V
`
`isualization Paradigms
`
`Chandrajit L. Bajaj
`
`University of Texas, Austin
`
`ABSTRACT
`
`A wide variety of techniques have been developed for the visualization of scalar,
`vector and tensor eld data. They range from volume visualization, to isocontour-
`ing, from vector eld streamlines or scalar, vector and tensor topology, to function
`on surfaces . This multiplicity of approaches responds to the requirements emerg-
`ing from an even wider range of application areas such as computational uid
`dynamics, chemical transport, fracture mechanics, new material development,
`electromagnetic scatteringabsorption, neuro-surgery, orthopedics, drug design.
`In this chapter I present a brief overview of the visualization paradigms currently
`used in several of the above application areas. A major objective is to provide
`a roadmap that encompasses the majority of the currently available methods
`to allow each potential userdeveloper to select the techniques suitable for his
`purpose.
`
` .
`
`Introduction
`
`Typically, informative visualizations are based on the combined use of multiple tech-
`niques. For example gure . shows the combined use of isocontouring, volume ren-
`dering and slicing to highlight and compare the internal D structure of three dierent
`vorticity elds. For a detailed description of each of the approaches we make reference
`to subsequent chapters in this book and previously published technical papers and
`books Bow , Cle , HU , KK , NHM , REE+ , Wat .
`
`DATA VISUALIZATION TECHNIQUES, Edited by C. Bajaj
`c  John Wiley & Sons Ltd
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`Bradium Technologies LLC
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`Figure . The combined display of isocontours, slicing and volume rendering used to
`highlight the D structure of vorticity elds.
`
`Figure . Two volume renderings showing snapshots of wind speed in a global climate
`model.
`
` . Volume Rendering
`
`Volume rendering is a projection technique that produces image displays of three-
`dimensional volumetric data see g. .. Its main characteristic is the production of
`view-dependent snapshots of volumetric data, rather than the extraction of geometric
`information such as isocontouring.
`Chapter  surveys alternate volume rendering algorithms reported in the literature.
`Two main classes of approaches that have been developed dier mainly on the order
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`Visualization Paradigms
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`of projection of the volume cells. Secondary distinctions arise from the dierences in
`color accumulation and composition techniques to produce the nal image.
`
`Forward projection techniques traverse the volume object space approach pro-
`jecting and display each volumetric cell or voxel. This approach takes advantage
`of graphics hardware acceleration by selecting appropriate drawing primitives to
`approximate the voxel image.
`Backward projection techniques traverse the image image space approach and
`cast through the data volume, one light ray per pixel, accumulating color intensities
`along the ray to determine the nal pixel color.
`
`Cell projection and splatting are both forward projection techniques. In cell pro-
`jection, the cells of the data volume are traversed and their images computed by
`subdivision into a polygonal approximation. In splatting, the samples of the volume
`are traversed and their contribution to the nal image is computed by convolution with
`a reconstruction kernel. Cell projection technique can be optimized by taking advan-
`tage of the spatial coherence of the volume cells both in the case of regular grids and
`in the case of unstructured meshes. Splatting has been shown to be a fast technique
`for hardware assisted scalar volume visualization, and was extended to vector elds
`see details in Chapter  Additional splatting techniques are developed for texture
`based visualization of velocity elds in the vicinity of contour surfaces see details in
`Chapter 
`Backward projection methods are accelerated by exploiting the coherence between
`adjacent rays. This idea has been implemented in a number of approaches using:
`i adaptive sampling along the rays depending on the importance" of dierent re-
`gionsii templating the paths of parallel rays through regular grids,iii bounding with
`simple polyhedra signicant regions that give the main contribution to the output im-
`age, or iv maintaining the front of propagating rays through irregular grids. The
`high computational cost of volume rendering in the spatial domain can sometimes be
`replaced by an asymptotically faster computation in the frequency domain Lev ,
`Mal , TL .
`
` .
`
`Isocontouring
`
`Isocontouring is the extraction of constant valued curves and surfaces from d and d
`scalar elds. Interactive display and quantitative interrogation of isocontours helps in
`determining the overall structure of a scalar eld see g. .  and its evolution over
`time see g. ..
`Chapter surveys the most commonly used isocontouring algorithms along with
`recent improvements that permit rapid evaluation of multiple isocontour queries, in
`an interactive environment. Traditional isocontouring techniques examine each cell of
`a mesh to test for intersection with the isocontour of interest. Accelerated isocontour-
`ing can be achieved by preprocessing the input scalar eld both in its domain the
`geometry of the input mesh and in its range the values of the sampled scalar eld.
`One the one hand, one takes advantage of the known adjacency information of mesh
`cells domain space optimization. Given a single cell c on an isocontour component
`one can trace the entire isocontour component from c, by propagating from cell to cell
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`using inter-cell adjacency. This reduces the search for isocontour components from a
`search in the entire input mesh to a search in a much smaller subset called the seed set.
`A seed set is a subset of the mesh cells which has at least one cell on each connected
`component of each isocontour. From this typically very small seed set of mesh cells
`one searches for starting cells for each component of the desired isocontour and then
`applies contour propagation through cell adjacencies.
`On the other hand, one independently optimizes the search for isocontours exploit-
`ing the simplicity of the range of the scalar eld range space optimization. The
`values of the eld are scalars that in range space form an interval. Within each cell
`of the mesh or of mesh cells of the seed set the scalar eld usually has a small
`continuous variation that can be represented in range space as a small subinterval.
`The isocontour computation is hence reduced in range space to the search for all the
`segments that intersect the currently selected isovalue w. This search can be optimally
`performed using well known interval tree or segment tree data structures.
`
`Figure . Skin and bone head models extracted as two dierent isocontours from the
`same volumetric MRI data of the Visible Human female.
`
`Figure . Three isocontours of wind speed that show the time evolution of air dynamics
`in a global climate model.
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`Visualization Paradigms
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` . Flow Visualization
`
`
`
`Visualization of vector elds is generally more complicated than visualizing scalar
`elds due to the increased amount of information inherent in vector data. Clearly
`vector data can be contracted to scalar quantities, for example by computation of
`vector magnitude, scalar product with a given vector, or magnitude of vorticity. In
`this case, scalar visualization techniques such as isosurfaces and volume rendering can
`be applied. Additional approaches to visualization of vector elds include iconogra-
`phy, particle tracking, and qualitative global ow visualization techniques. Chapter 
`reviews ow visualization techniques while Chapter  describes more in detail the ap-
`proaches designed to take advantage of currently existing graphics hardware to increase
`performance. For additional detail, refer to the papers cited in these two chapters.
`Particle tracking or advection techniques are based on following the trajectory of a
`theoretically massless particle in a ow. In its simplest form, the path traversed by a
`particle in a steady ow is called a streamline. If the ow is unsteady, or time-varying,
`the path followed by a particle over time is called a path line. A curve resulting from
`a number of particles emitted at regular or irregular intervals from a single source
`is called a streak line. Numerical techniques commonly used for evaluating the above
`equation include Euler and Runge-Kutta methods. In the case of incompressible ow, a
`single stream-function in D can be constructed such that the contours of the stream-
`function are streamlines of the vector eld. In D, a pair of dual stream functions
`is required, and streamlines will occur as the intersections of isocontours of the two
`functions
`KM .
`Particle tracking techniques may also be extended by grouping multiple particles
`together to form a stream ribbon, stream surface, stream tube or ow volume. Global
`techniques such as Line Integral Convolution present a qualitative view of the vector
`eld which presents intuitively meaningful visualizations for the user. Flow probes"
`may be placed at user-specied or computed locations to reveal local properties of the
`ow eld such as direction, speed, divergence, vorticity, etc. Properties are mapped to a
`geometric representation called an icon. The complexity of the icon increases with the
`amount of information that it is designed to represent. Representing curl vorticity,
`which is itself a vector eld with additional physical meaning, can be achieved by a
`cylindrical icon with candystriping to indicate both the direction and magnitude of
`vorticity.
`
` . Quantication
`
`In the quest for interrogative visualization Baj, in which the user can not only see
`the data, but navigate and query for increased understanding, the ability to quantify
`and perform volumetric measurements is vital. Another challenge to visualization is
`to give quantitative information concerning time dependent studies and time-varying
`structures e.g. ow. In the study of paralysis, researchers are constructing models of
`spinal cords and regions of damaged cord from histological samples. Figure . left
`is an example of a histological slice of an injured rat spine. In Figure . right,
`the damaged region has been reconstructed as a surface, and is visualized along with
`orthogonal slices of the D histological specimen. Traditionally, spinal damage has been
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`modelled as an expanding cylindrical region. The ability to more accurately dene the
`region of damage and measure the surface area and volume of the region are promising
`tools in developing a greater understanding of cysts and how they develop.
`Chapter  reviews tissue classication techniques using local reconstructions of band
`limited samplings, and Bayesian statistics. Such classication provides the means to
`accurately identify for isocontouring and volume rendering, and quantify the relevant
`substructures of three dimensional images.
`
`Figure . Left Histological sample of a rat spine. Right Reconstruced spinal lesion
`within slices of D histological volume.
`
`Chapter  reviews the shape analysis and visualization of free form surface mod-
`els used in computer aided geometric design and computer graphics. The analysis
`tools prove essential to detecting surface imperfections aswell as higher order inter-
`patch smoothness. Related research on free-form surfaces visualization are addressed
`in BR , BBB+ .
`Three special cases of volumetric quantication which are prevalent in data visual-
`ization applications apply to the following data types:
`
` Contours - surfaces which are created through isocontouring of scalar data
` Slices - surfaces which are formed by tiling multiple planar cross sections of objects
` Union of Balls - also known as the solvent accessible surface and common in
`molecular visualization
`
`Contour Quantication
`Bajaj, Pascucci and Schikore BPS  introduce the systematic quantication of met-
`ric properties of volumetric data and the relative isocontours. Given an isovalue w
`one can compute the surface area of the corresponding isosurface, the volume of the
`inside region or any other metric property also called signature function of w. The
`plot of the signatures gives rise to an interface that drives the user in the direct
`selection of interesting isovalues. Figure . shows the direct selection of noiseless iso-
`surfaces corresponding to skin and bone tissues which correspond to the maxima of
`the gradient-weighted area signature.
`Sliced Data
`Objects are frequently reconstructed from serial sections BCL b, BCL a. In this
`case, volume properties can be accurately computed using the following equation for
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`Visualization Paradigms
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`Figure . Three isosurfaces of the same volumetric MRI scan. The vertical bars in the
`spectrum interface top mark the selected isovalues.
`
`prismatoids, a triangular tiling of two parallel contours: V = h
` B + M + B where
`B is the area of lower base, B is the area of upper base, M is the area of the
`midsection joining the bases, and h is the separation between the contours. With n
`parallel slices of contours equally spaced, the composite volume computation results
`n(cid:0)
`n(cid:0)
`in: V = h
`Mi +  P
`Bi + Bn.
` B +  P
`
`
`Union of Balls
`The geometric, combinatorial and quantitative structure of the union of a set of balls
`has been presented by Ede , DE . The union of balls model is equivalent to the
`space lling model used to represent molecules where each atom is approximated by
`a ball with a relative van der Waals radius. Deeper insight on the properites of a
`molecule in solution is provided by the computation of the Solvent Accessible Surface
`and the Solvent Excluded Surface SSO . The two surfaces are dened by idealizing
`the solvent molecule e.g. water as a single ball and computing the boundary of the
`region that can be accessed by the solvent center Solvent Accessible oset of the Union
`of Balls model of the molecule or the boundary of the region that cannot be reached
`by any point of the solvent Solvent Excluded. On the basis of the union of balls model
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`exact representation of both the surfaces can be computed eciently BLMP  see
`g. ..
`
`Figure . top Union of Balls, Solvent Accessible and Solvent excluded surfaces of the
`Nutrasweet molecule with respect to the same solvent. bottom Solvent excluded surface of
`the Gramicidin molecule with respect to three increasing solvent radii.
`
` . Data Reduction
`
`Mesh reduction or simplication refers to a broad category of techniques designed to
`trade space and complexity for accuracy in representation of a surface or volume mesh.
`Like isocontouring mesh reduction is an algorithmic approach used to preprocess the
`input dataset to make more suitable for display or analysis queries. The dierence
`is that while isocontouring extracts an interesting feature like a particular isosurface
`from a volumetric dataset, mesh reduction is meant to generate a reduced version
`of the volume itself to speedup postprocessing. Figure . demonstrates mesh reduc-
`tion applied to D functional data, in this case a slice of MRI data. Related results
`come from several research communities, including Geographical Information Systems
`GIS, Computational Fluid Dynamics CFD, and Virtual EnvironmentsVirtual Re-
`ality VEVR. Each community has much the same goal for achieving interactivity
`with very large sets of data. An initial classication of techniques can be made by
`distinguishing between static simplication, in which a single resolution output is com-
`puted from a high-resolution input based on given simplication criteria, and dynamic
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`Visualization Paradigms
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`Figure . Original data left Reduced Triangulation center Reduced Image right.
`
`simplication, in which hierarchical triangulations are computed, from which reduced
`triangulations can be rapidly extracted depending on time dependent metrics such
`as distance from the viewer, location in the eld of view, and approximation error
`tolerance.
`Height-eld reduction: A driving application for reduction of height-elds is GIS. A
`wide range of techniques are based on extraction of key points or edges from the orig-
`inally dense set of points, followed by a constrained Delaunay triangulation DFP,
`FFNP, FL , PDDT , Tsa , WJ . Silva, et. al SMK  uses a greedy method
`for inserting points into an initially sparse mesh, reporting both better and faster re-
`duction compared to a freely available terrain reduction tool. A survey by Lee Lee 
`reviews methods for computing reduced meshes by both point insertion and point dele-
`tion. Bajaj and Schikore BS , BS a developed practical techniques for measuring
`the local errors introduced by simplication operations and bound the global error
`accumulated by multiple applications. Their techniques begin with simple scalar elds
`and extend easily to multi-valued elds and dened on arbitrary surfaces. Geometric
`error in the surface as well as functional error in the data are bounded in a uniform
`manner. Topology preserving, error-bounded mesh simplication have also been ex-
`plored BS . Figure . demonstrates geometric mesh reduction while Figure .
`demonstrates mesh reduction applied to D functional data.
`Geometry reduction: Geometric mesh reduction has been approached from several
`directions. In the reduction of polygonal models, Turk Tur  used point repulsion
`on the surface of a polygonal model to generate a set of vertices for retriangulation.
`Schroeder, et al. SZL  decimate dense polygonal meshes, generated by Marching
`Cubes LC, by deletion of vertices based on an error criteria, followed by local retri-
`angulation with a goal of maintaining good aspect ratio in the resulting triangulation.
`Errors incurred from local retriangulation are not propagated to the simplied mesh,
`hence there is no global error control. Rossignac, et al. RB  uses clustering and
`merging of features of an object which are geometrically close, but may not be topo-
`logically connected. In this scheme, long thin objects may collapse to an edge and
`small objects may contract to a point. Hamann Ham  applies a similar technique
`in which triangles are considered for deletion based on curvature estimates at the
`vertices. Reduction may be driven by mesh resolution or, in the case of functional
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`Figure . Three snapshots of geometric mesh simplication of an engine block
`
`HHK+  perform mesh reduction by
`surfaces, root-mean-square error. He, et al.
`volume sampling and low-pass ltering an object. A multi-resolution triangle mesh is
`extracted from the resulting multi-resolution volume buer using traditional isosur-
`facing techniques. Hoppe, et al. HDD+  perform time-intensive mesh optimization
`based on the denition of an energy function which balances the need for accurate
`geometry with the desire for compactness in representation. The level of mesh sim-
`plication is controlled by parameters in the energy function which penalizes meshes
`with large numbers of vertices, as well as a spring constant which helps guide the
`energy minimization to a desirable result.
`In Hop , Hoppe introduces Progressive Meshes, created by applying optimization
`with the set of basic operations reduced to only an edge contraction. Scalar attributes
`are handled by incorporating them into the energy function. Ronfard, et. al RR 
`also apply successive edge contraction operations to compute a wide range of levels-of-
`detail for triangulated polyhedra. Edges are extracted from a priority queue based on
`a computed edge cost such that edges of lesser signicance are removed rst. Cohen,
`et al. CVM+  introduce Simplication Envelopes to guide mesh simplication with
`global error bounds. Envelopes are an extension of oset surfaces which serve as an ex-
`treme boundary for the desired simplied surface. Lindstrom, et. al LKR+  impose
`a recursive triangulation on a regular terrain and compute preprocessing metrics at
`various levels of resolution which permits real-time adaptive triangulation for interac-
`tive y-through. Funkhouser et al. describe adaptive display algorithms for rendering
`complex environments at a sustained frame rate using multiple levels of detail FS .
`Delaunay techniques for static simplication have been extended to create hierar-
`chies of Delaunay triangulations from which a simplied mesh can be extracted on the
`y dBD . Successive levels of the hierarchy are created by deleting points from the
`current level and retriangulating according to the Delaunay criteria, giving the hier-
`archical structure of a directed acyclic graph DAG. Puppo improves on the approach
`of de Berg by augmenting the DAG with information on which triangles between suc-
`cessive resolutions are overlapping Pup . With this information, the problem of
`extracting a triangulation from the DAG is simplied and requires no backtracking.
`It is shown that for a given triangulation criteria, the optimal triangulation satisfying
`the criteria which is embedded in the DAG can be extracted in optimal time. Cohen
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`Visualization Paradigms
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`a
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`b
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`Figure . a Arrow plot of a two dimensional vector eld b Streamline along focus in
`the vortex core surrounded by nearby streamlines.
`
`and Levanoni adopt a tree representation for hierarchical Delaunay triangulation and
`demonstrate techniques for maintaining temporal coherence between successive trian-
`gulations COL . The technique is demonstrated for relatively sparse terrains, and
`it remains to be seen whether the constraints imposed by a tree representation will
`restrict simplication for dense triangulated terrains.
`Wavelets Mal , Dau  have been utilized for their multiresolution applications
`in many areas of computer graphics and visualization SDS , including image com-
`pression DJL a, surface description DJL b, EDD+ , CPD+ , tiling of con-
`tours Mey  and curve and surface editing FS , ZSS . A number of multiresolu-
`tion volume hierarchies have been proposed for developing adaptive volume rendering
`and isocontouring Mur , Mur , CDM+ , WV .
`
` . Topology
`
`Field topology refers to the analysis and classication of critical points and computa-
`tion of relationships between the critical points of eld data Del . Computation and
`display of eld topology can provide a compact global view of what is otherwise a very
`large set of data. Techniques such as volume rendering and line integral convolution
`provide qualitative global views of eld topology.
`Vector Topology: Given a continuous vector eld, the locations at which the vector
`becomes zero are called critical points. Analysis of the critical points can determine
`behavior of the vector eld in the local region. In a D vector eld, critical points are
`classied into sources, sinks, and saddles see Figure . a, with both spiral and
`non-spiral cases for sources and sinks HH , HH , GLL . In D, additional critical
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`points include the spiral-saddle HH , which is useful for locating vortex cores, as
`shown in Figure . .
`
`Spiral Source
`
`Saddle
`
`Source
`
`Maxima
`
`Minima
`
`Regular Saddle
`
`Spiral Sink
`
`Center
`
`Sink
`
`Degenerate Saddle
`
`Constant
`
`Figure . a Classication of Vector Field zeroes. b Critical point classication for
`Scalar Field.
`
`Figure .  Two examples of scalar topology of D left and D right scalar elds.
`The D case shows the scalar topology displayed over a color-map of the density in a pion
`collision simulation. The D case is that of the scalar topology diagram of the wave function
`computed for a high potential iron protein.
`
`Scalar Topology: Scalar eld topology can be viewed as a special case of vector eld
`topology, where the vector eld is given by the gradient of the scalar function BS b.
`Critical points in a scalar eld are dened by a zero gradient, and can be classied into
`maxima, minima, saddle points, and degenerate cases, as illustrated in Figure . b.
`Two examples are given in Figure . .
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`Volume Rendering
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` . Functions on surfaces
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` 
`
`Functions on surface visualization deals with the visual display of scalar functions
`whose domain is restricted to an arbitrary geometric surface in three dimensions. The
`surface may be the isosurface of another scalar eld, or simply a geometric domain
`with an associated function eld FLN+ , BOP , BX , BBX .
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`Figure . Visualiztion of functions on surfacea. top-left The electrostatic energy poten-
`tial shown on an isosurface of van der Waals interaction potenial energy. top-right Pressure
`distribution around the earth globe. bottom-left Pressure distribution on the surface of a
`jet engine modeled and displayed using tensor product surface splines. bottom-right Stress
`distribution on a human knee joint based on static loads.
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` E
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`cient Techniques for
`Volume Rendering of Scalar
`Fields
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`DATA VISUALIZATION TECHNIQUES, Edited by C. Bajaj
`c  John Wiley & Sons Ltd
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`Bradium Technologies LLC
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`Exhibit 2002
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` A
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`ccelerated IsoContouring of
`Scalar Fields
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`DATA VISUALIZATION TECHNIQUES, Edited by C. Bajaj
`c  John Wiley & Sons Ltd
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`Bradium Technologies LLC
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`Exhibit 2002
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`urface Interrogation:
`Visualization Techniques for
`Surface Analysis
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`DATA VISUALIZATION TECHNIQUES, Edited by C. Bajaj
`c  John Wiley & Sons Ltd
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`Bradium Technologies LLC
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`Exhibit 2002
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` V
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`ector Field Visualization
`Techniques
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`DATA VISUALIZATION TECHNIQUES, Edited by C. Bajaj
`c  John Wiley & Sons Ltd
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`Bradium Technologies LLC
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` A
`
`pplications of Texture
`Mapping to Volume and Flow
`Visualization
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`DATA VISUALIZATION TECHNIQUES, Edited by C. Bajaj
`c  John Wiley & Sons Ltd
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`Bradium Technologies LLC
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`Exhibit 2002
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` C
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`ontinuous Bayesian Tissue
`Classication for Visualization
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`DATA VISUALIZATION TECHNIQUES, Edited by C. Bajaj
`c  John Wiley & Sons Ltd
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`Bradium Technologies LLC
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`REFERENCES
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`References
`
`Baj
`
`BBX 
`
`C. Bajaj. Geometric modeling with algebraic surfaces. In D. Handscomb, editor,
`The Mathematics of Surfaces III, pages . Oxford Univ. Press, .
`BBB+  J. Bloomenthal, C. L. Bajaj, J. Blinn, M.-P. Cani-Gascuel, A. Rockwood,
`B. Wyvill, and G. Wyvill. Introduction to Implicit Surfaces. Morgan Kaufman
`Publishers, .
`C. L. Bajaj, F. Bernardini, and G. Xu. Automatic reconstruction of surfaces and
`scalar elds from D scans. In R. Cook, editor, SIGGRAPH  Conference Pro-
`ceedings, Annual Conference Series, pages  . ACM SIGGRAPH, Addison
`Wesley, August . held in Los Angeles, California, - August .
`BCL a C. L. Bajaj, E. J. Coyle, and K.-N. Lin. Surface and D triangular meshes from
`planar cross sections.
`In Proc. of the th International Meshing Roundtable,
`number SAND - UC- in Sandia Report, pages   , .
`BCL b C. L. Bajaj, E. J. Coyle, and K.-N. Lin. Arbitrary topology shape reconstruction
`from planar cross sections. Graphical Models and Image Processing, :
` , November .
`BLMP  C. Bajaj, H. Y. Lee, R. Merkert, and V. Pascucci. NURBS based B-rep models
`for macromolecules and their properties.
`In C. Homann and W. Bronsvort,
`editors, Proceedings of the th Symposium on Solid Modeling and Applications,
`pages  , New York, May   . ACM Press.
`R. Barnhill, K. Opitz, and H. Pottmann. Fat surfaces: a trivariate approach
`to triangle-based interpolation on surfaces. Computer Aided Geometric Design,
` :  , .
`J. E. Bowie, editor. Data Visualization in Molecular Science. Addison-Wesley
`Publishing Company, .
`C. L. Bajaj, V. Pascucci, and D. R. Schikore. The contour spectrum. In Pro-
`ceedings of IEEE Visualization ’ , pages   , October .
`C. Bajaj and A. Royappa. Triangulation and Display of Arbitrary Rational Para-
`metric Surfaces. In R. Bergeron, A. Kaufman, editor, Proc. of IEEE Visualization
`’  Conference, .
`C. L. Bajaj and D. R. Schikore. Decimation of d scalar data with error control,
` .
`C. L. Bajaj and D. R. Schikore. Error-bounded reduction of triangle meshes with
`multivariate data. In Proceedings of SPIE Symposium on Visual Data Exploration
`and Analysis III, .
`C. L. Bajaj and D. R. Schikore. Visualization of scalar topology for structural
`enhancement, .
`. Topology preserving data simplication with error bounds, .
`C. Bajaj and G. Xu. Modeling Scattered Function Data on Curved Surface.
`In J. Chen, N. Thalmann, Z. Tang, and D. Thalmann, editor, Fundamentals of
`Computer Graphics, pages   , Beijing, China, .
`CDM+  P. Cignoni, L. De Floriani, C. Montoni, E. Puppo, and R. Scopigno. Multiresolu-
`tion modeling and visualization of volume data based on simplicial complexes. In
`A. Kaufman and W. Krueger, editors,  Symposium on Volume Visualization,
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`DATA VISUALIZATION TECHNIQUES, Edited by C. Bajaj
`c  John Wiley & Sons Ltd
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`Bradium Technologies LLC
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`Exhibit 2002
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