Topology-based Transfer Function DesignGunther H. Weber, Gerik Scheuermann

AG Graphische Datenverarbeitung und Computergeometrie, FB Informatik,University of Kaiserslautern, Erwin-Schrodinger-Straße, 67663 Kaiserslautern, Germany

Abstract

Direct Volume Rendering (DVR) is commonly used to vi-sualize scalar fields. Quality and significance of renderedimages depend on the choice of an appropriate transferfunction that assigns optical properties (e.g., color andopacity) to scalar values. We present a method that au-tomatically generates a transfer function based on the topo-logical behavior of a scalar field. We find a set of criti-cal isovalues for which the topology of an isosurface,i.e.,a surface representing all locations where the scalar fieldassumes a certain valuev, changes. Then we generatea transfer function that emphasizes scalar values aroundthose critical isovalues. Images rendered using the result-ing transfer function reveal the fundamental topologicalstructure of a scalar data set.

1. Introduction

Direct Volume Rendering visualizes a three-dimensional(3D) scalar field by using atransfer function to mapscalar values to optical properties (e.g., color and opac-ity) and rendering the resulting image. This transfer func-tion presents a user with an additional parameter that in-fluences a resulting visualization. However, quality andsignificance of the resulting visualization hinge on a sen-sible choice of the transfer function. Transfer functions arecommonly determined manually by trial and error whichis time-consuming and prone to errors. Several attemptswere made to analyze a data set and generate appropriatetransfer functions automatically to aid a user in the visu-alization process. Pfisteret al. [9] give an overview overseveral techniques and compare results with manually cho-sen transfer functions.

Apart from DVR, isosurfaces are most commonlyused to visualize scalar fieldsf(x, y, z). An isosurface rep-resents all locations in 3D space, wheref assumes a givenisovaluev, i.e., wheref = v holds. By varying the iso-value v, it is possible to visualize the entire scalar field.Like choosing appropriate transfer functions, determiningisovalues where “interesting” isosurface behavior occurs isdifficult. Weberet al. [10] have considered the topologi-cal properties of a piecewise trilinear scalar field to deter-mine for which isovalues relevant isosurface behavior oc-curs. All fundamental changes are tracked: Closed sur-face components emerge or vanish at local minima or max-ima, and thegenus of an isosurface changes,i.e., holes ap-

pear/disappear in a surface component, or disjoint surfacecomponents merge at saddles. Values and locations wheresuch changes occur are determined and used to aid a userin data exploration.

Instead of using the resulting set of critical isovaluesas indicator for which isovalues expressive isosurfaces re-sult, it is possible to use them to construct a transfer func-tion that highlights topological properties of a scalar dataset. DVR commonly uses trilinear interpolation commonlywithin cells. Thus, critical isovalues extracted by Weberetal. [10] are also meaningful in a volume rendering context.We use the resulting list of critical isovalues to design trans-fer functions based on the work presented by Fujishiroetal. [3, 5]. We generate transfer functions that assign smallopacity to all scalar values except those close to critical iso-values. Colors are assigned using an HLS color model andvarying the hue component for different scalar values suchthat it changes more rapidly close to critical isovalues.

2. Related Work

Few authors utilize topological analysis for scalar field vi-sualization. Bajajet al. [2] developed a technique tovisualize topology to enhance visualizations of trivariatescalar fields. Their method employs aC1-continuous in-terpolation scheme for rectilinear grids, and it detects crit-ical points of a scalar field,i.e., points where the gradientof the scalar field vanishes. Subsequently, integral curves(tangent curves) are traced starting from locations close tosaddle points. These integral curves are superimposed ontovolume-rendered images to convey structural informationof the scalar field.

Fujishiroet al. [3] used ahyper-Reeb graph for ex-ploration of scalar fields. A Reeb graph encodes topologyof a surface. The hyper-Reeb graph encodes changes oftopology in an extracted isosurface. For each isovalue thatcorresponds to an isosurface topology change, a node ex-ists in the hyper-Reeb graph containing a Reeb graph en-coding the topology of that isosurface. Fujishiroet al. [3]construct a hyper-Reeb graph using “focusing with inter-val volumes,” an iterative approach that finds a subset of allcritical isovalues , which had been introduced by Fujishiroand Takeshima [4]. The hyper-Reeb graph can be used,for example, for automatic generation of transfer functions.Fujishiro et al. [5] extended this work and used a hyper-Reeb graph for exploration of volume data. In addition toautomatic transfer function design, their extended method

1

allows them to generate translucent isosurfaces betweencritical isovalues.

Critical point behavior is also important in the con-text of data simplification to preserve important features ofa data set. Bajaj and Schikore [1] extended previous meth-ods to develop a compression scheme preserving topolog-ical features. Their approach detects critical points of apiecewise linear bivariate scalar fieldf(x, y). In their ap-proach, “critical vertices” are those vertices for which the“normal space” of the surrounding triangle platelet con-tains the vector(0, 0, 1). Integral curves are computed bytracing edges of triangles along a “ridge” or “channel.” Ba-jaj and Schikore’s method incorporates an error measureand can be used for topology-preserving mesh simplifica-tion.

Gerstner and Pajarola [6] defined a bisection schemethat enumerates all grid points of a rectilinear grid in a tetra-hedral hierarchy. Using piecewise linear interpolation intetrahedra, critical points can be detected. Data sets aresimplified by specifying a traversal scheme that descendsonly as deep into the tetrahedral hierarchy as necessaryto preserve topology within a certain error bound. Thismethod incorporates heuristics that assign importance val-ues to topological features, enabling a controlled topologysimplification.

Kraus and Ertl [7] considered rectilinear data sets andpiecewise trilinear interpolation. Their technique partitionsrectilinear cells into tetrahedra and uses critical points oflinear interpolation to guide them in sub-sampling schemewhen attempting to preserve topology.

Weberet al. [10] use critical point analysis to help auser in exploring a scalar data set using isosurfaces. By pre-senting a user with a list of critical isovalues of a piecewisetrilinear scalar field they provide important “navigationalaids” for the visualization process. A user can explore adata set by examining isosurface behavior close to criticalisovalues.

Pfister et al. [9] review recent approaches to au-tomated transfer function design. They compare transferfunctions resulting from a variety of automated schemes tomanually chosen transfer functions.

3. Detecting Critical Isovalues

Our goal is to detectcritical isovalues of a piecewise tri-linear scalar field. For aC1-continuous functionf , criti-cal points occur where the gradient∇f assumes a value ofzero,i.e.,∇f = 0. The type of a critical point can be deter-mined by the signs of the eigenvalues of the Jacobian off .Piecewise trilinear interpolation when applied to rectilineargrids, in general, produces onlyC0-continuous functions.Therefore, we must define critical points differently.

Gerstner and Pajarola [6] considered piecewise lin-ear interpolation applied to tetrahedral grids, which alsoleads toC0-continuous functions. Considering piecewiselinear interpolation, critical points can only occur at meshvertices. Gerstner and Pajarola’s method classifies a mesh

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

Figure 1. (a) Around a regular pointx ∈ R3, the isosurfaceF−1(F (x)) divides space into a single connected volumeP with F > 0 (red) and a single connected volumeN withF < 0 (blue). (b) Around a minimum, all points inUhave a larger value thanF (x). (c) Around a maximum, allpoints inU have a smaller value thanF (x). (d) In case of asaddle, there is more than one separated region with valueslarger or smaller than the valueF (x).

vertex depending on its relationship with vertices in a lo-cal neighborhood. In the context of a refinement scheme,all tetrahedra sharing an edge that is to be collapsed de-fine a “surrounding polyhedron.” Vertices of this surround-ing polyhedron constitute the considered neighborhood ofa vertex. These vertices are marked with a “+” if their as-sociated function values are greater than the value of theclassified vertex; or they are marked with a “-” if their as-sociated function values are less than the value of the clas-sified vertex. Equal values are not considered. Edges ofthe surrounding polyhedron define an edge graph. In thisgraph, all edges connecting vertices of different polaritiesare deleted. A vertex is classified according to the numberof connected components in the remaining graph. If thisnumber is one, the classified vertex is a maximum or mini-mum (depending on the sign of the connected component).If it is two, the classified vertex is a regular point. Other-wise, the vertex is a saddle point. Connected componentsin an edge graph of a surrounding polyhedron correspondto connected components in a neighborhood of a vertex.This observation leads us to the following definition:

Definition 1 (Regular and Critical Points) LetF : Rd → R, d ≥ 2 be a continuous function. Apoint x ∈ Rd is called a (a) regular point, (b) minimum,(c) maximum, (d) saddle, or (e) flat point ofF , if forall ε > 0 there exists a neighborhoodU ⊂ Uε withthe following properties: If

⋃np

i=1Pi is a partition of thepreimage of [F (s),+∞) in U − {x} into connected

components and⋃nn

j=jNj is a partition of the preimage of(−∞, F (s))] in U − {x} into connected components, then(a) np = nn = 1 andP1 6= N1, (b) np = 1 andnn = 0,(c) nn = 1 andnp = 0, (d) np + nn > 2, or (e)np = nn = 1 andP1 = N1.

Remark 1 (a) – (d) See Figure 1 (e) All points inU havethe same value asF (x). It is possible to extend the con-

cept of being critical to entire regions and classify regionsrather than specific locations.

Remark 2 The casesnp = 2, nn = 0 andnp = 0, nn = 2are not possible ford ≥ 2.

We consider piecewise trilinear interpolation, whichreduces to bilinear interpolation on cell faces and to lin-ear interpolation along cell edges. All values that trilinearinterpolation assigns to positions in a cell lie between theminimal and maximal function values at the cell’s vertices(convex hull property). In fact, maxima and minima canonly occur at cell vertices. If two vertices connected byan edge have the same function value, the entire edge canrepresent an extremum or a saddle. It is even possible thata polyline defined by multiple edges in the grid, or a re-gion consisting of several cells, becomes critical. In thesecases, it is no longer possible to determine, locally, whethera function value is a critical isovalue. To avoid these typesof problem, we impose the restriction on a mesh that func-tion values at vertices connected by an edge must differ.Saddles can occur at cell vertices, on cell faces of a cell,and in a cell’s interior, but not on cell edges. This fact isdue to the restriction that an edge cannot have one constantfunction value.

A vertex can be classified by considering its edge-connected neighbor vertices. Analysis of the behavior oftrilinear interpolation, see [10],

Lemma 1 (Linear cell partition) Consider a cellC forwhich v := v0 6= v1, v2 6= v4. Then, for allε > 0 thereexists aδ < ε such that for the intersectionR of Uδ andCthe following statements hold: (a) Ifv > max{v1, v2, v4}thennn = 1 and N1 = R, i.e., all values in the regionare less thanv. (b) If there existi, j, k ∈ {1, 2, 4}, i 6=j 6= k, i 6= k, such thatv > max{vi, vj} andv < vk, thennn = np = 1 andR completely contains a surface dividingN1 andP1. Furthermore, all values on the trianglep0pipj

are less thanv. (c) If there existi, j, k ∈ {1, 2, 4}, i 6= j 6=k, i 6= k, such thatv < min{vi, vj} and v > vk, thennn = np = 1, and R completely contains a surface di-viding N1 andP1. Furthermore, all values on the trianglep0pipj are less thanv. (d) If v < max{v1, v2, v4}, thennn = 1 and N1 = R, i.e., all values in the region aregreater thanv.

Using theL1-norm1, the intersection of a neighborhoodwith a cell corresponds to a tetrahedron. According toLemma 1, this tetrahedron is partitioned in the same wayas a tetrahedron using linear interpolation (even when, as inour case, partitioning surfaces are not necessarily planar),see Figure 2. A vertex can be classified by consideringits edge-connected neighbor vertices. We treat these ver-tices as part of a local implicit tetrahedrization surround-ing a classified vertex, where the classified vertex and threeedge-connected vertices belonging to the same rectilinearcell imply a tetrahedron, see Figure 3.

1‖x‖1 =∑

i |xi|

v

(a)

v

(b)

v

(c)

v

(d)

Figure 2. When a small neighborhood is considered, a“tetrahedral region” havingv as a corner is partitioned inthe same way as a linear tetrahedron.

Figure 3. Edge-connected vertices as part of an implicittetrahedrization.

When applying Gerstner and Pajarola’s criterion [6]for connected components in an edge graph for the result-ing implicit tetrahedrization, we obtain a case table with26 = 64 entries that maps a configuration of “+” and “-” of edge-connected vertices to a vertex classification. (Itcan be shown that the connected components in an edgegraph correspond to connected components in a neighbor-hood.) We decided to generate this relatively small casetable manually.

On boundary faces trilinear interpolation reduces tobilinear interpolation. A bilinear interpolant can have asaddle point. This saddle is not necessarily a critical valueof the scalar field defined by piecewise trilinear interpo-lation. Using the following criterion we can determinewhether a face saddle is a saddle of piecewise trilinear in-terpolation.

Lemma 2 (Face saddle)Let p be a point on the sharedface of two cells, where the trilinear interpolant degenerateto the same bilinear interpolant. The pointp is a saddlepoint, when these two statements hold:

1. The pointp is a saddle point of the bilinear inter-

B−1

A1

D−1D1

B1

A−1

C1C−1y

zx

A

B

D

C

Figure 4. Vertex numbering scheme used in Lemma 2.

polant defined on the face.

2. With the notations of Figure 4, where, without loss ofgenerality, cells are rotated such thatA andC are thevalues on the shared cell face having a value largerthan the saddle value,C(A1 − A) + A(C1 − C) −D(B1−B)−B(D1−D) andC(A−1−A)+A(C−1−C)−D(B−1−B)−B(D−1−D) have the same sign.

Otherwise,p is a regular point of the trilinear interpolant.

We can detect face saddles of piecewise trilinear inter-polants effectively by considering all cell faces for a saddleof the bilinear interpolants on faces and checking whetherthe criterion stated in Lemma 2 holds.

Saddles of the trilinear interpolant in the interior of acell are also saddles of the piecewise trilinear interpolant.We compute these saddles by using the equations givenby Natarajan [8]. Inner saddles of a trilinear interpolantthat coincide with a cell’s boundary faces or vertices arenot necessarily saddles of a piecewise trilinear interpolant.Trilinear interpolation assigns constant values to locationsalong coordinate-axis-parallel lines passing through thesaddle. We currently rule out the possibility of an inter-nal saddle coinciding with a vertex or an edge. Otherwise,our requirement that edge-connected vertices differ in valuewould be violated. (Saddles of trilinear interpolants thatcoincide with cell faces are discussed in Lemma 2.)

4. Generating Transfer Functions

dmax

dmin

smin cv 0 smax

δh

cv n

Scalar Value

Hue

smin cv 0 smaxcv n

δo

ωo

α

Scalar Value

Opacity

Figure 5. Transfer function emphasizing topologicallyequivalent regions.

Given a list of critical isovalues we construct a corre-sponding transfer function based on the methods describedby Fujishiroet al. [5]. The domain of the transfer func-tion corresponds to the range of scalar values[smin, smax]occurring in a data set. Outside this range the transfer func-tion is undefined. Given a list of critical isovaluescvi, weeither construct a transfer function emphasizing volumescontaining topologically equivalent isosurfaces or a trans-fer function emphasizing structures close to critical values.

Figure 5 shows the construction of a transfer functionthat emphasizes topological equivalent regions. The colortransfer is chosen such that hue uniformly decreases withthe mapped value, except for a constant drop ofδh at eachcritical valuecvi. The opacity is constant for all valuesexcept for hat-like elevations around each critical valuecvi

having a width ofωo and a heightδo.

dmax

dmin

smin cv 0 smaxcv n

δh

hω

Scalar Value

Hue

smin cv 0 smaxcv n

δo

ωo

α

Scalar Value

Opacity

Figure 6. Transfer function emphasizing details close tocritical isovalues.

Figure 6 shows the construction of a transfer func-tion emphasizing details close to critical isovalues. The huetransfer function is constant except for linear descents of afixed amountδh within an interval with a widthωh centeredaround each critical isovaluecvi. The opacity is constantfor all values except in intervals with a widthωo centeredaround critical isovaluescvi where the opacity is elevatedby δo.

If several isovalues are so close together that intervalswith a width ωh or omegao would overlap, all isovaluesexcept the first are discarded to avoid high frequencies inthe transfer function that could cause aliasing artifacts inthe rendered image.

(a) Transfer function emphasizing topologically equivalent zones.

(b) Transfer function emphasizing structures close to critical isovalues.

Figure 7. “Nucleon” data set. Data set courtesy of SFB 382 of the German Research Council (DFG), seehttp://www.volvis.org .

5. Results

Figure 7 shows a data set obtained by simulating a two-body distribution probability of a nucleon in the atomic nu-cleus “16O” when a second nucleon is known to be po-sitioned atr′ = (2fm, 0, 0). This 413 data set is cour-tesy of the Sonderforschungsbereich (SFB) 382 of the Ger-man Research Council (DFG). It can be obtained athttp://www.volvis.org . Figure 7(a) emphasizes volumescontaining topologically equivalent isosurfaces. Detailsclose to these critical isovalues are better visible in Figure7(b).

6. Conclusions and Future Work

We have presented a method for the detection critical iso-values and transfer function design based on the resultingcritical isovalues. Resulting volume rendered images showthe topological structure of a scalar field and are an invalu-able help in examining volumetric data. Improvements toour method are possible. For example, it would be helpfulto eliminate the requirement that values at edge-connectedvertices of a rectilinear grid must differ. It is necessary toextend our mathematical framework and add the concept of“critical regions” and “polylines.” Considering the case ofa properly sampled implicitly defined torus, its minimumconsists of a closed polyline around which the torus ap-pears. Similar regions of a constant value can exist thatare extrema. These extensions will require us to considervalues in a larger region; and they cannot be implementedin a purely local approach. Some data sets contain a largenumber of critical points. Some of these critical points cor-respond to locations/regions of actual interest, but some arethe result of noise or improper sampling. We need to de-velop methods to eliminate such “false” critical points.

7. Acknowledgments

We thank the members of the AG Graphische Datenver-arbeitung und Computergeometrie at the Department ofComputer Science at the University of Kaiserslautern andthe Visualization Group at the Center for Image Process-ing and Integrated Computing (CIPIC) at the University ofCalifornia, Davis.

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