Hart2009 Oc Registration

8/6/2019 Hart2009 Oc Registration

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An Optimal Control Approach for Deformable Registration

Gabriel L. Hart, Christopher Zach

, Marc Niethammer

, #

Department of Computer Science, # Biomedical Research Imaging CenterUniversity of North Carolina at Chapel Hill

{hartg,cmzach,mn }@cs.unc.edu

Abstract

This paper addresses large-displacement-diffeomorphicmapping registration from an optimal control perspective.This viewpoint leads to two complementary formulations.One approach requires the explicit computation of coordi-

nate maps, whereas the other is formulated strictly in theimage domain (thus making it also applicable to manifoldswhich require multiple coordinate charts). We discuss their intrinsic relation as well as the advantages and disadvan-tages of the two approaches. Further, we propose a novel formulation for unbiased image registration, which natu-rally extends to the case of time-series of images. We dis-cuss numerical implementation details and carefully evalu-ate the properties of the alternative algorithms.

1. Introduction

Determining point-correspondences between image-pairs based on image information alone is a fundamen-tal problem and of critical importance in image analysis.Image-pairs are assumed to have similarities in structureas well as appearance, where the similarity can for exam-ple be the result of looking at temporal phenomena (e.g., attwo image frames of a movie) or of investigating relationsbetween two distinct images known a-priori to share large-scale structural similarities (e.g., comparing the brains of two subjects). The problem is usually termed image match-ing or optical ow in computer vision and image registra-tion in medical imaging 1 .

In this work, we propose nonlinear registrationschemes within the large-displacement-diffeomorphic map-ping (LDDM) setting, derived from optimal control princi-ples. This allows us to obtain an intrinsic (not relying onextrinsic coordinates) registration formulation and to showthe connection to previously proposed approaches, againstwhich we carefully compare different numerical implemen-tations. The viewpoint also leads to a novel way to make

1 We use image registration as our terminology of choice in this paper.

the estimation problem symmetric and is applicable to man-ifolds which require multiple coordinate charts.

Registration is based on simultaneously achieving sim-ilarity between corresponding image points and imposingspatial regularity on the correspondences. Similarities canbe measured by image-intensity differences, mutual infor-

mation, or cross correlation, to name but a few possibil-ities. Regularity is achieved by restricting the allowablespace of deformations, e.g., by imposing parametric mod-els of image transformation, such as rigid, similarity, orafne transformations, or by non-parametric modeling of image deformation requiring spatially smooth deformationelds. The non-parametric approach is the most exible,but poses the biggest challenges with respect to the estima-tion of point correspondences. The estimation problem isinnite-dimensional in the continuum and becomes high-dimensional by discretization.

A multitude of regularization approaches has been pro-posed for non-parametric registration [ 22, 33, 13]. The twomain classes are (i) regularizers operating on the displace-ment elds (elastic registration) and (ii) regularizers oper-ating on a velocity eld (uid registration). The estimationproblems associated with these two classes are

u = argminu us [u ] + 1 2 d [u, I 0 , I 1 ] d, and

v = argminv vs [v] d dt + 1 2 d [u, I 0 , I 1 ] d,

where u denotes the displacement eld ( (x ) = x + u (x )),I 0 and I 1 are the images to be registered, v is the time-dependent velocity eld ( t (x, t ) = v(x, t )), is the spa-tial domain, us and vs are the regularization measures, andd is the data attachment term. Choosing

us [u ] =d

i =1

u i2 and d [u ] = ( I t + DIu )2 (1)

yields Horn and Schunck (HS) optical ow [ 14], whichuses a linearized model for the image match ( d [u ], DI =( I )T ) and the Dirichlet energy for the regularization of


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displacements. HS optical ow is one of the simplestelastic registration methods. Since its inception, a vari-ety of improvements have been proposed for optical owapproaches, ranging from nonlinear data attachment termsto piecewise smooth regularization. See [ 3, 7] and refer-ences therein. Medical image registration frequently re-

quires bijective maps between images. This may be accom-plished explicitly by penalizing folds and singularities dur-ing the estimation of the deformation eld [ 12] or implic-itly by imposing sufcient regularity on estimated velocityelds [ 8, 20, 9, 29, 28]. Miller et al. [20, 21] have shownthat uid registration of the form

E (v) = 1

0v 2V dt +

1 2

I 0 1 ,0 I 12L 2 , (2)

yields diffeomorphic (smooth and invertible) maps 1 ,0 forsuitable norms V of the velocity eld v, while allowingfor large displacements (by penalizing v instead of u ). Here,1 , 0 is the velocity-induced image map from t = 0 to t = 1 .

Estimation problem 2 hinges on the estimation of 0 ,1 ,which is determined by v. In Sec. 2, we give an optimalcontrol interpretation of the estimation problem 2 indepen-dent of 0 ,1 and derive its associated optimality conditions.Sec. 3 discusses solution approaches to the optimizationproblem. Equivalence to the solution method by Beg etal. [4] is established in Sec. 4. Sec. 5 proposes a new ap-proach to render problem 2 symmetric with respect to theregistered images. Sec. 6 discusses numerical approachesto solve the image-based optimization problem. Results aregiven in Sec. 7, including a careful evaluation of the prop-erties of the alternative implementations. Sec. 8 concludesthe paper. Mathematical details are given in the Appendix.

2. Optimal Control Approach

The optimization problem 2 depends on the velocity eldv as well as the velocity-dependent map 0 ,1 . This formula-tion leads directly (upon explicitly deriving the variation of E (v) with respect to v) to the the map-based optimizationapproach proposed by Beg et al. [ 4].

This section shows how we can derive an equivalent re-sult directly in the image domain, without explicitly utiliz-ing the map 0 , 1 using an optimal control approach. An ad-vection equation and a scalar conservation law will be themain parts of the optimality conditions obtained. A global

coordinate map is no longer used, which is a cornerstonefor adaptations to manifolds which cannot be covered by asingle coordinate chart (such as the sphere). This is for ex-ample of particular importance for the analysis of corticalfunction and structure, where surfaces of spherical topolgyneed to be registered. While landmark-based methods ex-ist to compute large-displacement diffeomorphic maps onspherical surfaces, they do not readily extend to more gen-eral manifolds or require multiple coordinate charts [ 11 , 2];

and a large-displacement diffeomorphic registration schemeworking directly on images has not been investigated sofar [31].

Instead of minimizing Eq. 2, we look at image registra-tion from a dynamical systems point of view and solve it asa constrained minimization problem. The system dynamics

is the constraint imposed, the velocity eld v is regarded asa control input to the dynamical system. The optimizationproblem is to nd the optimal control v subject to the systemdynamics with state I (I (0) = I 0 ), such that image I (1)approximates I 1 . See [19] for background on the optimalcontrol of systems governed by partial differential equationsand the rich literature on data assimilation (as used for ex-ample in weather forecasting) for details [ 23, 10]. The en-ergy to be minimized is then

E (v) = 1

0v 2V dt +

1 2

I (1) I 1 2L 2 , (3)

s.t. I t + ( DI )v = 0 , I (0) = I 0 , (4)

which is an advection equation for I , governed by the ve-locity eld v. Source image, I 0 , and target image, I 1 , aregiven, the velocity eld v(x, t ) is sought. Here

f 2L 2 = f, f ; f, g := f g d,f 2V = f, f V ; f, g V := Lf,Lg = L Lf,g ,

where is the spatial solution domain, L is a differentialoperator (e.g., L = 2 + ) and L denotes its adjoint. Choosing an appropriate differential operator L en-sures a diffeomorphic transformation [9]. Minimizing en-

ergy 3 subject to constraint 4 is equivalent to minimizing

E (v, I , , ) = 1

0v 2V + , I t + ( DI )v dt

+ , I (0) I 0 +1

2I (1) I 1 2L 2 (5)

with respect to v(x, t ), I (x, t ), and the two Lagrangian mul-tipliers (x, t ), (x ) [27]. For a candidate minimizer of theunconstrained energy, its variations with respect to v, , I ,and need to vanish, which yields the optimality conditions(see Appendix A for a derivation)

I t + ( DI )v = 0 , (6) t div (v ) = 0 , (7)

2L Lv + ( DI )T = 0 , (8)I (0) = I 0 , (9)

(1) =2

2(I 1 I (1)) . (10)

The Lagrangian multiplier disappears due to the xed ini-tial condition for I . Eq. 6 is the prescribed state equation


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(the system dynamics), Eq. 7 is the adjoint or co-state equa-tion, Eq. 8 is the compatibility condition, and Eqs. 9 and 10are the initial condition for the image and the nal conditionfor the adjoint variable respectively. The nal condition (1) for the adjoint variable measures the error between thegiven image I 1 and image I 0 in the warped coordinate sys-

tem of time t = 1 . The adjoint equation 7 is a scalar conser-vation law, i.e., it ows the image error of Eq. 10 backwardin time while conserving the overall error measure. Since

E (v; dv ) = 1

02L Lv + ( DI )T ,dv dt,

the compatibility condition 8 equals the gradient of the en-ergy with respect to the velocity in the L 2 norm

L 2v E (v) = 2 L Lv + ( DI )

T .

As in the map-based solution approach by Beg et al. [4], orfor the evolution of Sobolev active contours [ 26], improve-ments in convergence by gradient descent in Sobolev spacemay be obtained for adjoint solution methods [ 25]. The gra-dient with respect to the norm V induced by L is

V v E (v) = 2 v + ( L L )

1 ((DI )T ). (11)

The overall algorithm proceeds by: (i) owing image I 0forward in time, (ii) computing the current error at time t =1, (iii) owing the error backward in time, subject to thescalar conservation law for the adjoint variable, and by (iv)performing a gradient descent step, where the gradient forany time t [0, 1] can be computed from the state and theadjoint variables. See Fig. 1 for the overall algorithm.

Borzi et al. [6] propose a similar approach for optical

ow computations. They use the same system dynamics,but a fundamentally different regularization of the veloc-ity eld v. Their solution method thus requires the solu-tion of an elliptic partial differential equation (across spaceand time) in addition to the solution of the state and the adjoint equations. In contrast, the approach in this paper doesnot explicitly regularize over time, which makes it easierto compute and allows (due to equivalence of the solutionto the solution obtained by the approach of Beg et al. [ 4],see Sec. 4) for diffeomorphic deformations if desired. Of note, the work by Papadakis and Memin [ 24] uses varia-tional data assimilation for the tracking of curves and uidmotions. However, while their approach is based on a weak

constraint formulation (using the control function to allowfor deviations from the nominal dynamics), we enforce thedynamical model strictly, and control the velocity v.

3. Solutions for the State and its Adjoint

Two equations play a central role in the optimal controlformulation for registration: The advection equation

I t + DIv = 0 , I (0) = I 0 (12)

and the scalar conservation law for the adjoint variable

t div (v ) = 0 , (1) = 1 . (13)

The former ows image intensities along the velocity eldv, the latter ows the matching error along v, while conserv-ing its overall mass. Given a current estimate of the velocity

eld v, the current estimate for the map t, 0 is governed by

t + D v = 0 , (0) = id,

an advection equation for . The solution to Eq. 12 is then

I (t ) = I 0 (t ).

Computing the inverse map 1 = t, 1 as

1t D 1 v = 0 , 1 (1) = id

we have for any innitesimal volume element dV

|D 1 (t)|dV (t ) = dV (1) .

Regarding as a mass density, conservation of mass implies

(1) 1 (t)dV (1) = (t )dV (t)

which is

(1) 1 (t)dV (1) = (t )1

|D 1 (t)|dV (1) ,

and allows expressing (t) with respect to (1) as

(t) = |D 1 (t )| (1) 1 (t ). (14)

Sec. 4 shows how these relations connect the image-basedapproach of Sec. 2 to the map-based approach by Beg et

al. [4]. Sec. 6 discusses the numerical ramications of choosing between the formally equivalent image-based andmap-based approaches. Of note, the same equivalence rela-tion can be applied to the approach by Borzi et al. [ 6].

4. Relation to Previous Approach

The map-based solutions of the advection equation andthe scalar conservation law given in Sec. 3 allow for thedesign of a hybrid solution approach. Here, the forward andthe inverse maps are solved for numerically. Once obtained,they provide an analytic solution to the state variable I andits adjoint , which allows for the computation of the energy

gradient with respect to the time-dependent velocity eld v:V v E (v) = 2 v + ( L L )

1 (DI (t)) T (t ) (15)

= 2 v +2

2(L L ) 1 (|D 1 (t )| (16)

(DI (t)) T (I 1 1 (t ) I (t))

= 2 v +2

2(L L ) 1 (|D 1 (t )| (17)

(D (I 0 (t))) T (I 1 1 (t) I (t )) ,


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Algorithm 1 : Image-based Image-to-image registrationData : I 0 , I 1 , , LResult : vInitialization: v = 0 , = 0 ;repeat

Flow image forward: I t + ( DI )v = 0 , I (0) = I 0 ;

Flow adjoint backward: t div (v ) = 0 , (1) =2

2 (I 1 I (1)) ;Update velocity elds: v(t ) = 2v(t ) + ( L L ) 1 (DI (t )) T (t) ;

until convergence ;

Figure 1. Image-based image-to-image registration and advection results (right). Right: Translat-ing a checkerboard square diagonally using different numerical methods. The 1st order accuratemethod blurs the image extensively. Methods with higher order numerical accuracy producemuch crisper images. WENO5 additionally suppresses undesirable oscillations.

Algorithm 2 : Map-based Image-to-image registrationData : I 0 , I 1 , , LResult : vInitialization: v = 0 , = 0 ;repeat

Flow map forward: t + ( D )v = 0 , (0) = id ;While computing and storing images: I (t) = I 0 (t);Compute nal condition: (1) = 2 2 (I 1 I (1)) ;Flow inverse map backward: 1t (D

1 )v = 0 , 1 (1) = id ;While computing and storing the adjoints: (t) = |D 1 (t)| 1 1 (t );Update velocity elds: v(t ) = 2v(t ) + ( L L ) 1 (DI (t )) T (t) ;

until convergence ;

Figure 2. Map-based image-to-image registration.

Source.

1st order.

Error comp.

WENO5.

aWENO5.

where Eq. 15 is the original gradient as obtained from theimage-based optimal control derivation, Eq. 16 is the gra-

dient as obtained by Beg et al. [ 4] directly from the map-based energy of Eq. 2, and Eq. 17 is the fully map-basedformulation used for the map-based implementation. Thisestablishes the formal equivalence between the image-basedand the map-based approaches. Of note, the image-basedscheme neither requires the explicit computation of the for-ward and the backward maps and 1 nor image inter-polation based on these coordinate transforms. While re-moving the explicit dependence on a map leads to a morecompact formulation, accurate solutions require care withrespect to the numerical method used. See Sec. 6 for de-tails. The overall algorithm is given in Fig. 2.

5. Unbiased Formulation

Image registration approaches are frequently not sym-metric with respect to the images to be registered. Thus,given two images A and B, registering A to B is not as-sured to give the inverse result to registering B to A. To re-move this registration bias, optimization problems need tobe symmetric. One solution is to simply replace the regis-tration energy by a sum of the original energy and its com-

plement. In the context of large displacement diffeomor-phic ows, approaches have been proposed by Avants [ 1],

Younes [32] and Beg [ 5]. The image mismatch can be com-puted at the midpoint in time [ 5, 32, 1]

I 0 12 , 0 I 1 12 ,1

2L 2 ,

which amounts to the estimation of two separate ow elds.Alternatively, reweighting the image difference [ 32]

|D 11 ,0 | (I 0 1 , 0 I 1 )2 dxor dening a matching cost over time [ 5]

1

0I 0 t, 0 I 1 t, 1

2L 2 dt (18)

symmetrizes the registration problem and thus makes it un-biased. Only the consistent integral cost of Eq. 18 readilyextends to the case of time-series (where no natural mid-point exists). However, the consistent integral cost leads to


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a gradient of the form [5]

V v E = 2 v +

2 2

(L L )(( I 1 t, 1 I 0 t, 0 )

[D (I 0 t, 0 )T 1

t|D t,u | du

+ D (I 1 t, 1 )T t0 |D t,u | du ])which is costly to compute due to the time integrals. Inour approach, we directly symmetrize the data attachmentterm, but combine this with an estimation of the optimalmatching template. While the original formulation was anoptimal control problem with free nal state subject to a -nal state penalty, the symmetrized version has a free initialas well as a free nal state subject to initial and nal statepenalties. The approach has the following properties: (i)it directly generalizes to time-series of images, (ii) the es-timation is performed in one continuous interval, and (iii)the formulation extends to data dened on a manifold thatcannot be covered with one coordinate chart. The optimalcontrol problem is then to minimize

E (v) = 1

0v 2V dt +

1 2

I (0) I 0 2L 2 +1

2I (1) I 1 2L 2 ,

subject toI t + DIv = 0 , I (0) = I T ,

with an unknown (free) initial state I (0) = I T . Due to theformal equivalence of the image-based and the map-basedapproaches this is identical to minimizing 2

E (v, I T ) = 1

0v 2V dt +

1 2

I T I 02L 2 +

1 2

I T 1 ,0 I 12L 2 .

The variation with respect to I T is

E (I T ; dI T ) =2

2I T I 0 + |D

1 |(I T I 1 1 ), dI T

which yields

I T =I 0 + |D 1 (0) |I 1 1 (0)

1 + |D 1 (0) |. (19)

The update equation constitutes a weighted average be-

tween the initial image I 0 and the warped image I 1 . Theweighting assures that differences are measured in their re-spective coordinate systems (for t = 0 and t = 1 ). Theoverall estimation scheme alternates between registration(using I T as the initial condition for the state equation) asbefore and an update of the template image I T according to

2 The same result can be obtained by strictly working in the image do-main. However, the derivation is more complicated, due to the interdepen-dency of image perturbations at different time points.

Eq. 19. To make the update independent of the map 1 ,we can make use of relation 14, to get

|D 1 (t)| = q(t), qt div (qv) = 0 , q(1) = 1 , (20) J t DJv = 0 , J (1) = I 1 .

The updated template image is then

I T =I 0 + q(0)J (0)

1 + q(0),

allowing for an image-based, unbiased registration.

6. Numerical Considerations

The image-based approach is potentially more desirablefor intensity-based image-matching, since (i) it extends tomanifolds without explicit construction of a coordinate sys-tem, (ii) it may require less computations (subject to the nu-

merics used), and (iii) it can serve as a proof of concept forimage-based algorithms with more complicated dynamics,where the map-based relations of Sec. 4 no longer hold.

Avoiding the computations of the maps and 1 re-moves the need for explicit image interpolations. Both theforward and the inverse map are smooth by construction.However, the image I and the adjoint will generally showsignicantly more local variation (creating edge informa-tion, which is the basis of registration itself). Preservingthis local variation on coarse grids is challenging.

To accurately capture the evolutions of the image andthe adjoint, requires numerical methods which show min-imal numerical diffusion (and thus minimal image blur-

ring). This can be achieved by nonlinear numerical dis-cretizations, e.g., using ux limiters [ 18], essentially non-oscillatory schemes [ 16], or schemes aiming at a diffusionreduction [ 30, 17], but may not always be necessary asshown in Sec. 7. Fig. 1 demonstrates the effect of differentnumerical schemes implementing the advection equation totranslate a synthetic checkerboard image. A rst order up-winding discretization blurs the image greatly. Higher orderaccurate numerical schemes (back and forth error compen-sation, the 5th order weighted essentially non-oscillatory(WENO5) scheme and its anti-diffusive (aWENO5) ver-sion) result in signicantly less image blurring. By design,the WENO schemes suppress image oscillations. The visu-

ally best result is achieved by aWENO5.Fig. 5 shows computation results for the determinant of

the Jacobian of an exemplary map and the respective adjoint versions according to Eq. 20. As is the case for theadvections (Fig. 1) the rst order accurate numerical solu-tion provides qualitatively correct, but blurry results. TheWENO5 result obtained by solving the scalar conservationlaw is visually indistinguishable from the direct computa-tion based on the deformation map.


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As a compromise between simplicity, computationalcomplexity, and accuracy, we use WENO5 for our regis-tration experiments: the Hamilton Jacobi scheme for theadvection equation [ 15] and a conservative scheme for theadjoint equation [ 16]. Time integration is achieved by 3rdorder total variation diminishing Runge Kutta. To compare,

we also compute the registrations using rst order upwind-ing with Euler forward. We use a rst order Roe upwindingscheme with entropy x for the adjoint equation [ 18].

7. Results

Since the image-based and the map-based approachesare theoretically equivalent (Eqs. 2 and 3), we compare re-sults by computing the resulting energies after convergenceof the optimization. To fairly compare the approaches, weassess the quality of the resulting estimated velocity eld,v, by computing the nal registered image for each methodby a single WENO5 map pass. We compare image- andmap-based approaches using the rst order upwinding Eu-ler forward and the WENO5 implementations.

We conducted experiments on a set of synthetic datasetsas well as for real brain images. For the synthetic tests, wecreated two scenarios: one corresponding to a translatingsquare (E cases) (see Fig. 3) and one where the source im-age had two additional small squares that were not presentin the target image and could thus not be matched exactly (Icases). For each scenario, we created three sub-cases to testthe implementations sensitivity to blurred edges and noise.We ultimately ran six synthetic tests: E1 and I1 had crispedges, E2 and I2 were blurred, and E3 and I3 were blurredand altered with random additive noise.

The synthetic results show that, as expected, both map-based cases perform slightly better when crisp edges arepresent. When blurring and additive noise are induced, thedisparities between the image-based and map-based resultsdiminish. When considering an impossible registration, thenal energy values are higher. The differences betweenthese energy values for the four methods, however, stay rel-atively constant. Fig. 7 shows an overview of the results.

We also tested our implementations on three pairs of cor-responding brain slices taken from two distinct brain vol-umes (Fig. 4). For this real data, the displacement eldsare signicantly more regular, and as such the disparity be-

tween the results for the different methods diminishes.To illustrate the unbiased registration approach we cre-

ated a synthetic test image with nonuniform appearance(Fig. 6). The unbiased registration correctly aligned the im-age, while estimating a meaningful template image, remi-niscent of the mean of the source and the target. The tem-plate image was initialized with the source image. Fig. 6shows the rst and the second template updates. No signif-icant change is noticeable after the second iteration.

Source. 1st order image. 1st order map.

Target. WENO5 image. WENO5 map.Figure 3. Synthetic registration results for test case E1 (see Fig. 7).For large synthetic distortions, the differences between the com-puted results become more apparent than for the real test images.

R1 R2 R3

Figure 4. Registration results for real test cases R1, R2, and R3(see Fig. 7). In descending order for each column: source, target,1st-order image, WENO5 image, 1st-order map, WENO5 map.

8. Conclusion and Future Work

We proposed an optimal control formulation for de-formable image registration. We showed the formal equiv-


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E1: 8.74 E2: 5.30 E3: 5.92 I1: 10.11 I2: 5.68 I3: 6.34 R1: 4.47e-1 R2: 3.91e-1 R3: 4.57e-11st Img 1.31e-1 1.64e-1 6.27e-1 1.49 5.65e-1 1.03 3.39e-1 2.73e-1 3.26e-15th Img 4.11e-1 3.15e-2 4.89e-1 1.78 4.16e-1 8.84e-1 3.87e-1 2.72e-1 3.25e-11st Map 8.36e-2 3.60e-2 4.95e-1 1.46 4.20e-1 8.96e-1 3.39e-1 2.74e-1 3.27e-15th Map 3.59e-2 3.05e-2 4.83e-1 1.41 4.15e-1 9.44e-1 3.39e-1 2.73e-1 3.27e-1

Figure 7. Final energy results for synthetic test cases E1, E2, E3, I1, I2, I3 (resolution 50 x 50), and real brain images R1, R2, and R3

(resolution 208 x 175). The E cases can be exactly matched, while the I cases are impossible to match perfectly. Cases E1 and I1 havecrisp edges, E2 and I2 are blurred, and E3 and I3 are blurred and modied with random additive noise. The initial energy for each test caseis given with the label. In the impossible cases, the nal energies all increase relative to the exact cases, but the discrepancies between themap-based and image-based registrations remain similar, showing that the relative error discrepancy gets smaller. The three R cases arecorresponding slices from two different brain volumes and hence cannot be matched perfectly.

Figure 5. Determinant of Jacobian of . Left to right: Warped

grid, , |D | map, |D | 1st-order, and |D | WENO5. WENO5produces a faithful result by solving a scalar conservation law. The1st-order approach matches qualitatively, but introduces blurring.

Source. Target.

Template 1st iteration. Template second iteration.

Warped Source. Warped Template.Figure 6. The unbiased registration approach aligns the images,while estimating a template image reminiscent of an image-mean.

alence between this approach and the registration methodproposed by Beg et al. [ 4], We demonstrated the practicalityof purely image-based registrations and conducted a care-ful analysis of numerical implementations, in comparisonto the standard map-based implementations. In comparisonto the map-based approach, the proposed image-based ap-proach can be readily extended to manifolds, which so farhad not been explored in the large-displacement diffeomor-phic image registration framework. We further proposed a

new method for unbiased image registration, which gener-alizes to image sequences. The work suggests the feasi-bility of similar image-based optimal control approaches todynamical systems of higher order, which we will explorein future work. Such higher-order dynamical systems areneeded to formulate registration as an initial value problem,

thereby reducing memory requirements.

A. Optimality Conditions

For a minimizer of energy 5, its variations with respectto v, , I , and need to vanish. Computing

E (v, I , , ; dv,dI,d,d ) =

E (v + dv, I + dI, + d, + d )| =0 ,

yields

E = 1

02 v,dv V + d,I t + ( DI )v

+ ,dI t +( DdI )v+( DI )dv dt + d,I (0) I 0 + ,dI (0)

+2

2I (1) I 1 , dI (1) .

Since

1

0,dI t dt =

1

0 t , dI dt + ,dI |

10 ,

and (by Greens theorem)

, (DdI )v = div (v ), dI +

dIv dS = div (v ), dI

assuming v vanishes on the domain boundary, we get

E = 1

02L Lv + ( DI )T ,dv + I t + ( DI )v,d

+ ( t + div (v )) , dI dt + ,dI |10

+2

2I (1) I 1 , dI (1) + I (0) I 0 , d + ,dI (0) .


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Collecting terms yields

E = 1

02L Lv + ( DI )T ,dv + I t + ( DI )v,d

+ ( t + div (v )) , dI dt +2

2(I (1) I 1 )+ (1) , dI (1)

+ I (0) I 0 , d + (0) , dI (0) . (21)

Since E needs to vanish for any dv , dI , d , d , fulllingthe boundary conditions (i.e., dI (0) = 0 ) of the problem,the necessary optimality conditions follow.

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