0 0 0 0 2 1 3 0 −4 0 2 2 0 −2 10 −6 1 1 3 0 −4 0 2 2 1 0 0 0 0 2

for i = 1, j = 2,

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

⎞ 0 0 ⎟ ⎟ 1 ⎟ 2 ⎟ 0 ⎠ 0

1 0 0 0 0 2 1 3 0 −4 0 2 2 0 −2 9 34 −6 1 1 0 −3 1 14 0 4 1 0 0 0 0 4

2 ≤ i, j ≤ n − 2,

for i = 1, j = 1,

for i = 0, j = 0,

⎞ ⎟ ⎟ ⎟ for i = 0, j = 1, ⎟ ⎠ ⎞ ⎟ ⎟ ⎟ for i = 0, j = 2, ⎟ ⎠

608


⎛ ⎜ ⎜ ⎜ ⎜ ⎝

0 0 0 0 0 0 0 0 0 0

⎛

0 0 ⎜ 0 0 ⎜ ⎜ 0 0 ⎜ ⎝ 0 0 0 0 and

⎛ ⎜ ⎜ ⎜ ⎜ ⎝

0 0 0 0 0 0 0 0 0 0

0 0

0 1 4

1 4

−1

0 0

0

0 0

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1 2

−2

0 0

1 2

0

0 0

0 1 2

1 2

−2

0 0

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⎞ 0 0 ⎟ ⎟ 1 ⎟ 4 ⎟ 0 ⎠ 0 ⎞ 0 0 ⎟ ⎟ 1 ⎟ 2 ⎟ 0 ⎠ 0 ⎞ 0 0 ⎟ ⎟ 1 ⎟ 2 ⎟ 0 ⎠ 0

for i = −1, j = 0,

for i = −1, j = 1,

for i = −1, j = 2.

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