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Group-wise Cortical Correspondence via SulcalCurve-Constrained Entropy Minimization

Ilwoo Lyu 1 , Sun Hyung Kim 2 , Joon-Kyung Seong 4 , Sang Wook Yoo 5 ,Alan C. Evans 6 , Yundi Shi 2 , Mar Sanchez 7 , Marc Niethammer 1 ,3 , and

Martin A. Styner 1,2

1 Dept. of Computer Science, University of North Carolina, Chapel Hill, NC, USA{ilwoolyu,styner}@cs.unc.edu

2 Dept. of Psychiatry, University of North Carolina, Chapel Hill, NC, USA3 BRIC, University of North Carolina, Chapel Hill, NC, USA

4 Dept. of Biomedical Engineering, Korea University, Seoul, South Korea5 Dept. of Computer Science, KAIST, Daejeon, South Korea

6 Montreal Neurological Institute, McGill University, Montreal, Quebec, Canada7 Yerkes National Primate Research Center, Emory University, Atlanta, GA, USA

Abstract. We present a novel cortical correspondence method employ-ing group-wise registration in a spherical parametrization space for theuse in local cortical thickness analysis in human and non-human primateneuroimaging studies. The proposed method is unbiased registration thatestimates a continuous smooth deformation eld into an unbiased averagespace via sulcal curve-constrained entropy minimization using sphericalharmonic decomposition of the spherical deformation eld. We initializea correspondence by our pair-wise method that establishes a surface cor-respondence with a prior template. Since this pair-wise correspondenceis biased to the choice of a template, we further improve the correspon-dence by employing unbiased ensemble entropy minimization across allsurfaces, which yields a deformation eld onto the iteratively updatedunbiased average. The specic entropy metric incorporates two terms:the rst focused on optimizing the correspondence of automatically ex-tracted sulcal landmarks and the second on that of sulcal depth maps.We also propose an encoding scheme for spherical deformation via spher-ical harmonics as well as a novel method to choose an optimal sphericalpolar coordinate system for the most efficient deformation eld estima-tion. The experimental results show evidence that the proposed methodimproves the correspondence quality in non-human primate and humansubjects as compared to the pair-wise method.

Keywords: Group-wise correspondence, Sulcal curves, Spherical har-monics, Entropy minimization, Cortical thickness

1 Introduction

Group analysis of cortical properties such as cortical thickness is an importanttask for monitoring brain growth, investigating anatomic connectivity, and dis-covering cortical disease patterns. A prerequisite to such tasks is to establish a

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Group-wise Cortical Correspondence with Sulcal Curve Constraints 3

SurfaceReconstruction

Curve Matching IndividualRegistration

Sulcal CurveExtraction

MR Images TriangulatedSurfaces

Pre-labeled Curves AutomaticallyLabeled CurvesPair-wise

Correspondence

Group-wiseCorrespondence

EntropyMinimization

Fig. 1. Schematic overview of the proposed method

smooth deformation eld representation, and 3) application in both non-humanprimate and human data. Figure 1 illustrates an overview of our pipeline.

2 Method

2.1 Preprocessing

Surface Reconstruction. Raw MR images are registered into the standard-ized stereotaxic space using a rigid transformation. The registered images arecorrected for intensity nonuniformity resulting from inhomogeneities in the mag-netic eld. The inner- and outer-surfaces are automatically extracted by the Con-strained Laplacian-based Automated Segmentation with Proximities (CLASP)algorithm [ 4]. We use the middle cortical surface models with a triangulatedmesh of 40,962 vertices in the native space. Each vertex of the surface modelsis then homeomorphically transformed onto the common unit sphere using adeformable surface model [4]. After surface reconstruction, we compute sulcaldepth maps via the geodesic distance from the gyral crowns as proposed in [ 12].

Landmark Correspondence. We use automatic sulcal curve extraction [13]and automatic sulcal curve labeling [7] to extract a set of labeled sulcal curves.In particular, the unlabeled sulcal curves (consisting of ordered sets of points)are extracted from the triangulated surface, and then pre-labeled sulcal curves inthe template are employed to label matched (corresponding) sulcal curves in thesubject, while discarding minor and extraneous curves. This labeling methodfurther establishes a point-by-point correspondence across these sulcal curvescalled sulcal landmarks in the remainder of this paper.

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2.2 Sulcal Curve-Constrained Pair-wise Correspondence

Problem Denition. For two given triangulated cortical surfaces (template

and subject), we denote V temp and V subj as the sets of the vertices of the templateand subject surfaces. Our goal is to estimate a continuous mapping function of the cortical correspondence M : IR3 IR3 such that

u = M (v) , (1)

where u V temp and v V subj are the corresponding points.

Consistent Displacement Encoding Scheme. To take advantage of the well-known spherical parametrization, we map all vertices of the cortical surfaces ontothe common unit sphere by using an invertible spherical mapping : IR3 S2established in the preprocessing stage. Note that this spherical mapping gener-ally does not establish an appropriate correspondence. We then locally encodethe deformation as change in local spherical polar angles of elevation andazimuth . It can be easily observed that the same length of geodesic dis-tances provides different angular differences close to the equator as comparedto closer to the pole. To avoid such inconsistency and thus to provide a consis-tent arclength encoding, we propose a locally normalized coordinate system. Forthis local normalization, let temp ( p) and subj (q ) be the mapped correspondingsulcal landmarks from the template and a subject, respectively. First, we nda rotation matrix R q with an angle ( 90) along the longitude circle passingthrough subj (q ) and the two poles, such that subj (q ) is exactly located on theequator. By applying R q to temp ( p) and subj (q ), we then compute normalizedangular displacements and . As sulcal landmarks are described by spher-ical polar coordinates, we denote an operator that rotates these landmarkswith a rotation matrix dened in a Cartesian coordinate system. Thus, the locallandmark displacement at a point i (

i,

i) on the unit sphere is represented as

a vector d i = R q i temp ( pi ) R q i subj (q i ) = [ i , i ]T (see Fig. 2a).Initial Deformation Field. To nd an initial deformation eld of the entiresurface, we compute least squares tting of spherical harmonics to displacementsof the sulcal landmarks established in the sulcal labeling step. This tting isstandard spherical harmonic decomposition of the spherical signals ( and ). At a point ( , ) on the sphere, the spherical harmonic basis function of degree l and order m (l m l) is given by

Y ml (, ) = 2l + 14 (l m )!(l + m )!P ml (cos )e im , (2)Y

m

l (,

) = ( 1)m Y m

l (,

),

(3)where Y ml denotes the complex conjugate of Y m

l and P ml is the associated

Legendre polynomial

P ml (x ) =(1)m

2l l!(1 x2)

m2

d( l+ m )

dx ( l+ m )(x2 1) l . (4)

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d i = [ i , i ]T

(a) (b)

Fig. 2. Displacement encoding (a) and estimated deformation eld (sampling of acontinuous representation) with sulcal depth maps (b). A spherical displacement isencoded as change in spherical angles after rotation onto the equator for arclengthpreservation, which avoids a distorted displacement representation. The deformationeld is estimated by extrapolation using spherical harmonic decomposition.

Since the spherical harmonic basis functions are dened in the complex domain,we use a real form of the functions dened by

Y l,m =

1 2 (Y m

l + ( 1)m Y ml ) m > 0 ,Y 0l m = 0 ,

1 2i (Y m

l (1)m Y ml ) m < 0 .(5)

Given spherical harmonic decomposition of degree k, we have (k + 1) 2 sphericalharmonic basis functions at a given point on the sphere. To avoid a rank decientproblem, we assume that the number of the landmarks n is larger than ( k + 1) 2 .The coefficients can then be estimated by standard least squares tting:

C = YY T 1

YD T , (6)

where D = [d 1 , d 2 , , d n ] and Y is a (k + 1) 2-by-n matrix that incorporatesthe spherical harmonic basis functions. Once the coefficients of the sphericalharmonic decomposition are determined, for v V subj , its deformed positionin the template space is easily reconstructed by the mapping function M :

u = M (v) = 1temp (R T v (R v subj (v) + C T Y v )) , (7)where Y v is a column vector of the spherical harmonic basis functions at subj (v)and R v is a rotation matrix that puts subj (v) on the equator. Figure 2b showsan example of the estimated deformation eld.

This spherical harmonic decomposition of the deformation eld is hierarchicaland orthonormal. We employ the hierarchy in that the initial deformation eldis computed via low degree ( k = 5) tting of sulcal landmarks and higher degreerepresentations are used in the optimization stage.

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Optimization. Since the initial coefficients are determined only by sulcal land-marks, the cortical correspondence is potentially biased to the specic sulcalfundic regions selected in the sulcal labeling step as well as affected by minormislabeling errors. For improved correspondence establishment, we further for-mulate a metric that incorporates sulcal landmark errors and differences betweensulcal depth maps via normalized cross correlation over the entire cortical sur-face. To regularize the impact of sulcal landmark errors, we dene a weightingfunction f under the Gaussian assumption. Specically, incorporating dmin asvoxel size, sulcal landmark errors below dmin are ignored and have the maximumat distance dmax , chosen 1020 times larger than dmin , as follows.

f (d) = 2 d

d min

I (d) 2 exp

12

x dmin

2

dx , (8)

I (d) = 1d dmin ,

0 otherwise ,(9)

where 6 = dmax dmin . Now, we dene L (, ) = f (arclen( , )) as a regularizedarclength, where is a ratio of the geodesic distance between two points on theunit sphere and on the template surface. In practice, it can be approximated by aratio of the triangle size under the assumption that the template surface consistsof uniform triangles. The resulting overall cost function like an M-estimator isthus formulated with a regularization factor w by letting an operator denotethe normalized cross correlation between two sulcal depth maps:

C = argminC

w1n

n

i =1

L ( pi , pi ) + (1 w)12

(1 S ({u}) S ({u})) ,(10)

where S () is a sulcal depth map reconstructed from a set of vertices. The opti-mization procedure employs the NEWUOA optimizer [11] for minimizing C . Inour experiment, we empirically set w = 0 .5.

Optimal Pole Selection. The proposed spherical polar coordinate system doesnot force the direction of displacements to be invariant to locations. Dependingon rotation to the equator, two identical displacements are likely to have adifferent sign in polar angles if they are computed on opposite sites with respectto the pole. This can yield a deformation eld with abrupt sign changes closeto the pole. A proper choice of the pole e can signicantly minimize this issueand yield smooth deformation elds. The presence of non-smooth deformationwill generally lead to high magnitude coefficients in the high-frequency harmonic

basis functions. Based on this observation, we employ a coefficient sum-basedmetric that weighs higher frequency coefficients stronger:

e = argmine

k

l=0

l

m = l(l + 1) |c l,m |+ |c l,m | , (11)

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where c and c are coefficients for elevation and azimuth displacements. As thismetric possibly has local minima, we initialize the optimization with multipleinitial guesses spread across the sphere and select the minimum as the nal pole.

2.3 Extension to Group-wise Correspondence

While our pair-wise method provides adequate registration results (see Sect. 3),we propose here the use of group-wise registration to further improve theseresults as well as to remove the template selection bias inherent to pair-wiseregistration. A group-wise correspondence is computed independently from thetemplate and thus is expected to perform more stably across a population of surfaces. We propose to use entropy minimization to establish a group-wise cor-respondence. Our group-wise correspondence method incorporates entropy termscomputed over the landmark distribution and sulcal depth maps.

Problem Denition. For N given triangulated cortical surfaces mapped ontothe unit sphere, each of which has the same number n of the common correspond-ing points, we let V i be the set of the vertices in the ith subject, i = 1 , , N .The goal is to estimate continuous mapping functions of the cortical correspon-dence M i : S2 S2 across subjects such that

M 1(v1) = M 2(v2) = = M N (vN ) , (12)where vi V i are the corresponding points across subjects. Let x (M j ) be acolumn vector of the corresponding points of a subject j deformed by M j , i.e.,x (M j ) = [M j (vj1 ), , M j (vjn )]T . As described in [9], we assume that x (M j )are instances of X drawn from a probability density function. The amount of the information in the random sampling is given by the entropy H [X ], and theminimization problem is then formulated as follows.

{M 1 , , M N }= argmin{M 1 ,,M N }H [x ] , (13)

which drives mapped/deformed corresponding points closer to each other.

Entropy of Landmark Errors. We employ the pair-wise correspondence(Sect. 2.2) as initialization for the proposed group-wise method. As the sulcal la-beling procedure yields varying parts of sulcal curves being labeled, we constrainthe set of sulcal landmarks only those that have a full correspondence across allcortical surfaces. A key step for entropy computation is the density estimationof corresponding sulcal landmarks. However, appropriate density estimation onthe sphere can be computationally demanding since it involves geodesic distancecomputation. Similar to [9] for efficiency, we assume that the initial mappingwell centralizes corresponding sulcal landmarks, which allows a mapping fromthe spherical space to the Euclidean space S 2 IR3 under the assumption of theproximity of the landmarks. We compute the average over corresponding sulcallandmarks, which is rescaled to the sphere, and the landmarks are projected ontothe tangential plane at that approximated mean to enable Euclidean statistics.

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Entropy of Sulcal Depth. Since sulcal landmarks are sparsely distributedover the sphere and also likely possess minor mislabeling, we employ additionalentropy computation over sulcal depth maps densely sampled across the surface.Let s () denote the sulcal depth at a given point and vj be the point in the j th subject such that u = M j (vj ), where u is a given point on the sphere.Given M j , v j has a correspondence with its corresponding points. Ideally, therewill be little difference in sulcal depth across the corresponding points if themapping is well established, that is, s (M 1(v1)) = = s (M N (vN )). By uniformicosahedron subdivision-based spherical sampling of u , sulcal depth agreementis straightforwardly plugged into the entropy minimization problem.

Entropy Minimization. We model x (M j ) as an instance of X such that

x (M j ) = proj v 1 (M j (vj1 )) , ,proj v n (M j (v jn )) , S ({M j (vj )})T

, (14)

where proj v () denotes the projection of a point onto the tangential plane at theapproximated mean v over the corresponding sulcal landmarks. For the densityestimation, we assume a multivariate Gaussian distribution with covariance and therefore, the entropy is obtained by

H [X ] 12

ln | | =12

ln , (15)

where are the eigenvalues of . By letting x be the sample mean and Z =[x (M 1) x , , x (M N ) x ], the sample covariance is given by 1N 1 ZZ

T . Inpractice, however, the eigendecomposition of the sample covariance is an in-tractable task for the large dimension of X ( N ). As stated in [9], we insteadcompute the eigenvalues of 1N

1 Z

T Z in the dual space. The optimization usesthe NEWUOA optimizer [ 11] for solving the entropy cost function.

3 Experimental Results

We applied the proposed method on both non-human primate and human subjects to evaluate the established correspondence quality. Since there exists noground-truth of the cortical correspondence, we made comparisons with the ini-tial spherical mapping and the pair-wise method via analysis on cortical thicknessas well as the agreement with manually labeled sulcal curves.

3.1 Macaque Cortical Surfaces

We used the same data set as that in [6]. 18-month-old macaques were im-aged under anesthesia at the Yerkes Imaging Center (Emory University, GA) ona 3T Siemens Trio scanner with an 8-channel phase array trans-receiving vol-ume coil. T 1-weighted scans were acquired using a 3D MP-RAGE sequence withGRAPPA at a high resolution of 0.6 mm 0.6 mm 0.6 mm (TR=3,000 ms ,

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Template Initial mapping Pair-wisecorrespondenceGroup-wise

correspondence

Fig. 3. Visual comparison of correspondence results. The colored template surface ( rst column ) is propagated to a selected, representative example surface via initial sphericalmapping ( second ), pair-wise correspondence ( third ), and group-wise correspondence( fourth ). The arrows indicate areas of visual differences across the correspondence

methods.

TE=3.33 ms , ip angle=8 , matrix=192 192). In the experiment, we ran-domly selected a single subject as a template with its 11 manually labeled sulcalcurves: central sulcus, arcuate sulcus, principal sulcus, superior temporal sulcus,intraparietal sulcus, lunate sulcus, inferior occipital sulcus, occipitotemporal sul-cus, cingulate sulcus, parieto-occipital ssure, and sylvian ssure.

We further established a sulcal curve-based color mapping across the corticalsurfaces to provide a visual quality assessment of the established correspondenceby propagation of the colorized template surface to other subjects. To generatethe reference colorized surface, a unique RGB color was assigned to each sulcalcurve, and then each RGB channel was extrapolated to the entire surface via

spherical harmonic decomposition. In Fig. 3, the proposed group-wise methodshows qualitative improvement over the pair-wise correspondence.

3.2 Human Cortical Surfaces

Pediatric 2-year-old subjects were acquired on a 3T Siemens Trio scanner ata resolution of 1.0 mm 1.0 mm 1.0 mm with T 1-(160 slices, TR = 2400 ms ,TE=3.16 ms , ip angle=8 , matrix=256 256) and T 2-weighted (160 slices,TR = 3200 ms , TE=499 ms , ip angle=120 , matrix=256 256) scans. Weused 10 subjects chosen at random from the scans acquired as part of the InfantBrain Imaging Study (IBIS) network 1 at four different sites (University of NorthCarolina, University of Washington, Washington University in Saint Louis, andThe Childrens Hospital of Philadelphia). We randomly selected a single subjectas a template, and an expert manually labeled 13 major curves on the templatesurface: superior temporal sulcus (STS), inferior temporal sulcus (ITS), collateralsulcus (ColS), central sulcus (CS), precentral sulcus (PrCS), postcentral sulcus

1 http://www.ibis-network.org
http://www.ibis-network.org/http://www.ibis-network.org/

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10 I. Lyu et al.

Initial mapping Pair-wise correspondence Group-wise correspondence

STS ITS ColS CS PrCS PoCS IFS SFS IPS CingS POF CalcF SylF

Fig. 4. Sulcal curve agreement by initial spherical mapping ( left ), pair-wise correspon-dence ( middle ), and group-wise correspondence ( right ). The arrows indicate improvedagreement in the group-wise as compared to the pair-wise correspondence.

(PoCS), inferior frontal sulcus (IFS), superior frontal sulcus (SFS), intraparietalsulcus (IPS), cingulate sulcus (CingS), parieto-occipital ssure (POF), calcarinessure (CalcF), and sylvian ssure (SylF) (see Fig. 4). Only left hemisphereswere used in the experiment. For a visual assessment, all major curves were alsomanually labeled on the 9 remaining subjects.

We applied a leave-one-out cross-validation technique for evaluation of theoptimal pole selection. We removed a single sulcus from each individual subject

during registration and measured landmark (reconstruction) errors between theremoved sulcus reconstructed by the deformation eld and its corresponding onein the template. Figure 5 shows the smaller average reconstruction errors for theoptimal pole and the reduced coefficient load of the azimuth displacement forthe high-frequency harmonic basis functions by the proposed pole selection.

For quantitative evaluation of the correspondence quality, we rst measuredcross-subject variance estimates of sulcal depth over all vertices of the entiresurface across subjects. However, such an evaluation is biased, as sulcal depthis employed in the cost function. We further used variance estimates of corti-cal thickness as well as a visual assessment of manually labeled sulcal curvesfor unbiased evaluation. Note that the manually labeled sulcal curves were notused during processing in our pipeline, which implies that the evaluation is in-dependent of the automatically labeled curves. In Table 1 the statistical anal-ysis indicates superior performance of our method for variance of sulcal depthand cortical thickness measures, with signicant differences to both the initialmapping and the pair-wise method, revealed by Students t-test ( p < 0.0001).Visually, the mapped major sulcal curves shows improved agreement in severalregions as compared to the pair-wise correspondence as shown in Fig. 4.

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Degree

R e c o n s .

E r r o r

5 7 9 11 13 15 17 190.08

0.09

0.1

0.11

0.12Standard Optimum

# of Basis Functions

C o e

f f .

L o a

d

1 6 11 16 21 26 31 360

0.20.4

0.6

0.8

1Standard Optimum

# of Basis Functions

C o e

f f .

L o a

d

1 6 11 16 21 26 31 360

0.20.4

0.6

0.8

1Standard Optimum

(a) (b) (c)

Fig. 5. Reconstruction errors (a) and cumulative coefficient load for elevation (b) andazimuth (c) displacements. No major differences are observed for the elevation dis-placements, whereas for the azimuth displacements, the total amount of the coefficientload signicantly decreases for the optimal pole selection.

Table 1. Statistical analysis of variance of cortical properties. The proposed methodshows better variance estimates of sulcal depth and of cortical thickness with signicantdifferences to both the initial mapping and the pair-wise method ( p < 0.0001).

Sulcal Depth ( mm 2 ) Cortical Thickness ( mm 2 )Method Mean SD Mean SDInitial 1.9234 1.1300 0.5003 0.3925Pair-wise 1.5014 0.9759 0.4948 0.3799Group-wise 1.3181 0.9446 0.4741 0.3616

4 Conclusion

We presented an automatic group-wise cortical correspondence method that esti-mates a smooth continuous deformation eld using entropy minimization of sul-cal landmarks and sulcal depth maps. In our experiment, the proposed method

outperformed the pair-wise method in non-human subjects via a visual assess-ment and in human subjects via quantitative analysis and visual comparisons.To enable enhanced spherical harmonic decomposition of the deformation eld,we also proposed a consistent displacement encoding scheme and an optimal poleselection strategy. In the experiment, the optimal pole selection showed smallerreconstruction errors and more efficient encoding of the deformation eld.

The proposed method allows the inclusion of additional information suchas DTI-based connectivity akin to [9]. Furthermore, any prior information suchas lobar labeling, surface colorization, etc. can be straightforwardly propagatedfrom the template. In this way, inter-subject variability of cortical properties de-ned in the template space could be incorporated into the entropy minimization.

It is noteworthy that our proposed method is sensitive to the quality of theautomatic labeling of sulcal curves, as mislabeled sulcal curve will negatively in-uence the established correspondence. In our experiment, the proposed methodhas (qualitatively) shown to be quite resilient to errors in sulcal labeling if thelarge majority of sulcal curves within the same class are correctly identied.In our future work, we will extend this method to include robust estimators of entropy for the sulcal curves to reduce the inuence of such errors.

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Acknowledgments. This work was funded by the National Institutes of Health(NIH) under Grant Nos. (R01 MH091645-02, U54 EB005149, P50 MH078105-01A2S1, P50 MH078105-01, P50 MH100029, and P30 HD003110).

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