A Magnification Lens forInteractive Volume Visualization

Eric LaMarCenter for Applied Scientific Computing,Lawrence Livermore National Laboratory

lamar1@llnl.gov

Bernd Hamann and Kenneth I. JoyCenter for Image Processing and Integrated Computing,

Dept. of Computer Science, University of California, Davis{hamann,joy}@cs.ucdavis.edu

Abstract

Volume visualization of large data sets suffersfrom the same problem that many other visualiza-tion modalities suffer from: either one can visual-ize the entire data set and loose small details orvisualize a small region and loose the context. Inthis paper, we present a magnification lens tech-nique for volume visualization. While the notion ofa magnification-lens is not new, and other tech-niques attempt to simulate the physical propertiesof a magnifying lens, our contribution is in devel-oping a magnification lens that is fast, can be im-plemented using a fairly small software overhead,and has a natural, intuitive appearance. The is-sue with magnification lens is the border, or tran-sition, region. The lens center and exterior have aconstant zoom factor, and are simple to render. Itis the border region that blends between the ex-ternal and interior magnification, and has a non-constant magnification. We use the “perspective-correct textures” capability, available in most cur-rent graphics systems, to produce a lens with atessellated border region that approximates linearcompression with respect to the radius of the mag-nification lens. We discuss how a “cubic” bordercan mitigate the discontinuities resulting from theuse of a linear function, without significant perfor-mance loss. We discuss various issues concerningdevelopment of a three-dimensional magnificationlens.

1 Introduction

To properly understand data, a person mustbe able to see the details while being able toplace these details in the larger context of thetotal data. Data sets are becoming so large thatrendering them to standard display devices re-sults in severe under-sampling. For example, itis not possible to view 20002 pixels on a 1280∗1024pixel display: the image resolution is simply toolarge. In this case, a user can use one of severalnon-intuitive schemes: closely examine, by mag-nification or perspective zoom, various regions,zooming in and out to determine location. Thesecond scheme is a dual-window design, whereone (local) window shows a zoomed region andthe second (world) window shows a severely sub-sampled version of the full data set, with a boxglyph showing the location of the zoom window.Both of these techniques are commonly found inpaint, picture-editing, and CAD programs. Thefirst option is poor as features may be larger thanthe zoomed region, or might not be viewed intheir entirety, and thus not be recognized by theuser. Also, to navigate to another location in thedata, the user must zoom out to determine loca-tion and zoom back to a new location or move insome direction in hope to find a way to the “cor-rect” location. The second approach is poor asit also has a similar issue with navigation - theuser must interpret translation commands in thezoom window to the world window.

In this paper, we will discuss a magnification-

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lens zoom technique for volume visualizationof very large data sets. We use hardware-accelerated, texture-based volume visualization,an extension of earlier work, see [8, 7, 6]. Wehave two requirements for our magnificationlens. First, it must have a “natural” or intuitiveappearance, in the sense that, as a user movesthe lens through a data set, the magnificationand transformation of the data seems proper.We are not interested in reproducing the prop-erties of a physical magnifying glass - significantother work of human interaction demonstratesthat glyphs and other artificial constructs aresufficient, and sometimes, better than any phys-ical or “real-world” transformation. We develop,compare, and contrast square and circular lensbases. We also develop, compare, and contrastperformance and visual characteristics of linearand cubic bases for warping between magnifiedand un-magnified regions of an image.

We discuss related work in Section 2. We de-velop the basic zooming texture techniques fortwo- and three-dimensional data in Section 3and discuss issues of geometry in Section 4. Per-formance results are covered in Section 5. Fu-ture work is discussed as part of the conclusionsin Section 6.

2 Related Work

Sarkar and Brown [10] developed a GlobalFish-eye technique for browsing large graphs (ofvertices and edges) by allowing local magnifica-tion of graph elements. Their magnification ba-sis is the movement of vertices away from somefocal area.

Sarkar and Brown [11] also introduced arubber-sheet-stretching metaphor for the localmagnification of large graphs. They defined or-thogonal and polygonal stretching bases. Theformer stretches different ranges of each axis in-dependently, and the latter stretches the entiredomain as a function of distance and orientationto a set of foci. The user can manipulate thestretching through handles. The downside of theorthogonal approach is that regions that are notin the set of foci are affected. They solve thisproblem with the polygonal basis. While moreexpensive, a polygonal basis generates a more in-tuitive magnification.

Bier et. al. [1] introduce the Magic Lensas a transparent interaction tool to overlay aworkspace. The Magic Lens provides two lev-

els of functionality. The first aspect is that theMagic Lens is represented in outline, as com-pared to opaque pop-up windows or menus, asto not cover or interfere with the workspace. Sec-ond, visualization of attributes of the data can beadded or removed. This is important with denseor multivariate data, as it is generally not pos-sible to comprehend an image with more thana few dimensions of data shown. For example,a magic lens could have four or six sub lenses,each showing a different set of variables, or thesame variables with different representations.The user can switch different views around by re-positioning the larger lens such that the smallerlens covers the region in question. While theirwork does not discuss “in-context” magnifica-tion, it is the basis of many other works devel-oping magnification techniques.

Ware and Lewis [13] discussed the “DragMagImage Magnifier,” a two-window approach to viewlarge (125002 pixel) images. The first windowshows the global view, while the second showsa magnified view. The second window appearsas a movable glyph in the global view, and theuser can navigate through either window. Theuser cannot modify the location of relative sizesof the two windows and must mentally fuse thelocal and global views.

Keahey [3] provides a good summary of the is-sues and techniques of the “Detail-in-Context”problem. He discusses the uses of glyphs, Level-of-Detail control of glyphs, embedded objects andnon-linear magnification of a montage of two-dimensional images.

Rauschenbach [9] developed a demand-driventransmission and magnification approach forlarge color images. He incorporates wavelet de-composition, transmission, and reconstruction.A simple orthogonal stretching method is used.

Keahey introduces volume warping in [4] asa method for easing the examination of a med-ical database with 11-dimensional records. Hedefines a non-linear magnification using multi-ple foci in a three-dimensional volume, allowingthe examination of clusters of records while pro-viding the larger context. This technique hasthe downside in the sense that the data is verysparse and glyphs, in the form of regular, volu-metric grid lines, must be added to the renderingto show the location and size of the foci.

Kurzion and Yagel [5] use volume (3D) tex-tures and adaptively tesselated proxy geometriesto warp and cut open volumetric objects. While

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they can produce interesting deformations, thebest use of their technique is for a natural“cutting-open-and-peeling” of an object. None ofthe images contains a large local stretch (thus,magnification) of the volume and they do not dis-cuss the issues of homogeneous-space textures.We can only assume that there will be significantartifacts in the imagery of highly warp or magni-fied regions.

Our technique differs from earlier techniquesin that we are developing a magnifying lens in thecontext of large, rectilinear data set volume visu-alization. We use hardware-texture based vol-ume visualization. We are not as interested indeveloping mathematical formulas for the defor-mation of space by a magnification locus, as weare in providing a “natural”, intuitive magnifica-tion lens that can render imagery very quicklyon conventional graphics hardware. We payparticular attention to the correct generation ofhomogeneous-space textures and how this af-fects the resulting imagery.

3 Textures

3.1 Zooming with Textures

Figure 1. Zooming with textures. Axis Gshows geometric space and axis T shows tex-ture space.

Figure 1 illustrates the concept of zoomingwith textures. Zooming is accomplished by mod-ifying the texture coordinates. The mandrill im-age is interpreted as a texture defined over a unitdomain. In figures 1(a), 1(b), and 1(c), two dif-ferent domains are shown: the lower-left axisshows the geometric space G, and the upper-right axis shows the texture space T. Figure 1(a)shows the mandrill drawn in geometric and tex-ture space. Figure 1(b) shows a small region andits corresponding geometric and texture coordi-nates. Figure 1(c) shows this region zoomed by

a factor of four by using the geometry from 1(a)and modifying the texture coordinates.

3.2 Warped Surface Textures

(a) 3d TCs (b) 4d TCs

Figure 2. Comparing zoom operations in thetop trapezoid: Image (a) shows a zoom oper-ation with texture coordinates (TCs) specifiedin affine space. Image (b) shows a zoom oper-ation with TCs projected into homogeneousspace.

If one wants to increase the scale of a texture,simulating a “zoom-in” operation applied locally,the simplest technique is to move the texture co-ordinates toward the center of interest. However,if different vertices of a polygon have differentzoom factors, a straight modification of texturecoordinates to affect a zoom does not work: thehardware is only capable of a linear transforma-tion and the zoom is not linear. Image 2(a) showsthe result - this result is clearly not desired.Graphics hardware can only perform linear in-terpolation of geometric, color, and texture val-ues across a polygon. The authors are not awareof any graphics hardware or graphics specifica-tion that allows anything other than linear in-terpolation. This is due to the fact that linearforward differencing is used during rasterizationto calculate values incrementally, from pixel topixel, across a polygon. Hence, the directionalderivative is constant.

Our solution is based on the use of “hardware-implemented, perspective correct textures.” Thistechnique, implemented in hardware, projectstexture coordinates (TCs) from affine space intohomogeneous space, iterates homogeneous tex-ture coordinates during polygon rasterization,and projects the TCs back into affine space forfinal texture lookup.1 We note that this tech-

1The OpenGL specification defines a texture coordinate by

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nique is a special case of projective textures andhomogeneous texture coordinates, see [12] for acomplete discussion. The center of the images inFigure 2 is magnified by a factor of four, so thetexture coordinates associated with the verticesof the center square of the lens are projected intohomogeneous space with weight 4. This leads toimage 2(b): we have a zoom effect, though with anon-linear compression through the lens border.One last step is necessary to achieve the effectwe want.

3.3 Multiple Segments

Affine HomogeneousLower-Left (0, 0, 0, 1) (0, 0, 0, 1)Lower-Right (1, 0, 0, 1) (1, 0, 0, 1)Upper-Left (.375, 1, 0, 1) (1.5, 4, 0, 4)Upper-Right (.625, 1, 0, 1) (2.5, 4, 0, 4)

Table 1. The affine-space texture coordi-nates of the warp trapezoid of Figure 3 pro-jected into homogeneous space with a weight(zoom) of four.

The last step is to tessellate a polygon into aseries of smaller polygons where the new vertices’geometric and texture coordinates linearly inter-polate the original polygon’s vertex’s geometricand texture coordinates. Image set 3(a) showsthe original, un-stretched, images with the re-gion to be warped delimited by a red trapezoidand the green arrows showing the direction anddegree of the stretch. Each image is stretchedat the top by a factor of four, with respect to thebottom edge, which corresponds to Q = 4. Table1 shows the texture coordinates of the vertices ofthe trapezoid in affine and homogeneous space.

Figure 3 shows one, two, four, eight, and 16segments for two data sets: a synthetic checkerboard, and a mandrill image. Column (2) is asynthetic image. We selected it as it demon-strates well the effects of different numbers ofsegments. Column (4) is the mandrill image andwas selected to show how the algorithm affects a

four scalars, or (S, T,R,Q). Thus, affine space correspondsto (S, T,R, 1), and to “project” this value into homogeneousspace by a “weight” Q results in (S ×Q,T ×Q,R×Q,Q). Ho-mogeneous space is equivalent to (S, T,R,Q), and the projec-

tion to affine space is(SQ, TQ, RQ, 1). When using textures of

lower dimension, the higher-order affine coordinates are ig-nored.

known image. Column (1) shows the number ofsegments used, and column (3) shows the seg-ment boundaries. For simplicity, we refer to thepairs of checker-board and mandrill images inrows 3(a) to 3(f) as “image set” (a) to (f). Eachsegment is the width of the original polygon andevenly tessellates the original polygon from topto bottom.

With one segment, see image 3(b), we obtainthe normal perspective image. With two andfour segments, see images 3(c) and 3(d), the exis-tence of the segments is still quite obvious. Witheight and 16 segments, see images 3(e) and 3(f),the segments disappear, and we obtain a fairlysmooth image. Typically, the lens border will besignificantly smaller than these images. How-ever, it is useful to illustrate the effects of thenumber of segments on the image.

3.4 Border Transitions

Figure 4. Border examples: warping the nor-mal (top) region into the zoomed (bottom) re-gion. Image (b) is simple; (c) is linear; and (d)is cubic.

We define a magnifying lens to have threeparts, see Figure 4(a), from top to bottom: nor-mal, warp, and zoom. Concerning images 4(b) to4(d), the normal and zoom regions are the same.What is different in the images is the warp re-gion.

Image 4(b) shows a single segment. The im-age is highly compressed near the normal-warpborder. In this region, small details can get lost,making it difficult to navigate to them. Also, thediscontinuity across the normal-warp border is,visually, very strong.

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(a)In

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(d)4

Seg

men

ts

(b)1

Seg

men

t

(e)8

Seg

men

ts

(c)2

Seg

men

ts

(f)16

Seg

men

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(1) (1)

Figure 3. Multiple Segments. The top border is stretched by a factor of four. The double-headedarrows show segment boundaries.

Image 4(c) uses the multiple-segment tech-nique with eight segments. The compression isconstant across the warp region. We have found,however, that the sharp edges at the normal-warp and warp-zoom borders can become quitedistracting when the magnification lens is movedinteractively. These sharp edges appear becausethe function across the regions is piece wise lin-ear and, thus, only C0-continuous.

Image 4(d) shows our solution to this problem:We generate the texture and geometric coordi-nates using a cubic Bezier curve, see [2], thatis defined to interpolate value and gradient atthe normal and zoom boundaries. This imageis composed of 16 segments. The compression isno longer uniform: The transition from the bor-der region to the fixed-magnification regions toeither side is much smoother. This lens seemsideal for artifact-free zooming of an image. Thistechnique requires more geometry than the lin-

ear technique (image 4(c)) which can become anissue for platforms where the CPU is the bottle-neck.

4 Geometry

Figure 5 shows annotated square and circulargeometry examples. Letters A-C denote separatetransformation regions. Region A is the magni-fied center of the lens (zoom); the texture coor-dinates are simply modified as discussed in Sec-tion 3.1; region B shows non-magnified, outerregion (normal); and region C shows the warp-magnified region that blends (warp) region (A) toregion (B).

Image 5(a) shows the square geometry, andimage 5(b) shows the circular geometry. Forsimplicity, this image shows only eight angularsegments: all other images use 32 angular seg-ments. The square geometry has the advantage

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(a) Square lens

(b) Circular lens

Figure 5. Local geometric transformations:Letters A-C denote separate transformations:A: Magnified, center region of the lens;B: Non-magnified, outer region; C: Warp-magnified region that blends the inner, mag-nified, region (A) to the outer, non-magnified,region (B).

that there is much less geometry, both to gen-erate and to clip. However, the clipping is onlysimple if the geometry is aligned with the basedata, e.g., a rectilinear data set. The significantdisadvantage of this lens geometry is its unnat-ural appearance: the squareness, particularly atthe corners, is a distracting artifact and can in-fluence the interpretation of an image. The cir-cular geometry allows us to approximate a circle(when tessellated to 32 angular segments). Whilethis produces significantly more geometry, whichmust also be clipped, the overall visual qualityseems much better and much more natural. Oneno longer sees the lens - one just sees the mag-nification.

4.1 Bounded Lens

Figure 6 shows that the circular tessellationfor the lens does not need to be global: onecan enclose the lens geometry inside a boundingsquare.

Figure 6. Circular geometry need not beglobal: a lens geometry can be “embedded”into a larger polygon (here, a square) or “sten-cil.”

4.2 Clipping

If the lens or border is at the edge of an im-age tile, then the lens must be clipped in im-age/texture space, not the window. For the fol-lowing discussion, we define the thicker lines inimages 5(a) and 5(b) to be the tile boundariesof an image broken into 2x2 tiles. When usingthe square lens basis (Figure 5) that is alignedto the tile boundaries, clipping is very simple.However, when using a square lens basis that isnot aligned with the tile boundaries, or when us-ing the circular lens basis (image 5(b)), clippingbecomes much more complicated and expen-sive. Examining the tile boundaries in image 5(b)shows that the tile boundaries that become con-cave. On the other hand, using the bounding-polygon technique discussed in the last sectioncan significantly simplify the clipping operationby testing the bounding polygon for intersectionwith tile boundaries.

4.3 3D Geometry

Figure 7 shows how the geometry is generatedfor a volume. To render a volume, one rendersa stack of textured polygons from back to front,where the viewing direction would ideally be per-pendicular to the planes. For this image, weshow the stack from the side to better illustrateits structure.

Adding a magnification lens is fairly simple.We use this terminology: the volume lens hasnormal, warp, and zoom regions. The normal,warp, and zoom regions in a plane are those re-gions of the plane that intersect the correspond-ing regions of the volume lens. The dashed lines

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Figure 7. Stacking a square-lens geometry in3D space.

in Figure 7 show the volume boundaries. Thelens is located in the front-top-left corner of thevolume. For simplicity, we show a square lenswith 16 planes intersecting the volume. Whenplanes do not intersect the warp or zoom regionsof the lens (i.e., the normal regions), we use asingle polygon. These are shown as solid greenlines. When planes that intersect the warp re-gion, we blend the volume’s warp and zoom re-gions into the plane’s warp and zoom regions.These are shown as solid blue lines. Whenplanes that intersect the zoom region, we againblend the volume’s warp and zoom regions intothe plane’s warp and zoom regions. These areshown as solid red lines.

Planes are not warped “out of the plane.” Thisapproach is simpler, and any warping in the viewdirection would be removed by orthographic pro-jection. This aspect may be important for per-spective projections, but we have not found thisto be a an issue.

5 Results

5.1 Comparisons of Geometry and Texture Basisin 2D

Figure 8 shows examples generated with thesquare and circular lenses. This figure contraststhe simple, linear and cubic approaches. Ta-ble 2 summarizes the rendering performance, inframes per a second, for 2D imagery and tex-tures. The window size is 5002 pixels, with a total

Images/SecondOnyx/ P3/ Onyx/ P3/

Basis IR GeForce IR GeForceBaseline 456 1067

Square CircularSimple 425 1035 380 900Linear 390 1012 357 908Cubic 374 987 332 907

Table 2. Rendering performance in images-per-second for 2D imagery.

lens size of 2502 pixels, the interior lens size is ap-proximately 1862 pixels, with a border size of 32pixels. The square geometries have four angu-lar segments, one for each face, and four linearor eight cubic radial (perpendicular to angular)segments. The circular method uses 32 angularsegments, and four linear or eight cubic radialsegments.

We used an SGI Onyx2 with four 195MHzMIPS R10K processors (only one used) and an In-finite Reality (IR) graphics subsystem; the secondsystem that we used was a PC with an 866MHzIntel Pentium 3 processor with a GeForce2 GTSgraphics card.

The texture was loaded into texture memoryonce, and then the textured geometry was ren-dered 100 times, and the average time computed.We have measured just the intra-frame rates,that is, the elapsed time for rendering all tex-tured geometry.

The row titled “Baseline” refers to an un-zoomed image, rendered with two triangles, toprovide a maximum rendering rate. It is impor-tant to notice that the speeds are much closer toeach other for the GeForce than for the IR - whichis probably due to processor speed: The genera-tion of the base geometry is strictly a function ofthe base processor, and the MIPS R10K/195MHzprocessor is much slower than the PIII/866MHzprocessor.

5.2 Comparisons of Geometry and Texture Basisin 3D

Figures 9 and 10 show the magnification-lenstechnique applied to two volumetric data sets.Figure 9 show the effect of square vs. circular ba-sis and linear vs. cubic basis. Figure 10 showsthe integration of the magnification lens with a

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(1)S

quar

e(2

)C

ircu

lar

(a) Simple (b) Linear (c) Cubic

Figure 8. Zoom comparisons: Square/Circular and Simple/Linear/Cubic. Square-Linear is the sim-plest to process; Circular-Cubic produces the best quality; and Square-Cubic and Linear-Circular areshown for completeness.

SkullBaseline 5.9Square, Simple 4.8Square, Linear 4.6Square, Cubic 4.4Circular, Simple 4.3Circular, Linear 3.9Circular, Cubic 3.6

Table 3. Rendering performance in frames-per-second for skull data set (Figure 9).

clipping plane.Table 3 summarizes the performance for the

skull data set, shown in Figure 9. For the lensesused in Figure 9, the linear basis uses four ra-dial segments while cubic basis uses eight radialsegments; the circular basis use 16 angular seg-ments. The texture was loaded to texture mem-

ory once, and then the textured geometry ren-dered several times, and the average time com-puted. We have measured just the intra-framerates, that is, the elapsed time for rendering alltextured geometry. These time are measured onan Onyx2 with InfiniteReality graphics with oneRaster Manager. This test was not done on theGeForce, as our algorithm uses only 3D textures,and the GeForce2 does not support 3D textures.

6 Conclusions and Future Work

We have presented an algorithm for perform-ing a magnification-lens technique for volume vi-sualization. We use homogeneous texture coor-dinates and special geometries to implement amagnifying glass to provide a user with the abil-ity to zoom in on small regions of a very largedata set, while providing a smooth transitionto un-zoomed regions. This technique is quite

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(a) Original, (b) Square lens,no lens or warping linear warping

(c) Circular lens, (d) Circular lens,linear warping cubic warping

Figure 9. Skull data set rendered with a lenslocated in the right orbit.

effective for zooming into large data sets, pro-vided that the system used for rendering is notgenerally compute-bound (in contrast to fill-ratebound). When a clipping plane is added, we havea very powerful technique for examining smalldetails of a very large data set.

We plan to apply this method to our mul-tiresolution volume visualization system [8, 7, 6].However, a significant hurdle is the efficient clip-ping of the lens against individual tiles. If thethick lines in Figures 5(a) and 5(b) delimit tile(individual texture) boundaries, hardware-basedclipping becomes very difficult.

A software technique that clips based on tex-ture coordinates might work, but then efficiencybecomes a concern. Second, new graphics cards(e.g., Nvidia’s GeForce3) allow dependent texturelook-ups: One could implement the schemesdiscussed here by defining an intermediate tex-ture that performs the warping. This wouldvastly reduce the amount of geometry required.Third, blending between two images, or datasets, through the warp region will be necessarybecause the normal and zoom regions typicallyuse different images, one being a higher resolu-tion version of the other. We also would have to

(a) (b)

(c) (d)

Figure 10. Clipped CT data set. Image (a)shows the original, undistorted head; (b)shows the head with clipping plane expos-ing the nasal passages; (c) shows the headwith a circular-cubic lens; and (d) shows thehead with clipping plane and magnification,exposing and enlarging the nasal passages.

extend this method to warp hierarchies that canblend between several levels of resolution.

Acknowledgements

This document was prepared as an accountof work sponsored by an agency of the UnitedStates Government. Neither the United StatesGovernment nor the University of California norany of their employees, makes any warranty, ex-press or implied, or assumes any legal liabil-ity or responsibility for the accuracy, complete-ness, or usefulness of any information, appa-ratus, product, or process disclosed, or repre-sents that its use would not infringe privatelyowned rights. Reference herein to any specificcommercial product, process, or service by tradename, trademark, manufacturer, or otherwise,does not necessarily constitute or imply its en-dorsement, recommendation, or favoring by theUnited States Government or the University ofCalifornia. The views and opinions of authors ex-

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pressed herein do not necessarily state or reflectthose of the United States Government or theUniversity of California, and shall not be used foradvertising or product endorsement purposes.

This work was supported by the National Sci-ence Foundation under contracts ACI 9624034(CAREER Award), through the Large Scien-tific and Software Data Set Visualization (LSS-DSV) program under contract ACI 9982251, andthrough the National Partnership for AdvancedComputational Infrastructure (NPACI); the Officeof Naval Research under contract N00014-97-1-0222; the Army Research Office under contractARO 36598-MA-RIP; the NASA Ames ResearchCenter through an NRA award under contractNAG2-1216; the Lawrence Livermore NationalLaboratory under ASCI ASAP Level-2 Memoran-dum Agreement B347878 and under Memoran-dum Agreement B503159; the Lawrence Berke-ley National Laboratory; the Los Alamos NationalLaboratory; and the North Atlantic Treaty Or-ganization (NATO) under contract CRG.971628.We also acknowledge the support of ALSTOMSchilling Robotics and SGI. We thank the mem-bers of the Visualization and Graphics ResearchGroup at the Center for Image Processing andIntegrated Computing (CIPIC) at the Universityof California, Davis, and the members of theData Analysis and Exploration thrust of the Cen-ter for Applied Scientific Computing (CASC) atLawrence Livermore National Laboratory.

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