MULTISCALE MODEL. SIMUL c - am.uni-duesseldorf.dehenn/html/artikel/HW-MMS05.pdfMULTISCALE MODEL. SIMUL. c 2005 Society for Industrial and Applied Mathematics Vol. 4, No. 2, pp. 584–609

MULTISCALE MODEL. SIMUL. c© 2005 Society for Industrial and Applied MathematicsVol. 4, No. 2, pp. 584–609

IMAGE REGISTRATION BASED ON MULTISCALE ENERGYINFORMATION∗

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 con-firmed to be mathematical ill-posed due to the fact that determining the unknown components ofthe displacements merely from the images is an underdetermined problem. The approach presentedhere utilizes an auxiliary regularization term based on the energy of a plate with free edges, whichincorporates smoothness constraints into the deformation field. One of the important aspects of thisapproach is that the energy does not penalize affine-linear functions. Consequently, the kernel of theEuler–Lagrange equation is spanned by all rigid motions. Hence, the presented approach is invariantunder planar rotation and translation. In order to find an optimal deformation, we solve a sequenceof subproblems with decreasing regularization parameter. In this framework the regularization pa-rameter 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, regulariza-tion, 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 onimages. In the past two decades many methods in image processing have raised astrong interest in the mathematical community. As the field requires higher levelsof reliability and efficiency, classical mathematical scopes including powerful tools tosolve PDEs, variational methods, and multiscale approaches become relevant to an-swer fundamental questions in image processing. The mathematical point of view hasopened new ways to handle classical image processing issues (restoration, segmenta-tion, optical flow computation, registration, etc.).

1.1. The image registration problem. In this paper we consider the so-calledimage registration problem. Image registration, also known as image matching or im-age mapping, is one of the most challenging problems in image processing. Theimportance of the problem can be seen at a glance in the following probably incom-plete list of applications: medical imaging, geophysics, virtual reality, robotics, andmeteorology; see, e.g., [3, 4, 9, 12, 13, 17, 20, 21, 22, 37]. A good survey of somepractical applications is given in [5, 28] and the references therein.

Mainly two different registration techniques for medical applications have beendiscussed in the literature. On the one hand, Grenander and his coworkers haveintroduced the generation of diffeomorphisms as flows (solutions of an ODE) in aframework which guarantees smoothness and consistency; see [5, 18]. Here the in-troduction of a time-dependent vector field (restricted to the case of homogeneousnorms) 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-Universitat Dusseldorf, Universitatsstraße 1, D-40225

Dusseldorf, Germany ([email protected], [email protected]).

584

IMAGE REGISTRATION BASED ON MULTISCALE INFORMATION 585

way to define a group, which shares many of the properties of finite-dimensional Liegroups.

On the other hand, so-called variational registration techniques deal with a directregularization of the problem, typically adding a gradient-based convex regularizationfunctional 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 resultingfrom the deformation of the images. This approach is related to the well-known clas-sical Tikhonov regularization of the originally ill-posed problem; see [38, 39]. Takinginto account a time-step discretization this methodology is closely related to iterativeTikhonov regularization methods [21, 35].

1.2. Multiscale nature of image registration. Image registration problemsare often multiscale problems in nature; namely, the reason for a displacement isgoverned by effects on different scales. This phenomenon is given, e.g., in human brainmapping. Here the displacements often come from global transformations (translationand rotation) as well as from the different morphology of complex neuroanatomicalshapes of the underlying brains.

The main aim of this paper is to propose a novel variational multiscale approachfor the image registration problem, which is attractive both from a theoretical anda practical point of view. Here, from an abstract point of view, we encounter asolution which illustrates more precisely the link between scale space and image reg-istration. 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 onconventional multiscale approaches which are widely known and used concepts inimage analysis; cf. [1, 2, 4, 11, 15, 24, 25, 32, 33, 36, 40, 42]. We emphasize thatstandard multiscale techniques, such as the multigrid method [11, 21], the waveletmethod [3], multilevel methods [4, 21], and scale space approaches for the underlyingimages [2, 13, 15] have been used in order to speed up the computations and to findlarge nonlinear displacements. From a practical point of view, the proposed multiscaleframework unifies existing variational approaches, which are normally classified dueto the underlying transformations into either affine-linear or nonlinear approaches.Therefore the proposed approach is a flexible image registration scheme, which treatsthe deformations on different spatial scales. This is an attractive option in the situa-tion where no a priori knowledge of the displacements is available.

1.3. A variational formulation. A general formulation of the image registra-tion 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 thatin continuous variables the images can be represented by compactly supported func-tions T,R : R

2 → R and that the template is distorted by a deformation x−u(x) witha displacement field u : R

2 → 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 trans-formed 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. Forsuch 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 min-imization determines the optimal transformation. Several such functions have beenproposed, e.g., in [13, 22, 23, 26, 41], each characterized by a different set of properties.

586 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:

DLSQ[R, T, u(x)] =

∫Ω

(T (x− u(x)) −R(x)

)2

dx.(1.1)

This is a common criterion (see, e.g., [3, 7, 9, 17, 20, 21, 29]) in the case wherethe images are recorded with the same imaging machinery. In general, if the imagesare 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 thisdifficulty, 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 be-tween T (x − u(x)) and R, the image registration problem can be identified, in thatway, with a minimization problem:

Find u ∈ X , such that D[R, T, u] is minimal.(1.2)

Unfortunately, this problem is not well-posed: Solutions, if they exist, are in generalneither unique nor stable. Different solutions can give very similar outputs, and smalldata 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 isill-posed, we have to apply a regularizing technique to solve the problem in a stableway. This leads us to solve the following nonlinear minimization problem:

minu∈X

{D[R, T, u] + εR[u]

}(1.3)

with a regularization parameter ε > 0, which controls the quality of the fit of thedata, as measured by D[R, T, u], and the variability of the approximate solution, asmeasured by the regularization term R[u].

1.4. Review of common regularization techniques. The choice of R de-pends crucially on the underlying application. Many regularization methods are dis-cussed in the literature. They incorporate desired features of the displacements intothe minimization problem and determine what part of X is preserved and what partis eliminated.

For instance, so-called elastic registration techniques deal with a regularizationterm given by the energy of an elastic deformation (see, e.g., [9, 21, 23, 15]). Herethe minimization over H1

0 (Ω) × H10 (Ω) ⊂ X may be interpreted physically as the

deformation of a clamped elastic membrane. This model, of course, allows only elasticdeformations and penalizes others, in particular affine-linear ones. The minimizationof the same problem over H1(Ω) ×H1(Ω) admits constant displacements.

Solutions of the approach presented in [17] may be interpreted physically as thedisplacement of a plate subject to a load. Here the authors propose a regularizationterm R[u] =

∑2l=1

∫Ω

Δ2ul dΩ, which involves higher-order derivatives of u. Althoughthe regularization term is neutral with respect to affine-linear displacements, the func-tional is not H2(Ω)×H2(Ω)-coercive (see, e.g., [19]), the kernel is spanned by infinitelymany elements, and the question of existence of minima is unsolved. To overcomethis problem the approach is restricted to the space{

ui ∈ H2(Ω),∂ui

∂n=

∂Δui

∂n= 0, i = 1, 2

}⊂ H2(Ω) ×H2(Ω)(1.4)


of displacements. As a consequence the affine-linear displacements are penalized bythe underlying function space. Other fourth-order approaches are proposed in [3, 31].They add a Helmholtz term, yielding a positive definite operator penalizing affine-linear transformations. However, explicitly preserving affine-linear transformationsrequires an energy term which uses only second-order derivatives on the boundary.

This paper mainly addresses the issue of developing a novel regularization func-tional and a multiscale image registration approach. In order to regularize (1.2), weuse for each component of the displacement field u an approximation for the sum ofthe squared principal curvatures. It turns out that this approach is equivalent to us-ing the energy of a thin plate with free edges. This particular choice contains variousimportant aspects:

(a) the proposed energy is H2(Ω) × H2(Ω)-coercive, yielding the existence ofminima by standard Hilbert space existence theory;

(b) the proposed energy is described by a symmetric positive semidefinite bilinearform;

(c) the kernel of the proposed energy consists only of the affine-linear displace-ments, and consequently the energy is invariant under planar rotation andtranslation;

(d) the underlying regularization parameter ε 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 bemade robust to noise, because it is based on solving the registration problemat multiple scales;

(f) the proposed multiscale framework unifies existing image registration ap-proaches, which are normally classified with respect to the underlying trans-formations 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 thissection we propose a novel regularization energy for the image registration problembased on an approximation of the sum of squared principal curvature for each com-ponent 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 curvaturestructure of the lth component ul (l = 1, 2) of the displacement field u. Consider thesum of squared principal curvature of ul

κ21(ul) + κ2

2(ul) = (κ1(ul) + κ2(ul))2 − 2κ1(ul)κ2(ul) = H2(ul) − 2K(ul)

with mean curvature

H(ul) = κ1(ul) + κ2(ul) =(1 + ul

2y)ulxx − 2ulxulyulxy + (1 + ul

2x)ulyy

(1 + ulx + uly)3/2

and Gaussian curvature

K(ul) = κ1(ul)κ2(ul) =ulxxulyy − ul

2xy

(1 + ulx + uly)2.


For the case where the nonlinear deflection is small, i.e., ∇ul ≈ 0, the sum of squaredprincipal curvature can be approximated by

κ21(ul) + κ2

2(ul) ≈ Δ2ul − 2(ulxxulyy − ul2xy).

Based on this approximation, we propose the regularization energy

G[u] =2∑

l=1

B[ul, ul],(2.1)

which measures the curvature on a domain Ω for a displacement field u, where thebilinear form B is given by

B[ul, vl] =1

2

∫Ω

(∂2ul

∂x2+

∂2ul

∂y2

)(∂2vl∂x2

+∂2vl∂y2

)

+

(2∂2ul

∂x∂y

∂2vl∂x∂y

−(∂2ul

∂x2

∂2vl∂y2

+∂2vl∂x2

∂2ul

∂y2

))dΩ.

For each component of a displacement field u this energy has its physical origin in thevariational formulation of the plate problem with free edges [10]

P[ul] = εB[ul, ul] − 〈f, ul〉,(2.2)

which models the equilibrium position of a plate of constant thickness under the loadof a transverse force f and flexural rigidity ε. The use of the proposed energy term Ghas important computational aspects, which are investigated in the following.

2.2. The Euler–Lagrange equation of the approach. The bilinear form Bis symmetric and positive semidefinite over H2(Ω) and positive definite over

V =

{v ∈ H2(Ω),

∫Ω

vdx =

∫Ω

x1vdx =

∫Ω

x2vdx = 0

}⊂ H2(Ω).

Consequently, the Lax–Milgram theorem can be used to prove the existence anduniqueness 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

εB[v, φ] = 〈f, φ〉 for every ϕ ∈ V.(2.3)

Referring to the Riesz theorem of the representation of a linear functional, one canwrite

εB[v, φ] = 〈εLv, φ〉 = 〈f, φ〉 for every φ ∈ V.(2.4)

Here the linear operator L is given by

(Lv)(x1, x2) =

⎧⎪⎨⎪⎩

ΔΔv(x1, x2) for (x1, x2) ∈ Ω,

B1[v(x1, x2)] for (x1, x2) ∈ ∂Ω,

B2[v(x1, x2)] for (x1, x2) ∈ ∂Ω,

(2.5)


with

B1[v(x1, x2)] = − ∂

∂nΔv(x1, x2) −K[v(x1, x2)],

B2[v(x1, x2)] =∂2v(x1, x2)

∂n2,

K[v(x1, x2)] =∂

∂s

[∂2v(x1, x2)

∂x1∂x2(n2

x1− n2

x2) +

(∂2v(x1, x2)

∂x22

− ∂2v(x1, x2)

∂x21

)nx1nx2

],

where n = (nx1 , nx2) stands for the normal component in outward direction, ands stands for the tangential component vertical to n. The classical solution u∗ ∈ C4(Ω)of (2.3) is characterized by the Euler–Lagrange equation

ε(Lv)(x1, x2) =

⎧⎪⎨⎪⎩

εΔΔv(x1, x2) = f(x1, x2) for each (x1, x2) ∈ Ω,

B1[v(x1, x2)] = 0 for each (x1, x2) ∈ ∂Ω,

B2[v(x1, x2)] = 0 for each (x1, x2) ∈ ∂Ω.

(2.6)

The set of solutions of (2.3) over H2(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 nontrivialkernel containing only affine-linear transformations, i.e.,

G[M · (x1, x2)t + b] = 0 for all M ∈ R

2×2 and b ∈ R2.

Consequently, G is invariant under planar rotation and translation.

3. A multiscale minimization approach. As mentioned above, we have tominimize the functional

D[u(x)] + εG[u(x)] = D[u(x)] + ε2∑

l=1

B[ul(x), ul(x)]

︸︷︷︸=:Jε[u(x)]

� min.!(3.1)

over H2(Ω) ×H2(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) ischaracterized by the necessary condition

εB[ul(x), ϕl(x)] + Du[ϕl(x)] = ε〈Lul(x), ϕl(x)〉 + 〈fl(u(x)), ϕl(x)〉

= 〈εLul(x) + fl(u(x)), ϕl(x)〉 != 0

for all ϕl(x) ∈ H2(Ω), l = 1, 2, with the Gateaux derivative

Du[v(x)] = lims→0

D[u(x) + sv(x)] −D[u(x)]

s

=:(〈f1(u(x)), v1(x)〉, 〈f2(u(x)), v2(x)〉

)tof D and v(x) = (v1(x), v2(x))t ∈ L2(Ω) × L2(Ω). Classical solutions fulfill

εLul(x) − fl(u(x)) = 0 for l = 1, 2,(3.2)


which is equivalent to

(εL + αI)ul(x) = −fl(u(x)) + αul(x) for l = 1, 2,(3.3)

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 linearsubproblems given by

(αI + εL)u(k+1)l (x) = −fl(u

(k)(x)) + αu(k)l (x) for l = 1, 2

or, equivalently, (I +

ε

αL)

︸︷︷︸=:C(α,ε)

u(k+1)l (x) = − 1

αfl(u

(k)(x)) + u(k)l (x)︸︷︷︸

=:b(k)l,α(x)

.(3.4)

This equation can be seen as the semi-implicit time stepping, with 1α > 0 the length of

the time step, of the fourth-order diffusion equations given by the following parabolicPDE:

∂ui(x,t)∂t + εΔ2ui(x, t) = −f(u(x, t)) on Ω × (0, T ),

ui(x, 0) = u∗i (x) on Ω

(3.5)

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, ε). In order to express the sensitivity ofa 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)

u(k+1)l (x) =

∫Ω

Gε(x, ξ) ∗ b(k)l,α(ξ)dξ(3.6)

for a domain with infinite extent. The associated Green’s function (see [6]) is givenby the sum of modified Bessel functions of the second kind.

Here the regularization parameter ε can be used to adjust the extent of the in-

fluence of b(k)α (x). On the one hand, we have a rapid decay of Gε for ε → 0 (see

Figure 1). This means that the transformations are highly localized and thereforesensitive to discontinuities or noise in the images. On the other hand, for ε → ∞the width of the influence becomes large. Consequently, a huge parameter ε penalizeshigher oscillations in the solution and prefers global transformations. This behaviorpoints out the role of ε as a scale-selection parameter. This is analyzed in the nextsection. Of course, the explicit description of a solution (3.6) could only be of use fortheoretical considerations. For the purpose of the numerical solution of (3.4) we haveto approximate the displacements on a finite-dimensional space.

4. Iterative multiscale image registration. In general the Euler–Lagrangeequation (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 mainidea is to decompose the solution into components with different spatial scales.


Fig. 1. The extent of influence for Gε.

4.1. Energy-based scale space decomposition. In this section, we assumethat Sh = (ΔhΔh, B1,h, B2,h) is a symmetric discretization of the differential opera-tor 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, ε) = (Ih + εαSh) be the discretization of C(α, ε); then it follows directly

that (λ∗, Vi), with λ∗ = 1 + λiεα ≥ 1, is an eigensystem of Ch(α, ε). The next result

relates the solution of the kth iterate of

Ch(α, ε)u(k+1)l (x) = b

(k)l,α(x)(4.1)

to the eigensystem of Sh.

Theorem 4.1. The solution u(k+1)l (x) of the kth step of (4.1) is given by

u(k+1)l (x) =

n∑i=1

α

α + λiε〈Vi, b

(k)l,α(x)〉Vi(4.2)

and ||u(k+1)l (x)|| ≤ ||b(k)

l,α(x)|| < ∞ when α > 0.Proof. Consider the eigendecomposition of the matrix

Ch(α, ε) =(Ih +

ε

αSh

)= V ΛV T ,

with a diagonal matrix Λ = diag(λ∗(α, ε)), with the positive eigenvalues

λ∗i (α, ε) = 1 +

λiε

α≥ 1,(4.3)


and the eigenvalues λi of Sh. Consequently, the inverse of Ch(α, ε) is given by

C−1h (α, ε) = V diag(1/λ∗

i (α, ε))VT = V diag

(1

1 + λiεα

)V T .

Therefore, the solution of (4.1) is given by

u(k+1)l (x) = C−1(α, ε)b

(k)l,α(x) = V diag

(1

1 + ελi

α

)V T b

(k)l,α(x) =

n∑i=1

α

α + ελi〈Vi, b

(k)l,α(x)〉Vi.

Since ||b(k)l,α(x)|| < ∞ for α > 0, from (4.2) it follows that ||u(k+1)

l (x)|| < ∞ di-rectly.

The multiscale nature of the proposed approach can be seen by the followingresult.

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)] =

2∑l=1

(n∑

i=4

(α

α + ελi

⟨Vi, b

(k)l,α(x)

⟩)2

λi

).

Proof.(a) Recall from (2.1) that

G[(Vi, Vi)t] =

2∑l=1

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

(b) By Theorem 4.1, we get

G[u(k+1)] =

2∑l=1

B

[n∑

i=1

α

α + ελi

⟨Vi, b

(k)l,α(x)

⟩Vi,

n∑j=1

α

α + ελj

⟨Vj , b

(k)l,α(x)

⟩Vj

]

=

2∑l=1

⟨L

n∑i=1

α

α + ελi

⟨Vi, b

(k)l,α(x)

⟩Vi,

n∑j=1

α

α + ελj

⟨Vj , b

(k)l,α(x)

⟩Vj

⟩

=

2∑l=1

(n∑

i=1

α

α + ελi

⟨Vi, b

(k)l,α(x)

⟩ n∑j=1

α

α + ελj

⟨Vj , b

(k)l,α(x)

⟩⟨LVi, Vj

⟩)

=

2∑l=1

(n∑

i=1

(α

α + ελi

⟨Vi, b

(k)l,α(x)

⟩)2

λi

).

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


Fig. 2. Three-dimensional plot of the first four eigenvectors (with nonzero eigenvalue) of thebiharmonic 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 for18 × 18 grid points.

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


The major difficulty in the proposed image registration approach is to choosea suitable value for ε that gives an essential reduction of the similarity functionalD[R, T, u]. The role of the parameter ε 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 ofthe similarity functional D[R, T, u] is performed.

4.2. Iterative multiscale minimization. The set of eigenvectors of Ch(α, ε)(resp., Sh) can be regarded as a scale space. The representation (4.2) explicitly showsthe role of ε as a scale constant. For large values of ε we have the following result.

Theorem 4.3. Let ε 0; then the solution u(k+1)(x) of the kth step of (4.1) isgiven by

u(k+1)l (x) ≈

3∑i=1

〈Vi, b(k)l,α(x)〉Vi.(4.4)

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

u(k+1)l (x) =

3∑i=1

〈Vi, b(k)l,α(x)〉Vi +

n∑i=4

α

α + ελi〈Vi, b

(k)l,α(x)〉Vi

and, consequently,

limε→∞

u(k+1)l (x) =

3∑i=1

〈Vi, b(k)l,α(x)〉Vi.

First, it follows directly from representation (4.2) that decreasing ε incorporatesfunctions with higher curvature. Second, the parameter ε allows us to balance theinfluence of both terms in the functional Jε[u(x)].

Theorem 4.4. Assume it is possible to calculate solutions uεk(x) of the energy

Jεk [u(x)] = D[u(x)] + εkG[u(x)]

for a strictly monotone decreasing sequence {εk}k∈N. Then for all εn > εm it yields

G[uεn(x)] ≤ G[uεm(x)] and D[uεm(x)] ≤ D[uεn(x)].(4.5)

Proof. Consider

Jεm [uεm(x)] = D[uεm(x)] + εm G[uεm(x)]

≤ D[uεn(x)] + εm G[uεn(x)] = Jεm [uεn(x)](4.6)

and

Jεn [uεn(x)] = D[uεn(x)] + εn G[uεn(x)]

≤ D[uεm(x)] + εn G[uεm(x)] = Jεn [uεm(x)].(4.7)

With (4.7) and (4.6), (εn − εm)G[uεn(x)] ≤ (εn − εm)G[uεm(x)] follows, and we get

G[uεn(x)] ≤ G[uεm(x)] and D[uεm(x)] ≤ D[uεn(x)].


As a consequence of the last theorem, we embed the minimization of (3.1) intoa scale space framework, which treats different scales efficiently. We start the mini-mization with initial displacement u(0) = 0 and a large initial scale parameter, i.e.,ε0 0. As a consequence of Theorem 4.3 the minimization is aimed at affine-lineartransformations 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 εs := τεs−1 with some decay rateτ ∈ (0, 1). This is done for a number of scale parameters εs < εs−1 < · · · < ε1 < ε0,where the coarse-scale solution serves as initial data for solving the problem at finerscales. Here for smaller ε the iteration incorporates more and more finer-scaled func-tions. Theorem 4.4 illustrates that decreasing ε should give a better match of theimages. When ε is decreased, (3.5) shows that the underlying parabolic equationbecomes stiffer, and therefore the time step 1

α has to be decreased; i.e., α has to beincreased. This can be done using a step-size control. Another option is to choosethe parameter α so as to impose D[u(k+1)(x)] < D[u(k)(x)], but in all our examplesan increase of α by a factor of 2 was sufficient when ε was decreased by a factorof 10. This stabilizes the iteration sufficiently. In practice, decreasing ε too muchleads to strong artifacts due to the influence of high-frequency structures in the imagedata, but this is indicated by an increase of the registration energy. In that case theiteration is stopped as described in (4.8).

5. Numerical results. In order to illustrate the advantages of the proposedapproach, we present numerical results for four experiments. In all examples, we setε0 = 106 and α0 = 1

10 and apply iteration (4.1). In order to measure the distancebetween the images, we use the L2(Ω)-distance of the images; see (1.1). The Gateauxderivative of DLSQ[R, T, u(x)] at u(k)(x) is given by

fLSQ(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

fLSQ(u(k)(xi, xj)) =1

h

(Ti+1,j − Ti−1,j

Ti,j+1 − Ti,j−1

)(Ti,j −Ri,j),

where

Ti,j = T(xi − u

(k)1,h(xi, xj), xj − u

(k)2,h(xi, xj)

),

Ri,j = R(xi, xj).

The focus of the present paper is not on fast numerical solvers of (4.1); therefore weignore complexity issues and simply use a Cholesky factorization of Ch(α, ε). Thismakes sense when the image size is moderate (here (128×128)) and the linear systemhas to be solved many times with the changing right-hand side. Then the Choleskyfactorization can be computed once and used for all further iteration steps (with thesame parameter setting). To illustrate the advantages of the new multiscale approach,we present the following synthetic example, which illustrates the properties of theproposed regularization.


20 40 60 80 100 120

20

40

60

80

100

120

(a) Reference

20 40 60 80 100 120

20

40

60

80

100

120

(b) Template

Fig. 4. Synthetic example.

5.1. A synthetic example. In order to study the characteristics of the pre-sented multiscale approach we start with a synthetic example. Figure 4(a) displaysthe reference image R given by overlying three squares with decreasing size and grayvalues positioned in the middle of the image domain Ω. In contrast the templateimage T is a rotated and translated version of the reference. This means that anappropriate affine-linear transformation would produce a perfect registration resultin this example. We apply iteration (4.1) with the operator as proposed in (2.5). Asa consequence of Theorem 4.3 the multiscale minimization process is performed onthe coarsest scale, i.e., on the affine-linear functions (see Figure 5(b)). Hence, theiteration succeeds in completely matching these two images (except for interpolationartifacts resulting from the bilinear image transformation), without changing the scaleparameter ε; see Figure 5(a).

In contrast, iteration (4.1) with the same parameter setting and the biharmonicoperator with double Neumann boundary conditions (as introduced in [17]) findsa constant displacement (see Figure 5(d)). The result presented in Figure 5(c) isoptimal, 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 thedecreasing 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 thecurvature regularization. Here by the 15th iteration the image difference is reduceddown to 60 and cannot be decreased by the following steps. The impact of theproposed approach can be seen in the solid line. It also reduces the image differenceby a translation down to 60 within the first 15 iterations. Then the translation isrecognized as being optimal, and hence the image difference is decreased by rotatingthe template in the following iterations.

Although the proposed multiscale minimization approach (see section 4.2) canimprove 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 isobvious from the displacement field displayed in Figure 5(f), which is the transforma-tion from the image depicted in Figure 5(c) to the image (after applying multiscaleminimization) in Figure 5(e).


20 40 60 80 100 120

20

40

60

80

100

120

(a) Result proposed approach (b) Associated displacement field

20 40 60 80 100 120

20

40

60

80

100

120

(c) Result curvature approach [17] (d) Associated displacement field

(e) Result multiscale minimization byusing the operator as proposed in [17]

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

Fig. 5. Results of the synthetic example.


10 20 30 40 50 60 700

10

20

30

40

50

60

70

80

90

100

Iteration (k)

DLS

Q[u

(k) ]

curvature approachproposed approach

Fig. 6. History of DLSQ during the multiscale iteration for the example depicted in Figure 4.

20 40 60 80 100 120

20

40

60

80

100

120

(a) Reference (b) Template

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

5.2. Registration of magnetic resonance (MR) images of a human brain1. As a second example, we used two MR images of a human brain. Figure 7 showsa typical alignment of a brain data set with a standard brain of a human brain at-las (different individuals). Here the MR image displayed on the right-hand side isassumed to be the deformed template and should be matched onto the correspond-ing reference MR image (Figure 7(a)). Notice that the template image includes anonlinear deformation, a rotation, and a translation. In this example, the templateimage T (x) contains artificial white Gaussian noise, so that ||T − T δ||2 = ||δ(x)||2equals 36% of ||T ||2. This is a serious problem in practical applications, where thedeformations must be estimated using noisy data.

The impact of the proposed approach can be seen in Figure 8. To give an idea ofthe effect from the displacements, we display the transformed template image duringthe iteration with the superimposed contour of the reference image. Since the iterationis started with a large ε0, only affine-linear transformations are taken into account.By decreasing ε the iteration incorporates more and more functions with smaller


(a) k = 1 (b) k = 25

(c) k = 50 (d) k = 100

(e) k = 125 (f) k = 175

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 thesuperimposed reference contour.

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


0 20 40 60 80 100 120 140 160 180 20010

15

20

25

30

35

40

45

50

Iteration (k)

||R(x

) -

T δ (x

- u

(k) (x

))||2 2

||δ|| 22

ε=106

ε=103

ε=100


needs 145 steps to determine the underlying affine-linear transformation. Then theparameter is recognized as being too large and is reduced down to 103 (first arrowin Figure 9). In the following 10 iteration steps the value of the functional decreasesdrastically. Then the parameter is reduced again down to 100. In this example thevalue 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 ex-amples were presented to demonstrate the principle and reliability of the proposedapproach. Most work is done on the coarsest scales, i.e., by the affine-linear func-tions. To give an idea of the full multiscale potential of the proposed approach, wepresent 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 ε = 106) the template is transformed only bycoarse-scale basis functions (rigid motion). During the iteration the parameter ε isdecreased successively by ε ← τε, as described in section 4.2. Consequently, theresulting displacements incorporates more and more finer-scaled functions. In orderto illustrate the effect of calculated displacements, we show the transformed templateT (x − u(k)(x)) after k = 27, 33, 104, 113, 202, 215, 235, and 242 iteration steps inFigure 11(a)–(h).

In this example, we demonstrate how the presented multiscale minimization ap-proach separates between different features of the displacement field. Figure 12 givesa multiscale representation of the first component u1 (abs) of the displacement field.

The larger values of the scale parameter ε correspond to coarse scales of thedisplacement field, while the finer scales are detected with decreasing ε. As canbe seen in Figure 12(a), for large ε the displacements are given by an affine-linearfunction corresponding to the coarsest scale. If we continue the iteration into finerscales (smaller ε), mainly highly located displacements, due to the high-frequencystructures of the images, are detected.

These findings are stressed by Figure 13. Here we observe a strong decay of D asthe scale parameter ε decreases. The arrows give the link to the corresponding resultin Figures 11 and 12.


20 40 60 80 100 120

20

40

60

80

100

120

(a) Reference

20 40 60 80 100 120

20

40

60

80

100

120

(b) Template

Fig. 10. MR image example 2: Both images are superimposed with the contour of the referenceimage.

20 40 60 80 100 120

20

40

60

80

100

120

(a) ε = 106, k = 27

20 40 60 80 100 120

20

40

60

80

100

120

(b) ε = 105, k = 33

20 40 60 80 100 120

20

40

60

80

100

120

(c) ε = 104, k = 104

20 40 60 80 100 120

20

40

60

80

100

120

(d) ε = 103, k = 113

Fig. 11. MR image example 2: All images are superimposed with the contour of the referenceimage.


20 40 60 80 100 120

20

40

60

80

100

120

(e) ε = 102, k = 202

20 40 60 80 100 120

20

40

60

80

100

120

(f) ε = 101, k = 215

20 40 60 80 100 120

20

40

60

80

100

120

(g) ε = 100, k = 235

20 40 60 80 100 120

20

40

60

80

100

120

(h) ε = 10−1, k = 242

Fig. 11 (cont’d.). MR image example 2: All images are superimposed with the contour of thereference image.

5.4. Registration of X-rays of a human hand. For the last example weconsider Figure 14. Figure 14(a) displays the reference R, whereas the template T isdepicted in Figure 14(b). Notice that the transformation from the template to thereference image includes a rotation, a translation, and nonlinear deformations.

To underline the role of the proposed approach, we superimpose in Figure 15 thetransformed template image by the contour of the reference during the iteration. Ineach step the template was altered by the computed displacements. One can easily seehow the deformation fields produces an image becoming more and more similar to thereference image. As a consequence of the large regularization parameter the iterationfirst determines the affine-linear transformations. Then the value of ε is recognized asbeing too large, and hence decreased in the following iterations down to ε = 1.

Figure 16 shows the graph of the decreasing difference between the images aftereach step of the iteration. After 12 iteration steps the parameter ε is reduced toε1 = ε0 ·10−3 = 103 (first arrow in Figure 16). Using this parameter setting, the valueof DLSQ decreases drastically during the following iteration steps, until the parameteris again reduced after the 35th iteration step.


(a) ε = 106 (b) ε = 105

(c) ε = 104 (d) ε = 103

(e) ε = 102 (f) ε = 101

(g) ε = 100 (h) ε = 10−1

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


50 100 150 200 2505

10

15

20

25

30

35

40

45

Iteration (k)

DS

SD

[u(k

) ]

ε=106

ε=105

ε=104

ε=103

ε=102

ε=101

ε=100

ε=10-1

a)

b)

c)

d)

e)

f)

g) h)


20 40 60 80 100 120

20

40

60

80

100

120

(a) Reference

20 40 60 80 100 120

20

40

60

80

100

120

(b) Template

Fig. 14. Human hand example: Both images are superimposed with the contour of the referenceimage.

6. Conclusion. In this paper we have proposed a novel image registration ap-proach. Image registration strategies currently used are normally classified due tothe underlying transformations into either affine-linear (the whole image undergoesthe same type of motion) or nonlinear approaches; e.g., the transformation reflectsphysical properties of an elastic deformation. The presented multiscale frameworkunifies these existing approaches.

The presented approach may be interpreted as the deformation of a plate withfree edges and is consequently neutral with respect to translations and rotations. Thisis an important aspect, since the presented approach allows a rigid alignment of theunderlying images. Other approaches penalize affine-linear transformations by usingDirichlet or Neumann boundary conditions (see, e.g., [4, 9, 21, 17]) or by adding aHelmholtz term [3, 31].

In a theoretical analysis of the proposed model it turns out that the regularizationparameter ε can be used as a scale parameter. Experimental results indicate that the


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 withthe 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 approachfor three-dimensional problems is currently under investigation. In this case, thedeveloped multiscale approach has to be combined with fast multigrid solvers for(3.4).


10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

Iteration (k)

DS

SD

[u(k

) ]

ε=106

ε=103

ε=100


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, andh = 1

N . We abbreviate ui,j and fi,j . We identify the vectors u and f as either atwo-dimensional (nx, ny) vector or a (nxny, 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 Dirich-

let or Neumann boundary conditions have been considered in the literature. Applyingthese 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 Bh of B the discretization matrix Lh of L is charac-terized by

Bh[uh, uh] = 〈Lhu, u〉 =

n∑k=0

n∑l=0

Lk,lukul

and therefore

Lhk,l =

∂2

∂uk∂ulBh[uh, uh] for all 1 ≤ k, l ≤ n.

For example, the well-known 13-point approximation with truncation error of order h2

1

h2

(20ui,j − 8(ui+1,j + ui−1,j + ui,j+1 + ui,j−1) + 2(ui+1,j+1 + ui−1,j−1

+ 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


• one line of ghost points at the boundaries (without the corners),

• standard second-order approximations for ∂2u∂x2 and ∂2u

∂y2 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 secondintegral in (2.2),

• the 4-point approximation to approximate ∂2

∂ui∂ujB[u, u], which is exact in

this case.

The entries Lk,l can most comfortably be written as two-dimensional point dependentdifference stars with at most 5 × 5 entries. Exploiting the obvious symmetries in theproblem (x ↔ 1 − x, y ↔ 1 − y, and x ↔ y) approximately 1

8 of the stencils aresufficient to determine all. These are, for instance,

⎛⎜⎜⎜⎜⎝

0 0 1 0 00 2 −8 2 01 −8 20 −8 10 2 −8 2 00 0 1 0 0

⎞⎟⎟⎟⎟⎠ for all 2 ≤ i, j ≤ n− 2,

⎛⎜⎜⎜⎜⎝

0 0 1 0 00 3

2 −8 2 012 −6 19 1

2 −8 10 3

2 −8 2 00 0 1 0 0

⎞⎟⎟⎟⎟⎠ for i = 1, j = 2,

⎛⎜⎜⎜⎜⎝

0 0 1 0 00 3

2 −8 2 012 −6 19 −8 10 1 −6 3

2 00 0 1

2 0 0

⎞⎟⎟⎟⎟⎠ for i = 1, j = 1,

⎛⎜⎜⎜⎜⎝

0 0 12 0 0

0 12 −3 1 0

0 −1 5 −3 12

0 0 −1 12 0

0 0 0 0 0

⎞⎟⎟⎟⎟⎠ for i = 0, j = 0,

⎛⎜⎜⎜⎜⎝

0 0 12 0 0

0 12 −4 3

2 00 −2 9 3

4 −6 10 1

4 −3 1 14 0

0 0 14 0 0

⎞⎟⎟⎟⎟⎠ for i = 0, j = 1,

⎛⎜⎜⎜⎜⎝

0 0 12 0 0

0 12 −4 3

2 00 −2 10 −6 10 1

2 −4 32 0

0 0 12 0 0

⎞⎟⎟⎟⎟⎠ for i = 0, j = 2,

608 STEFAN HENN AND KRISTIAN WITSCH⎛⎜⎜⎜⎜⎝

0 0 0 0 00 0 0 1

4 00 0 1

4 −1 14

0 0 0 14 0

0 0 0 0 0

⎞⎟⎟⎟⎟⎠ for i = −1, j = 0,

⎛⎜⎜⎜⎜⎝

0 0 0 0 00 0 0 1

2 00 0 1

2 −2 12

0 0 0 12 0

0 0 0 0 0

⎞⎟⎟⎟⎟⎠ for i = −1, j = 1,

and ⎛⎜⎜⎜⎜⎝

0 0 0 0 00 0 0 1

2 00 0 1

2 −2 12

0 0 0 12 0

0 0 0 0 0

⎞⎟⎟⎟⎟⎠ for i = −1, j = 2.

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MULTISCALE MODEL. SIMUL c - am.uni-duesseldorf.dehenn/html/artikel/HW-MMS05.pdfMULTISCALE MODEL. SIMUL. c 2005 Society for Industrial and Applied Mathematics Vol. 4, No. 2, pp. 584–609