Visualizing a 3D scalar function is animportant aspect of data analysis inscience, medicine, and engineering.However, many software systems for

3D data visualization offer only the default ren-dering capability provided by the computer’sgraphics card, which implements a graphics li-brary such as OpenGL1 at the hardware level torender polygons using local illumination. Localillumination computes the amount of light thatwould be received at a point on a surface from aluminaire, neglecting the shadows cast by anyintervening surfaces and indirect illuminationfrom light reflected from other surfaces. Thissimplified version of light transport is nonphys-ical, but it produces images that look 3D andthat graphics hardware can generate at rates ofmillions of triangles per second.

By comparison, more realistic global illumina-tion results from solving (or at least approximat-

ing) the equation for light transport. Generally,renderers that solve the light transport equationare implemented in software and are conse-quently much slower than their hardware-basedcounterparts. So a user faces the choice betweenfast-but-incorrect and slow-but-accurate displaysof illuminated 3D scenes. Global illuminationmight make complicated 3D scenes more com-prehensible to the human visual system, but un-less we can produce it at interactive rates (fasterthan one frame per second), scientific users willcontinue to analyze isosurfaces of their 3D scalardata rendered with less realistic, but faster, localillumination.

We propose a solution to this trade-off that in-volves precomputing global illumination andstoring the values in a 3D illumination grid. Wecan perform this task as a batch process on mul-tiple processors. Later, when the user applies avisualization tool to sweep through level sets ofthe scalar function, the precomputed illumina-tion is simply texture mapped at interactive ratesonto the isosurfaces. Consequently, users canperform arbitrarily complex illumination calcu-lations (for example, to capture subsurface scat-tering or caustics) and display the results in realtime while examining different level sets. Thisstrategy lets users apply realistic illumination atinteractive speeds in a data visualization softwaresystem.

48 THIS ARTICLE HAS BEEN PEER-REVIEWED. COMPUTING IN SCIENCE & ENGINEERING

Fast Global Illuminationfor Visualizing Isosurfaceswith a 3D Illumination Grid

A N A T O M I C R E N D E R I N GA N D V I S U A L I Z A T I O N

Users who examine isosurfaces of their 3D data sets generally view them with localillumination because global illumination is too computationally intensive. By storing theprecomputed illumination in a texture map, visualization systems can let users sweepthrough globally illuminated isosurfaces of their data at interactive speeds.

DAVID C. BANKS

University of Tennessee / Oak Ridge National LaboratoryJoint Institute for Computational ScienceKEVIN BEASON

Rhythm and Hues Studios

1521-9615/07/$20.00 © 2007 IEEE

Copublished by the IEEE CS and the AIP

JANUARY/FEBRUARY 2007 49

Global IlluminationDetermining the spatial arrangement of one fea-ture with respect to another is a basic concern fora user analyzing 3D data. However, if a scene iscomplex, with many objects obstructing others, itcan be tedious or even impossible for a user to se-lect a viewpoint that clearly discloses the differentobjects’ relative pose. As data sets grow in size, theirgeometric complexity generally grows as well. Fig-ure 1 illustrates a complicated isosurface renderedwith local and global illumination. This data set,imaged ex vivo using confocal microscopy, shows aneuron from a mouse hippocampus with severallong cylindrical dendrites emanating from the cellbody. Global illumination in Figure 1b disam-

biguates the dendrites’ relative depths at crossingpoints in the image.

Global illumination is also useful for displayingdepth in isosurfaces with complicated shapes. Fig-ure 2 shows an isosurface of the brain from magneticresonance imaging (MRI) data. Global illuminationin Figure 2b makes the central commissure (sepa-rating the cerebral hemispheres) readily apparent.We presented both images in Figure 2 to one of ourcollaborators (a neurosurgeon), who remarked thatthe globally illuminated image in Figure 2b “is whata brain actually looks like.” Although making 3Dsurfaces look real might not be data visualization’smost important task, users deserve to have thischoice available when they’re analyzing data.

(a) (b)

Figure 1. A mouse neuron, imaged with confocal laser microscopy, has numerous long thin dendrites projecting outwardfrom the cell body. (a) With local illumination, it’s difficult to discern the dendrites’ relative depths. (b) With globalillumination, shadows near a crossing (such as the X shape on the lower left) indicate that the two dendrites are nearlytouching.

(a) (b)

Figure 2. Two hemispheres of a human brain separated by the cleft of the central commissure. (a) With local illumination,even the cleft’s deepest section (the bright crescent at the bottom center) is as bright as the cortex’s outermost surface.(b) With global illumination, the cleft is immediately evident as a dark valley.

50 COMPUTING IN SCIENCE & ENGINEERING

IsosurfacesExamining scalar-valued data on a 3D grid is acommon starting point for 3D visualization. Suchdata sets arise, for example, from a patient’s MRI,in confocal microscopy of cells and tissue, in thesolution of heat flow on a 3D domain, or fromcomputing vorticity magnitude in a turbulentflow. The scalar function h(x): �n � � assigns areal value to each point x in �n. A level set Lc of his the locus of points x satisfying Lc = {x: h(x) – c =0}, which corresponds to points x lying on the in-tersection of the graph of h and the hyperplane atheight c. Sweeping through level sets is a standardway for a user to examine such a scalar function.In the 3D case (n = 3), the level set forms an iso-surface in �3.

A typical 3D data set contains the values of h onlyat discrete points xa on the grid, so we must inferthe values of h at nongrid points from the grid val-ues by some interpolation scheme. Researchershave widely used the marching cubes (MC) algo-rithm2 for interpolating the data and constructinga polygonal mesh approximating the isosurface.MC uses linear interpolation to locate roots of h(x)– c at points along edges of cubes tiling the domainand then connects the points into polygons. Figure3 illustrates a version of the MC algorithm in twodimensions, called marching squares, for a scalarfunction defined on a grid in the plane whose levelsets are curves. Figure 3a shows the values at gridpoints in the plane’s positive quadrant of the func-tion h(x, y) = x2 + y2, which has level sets Lc = {(x, y):x2 + y2 = c} that are circles of radius . The gridpoints define squares that tile the domain. Figure3b shows the evaluation of h – c when the isovaluec happens to be seven.

A typical software system for generating iso-surfaces—the isosurface engine—connects the iso-value c with a GUI component that the user sees.The user moves a slider bar to dynamicallychange the isovalue, and then the isosurface en-

gine generates a polygonal mesh representing Lc.Finally, the graphics card displays the mesh (us-ing local illumination). For scalar data sets of size256 � 256 � 256, an ordinary desktop computercan perform this pipeline of tasks at interactiverates. Running a global illumination code tomake the isosurface look more realistic, however,can add tens of minutes to the display time whenthe isovalue c is updated.

To achieve an interactive display rate while ren-dering globally illuminated isosurfaces, the visual-ization system can absorb the calculation of lighttransport into a precomputing stage. In the mostnaive approach, the system would repeatedly ex-tract isosurface after isosurface of the function h,compute global illumination for each isosurface,and then store the illuminated meshes—all beforethe user even begins an interactive session to sweepthrough the data. As the user moves a slider bar inthe GUI, the visualization system would simply re-trieve the mesh with precomputed illumination.The storage cost, however, for archiving an unlim-ited number of isosurfaces is unreasonably high. Amore prudent tactic is to store only the illumina-tion values and then “paint” them onto isosurfaces,a generic technique known as texture mapping,which we describe later.

Illumination as a Texture MapTexture mapping (see Figure 4) assigns a coordi-nate chart to a surface and then applies colors froma 2D image onto the surface according to the tex-ture coordinates. With 3D texture mapping, we as-sign each vertex of the mesh a triple (i, j, k) oftexture coordinates that index into a 3D volume ofcolors. We can exploit 3D textures to paint realis-tic lighting onto polygons much faster than thelighting can actually be computed (in millisecondsrather than minutes).

During the precomputation phase, a series ofisosurfaces is constructed and globally illuminated

c

25181310

13 5 4 20

1710 5 2 1

1 4 16

9

8

0 9

−3

−7 −6 −3

−6 −5 −2 10

−2 6 13

3 11 186

3

92

2

1 −3

−3 2

−2 3

−2 1

32

−3

2

−3 2

−2 3

−2 1

32

(a) (b) (c) (e)(d)

Figure 3. Level set Lc of the function h(x, y) = x2 + y2. The isovalue c = 7 yields a circle of radius centered at the origin.(a) h evaluated at grid points. (b) h – 7 evaluated at grid points. (c) Squares and edges that straddle negative andpositive values of h – 7. (d) Roots (yellow) of h – 7. (e) The level set L7 approximated by a polyline (gold).

7

JANUARY/FEBRUARY 2007 51

using a numerical solver for light transport. Thisbatch-processing step might take minutes or hoursdepending on the data’s size and the sophisticationof the rendering algorithm used. Figures 5athrough 5d demonstrate four steps in this process.At many sample points (x, y, z) on a given level set,the incoming light is computed and saved by thenumerical solver. Figure 5e depicts thousands ofsamples of incident light on several isosurfaces.These scattered samples are interpolated by a sub-sequent process onto a 3D uniform grid, or illumi-nation grid (see Figure 5f).

Where the isosurfaces are spaced far apart fromeach other, they produce visible undersamplingartifacts in the illumination grid. A high-qualityillumination grid requires a large number ofclosely spaced level sets during the precomputingphase to produce a dense collection of illumina-tion samples. Ideally, the level sets would beuniformly spaced in the volume, and the precom-puting phase would simply increment one isovaluecm by a constant amount � to produce the next iso-value cm+1 = cm + �. But the spacing between levelsets depends on the increment � and the gradientof the scalar function h; when the gradient mag-nitude is small near a level set Lc, neighboring iso-values produce level sets that are far apart. KevinBeason and his colleagues3 (who developed thetechnical basis for the work we describe in this ar-ticle) describe different strategies for extractinglevel sets for use in precomputed global illumina-tion, including an adaptive sampling method thatconsults the gradient to produce roughly evenlyspaced isosurfaces.

Figure 6 shows a volumetric rendering of the il-lumination grid (viewed directly from one side)that results from a dense set of isosurfaces. We caninterpret the illumination grid in the figure as fol-lows: the isosurface nearest the luminaire (bottomof the box) receives a great amount of incident lightover a large region. As the isosurface recedes fromthe light, the direct illumination from the luminairebecomes less intense. Taken together, this sweep ofdirectly illuminated isosurfaces produces a kind of“light bubble” at the bottom of the illuminationgrid. Where an isosurface passes through the bub-ble, it’s necessarily in close proximity to the lumi-naire. On the left and right sides of the illuminationgrid, we can see light spillage from the red andgreen walls. The colors aren’t due to red and greenlight scattering through a participating medium;rather, they mark spots on isosurfaces where lighthas bounced off a colored wall and onto the isosur-face. Likewise, the illumination grid’s dark portionsmark spots where an isosurface is in a shadow.

(a)

(b)

(c)

(d)

Figure 4. 2D texture mapping. (a) Each vertex of apolygonal mesh has 3D spatial coordinates (x, y,z). (b) Each vertex is assigned additionalcoordinates (i, j) that index into a 2D texturegrid. (c) The image texture provides the 2D gridof colors. (d) The color at position (i, j) in theimage texture is applied to points having texturecoordinates (i, j) in the scene.

52 COMPUTING IN SCIENCE & ENGINEERING

Thus, the illumination grid stores, at each point x,the light that would be received if the only surfacein the box were the level set Lc passing through x.Of course, that level set is the one with isovalue c =h(x). If the illumination grid were a block of solidmaterial, we could carve it away to expose any iso-surface, revealing a layer that was precolored in ex-actly the right way to mimic the effect of beingsituated inside the illuminated box.

Once the illumination grid has been created bythe interpolation step, we can use it as a 3D texturemap to provide real-time global illumination of iso-surfaces. As the user sweeps through isovalue c, theisosurface engine generates polygonal meshes asusual. In addition to the spatial coordinates (x, y, z)defining each vertex, the polygonal mesh is en-dowed with a 3D texture index (i, j, k) at each ver-tex. If the illumination grid is scaled to match thespatial dimensions of the domain of function h,then the spatial coordinates are the same as the tex-ture coordinates—namely, (i, j, k) = (x, y, z).

Some data sets contain two separate scalar quan-

(a)

(e) (f)

(b) (c) (d)

Figure 5. 3D texture constructed from globally illuminated isosurfaces. The surrounding scene includes a box with aluminaire at the bottom, a red wall on the left, and a green wall on the right. (a) through (d) Successive level sets areplaced in the scene and globally illuminated. Shadows are evident on the top wall and on the largest blob of theisosurfaces; indirect illumination is evident in the red and green tint on the sides of the isosurfaces. (e) The incominglight is computed at sample points on each level set. (f) The scattered samples are interpolated onto a 3D illuminationgrid that serves as a 3D texture for later use.

Figure 6. Densely sampled illumination grid.Many isosurfaces, spaced closely within thevolume, are each globally illuminated. Weinterpolated the resulting illumination valuesonto the 3D uniform grid.

JANUARY/FEBRUARY 2007 53

tities h1 and h2 that the user wants to inspect si-multaneously. Each scalar function possesses itsown level sets, but when the level set L1, c for h1 isdisplayed together with the level set L2,c for h2, lightcan inter-reflect between the two isosurfaces.

To manage this situation, we construct two sep-arate illumination grids g1 and g2 during the pre-processing stage. Each grid contains the averagereflected illumination at points on its corre-sponding level sets. Later when the user selects

isovalue c, the isosurface engine constructs apolygonal mesh for L1, c and applies the precom-puted illumination texture from g1. Then, it con-structs the mesh for L2, c and applies the texturefrom g2.

Figure 7 illustrates this use of two illuminationgrids for isosurfaces of the density of protons (red)and neutrons (white) in a computational simulationof the arrangement of nucleons in a neutron star.The simulation predicts that protons and neutrons

(a) (b) (c)

Figure 7. Nucleon arrangement in a neutron star. At high densities, protons (red) and neutrons (white) form intoclusters. We applied global illumination to the entire scene and stored the results in individual textures for the protonand neutron densities. The user sweeps, from (a) to (c), through isodensity values from high to low density.

(a) (b) (c)

Figure 8. Laser-assisted particle removal. A nanoparticle of dirt (brown sphere) adheres to a substrate (white floor in thescene) covered by a thin layer of fluid (translucent white material). Rapidly heating the substrate causes the fluid tovaporize upward, lifting the nanoparticle. The user sweeps, from (a) to (c), through different isodensity values of thefluid, from high to low density.

54 COMPUTING IN SCIENCE & ENGINEERING

form these popcorn-shaped clusters at subnucleardensities of 1014 g/cm3.4 The figure shows an inter-active session in which the user moves the 3D wid-get (bottom of scene) to select different isovalues c.

The illumination grid stores the reflected lightat each point on an isosurface, averaged over inci-dent directions. For a diffusely reflecting surface,this single color is sufficient to later display a real-istic image. If the isosurface is shiny or translucent,however, a single color value is insufficient to ex-hibit the directional (view-dependent) values of thereflected or transmitted light. (Beason and his col-leagues3 describe how local illumination providedby the computer’s graphics card can approximatethis view-dependent component for shiny andtranslucent surfaces, but a more accurate ap-proach, using spherical harmonics, is availableelsewhere.5)

Figure 8 illustrates this use of texture-mapped il-lumination together with hardware-assisted light-ing for translucent isosurfaces. We used moleculardynamics to simulate a nanoparticle of dirt, coveredby a thin film of fluid, resting on a substrate. Whenwe apply heat to the substrate, the fluid layerrapidly changes phases and explosively lifts the dirtfrom the substrate, a process called laser-assisted par-ticle removal.6 The fluid particles’ point masses wereaveraged by convolution with a point-spread func-tion over finite volumes to produce a scalar densityfunction h. The figure shows different isosurfacesof h, from a single animation frame of time-seriesdata, used to visualize the fluid’s wake as it movespast the nanoparticle.

These examples illustrate interactive sessions ofa user examining 3D scalar data sets. The precom-puted 3D illumination grid lets hardware- or soft-ware-based texture mapping apply realistic globalillumination to isosurfaces at interactive speeds.The bottleneck for the visualization system be-comes the isosurface engine rather than the global-illumination solver.

We’re actively pursuing better tech-niques for sampling the radiancethroughout the 3D volume. Tolimit the maximum error when

the illumination grid is texture-mapped to an arbi-trary level set, we must run the preprocessing stepon many closely spaced isosurfaces. We’re investi-gating how to decouple the process of simulatinglight transport from the process of extracting iso-surfaces prior to generating the illumination grid.The goal is to optimally distribute photon sampleswithout constraining them to lie on a fixed set of

isosurfaces, thereby accelerating the costly prepro-cessing step.

AcknowledgmentsThe neuron data set in Figure 1 was provided byCharles Ouimet and Karen Dietz, Department ofBiomedical Sciences, Florida State University. The braindata set in Figure 2 comes from Colin Holmes at theBrain Imaging Center, Montreal Neurological Institute,McGill University. The laboratory scene in Figure 4c wasmodeled by Yoshihito Yagi, Department of ComputerScience, Florida State University. The nucleon data setwas provided by Jorge Piekarewicz, Department ofPhysics, Florida State University. The nanoparticle dataset in Figure 8 was computed by Simon-Serge Sablinand M. Yousuff Hussaini, School of ComputationalScience, Florida State University. Josh Grant, PixarStudios, developed the isosurface engine. The inter-active 3D display tool used for Figures 7 and 8 wasdeveloped by Brad Futch, Department of ComputerScience, Florida State University. US National ScienceFoundation grant numbers 0101429 and 0430954supported this work.

References1. D. Shreiner et al., OpenGL Programming Guide: The Official Guide

to Learning OpenGL Version 2, 5th ed., OpenGL Architecture Rev.Board, Addison-Wesley Professional, 2005.

2. W.E. Lorensen and H.E. Cline, “Marching Cubes: A High Resolu-tion 3D Surface Construction Algorithm,” Computer Graphics(Proc. ACM Siggraph 1987), vol. 21, July 1987, pp. 163–169.

3. K.M. Beason et al., “Pre-Computed Illumination for Isosurfaces,”Proc. IS&T/SPIE Int’l Symp. Electronic Imaging, Conf. Visualizationand Data Analysis (EI10), R.F. Erbacher et al., eds., vol. 6060, Int’lSoc. Optical Eng. (SPIE), pp. 6060B:1–11.

4. C.J. Horowitz et al., “Dynamical Response of the Nuclear ‘Pasta’in Neutron Star Crusts,” Physics Rev. C, vol. 72, 2005, 031001.

5. C. Wyman et al., “Interactive Display of Isosurfaces with GlobalIllumination,” IEEE Trans. Visualization and Computer Graphics,vol. 12, no. 2, 2006, pp. 186–196.

6. K.M. Smith et al., “Modeling Laser-Assisted Particle Removal Us-ing Molecular Dynamics,” Applied Physics A: Materials Science andProcessing, vol. 77, no. 7, 2003, pp. 877–882.

David C. Banks is a member of the University ofTennessee/Oak Ridge National Laboratory (UT/ORNL)Joint Institute for Computational Science and is a visitingprofessor of radiology at Harvard Medical School. His re-search interests include visualizing scalar and vector-valued data sets that arise in science and medicine. Bankshas a PhD in computer science from the University ofNorth Carolina at Chapel Hill. Contact him at banks@bwh.harvard.edu.

Kevin Beason is a graphics software developer at Rhythmand Hues Studios, developing rendering software for pro-ducing animations and special effects for film and TV.Contact him at beason@rhythm.com.