Computerized Medical Imaging and Graphics · cerebral cortex in real time during generation of the surface using information from parcellated 3D ... warps a pre-labeled ... A manual

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Computerized Medical Imaging and Graphics 35 (2011) 206–219

Contents lists available at ScienceDirect

Computerized Medical Imaging and Graphics

journa l homepage: www.e lsev ier .com/ locate /compmedimag

n improved representation of regional boundaries on parcellated morphologicalurfaces

uejun Hao ∗, Dongrong Xu1, Ravi Bansal2, Jun Liu3, Bradley S. Peterson4

RI Unit, Psychiatry Department, Columbia University & the New York State Psychiatric Institute, United States

r t i c l e i n f o

rticle history:eceived 7 October 2008eceived in revised form7 September 2010ccepted 8 November 2010

eywords:

a b s t r a c t

Establishing the correspondences of brain anatomy with function is important for understanding neu-roimaging data. Regional delineations on morphological surfaces define anatomical landmarks and helpto visualize and interpret both functional data and morphological measures mapped onto the corticalsurface. We present an efficient algorithm that accurately delineates the morphological surface of thecerebral cortex in real time during generation of the surface using information from parcellated 3D data.With this accurate region delineation, we then develop methods for boundary-preserved simplificationand smoothing, as well as procedures for the automated correction of small, misclassified regions to

egion delineationarcellationarching-cubes

urface constructionurface simplification-spline curve

improve the quality of the delineated surface. We demonstrate that our delineation algorithm, togetherwith a new method for double-snapshot visualization of cortical regions, can be used to establish a clearcorrespondence between brain anatomy and mapped quantities, such as morphological measures, acrossgroups of subjects.

© 2010 Elsevier Ltd. All rights reserved.
isualizationurface rendering
. Introduction

The primary goals of human brain mapping are to integraterain anatomy with measures of brain function and behavior and tostablish correspondences between them. Sulci and gyri on the sur-ace of the cerebrum, for example, are anatomical landmarks thatelp to define the locations of major functional areas. Labeling and,ltimately, parcellation of geometric features of the cortical sur-ace is thus important for analyzing and visualizing both functionalnd structural neuroimaging data. Mapping of functional activa-ions or morphological measures onto parcellated cerebral cortexcross ages, for example, can improve understanding of brain devel-pment.

Individual variability in the folding patterns of each individual
ortex makes the automated identification and labeling of corticaltructures challenging. Various techniques have been developedo address this difficulty, especially the automated delineation ofulci. They can be categorized broadly as either a surface-based
∗ Corresponding author at: Columbia College of Physicians & Surgeons, Columbianiversity and the New York State Psychiatric Institute, 1051 Riverside Drive, Unit4, New York, NY 10032, United States. Tel.: +1 212 543 1905; fax: +1 212 543 0522.

E-mail address: [email protected] (X. Hao).1 Tel.: +1 212 543 5495; fax: +1 212 543 0522.2 Tel.: +1 212 543 6145; fax: +1 212 543 0522.3 Tel.: +1 212 543 5207; fax: +1 212 543 0522.4 Tel.: +1 212 543 5330; fax: +1 212 543 0522.

895-6111/$ – see front matter © 2010 Elsevier Ltd. All rights reserved.oi:10.1016/j.compmedimag.2010.11.002

[1–8], graph-based [9,10], or volumetric-region-based approach[11]. One algorithm, for example, warps a pre-labeled brain atlasonto the surface model of each participant’s brain to establish thecorrespondences between them that permit labeling of each voxelas a particular tissue class and a specific region of interest (ROI)[2]. Another algorithm models the sulci as vertices and the relationbetween sulci as arcs, and then assigns labels to the identified sulciusing a manually generated training set [10]. Still another appliesa watershed algorithm on the cortical surface and then manuallylabels identified sulci [5].

The automatic labeling of tissues and parcellation of the brain,however, requires quantitative knowledge of the geometric varia-tion in brain anatomy, as well as its critical functional interfaces.A manual parcellation of the cortex by an experienced neu-roanatomist can take advantage of much additional information,such as cytoarchitectonic labeling properties and knowledge ofstructure–function relationships. The inclusion of such prior infor-mation is critically important for the success of any algorithm forcortical parcellation [6].

Anatomical correspondence is especially important at func-tional interfaces and cytoarchitectonic boundaries [1,12]. Aparcellated cortical surface that has a clear delineation of regional
boundaries can be used for mapping onto the cortex and regionallydelineating functional imaging data and measures for morpholog-ical differences, thus revealing the correspondences and spatialrelationships between brain structure and function. The generalapproaches to this problem, however, focus on the parcellation
dx.doi.org/10.1016/j.compmedimag.2010.11.002

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dx.doi.org/10.1016/j.compmedimag.2010.11.002

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tself, rather than on generating and manipulating the boundariesf the regional delineation of the cortical surface that the parcella-ion produces. These approaches generally aim to associate a labelo each surface element (e.g. a triangle or quadruple) on an existingurface without adjusting the surface geometry. The region delin-ations thus generated, though satisfactory for many purposes, cane imprecise because the assumption may not always be true thathe actual regional boundary coincides with the existing discreteurface geometry. Moreover, these approaches make difficult theontrol of precision for further processing of the delineated surface,uch as smoothing of a regional boundary and surface simplifica-ion.

We propose an efficient algorithm for the delineation of corticalurfaces using manually parcellated volumes generated previouslyy experienced neuroanatomists. Our program takes as input twoD volumetric data – one that is a scanned intensity volume, andhe other that is a parcellated categorical volume coinciding spa-ially with the first and that encodes all the expert knowledge ofhe parcellation. Our algorithm integrates surface delineation withurface generation and guarantees precise coincidence of regionaloundaries with surface geometry. We extend the marching-cubeslgorithm [13] and generate the cortical surface while simul-aneously delineating cortical regions by adaptively subdividingurface triangles that are located on regional boundaries. We thenevelop various automatic algorithms for surface processing, suchs boundary-preserved surface simplification and smoothing ofegional boundaries under user-specified error estimates. In addi-ion, we design a new visualization algorithm that displays regionaloundaries clearly, even when portions of the curves that define theegion are buried inside sulci and are not visible from a particulariewing perspective.

. Methods

Parcellation of a surface requires the accurate partitioning ofhat surface into disjoint regions according to the anatomical orunctional sub-structures that each of the regions represents, andith computationally efficient visualization of the regions that are

hus generated. A surface is normally represented graphically as aesh of triangles. Region delineation thus partitions the triangularesh into a set of disjoint sub-meshes. This partition then can be

isualized by assigning different colors to different sub-meshes ory superimposing onto the mesh curves that mark the boundariesetween regions.

.1. Partition of a surface into regions

A surface is defined in a 3D volumetric data space V by specifyinghe conditions that points on the surface must satisfy. For example,he cortical surface of a brain can be generated from the iso-valueontour of gray matter that has been defined on an MR image. Theurface thus generated is called the iso-surface.

Let V be a continuous 3D space, S(a) be an iso-surface havinghe iso-value a, x be a point in V, and the volumetric data definedontinuously over V be intensity I(V), then S(a) is the collection ofoints that have the intensity value a:

(a) = {x|x ∈ V, I(x) = a}

An intensity volume such as I(V) is usually defined by a set ofontinuous values that have not been classified previously into
ifferent sub-volumes or subregions. This classification can be per-ormed using a wide range of procedures, such as the manual orutomatic, mathematically based segmentation of I(V). Let C(V) benother volumetric data space defined in the same space V that par-itions V categorically into the collection of disjoint sub-spaces V1,
ng and Graphics 35 (2011) 206–219 207

V2, . . . Vn, such that the points in each sub-space Vi have the samecategorical value Ci. The delineation of a surface S(a) according toC(V) maps each point x on the surface to a categorical value Ci. Thisprocess defines the segmentation of V.

The assignment of categorical values to a surface is straightfor-ward if volumetric data I(V) and C(V), as well as the surface S, arecontinuous. One needs only to determine in which sub-space Vieach point x on S belongs. If either one or both of I(V) (and thusC(V)) and S are discretely sampled, however, and if they produce adiscontinuity of the sampled points on S with the sampled points onV, then the mapping of Ci to x will not be obvious and will need to bemore precisely defined. Here we assume that the set of values of I(V)and C(V) are given at the same discrete locations in the space wherethey overlap. This is the case when C(V) is the segmented versionof I(V), which is the consequence of manual or automatic segmen-tation approaches. If C(V) is generated by a different procedurethan is I(V), however, as when C(V) is defined by cytoarchitectonicmaps of one subject and I(V) is defined on a T1-weighted MR image,then these two volumes must be coregistered into the same space,where they will be discontinuous with one another. The points inthe region where these volumes overlap must be sampled before asegmented surface S(a) of I(V) can be delineated by C(V).

2.1.1. Extension of the marching-cubes algorithmTo delineate a surface S(a) that is represented as a collection

of triangles, we must map each triangle within S(a) to a categoricalregion Ci. Assigning categorical regions to a triangle’s three verticesis relatively easy if we map those vertices to the categorical labelof the nearest voxel, as defined by volume C(V). A triangle T, how-ever, can have vertices that belong to differing categories – i.e., thathave differing values of Ci. In that instance, the segmentation of Tcannot be determined from this simple mapping. Moreover, evenif all three vertices of T do belong to a single category, a portion ofT still could belong to another category.

A voxel in a discretely sampled space represents a unit volumeof that space. The shape of this unit volume depends on the methodused to sample the space. With a homogeneous orthogonal sam-pling, the space is divided evenly into unit cubic volumes. We assignsampled values at each unit volume to its center and call it a voxel.Thus a voxel of C(V) having a categorical label Ci represents a unitvolume of space (Fig. 1a) that belongs to subspace Vi, and all vox-els of C(V) together impose upon the space a discrete partition. Toidentify the category to which a triangle belongs, one needs only toquery the voxels of C(V) that enclose the triangle. If the triangle isfully enclosed by one or more voxels having the same value Ci, thetriangle simply is assigned this single categorical label Ci. If a por-tion of the triangle crosses voxels of differing Ci (Fig. 1b), however,then we cannot assign its category unambiguously, and so we willneed to subdivide this triangle into smaller ones until each of themlies entirely in a single subspace.

Marching-cubes is a popular algorithm used to generate an iso-surface S(a) from a rectangular sampled volume I(V), in which eachsampled point is called a voxel. The algorithm is computation-ally efficient, employing a divide-and-conquer approach in treatingseparately each unit rectangular cube that composes the volumeand that is defined by eight neighboring voxels positioned at itscorners. The algorithm assesses whether the sampled value in eachcube matches the iso-value of the surface to determine whetherthe cube intersects the iso-surface. The algorithm has an efficientlinear O(N) complexity, where N is the number of voxels in the datavolume. One advantage of using the marching-cubes algorithm for
delineation of a surface is that each triangle is guaranteed to lie inthe cubic space enclosed by the eight voxels at the corners of thecube (Fig. 1b). Thus, we need only to identify the labels Ci at theseeight voxels to determine the label at the triangle. For surfaces gen-erated using other algorithms, even though our method remains

208 X. Hao et al. / Computerized Medical Imaging and Graphics 35 (2011) 206–219

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ig. 1. Voxel representation and triangle labeling. (a) A voxel P1 represents the unitoints at (Pa, Pb, Pc) is categorically labeled according to the categorical values at voxo category C1, while the other five voxels belong to category C2. We show later thatabel. (For interpretation of the references to color in this figure legend, the reader i

pplicable, more voxels may need to be assessed, the consequenceeing a longer computational time.

The eight voxels at the corners of the unit cubic volume of thearching-cubes algorithm inherently divides the cubic volume into

ight subregions, each of which belongs to the corresponding voxelFig. 2). The division occurs at the boundaries of the unit volumehat is represented by these voxels, which are three orthogonallanes positioned midway between the voxels, dividing the space

nto eight equally sized portions. If the labels differ across theseight voxels, then these three planes can be used to subdivide ariangle into several smaller triangles, each belonging to a singleoxel and thus being assigned a single segmentation label.

We extend the marching-cubes algorithm by incorporatingurface delineation through dynamically labeled triangles. Ourxtension and other contributions in this report can be briefly sum-arized as follows:

ig. 2. Subdivision of unit cubic space. A unit cubic space enclosed by eight adjacentoxel centers is divided into eight disjoint sub-volumes. The division happens athree planes perpendicular to each other and each cuts the cubic space into twoqual-sized halves.

e centered at P1 (its boundary is marked by blue lines); (b) a triangle with corner, P2, . . ., P8 enclosing the space where the triangle locates. Here P1, P4, and P5 belongiangle needs to be sub-divided in order to unambiguously determine its categoricalrred to the web version of the article.)

(1) Dynamic assignment of labels to surface triangles using trian-gular subdivisions, if required, during surface generation.

(2) Simplification of the surface thus generated so as to reduce thenumber of triangles while preserving a valid regional interface.

(3) Smoothing of the regional interface using B-spline curves underuser-controlled error thresholding and smoothing.

(4) Redistributing vertices during the smoothing of the regionalinterface.

In addition, we develop a double-snapshot algorithm for visu-alizing the labeled surfaces with other information superimposed.Our algorithm is implemented in C++, and visualization is realizedusing OpenGL.

An iso-value triangle generated from a unit cubic volume in I(V)is used to identify the label Ci for each of the eight voxels that definethe corresponding cubic space in C(V). As noted above, if all eightvoxels belong to the same category, we need no further processingand simply assign this categorical value to the triangle. If, on theother hand, these labels are not all the same, then the triangle ison the regional boundary and must be subdivided. We subdividea triangle on a boundary by identifying how it intersects the threeplanes that equally divide the unit cubic volume enclosing the tri-angle. Each possible intersection splits the triangle into one or moresmaller triangles (Fig. 3). Then we assign the corresponding labelsat the eight voxels to the subdivided triangles.

A non-boundary triangle may need to be re-tessellated if it liesadjacent to one or more boundary triangles that have been subdi-vided (Fig. 4). The re-tessellation of these non-boundary trianglesdepends on how many edges they share with boundary trianglesand how those shared edges are subdivided.

Our adaptation of the marching-cubes algorithm uses the samedivide-and-conquer approach as the original algorithm. It there-fore has a similar linear O(N) complexity in terms of the number ofvoxels, adding only a constant factor to that efficiency associatedwith subdividing boundary triangles and re-tessellating trianglesthat are adjacent to subdivided boundary triangles. As an algorithmthat needs to scan all its input to determine which operation to per-
form on its input will require a running time no better than a linearfunction of N, as each input has to be at least read once. Our algo-rithm has a worst-case running time that is a linear function of itsinput size (the number of voxels), thus is computationally efficient,and requires no more than a few seconds of CPU time. The memory

X. Hao et al. / Computerized Medical Imaging and Graphics 35 (2011) 206–219 209

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ig. 3. Subdivision of a boundary triangle. A triangle is subdivided according to its ina) A plane subdivides triangle (Pa, Pb, Pc) into three smaller triangles: (Pa, Pb, Pbc),nes: (Pa, Pb, Pbc) and (Pa, Pbc, Pc) since vertex Pa coincides with the intersecting pla

equirements of our adapted algorithm, however, are higher thanhose of the original. Additional memory is needed for the load-inf the volume data C(V), whereas the original algorithm requireshe loading of I(V) alone.

.1.2. Simplification of the boundary-preserved surfaceThe surface generated by our adapted marching-cubes algo-

ithm will have many new vertices and new triangles comparedith the surface generated by the original marching-cubes algo-

ithm. The increased number of triangles slows the speed of surface
endering, which may be undesirable for applications that requireser interaction in real time. To improve interactivity of the algo-ithm during display, we must reduce the overall number ofertices, and thus also the number of representational triangles, tohe number present if no regional subdivisions were required. It is
ig. 4. Re-tessellation of a non-boundary triangle. Triangle (Pb, Pc, Pd), thoughnclosed by voxels (P1, P2, . . ., P8) having the same categorical value C1, has to beplitted into triangles (Pc, Pbc, Pd) and (Pd, Pbc, Pb) since it is adjacent to a bound-ry triangle (Pa, Pb, Pc) and the common edge between them, (Pb, Pc) has beenubdivided.

tions with the three planes that subdivide the cubic space containing the triangle.c, Pac), and (Pc, Pac, Pbc); (b) a plane subdivides triangle (Pa, Pb, Pc) into two smaller

preferable if this number can be reduced further without sacrificingthe quality of visualization.

Simplification of the graphical representation of a surface hasbeen studied extensively by researchers in interactive graphics[14]. Surfaces are generally simplified by reducing the number ofrepresentational triangles, which requires reducing the number ofvertices using methods such as edge collapse (Fig. 5). The numberof triangles is reduced while preserving the quality of surface dis-play according to criteria such as the retention of overall shape andcurvature of the surface. An additional important criterion for ourpurposes is to preserve the regional boundaries that were gener-ated from the adapted marching-cubes algorithm.

Our algorithm for surface simplification is based on edge col-lapse using quadric error metrics (QEM) [15]. QEM compute thearea-weighted sum of squared distances of a new vertex position toa set of planes neighboring the edge that is collapsed. Simplificationof an algorithm using QEM can produce high quality approxima-tions while retaining the overall shape and curvature of the originalsurface, as the quadric error relates directly to surface curvature[16].

To preserve the regional boundaries, we enforce the followingtwo rules during the simplification process:

(1) Only non-boundary vertices can be merged into a boundaryvertex, but not vice versa.

(2) A boundary vertex P may be merged into another boundaryvertex, provided that the merge will not change the local shapeof the regional boundary involving P. As an example, if Pa, Pb,and Pc are three consecutive vertices on a boundary that sepa-rates two regions, and if they are collinear, then vertex Pb canbe merged into either vertex Pa or Pc.

One can easily see that rule (1) does not change boundary ver-tices at all, whereas rule (2) removes boundary vertices withoutaffecting the shape of the boundary. In this way we maintain a highresolution mesh around the regional boundary while lowering res-olution of the mesh in regions where the quality of visualization isaffected little by the simplification.

2.1.3. Perturbation of the vertex location for the smoothing ofregional boundaries

A regional boundary is a curve that separates regions belong-ing to differing segmentation categories. This boundary will be asmooth curve if the surface is continuous. For discrete surface that


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of each vertex on the boundary toward this curve by moving eachvertex onto the curve (Fig. 7a and b). With the local control property,we need only test against the curve segments that have this vertexas a control point for moving a particular vertex, and then we ensure

ig. 5. Surface simplification through edge collapse. An edge (Pe, Pf) is removed whemoval of two triangles (colored in dark blue) sharing the edge (Pe, Pf). (For interpersion of the article.)

re represented as a collection of triangles, however, a regionaloundary is defined as a series of line segments, each of which isn edge shared by two triangles belonging to differing segmenta-ion categories. The regional boundaries thus defined are usuallyigzag in contour rather than smooth (Fig. 6), a direct consequencef the discrete sampling of C(V). We can improve the smoothnessf these regional boundaries, however, through a perturbation ofhe locations of vertices on the boundary that is based on theverall curvature of the line segments that compose the bound-ry. With the accurate region boundaries defined in Section 2.1.1,ur smoothing algorithm can provide a user complete and preciseontrol over the smoothed curves.

The smoothing procedure consists of two steps. First, we con-truct a smooth, curvilinear approximation of the boundary to bemoothed. Second, we locally perturb the vertices of the boundarynd then move them toward this approximation curve.

We have selected to use one kind of cubic polynomial—a uni-orm, non-rational B-spline [17], to approximate the boundaryurve. Cubic polynomial representations of curves offer tangentontinuity (i.e., curves at the joint point share the same tangents, ordentical first parametric derivatives) and flexibility in controllinghe shape of the representation. In addition to tangent continu-
ty, B-splines provide continuity of curvature (i.e., identical secondarametric derivatives) at the joint point and thus are inherentlymoother than general cubic polynomial curves.
Another important property of B-splines is the local controlhat they provide, in that B-splines are curvilinear segments whose

ig. 6. Region boundary. A region boundary is a series of joint line segments, eacheing an edge shared by two triangles that belong to different categories. Here theed line segments represent the boundary between two regions marked by bluend green respectively. The boundary has a zigzag appearance due to the discreteature of surface, and thus discrete curve representation. (For interpretation of theeferences to color in this figure legend, the reader is referred to the web version ofhe article.)

rtex Pe is merged with vertex Pf that results in the removal of vertex Pe and theon of the references to color in this figure legend, the reader is referred to the web

polynomial coefficients depend on just a few local control points, sothat moving a control point affects only a small portion of the curve.This local control property greatly reduces the time required tocompute the polynomial coefficients. More importantly, this localcontrol provides better constraints on the error induced duringsmoothing of the curve. Because each segment of a B-spline curveis governed by four control points, their blending functions sum toone and are everywhere nonnegative; thus the curve segment isconstrained to the convex hull of its four control points.

Following the construction of B-spline curves using the verticeson the boundary as control points, we then perturb the location

Fig. 7. Perturbation of boundary vertices’ positions. Vertices are moved toward theB-spline fitted curve of a regional boundary. (a) B-spline fitting (curve in dark blue)of the boundary line segments (in red) and boundary vertices are moved toward(yellow arrows) the B-spline curve; (b) boundary vertices are now on the B-splinecurve and the boundary (in red) is much smoother after the perturbation. (For inter-pretation of the references to color in this figure legend, the reader is referred to theweb version of the article.)

X. Hao et al. / Computerized Medical Imagi

Fig. 8. Turning angle at a boundary vertex. Boundary vertices Pa, Pb, and Pc are consec-utive end points along a boundary line segments (in green) that discretely sampleabit

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smooth boundary curve (in blue). The turning angle ϕ at vertex Pb is the angleetween left tangent line (PaPb) and right tangent line (PbPc, both lines in red). (For

nterpretation of the references to color in this figure legend, the reader is referredo the web version of the article.)

hat this movement will not flip the orientation of any triangles thusnvolved, so that local topology will not be altered.

We estimate the smoothness of a curve using the turning anglest vertices along the curve (i.e., the angle formed between the leftnd right tangent lines). The turning angle will equal zero for aontinuous curve with tangent continuity, as these two lines willlways coincide. For a discrete curve consisting of line segments,he left tangent line is approximated by the line that connects theertex to the vertex’s predecessor along the boundary curve, andhe right tangent line is approximated by the line connecting theertex to its successor (Fig. 8). The angle will then be nonzero, andts value will indicate the local smoothness of the boundary curvet the measured vertex.

Multiple iterations of the B-spline smoothing algorithm providerogressively improved smoothing of regional boundary curves.evertheless, the overall smoothness of a boundary curve might

till be less than ideal at locations where the curve takes sharp turns,egardless of the number of iterations of the algorithm, because theesirable movement of vertices at these locations for ideal smooth-

ng might flip the orientation of neighboring triangles (Fig. 9).One solution to this flipping difficulty is to adjust the position

f neighboring vertices of a boundary vertex during the smooth-ng process. For example, one can treat each vertex as a positiveoint charge. The movement of a boundary vertex will then produceewly unbalanced repulsion forces among neighboring verticesnd produce a redistribution of their positions. This redistributionreates room for further movement of the boundary vertex dur-ng the next iteration of smoothing. The redistribution, however, islso constrained in that it will not change the position of a bound-ry vertex – i.e., movement of a boundary curve will redistributets neighboring non-boundary vertices, but not the vertices on theurve itself, nor vertices on other boundary curves. Thus the redis-ribution is a local property of our algorithm. Our implementationf redistribution in this report has adopted a Laplacian operator18], which will redistribute an affected vertex to the barycentricosition of all its neighboring vertices.

To enable complete and precise control over the smoothing pro-edure, we assess at each iteration of the perturbation the distancef the perturbed vertex to the vertex’s nearest regional boundarylane, as defined in Section 2.1.1. Our algorithm stops perturbingvertex when the error induced at the vertex approaches a user-efined, preset threshold. The preset threshold is determined byhe precision required by each particular application and usuallyoverned by the sizes of interesting areas or regions. Our programakes the size of half voxel as the default threshold if no value isrovided by the user. This default threshold guarantees that theoundary curve lies between voxels belonging to different cate-ories, instead of cutting space of voxels belonging to the sameategory.

The total number of iterations for smoothing is thus determined
y the balance between the smoothness of the boundary (in termsf its average turning angle) and the increase in error that is inducedy the perturbation of the positions of the boundary vertex. Bothf these criteria are pre-established by the user. For example, inur first sample of smoothing of the cortical surface shown in Sec-
ng and Graphics 35 (2011) 206–219 211

tion 3, we stipulated the maximum error to be less than 0.5 mm(half the width of a voxel) and the desired smoothness to be 15◦

(approximately 0.27 rad). These pre-established thresholds yieldeda termination of the smoothing algorithm after 50 iterations.

The final output of our algorithm for representing a surface par-cellation thus consists of a mesh of triangles on that surface, eachof which is assigned a unique segmentation label, with smoothregional boundaries represented by lists of edges.

2.1.4. Automated correction of small, misclassified regionsHuman error in the manual parcellation of a brain is unavoid-

able. The brain is a highly complicated structure, and thesignal-to-noise ratios of imaging data are usually not ideal. To makethe parcellation task even more error-prone, the 3D viewing angleprovided by interactive editing tools to neuroanatomists for man-ual parcellation is usually limited to only a few directions. Errorsin parcellation produce a misclassification of small regions on theparcellated surface that range from several to tens of voxels in size.

We have included in our algorithm an automatic procedureto correct these small misclassified surface areas. From the con-structed curves of regional boundaries, we identify those curvesthat are self-closed. The curve lengths are then compared against apre-established threshold, below which the curve-enclosed area islabeled as misclassified and its categorical value is changed to thesame value as the area outside of the curve. This threshold must bedesignated on an ad hoc basis by a knowledgeable user. For exam-ple, in the thalamic surface shown in Section 3, a threshold of 10 mmwas used to remove small, misclassified regions, assuming that nofunctional sub-regions have sizes smaller than 3 mm in diameter.Another decision of a user is the number of delineated regions torequire. The algorithm will rank the regions according to their sur-face area and remove the excessive number of small, self-loopedregions with ranks below the user-defined number.

2.2. Visualizing the parcellated surface

To visualize the final parcellation, different colors can beassigned to regions defined by differing segmentation labels(Fig. 15). The colored surface thus represented will have a cleardelineation of the parcellation units [6] if the colors between eachpair of neighboring regions have discernable contrast. This methodis simple and effective if displaying only the final parcellation.When used together with the display of other measures on themorphological surface, however, as when representing functionalactivations or other statistical parameters using color encoding, thiscolored regional display will likely confuse the visual interpretationof those measures. In this case, an alternative is needed to displaythe parcellated surface. One option is to display the regional bound-aries using a color that is located outside of the color gamut usedfor display of the statistical parameters and that also contrasts wellwith the representation of the background surface.

The explicit display of the regional boundaries can createanother problem, however. A boundary curve usually is not visi-ble in its entirety, even though the two regions that it delineatesare clearly visible, because a portion of the boundary curve might insome 3D viewing perspectives be located behind one of the regionsthat it delineates. This is especially true for boundary curves on thecortical surface that are located deep within sulci. Surface flatten-ing provides one solution to this problem of boundary visualization[19]. Flattening procedures, however, are computationally com-plex, and interpretation of the flattened surfaces is non-intuitive.
An alternative solution for surfaces that have not been flattened issometimes desirable.
We have developed a double-snapshot display algorithm thatcombines the advantages of a colored display of surface regionswith the display of boundary curves to represent parcellation


Fig. 9. Adjustment of non-boundary vertex position. After perturbation of a boundary vertex, its neighboring non-boundary vertices’ positions get adjusted to improve smoothingresults in the next smoothing iteration. Two different regions are shown in blue color and green color respectively. Red line segments mark the regional boundary. (a) Beforep ertexr otherc eferena

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The boundary curve on the ellipsoid surface separates regionone and region two. Essentially it is the intersection of the ellip-

erturbation of vertex Pa. (b) After the perturbation, Pa is close to a non-boundary vepulsion of Pb away from Pa makes room for further movement of Pa. (d) After anurve (in red) is much smoother than the curve in (b). (For interpretation of the rrticle.)

nits. A colored regional display uses different colors to providen implicit definition of boundary curves. We make this boundaryxplicit by tracing the implicit boundary curves and superimposinghem onto a uniformly colored surface to provide a complete delin-ation of the parcellated surface. The algorithm works as follows:

For every possible viewing perspective,

1) Generate a 2D snapshot of the surface in which each triangleis displayed with a uniform intensity that is proportional to itslabel value. No lighting is yet applied to the surface represen-tation.

2) Obtain the explicit boundary curves from the 2D snapshot bytracing out the individual segments of the implicit curve thatdelineates two adjacent regions having different intensity val-ues.

3) Generate another lighted 2D snapshot of the surface in whichthe other quantitative measures (in our example, statisticalparameters) are mapped onto the surface as colored regions.

4) Superimpose the boundary curves from step (2) onto thesnapshot from step (3), replacing the values of pixels on theboundary with a new, pre-selected color value to mark theboundary.

5) Display the superimposed snapshots from step (4).

Display of the parcellated surface using this double-snapshotlgorithm requires approximately twice the computational times a conventional display, as the same surface is displayed twice,n steps (1) and (3). Steps (2) and (4), however, require very littleomputational time.

.3. Performance on a simulated dataset

Application of our algorithm to a simulated dataset providesknown ground truth to evaluate its performance and to justify

he choice of its operational parameters, such as the number ofmoothing iterations to employ. Moreover, we sample the simu-

Pb and cannot move further without flipping the orientation of any triangle. (c) Theiteration of smoothing, Pa moves closer to its desirable location, and the boundaryces to color in this figure legend, the reader is referred to the web version of the

lated dataset at various resolutions and investigate the effects ofpartial volume averaging on the performance of our algorithm.

We generate an ellipsoid of size comparable to a human brain:

x2

a2+ y2

b2+ z2

c2≤ 1; (a = 72.5 mm, b = 40.5 mm, c = 91.5 mm)

where the surface is defined by the equal sign of the equation. Theportion of the ellipsoid inside a half cone is classified as region oneand the portion that is outside of the cone is classified as regiontwo. The cone is given by:

(x − a1)2 + (y − b1)2 = [r(z − c1)]2;

Fig. 10. Simulated dataset. An ellipsoid is generated for algorithm evaluation. Thetop half of the ellipsoid is outside the cone and is classified as region one. The bottomhalf of the ellipsoid is inside the cone and classified as region two. A boundary curvethat defines different regions is thus delineated on the surface of the ellipsoid.


F re comv ) maxt the t

sc

rtouIloia

f

F(

ig. 11. Parcellations on the simulated dataset at 1 mm resolution. Different measures aertices, as functions of smoothing iterations: (a) average turning angles (in rad); (bhe true boundary (in mm); (d) maximum distance of the smoothed boundary from

oid surface with the cone surface (Fig. 10). Its curvature variesontinuously from 7.13 m−1 to 47.30 m−1.

Volume I(V) is generated at resolutions of 1, 2, 3, 4, 6, and 8 mm,espectively. Each voxel of the volume is assigned an intensity valuehat measures the distance from the ellipsoid surface, with a valuef zero defining the surface of the ellipsoid. The classification vol-me C(V), generated at the same resolution as I(V), coincides with

(V). A voxel of C(V) inside the ellipsoid (with an intensity valueess than or equal to zero) is assigned a classification value based
n whether or not more than half of the volume of the voxel isnside the cone. Voxels of C(V) outside of the ellipsoid are assignedclassification value of zero.
Given I(V) and C(V) as inputs, our algorithm generates the sur-ace of a regionally delineated ellipsoid. We measure for each

ig. 12. Curvature-related parcellation on the simulated dataset at mm resolution. The resulta) Average distance from true boundary is displayed as a function of curvature; (b) avera

pared between approaches of smoothing with and without adjusting neighborhoodimum turning angle (in rad); (c) average distance of the smoothed boundary from

rue boundary (in mm).

point on our generated regional boundary the distance to the trueregional boundary as defined by the intersection of the surfaces ofthe ellipsoid and cone. We then evaluate the performance of ouralgorithm under differing values of its operational parameters.

3. Results and discussion

3.1. Computer simulations

For volumes at a resolution of 1 mm, performance of our algo-rithm that redistributed neighboring vertices during smoothingiterations, compared with performance of our algorithm withoutredistribution, was superior on all categories tested: average dis-

s are generated using smoothing with neighborhood adjustment with 50 iterations.ge turning angle is displayed as a function of curvature.


F ent mu rnings smoo

tabnaagsevfTbaaaaisonica

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tsbaa

ig. 13. Comparing parcellations on the simulated dataset at varying resolutions. Differsing smoothing with neighborhood adjustment with 50 iterations. (a) Average tumoothed boundary from the true boundary (in mm); (d) maximum distance of the

ance of smoothed boundary to the true boundary, average turningngle, maximum distance of the smoothed boundary to the trueoundary, maximum turning angle (Fig. 11). With an increasingumber of smoothing iterations, the average turning angle andverage distance of the smoothed boundary to the true bound-ry decreased monotonically. Moreover, the boundaries that wereenerated using a redistribution of neighboring vertices duringmoothing converged to lower values than did the boundaries gen-rated without redistribution (average turning angles of 0.018 rads. 0.055 rad, and an average distance of the smoothed boundaryrom the true boundary of 0.092 mm vs. 0.107 mm, respectively).he maximum distance of the smoothed boundary from the trueoundary generated using the redistribution approach decreaseds iterations increased, up to a value of approximately 50 iterations,nd thereafter remained constant. Both the maximum turningngle and the maximum distance from the true boundary gener-ted without redistribution of voxels decreased for the first fewterations and thereafter remained constant. As noted earlier, idealmoothing cannot be achieved at some sharp turning points with-ut flipping the orientation of their neighboring triangles, unlesseighboring vertices are redistributed. Thus these simulation find-

ngs suggest that the redistribution schema is useful for smoothingurve segments that take sharp turns and that an iteration count ofpproximately 50 provides optimal performance.

We also note that the simulation suggests that performancef our algorithm does not depend on the boundary’s curvatureFig. 12).

Based on our findings using the simulated dataset at a resolu-
ion of 1 mm, we assessed the performance of our algorithm onimulated datasets at varying spatial resolutions using the redistri-ution of neighboring voxels during smoothing of the boundariesnd an iteration count of 50 (Fig. 13 and Table 1). Both the aver-ge and maximum distances of the smoothed boundary from the
easures are compared as functions of volume resolutions. The results are generatedangles (in rad); (b) maximum turning angle (in rad); (c) average distance of thethed boundary from the true boundary (in mm).

true boundary increased nearly linearly with decreasing resolu-tion, as expected. However, the distances at a resolution of 4 mmapproximated those generated using the original marching-cubesalgorithm at a resolution of 1 mm. Moreover, both the average andmaximum turning angle generated by our algorithm at a resolu-tion of 8 mm are much smoother than the boundaries generatedby the original marching cube at a 1 mm resolution (Fig. 14 andTable 1). One unexpected finding is that the maximum turningangle decreased as resolution increased from 1 to 8 mm (Fig. 13b).This finding can be visualized by noting that with increasing voxelsize, the surface has fewer vertices and each of those vertices hasmore unoccupied adjacent space to adjust their positioning dur-ing boundary smoothing. Similarly, the misclassified surface areagenerated by our algorithm at a resolution of 4 mm approximatedthose generated using the original marching-cubes algorithm at aresolution of 1 mm (Table 1).

3.2. Real-world MRI datasets

We next show the output of our algorithm on a single subjecttemplate brain with its associated demarcated parcellation labelsfrom the International Consortium for Brain Mapping (ICBM) [20].This template is the average of 27 high resolution, T1-weighted MRIacquisitions from a single subject, aligned within the stereotaxicspace of the ICBM average template. The 3D parcellation of corticalgyri, subcortical nuclei, and cerebellum was defined manually byan expert neuroanatomist.

Approximately 611,000 triangles were needed to completely
represent the cortical surface using the original marching-cubesalgorithm, with roughly 1/5 of those triangles (i.e., 135,466) locatedat regional boundaries. In contrast, approximately 1,440,376 tri-angles were required using our adapted marching-cubes approachthat subdivides triangles along regional boundaries, a 140% increase


Table 1Comparison of results on simulated dataset at various resolutions using our adapted marching-cubes algorithm with those obtained by original marching-cubes algorithm,in terms of average distance from true boundary, maximum distance from true boundary, average turning angle along the boundary curves, maximum turning angle alongthe boundary curves, and percentage of misclassified area.

Resolution Measurement Original marching-cubes Adapted marching-cubes

Non-smoothed Smoothed

1 mm

Average distance from true boundary (mm) 0.389 0.181 0.096Maximum distance from true boundary (mm) 0.994 0.499 0.316Average angle (rad) 1.616 0.243 0.028Maximum angle (rad) 2.995 2.417 0.352Percentage of misclassified area 0.206% 0.103% 0.056%

2 mm


3 mm


4 mm


6 mm


8 mm

Average distance from true boundary (mm) 2.911 1.476 1.055Maximum distance from true boundary (mm) 7.211 3.916 2.664Average angle (rad) 1.638 0.306 0.044

oaornwcorpX

avo

Fb1v

Maximum angle (rad)Percentage of misclassified area

ver the number required using the original algorithm. We thenpplied our algorithm for the boundary-preserved simplificationf the newly generated mesh, and a standard simplification algo-ithm to the original mesh, so that the both meshes had equalumber of triangles following application of the algorithms, whichas approximately half of the count from the original marching-

ubes algorithm. In addition, we applied the B-spline smoothingf the regional boundary curves and corrected small misclassifiedegions. Less than 1 min was required to generate, delineate, sim-lify, and smooth the parcellated surface when running on an Inteleon 3.0 GHz PC with 2 GB RAM.

The regional delineation using our algorithm is clearly moreccurate than are approaches that simply assign labels to a pre-iously generated surface mesh without an adaptive subdivisionf that mesh (Fig. 15). The report on average turning angle over

ig. 14. Comparing regional boundaries generated by different methods at varying resolutionsoundary generated by adapted marching-cubes algorithm without smoothing; bottom romm resolution; (b) 2 mm resolution; (c) 3 mm resolution; (d) 4 mm resolution; (e) 6 mmiew shown in (a)–(f) marked.

3.079 1.603 0.2371.08% 0.793% 0.387%

all boundary vertices, as well as the maximum and average dis-tance over all boundary vertices to their nearest region boundaryplane as defined in Section 2.1.1 confirms the advantage of our algo-rithm (Table 2). The average turning angle is reduced from 1.636 radto 0.908 rad using our accurate region delineation method, andfurther down to 0.271 rad after smoothing. It represents a fac-tor of six improvements. In addition, both the maximum andaverage distances are improved by a factor of approximatelytwo.

Moreover, our double-snapshot algorithm achieved a clear dis-play of the improved regional parcellations. Regional boundary
curves generated by the first snapshot (Fig. 16a) clearly demarcatethe parcellation units (Fig. 16b and c). The transfer of the bound-ary curves back to the second lighted snapshot (Fig. 16d) gives afinal clear visualization of the parcellation units (Fig. 16e), contrasts
. Top row: boundary generated by original marching-cubes algorithm; middle row:w: boundary generated by adapted marching-cubes algorithm with smoothing. (a)resolution; (f) 8 mm resolution; (g) ellipsoid with selected region whose enlarged


Fig. 15. Comparing parcellations on a cortical surface. The cortical surface is displayed as colored regions. (a) Surface generated by original marching-cubes algorithm withlabel assigned to each triangle from the voxel that contains the largest portion of the triangle; (b) surface generated using our adaptive subdivision with boundary-preservedsimplification and B-spline smoothing. Both surfaces have equal number of triangles. On the top left side of each image is an enlarged fragment of the view. (For interpretationof the references to color in this figure legend, the reader is referred to the web version of the article.)

Table 2Comparison of results on a cortical surface using our adapted marching-cubes algorithm with those obtained by original marching-cubes algorithm, in terms of trianglecount, maximum distance, and average distance of boundary vertices to their nearest region boundary plane as defined in Section 2.1.1, as well as the average turning anglealong the boundary curves.

Initial triangles Triangles after simplification Maximum distance (mm) Average distance (mm) Average angle (rad)

Original marching-cubes 611,056 300,852 0.900 0.252 1.636

w(

mpidf

vcsap

TCta

Adaptedmarching-cubes

Non-smoothed 1,440,376 300,852Smoothed 1,440,376 300,852

ell with the results generated by a regular visualization methodFig. 16f).

Our algorithm achieved similar improvements on right thala-us (Fig. 17). Thalamus surface was delineated using volumetric

arcellated histological maps [21]. As before, the average turn-ng angle is largely reduced (a factor of 12), accompanied by theecrease in the maximum, as well as the average distance deviatedrom the actual boundaries (Table 3).

We applied our complete algorithm for region delineation andisualization to the display of results from a previous morphologi-
al study [22]. At each point on the cerebral surface, the statisticalignificance (probability values) of differences in cortical thicknesscross groups (high vs. low risk in major depressive disorder) inarticipants from 2nd and 3rd generations are color coded and
able 3omparison of results on a right thalamus surface using our adapted marching-cubes ariangle count, maximum distance, and average distance of boundary vertices to their nearngle along the boundary curves.

Triangles Maxim

Original marching-cubes 74,804 0.334Adaptedmarching-cubes

Non-smoothed 74,804 0.000Smoothed 74,804 0.221

0.000 0.000 0.9080.493 0.130 0.271

displayed with and without overlay of the delineation of regionalboundaries (Fig. 18). The statistical models accounted for the degreeof familial relatedness, as well as for the age and sex, of all partici-pants. The p-values were thresholded at p < 0.05 after correction formultiple comparisons using the theory of Gaussian Random Fieldson a two-dimensional manifold [23]. The images that display theregional delineation provide exquisite anatomical localization ofgroup differences.

We have presented results for the delineation and visualizationof parcellation units on the cortical surface and thalamus surface.
Our algorithm, however, can be applied to the representation ofa parcellation scheme on any surface, as long as that parcellationhas been previously defined. Our algorithm can also be appliedto the representation of surfaces generated by means other than
lgorithm with those obtained by original marching-cubes algorithm, in terms ofest region boundary plane as defined in Section 2.1.1, as well as the average turning

um distance (mm) Average distance (mm) Average angle (rad)

0.054 1.8250.000 0.7070.034 0.154


Fig. 16. Delineation display using double-snapshot method. (a) The first snapshot generates an image from labeled markers of triangles without lighting; (b) region boundaries(in red) are traced from the above image by pixels that are on the boundary of regions wthe lighted surface, at the same viewing angle as in (a); (e) region boundaries from (c) arresults; (f) result of traditional display, in which some of the boundary curves are buried ilegend, the reader is referred to the web version of the article.)

Fig. 17. Comparing parcellations of the right thalamus. The surface of right thalamus isdisplayed as colored regions. (a) Surface generated by original marching-cubes algo-rithm with label assigned to each triangle from the voxel that contains the largestportion of the triangle. White circle marks a misclassified region; (b) surface gen-erated using our adaptive subdivision with boundary-preserved simplification andB-spline smoothing. Both surfaces have equal number of triangles. (For interpreta-tion of the references to color in this figure legend, the reader is referred to the webversion of the article.)

ith different intensity values; (c) region boundaries alone; (d) second snapshot ofe super-imposed onto image from (d) to give the final visualization of delineationnside sulci and invisible. (For interpretation of the references to color in this figure

the marching-cubes algorithm, as our algorithms for region defini-tion and visualization are independent of the particular methodsfor graphical representation that is used to generate the surfacedisplay.

Our delineation algorithm, however, requires that a 3D volumeparcellation is previously defined over the same space where thesurface is to be generated. If the parcellation exists in another space,then it has to be transformed into the space of surface definition andtransformation errors might be induced and degrade the accuracyof the delineation results.

Our paper assumes an accurate parcellation of the volumetricspace where the surface is generated. The assumption could be alimitation of our algorithm. In addition to the deterministic atlasesour algorithm is designed for, there exist probabilistic atlases thatassign each voxel some probability values to specify the chances ofthe voxel belonging to each category. For such data, the optimumseparation of regions will not be at the middle plane separatingvoxels. Moreover, the optimum separation will not even be a planeat all. To extend our algorithm for these atlases, a curved surfacecutting schema will be required at the surface generation stage.Moreover, to accurately estimate the error will be more difficultduring the simplification and smoothing stages as a collinearity ofpoints along the boundary curve is no longer a quality guarantee.We plan to explore the problem in the future. In addition, different
expert neuroanatomists might parcellate data slightly different andrender the surface regions thus defined not identical. We would alsolike to investigate the reproducibility of surface delineation limitedby the volumetric-parcellation reproducibility in the future.


Fig. 18. Maps of group differences in cortical thickness. Statistical significance of differences in cortical thickness of right hemisphere across groups (high vs. low risk in majordepressive disorder) in participants from 2nd and 3rd generations are color coded, with warm colors (yellow, orange, and red) representing significantly thicker cortices inthe high risk group and cooler colors (blue and purple) representing thinner cortices in that group. The color bar indicates the color-coding of p-values for testing of statisticals on bouM or parg late; Pc in this

4

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ignificance at each point in the brain. Upper panel: results displayed without regiedial views: (b, d). AC: anterior cingulate; IOG: inferior occipital gyrus; IP: inferi

yrus; MTG: middle temporal gyrus; OF: orbitofrontal cortex; PC: posterior cinguortex; STG: superior temporal gyrus. (For interpretation of the references to color

. Conclusion

We have presented an algorithm for the improved delineationf parcellation units on a 3D morphological surface. Our algorithms both accurate and computationally efficient. Moreover, our newlgorithm for visualization of regional boundaries using a double-napshot technique helps to display clearly and unambiguouslyoth the parcellation units and other quantitative measures of

nterest, while avoiding the computational demands and interpre-ive difficulties associated with surface flattening algorithms. Ourlgorithms can be applied easily to the representation of parcella-ion schemes across a wide variety of morphological surfaces to aidhe representation and interpretation of data.

cknowledgments

This work was funded in part by NIMH grants K02 74677,H089582, MH068318, 1P50MH090966, NIEHS grant ES015579,IDA grants DA027100 and DA017820, and a grant from theational Alliance for Research in Schizophrenia and Affective Dis-rders (NARSAD).

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Imagi

[

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X&r2

Ph.D. in Statistics from Rutgers University in 2004.

Bradley S. Peterson is Suzanne Crosby Murphy Professor, Director of Child & Ado-

X. Hao et al. / Computerized Medical

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faces using Gaussian Random Fields on 2-D manifolds. IEEE Trans Med Imag2007;26(1):46–57.
uejun Hao is an associate research scientist at Columbia College of PhysiciansSurgeons, Columbia University and the New York State Psychiatric Institute. He

eceived his Ph.D. in Computer Science from University of Maryland, College Park in004.

ng and Graphics 35 (2011) 206–219 219

Dongrong Xu is an associate professor at Columbia College of Physicians & Sur-geons, Columbia University and the New York State Psychiatric Institute. Prior tojoin Columbia University, he had extensive research experience at Stony Brook Uni-versity, Johns Hopkins University, and University of Pennsylvania. He obtained hisPh.D. in Computer Science from Zhejiang University in 1995.

Ravi Bansal is an associate professor at Columbia College of Physicians & Surgeons,Columbia University and the New York State Psychiatric Institute. He obtained hisPh.D. in Electrical Engineering from Yale University in 2000.

Jun Liu is an assistant professor at Columbia College of Physicians & Surgeons,Columbia University and the New York State Psychiatric Institute. He obtained his

lescent Psychiatry, and Director of MRI Research at Columbia College of Physicians& Surgeons, Columbia University and the New York State Psychiatric Institute.He obtained his M.D. from University of Wisconsin-Madison Medical Schoolin 1987.

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Computerized Medical Imaging and Graphics · cerebral cortex in real time during generation of the surface using information from parcellated 3D ... warps a pre-labeled ... A manual