EUROGRAPHICS 2006 / D. W. Fellner and C. Hansen Short Papers

Connectivity-aware sectional visualization of 3D DTI volumesusing perceptual flat-torus coloring and edge rendering

Çagatay Demiralp1 Song Zhang1 David F. Tate1 Stephen Correia1 David H. Laidlaw1

1Brown University, Providence, RI, USA

AbstractWe present two new methods for visualizing cross-sections of 3D diffusion tensor magnetic resonance imaging(DTI) volumes. For each of the methods we show examples of visualizations of the corpus callosum in the mid-sagittal plane of several normal volunteers. In both methods, we start from points sampled on a regular grid on thecross-section and, from each point, generate integral curves in both directions following the principal eigenvectorof the underlying diffusion tensor field. We compute an anatomically motivated pairwise distance measure betweeneach pair of integral curves and assemble the measures to create a distance matrix. We next find a set of pointsin a plane that best preserves the calculated distances that are small—each point in this plane represents one ofthe original integral curves. Our first visualization method wraps this planar representation onto a flat-torus andthen projects that torus into a visible portion of a perceptually uniform color space (L*a*b*). The colors for thepaths are used to color the corresponding grid points on the original cross-section. The resulting image showslarger changes in color where neighboring integral curves differ more. Our second visualization method lays outthe grid points on the cross section and connects the neighboring points with edges that are rendered accordingto the distances between curves generated from these points. Both methods provide a way to visually segment 2Dcross sections of DTI data. Also, a particular contribution of the coloring technique used in our first visualizationmethod is to give a continuous 2D color mapping that provides approximate perceptual uniformity and can berepeated an arbitrary number of times in both directions to increase sensitivity.

1. Introduction

Diffusion-Tensor Magnetic Resonance Imaging (DTI) en-ables the exploration of fibrous tissues such as brain whitematter and muscles non-invasivelyin-vivo. It exploits thefact that water in these tissues diffuses at faster rates alongthe fibers than orthogonal to them. However, the multival-ued nature of DTI data poses challenges in visualizing andunderstanding the underlying structures. Integral curves thatrepresents neural pathways by showing paths of fastest diffu-sion are among the most common information derived fromDTI volumes. They are generated by tracking the principaleigenvector of the underlying diffusion tensor field in bothdirections. They are often visualized with streamlines orvariations of streamlines (streamtubes and hyperstreamlines)in 3D. In this paper, we present two new methods for visual-izing cross sections of DTI volumes that incorporate the 3Dout-of-plane connectivity information typically conveyed bythe integral curves (see Fig. 1). Slice-based 2D visualiza-tions of scientific data are generally effective, fast and syn-optic [CM02,SWD04]. Also, looking at 2D cross-sections isstill the most common practice by far among scientists andphysicians for data exploration. Furthermore, there is someanecdotal evidence that incorporation of 2D cross-sections

in 3D visualizations of medical data sets data is preferred bythe same group [DJK∗06].

For each of our visualization methods we show exam-ples of visualizations of the corpus callosum in the mid-sagittal plane of three normal volunteers. The corpus cal-losum is the largest white matter fiber bundle in the brainand a target for clinical and neuroscience research into nor-mal developmental vs. pathological changes in white matterintegrity across the lifespan and the functional correlates ofthose changes. Distinct cross-sectional regions of the corpuscallosum may contain fibers that subserve specific cognitiveor behavioral functions mediated by the cortical regions towhich they project. Proxy measures (e.g., thickness, volume,area, shape) of the health of these cross-sectional regionsmay correlate with measures of the cognitive and behav-ioral functions they subserve. In fact, the corpus callosumhas been shown to differ on such measures by handedness,gender, and age as well as in disorders such as Alzheimer’sdisease and schizophrenia [HJL∗04].

2. Related WorkMapping colors to data values is a fundamental operationin scientific visualization. Previous work based on empiri-cal studies addressed the problem of generating perceptu-

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Figure 1: Flat-torus coloring (left) and edge rendering (right) visualizations of the mid-saggital plane of the corpus callosumin a normal person’s brain

ally effective colormaps [War88,Hea96,LH92,KRC02]. Weuse the CIE L*a*b* color space that is a perceptually uni-form (approximately) color space proposed by the Commis-sion Internationale de l’Eclairage (CIE) in 1976. A colorspace is said to be perceptually uniform if the perceptualdifference between any two colors in just noticeable dif-ference (JND) units is equal to the Euclidean distance inthat color space. Several different geometric models, includ-ing line, plane, cone, cylinder, and B-spline surfaces havebeen proposed for univariate, bivariate or trivariate colormapping [Pha90, Rob88]. We extend the earlier models byintroducing the flat-torus model to give a continuous 2Dcolor mapping that is approximately uniform and that canbe repeated an arbitrary number of times in both directionsto increase sensitivity. Integral curves generated from DTIvolumes have been visualized generally with streamlines in3D with different geometric (i.e., hyperstreamlines, stream-tubes, etc.) and coloring combinations. In a work that is con-ceptually partly similar to our work, Brunet al.colored DTIintegral curves by embedding them in 3D RGB space using anon-linear dimensionality reduction technique [BPKW03].Volume visualization of DTI data included isosurface ex-traction and volume rendering. Previous cross-sectional vi-sualizations of DTI mapped glyphs (box, ellipsoid and su-perquadratic) and colors to tensor voxels [ZKL04]. Paje-vic et al.proposed methods to colormap DTI cross-sectionsaccording to principle eigenvectors of tensor voxels usingdifferent color spaces, including perceptually uniform CIEL*u*v color space. The authors point at the potential limi-tations due to the irregularity of the L*u*v space. Our flat-torus model addresses some of the limitations discussed inthis work [PP99].

3. MethodsIn both visualization methods presented here, we start frompoints (seeds) sampled on a regular grid on the cross-sectionand, from each point, generate integral curves in both di-rections following the principal eigenvector of the underly-ing diffusion tensor field. We compute an anatomically mo-tivated pairwise distance measure between each pair of inte-gral curves and assemble the measures to create a distance

C2

iC1

i

C2

jC1

j

Cnj

Cn−1

j

Cmi

Cm−1

i

Figure 2: Polyline representations of two integral curves Ciand Cj

matrix. The distance matrix is utilized by the both methodsto convey the out-of-plane connectivity information. We ex-plain how we measure distances between the integral curvesto construct the distance matrix in the next section.3.1. Distance Measure Between Integral CurvesIntegral curves generated from DTI volumes are solutions tothe first-order differential equationdC

ds = ~v1(C(s)), wheresparameterizes the curve andv1 is the principal eigenvectorat the pointC(s) = (x(s),y(s),z(s)). We compute the inte-gral curveC(s)passing through a given seed pointC(0) (ini-tial conditions) by integrating the above equation fors> 0ands< 0 (i.e., both directions from the seed point). Therehave been different distance measures proposed for integralcurves generated from DTI volumes [MVvW05]. In the cur-rent work we adapt a measure proposed by Zhanget al.witha slight modification [ZDL03]. The measure is anatomicallymotivated in that it is designed to increase whenever one pathhas points that are not near the other path. Note that our mea-sure does not necessarily satisfy triangle inequality, there-fore, it is not a metric. Given any two integral curvesCi andCj that are represented as polylines withmandn vertices re-spectively (like the ones shown in Fig. 2), we first find meandistancesdi j anddji then, determine the maximum of thesetwo distances as the distanceDi j between the two curves:

di j =∑m

k=1dist(Cki ,Cj)

m, dji =

∑nk=1dist(Ck

j ,Ci)

n(1)

Di j = Dji = max(di j ,dji ) (2)

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Demiralp et al. / Connectivity-Aware Sectional Visualization of 3D DTI Volumes 120

Figure 3: Flat-torus coloring of the mid-sagittal of the cor-pus callosum in a normal person’sbrain

distance fitting

Figure 4: Seed points are adjusted so that Euclidean dis-tances between the points on the plane reflect the distancesbetween their associated integral curves.

The function dist(p,C) returns the shortest Euclidean dis-tancebetween thepoint p and curveC. Wecomputedistancebetween each pair of integral curves as we denoted and as-semble the measures to create a distance matrix. The dis-tance matrix is a real positive symmetric matrix with zerosalong thediagonal.

3.2. Flat-TorusColor ingThe goal of our first method is to reflect the boundaries indistance differences in the data as perceptual boundaries.For this, we lay out the seed points on a plane and adjusttheir positions using a simple mass-spring-based optimiza-tion algorithm so that the calculated distances between theirassociated integral curves are best preserved locally. Fig. 4illustrates how seed point coordinates change after runningtheoptimization algorithm. Wecoordinate-transform thead-justed points using principle component analysis (PCA) tohave a succinct representation. Finally, we wrap this planarrepresentation onto a flat torus and then projects that torusinto a visible portion of the CIE L*a*b* space. A flat-torusin 4-space is a Cartesian product of two circles in R2. It canbeobtained by amapping W : R2 → R4 such that

W(x,y) = (u,v,s, t) = (r1cosx, r1sinx, r2cosy, r2siny) (3)

where r1 and r2 are the radii of the circles. The flat-torushas 0 Gaussian curvature everywhere (i.e., is a developablesurface), thereforeaplanecan bewrapped around it withoutdistortion [dC76].

Weproject theflat-torus to thevisible partition of L*a*b*color space, centered at (Lo,ao + r1,bo) as follows:

L∗ = Lo + t, a∗ = ao + r1 +u+s, b∗ = bo +v (4)

Note that this projection is not isometric. It has two linesof self -intersection (where different (x,y) points map to the

Figure 5: Edge rendering visualization of the mid-sagittalplaneof thecorpus callosum in a normal person’s brain.

samecolors) aswell asdistorting theanglesbetween theco-ordinatedirections. Wediscuss this further in Sec. 4. For theexamples shown in this paper, we locate the projected toricin theL*a*b interval I = (IL, Ib, Ic), where IL = [60,80], Ia =[−50,30], Ib = [−20,60] and use r1 = 30, r2 = 10. The re-sulting images show larger changes in color where neigh-boring integral curves differ more. One of the advantages ofusing this flat-torus projection is that we can adjust the sen-sitivity of the color mapping by rescaling point plane andwrapping around the two circles continuously.

3.3. EdgeRender ingOur second visualization method lays out the grid pointson the cross section and connects the neighboring pointswith edges that are rendered according to the distances be-tween curvesgenerated from thesepoints. Notethat wesam-ple seed points on a rectilinear grid where the vertical andhorizontal distances between the grid points are equal toδ . We define the seed points Xi and Xj to be neighbors if||Xi − Xj ||2 = δ or ||Xi − Xj ||2 = δ

√2 (i.e., a seed point

can have maximum 8 neighbors). Edges are drawn redderin color and thicker where neighboring seed points’ integralcurves differ more.

4. Results and DiscussionFigs. 3 and 5 show the visualizations of the same normalperson’s corpus callosum with close-up views of the sameregion. Notice the correspondance between regions in flat-torus coloring and edges in edge rendering. Other resultsfrom two DTI brain data sets are shown in Fig. 6. It is im-portant to note that the perceptual uniformity in our colormapping is an approximation, because the flat-torus cannotbemapped to threedimensions isometrically. Our projectioncan deemphasizechanges in certain regionsof theflat torus.There are other projections that may be closer to isometric,and it also may bepossible to add afourth perceptual dimen-sion like texture to the threecolor dimensions, removing theneed for aprojection and preserving thepropertiesof theflattorus.

5. ConclusionsWe have presented two new cross-sectional visualizationmethods. Theprimary strength of both methods is providinga compact and contextual visualization by bringing higherdimensional connectivity information onto a2D planewhichis effective and familiar to practitioners. We have applied

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Demiralp et al. / Connectivity-Aware Sectional Visualization of 3D DTI Volumes 121

Figure 6: Flat-torus coloring (left) and edge rendering (right) visualizations of the mid-saggital plane of the corpus callosumin two normal persons’ brains.

them to visually segment the mid-saggital cross-section ofthe corpus callosum in the brain. Feedback from neurosci-entist collaborators suggests that our visualization methodscan be useful in identification of smaller caliber anatomi-cally or functionally related white-matter structures, partic-ularly those that are contained within large bundles or fas-ciculi that project to multiple areas. Flat-torus coloring is anew geometric model for bivariate color mapping that is ap-proximately uniform perceptually and that can be repeatedan arbitrary number of times in both directions. The under-lying idea of this work can be extended to visualization ofother vector, tensor or multi-scalar data volumes.

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