Multi-dimensional Transfer Functions for

Interactive 3D Flow Visualization

Sung W. Park∗ Brian Budge∗ Lars Linsen∗ Bernd Hamann∗ Kenneth I. Joy∗

Institute for Data Analysis and Visualization (IDAV)†

Department of Computer ScienceUniversity of California, Davis

One Shields Avenue, Davis, CA 95616, U.S.A.

Abstract

Transfer functions are a standard technique used in vol-ume rendering to assign color and opacity to a volumeof a scalar field. Multi-dimensional transfer functions(MDTFs) have proven to be an effective way to extractspecific features with subtle properties. As 3D texture-based methods gain widespread popularity for the visu-alization of steady and unsteady flow field data, there isa need to define and apply similar MDTFs to interac-tive 3D flow visualization. We exploit flow field proper-ties such as velocity, gradient, curl, helicity, and diver-gence using vector calculus methods to define an MDTFthat can be used to extract and track features in a flowfield. We show how the defined MDTF can be applied tointeractive 3D flow visualization by combining them withstate-of-the-art texture-based flow visualization of steadyand unsteady fields. We demonstrate that MDTFs can beused to help alleviate the problem of occlusion, which isone of the main inherent drawbacks of 3D texture-basedflow visualization techniques. In our implementation, wemake use of current graphics hardware to obtain interac-tive frame rates.

1. Introduction

Flow visualization has always been a significant area ofscientific data visualization; it has also been one of themost challenging, especially when looking at volumetricdata. A rich variety of applications attach high impor-tance to the visual exploration of 3D flow fields. Many

∗ {sunpark|bcbudge|llinsen}@ucdavis.edu,{hamann|joy}@cs.ucdavis.edu

† http://graphics.cs.ucdavis.edu

engineering and scientific disciplines including mechan-ical engineering, physics, chemistry, meteorology, geol-ogy, and medicine make use of 3D flow visualization forapplications such as aero and fluid dynamics. The mea-sured or simulated flow field can be static or time vary-ing.

Early attempts to visualize flow field data in-clude hedgehog plots, particle tracing, and stream-lines. Streamlines have been elaborated to streamribbons and stream tubes for steady fields [1] and path-lines, timelines, and streaklines for unsteady fields.Texture-based approaches have gained the most popu-larity recently. This is mostly due to the tremendousprogress in graphics hardware, which is now highlyamenable to texture-based approaches. The new hard-ware supports storage and processing of 2D and 3Dtextures of steadily increasing size as well as pro-grammability, which makes possible the extension ofthe graphics processing unit to general purpose com-putation. In Section 3, we review 3D flow visualizationtechniques and describe the approaches we inte-grated with to our new feature-extraction method.

Volume rendering approaches for 3D scalar fieldshave made use of 3D textures in graphics hardwareeven before hardware-assisted flow visualization meth-ods emerged. In these scalar field volume rendering ap-proaches, color and opacity are assigned to scalar val-ues using transfer functions. The color and opacity as-signments can be further refined and improved by usingmulti-dimensional transfer functions (MDTFs), whichallow for selection and extraction of very distinct fea-tures without having them occluded by features withsimilar, yet slightly different characteristics.

We present MDTFs for 3D flow visualization. Thetransfer functions are based on vector field propertiesderived from vector field calculus. The dimensions of

our transfer functions are given by velocity magnitude,velocity gradient tensor determinant, curl magnitude,helicity, and divergence of 3D flow fields [2]. In Sec-tion 4, we explain and discuss the use of these vectorfield properties, and in Section 5, we combine the de-rived properties to form an MDTF. The MDTF ap-proach can be combined with many 3D flow visual-ization techniques. In Section 6, we describe how ourtransfer function can be applied to GPU-based 3D tex-ture advection flow visualization methods.

Using our MDTF, we can interactively explore 3Dflow fields and extract/select features of well-definedbehavior. Our approach helps to overcome the oc-clusion problems that texture-based flow visualizationtechniques generally suffer from. In Section 7, we showresults of our approach for steady as well as unsteady3D flow fields. In particular, we show how significantfeatures in an unsteady data set can be tracked andhow they evolve over time.

2. Related Work

Our work utilizes transfer functions to aid in featureextraction. Transfer functions are crucial componentsof volume visualization techniques [3, 4, 5]. In recentyears, a large number of improvements have been madeto make transfer functions produce higher quality im-ages as well as making them easier to use.

Bergman et al. [6] described making colormap se-lection an interactive process. The colormaps providedto the user are decided by the rule-based approachdepending on the type of data being visualized. Heet al. [7] and Kindlmann and Durkin [8] introducedimproved methods for generating transfer functions.He’s method generates transfer functions by use of astochastic optimization process. The process relies ona user defined fitness function, or user input in orderto decide the quality of transfer functions. Kindlmann’smethod automatically attempts to search out isovalueswhich might describe material boundaries.

Specifically, our work makes use of multi-dimensional transfer functions, or MDTFs. MDTFscan be a very powerful tool, yet can be trickyto deal with. Kniss et al. [9] described a wid-get based method for dealing with multi-dimensionaltransfer functions. Their method supports inter-active exploration of multivariate volumes. Morrisand Ebert [10] used multi-dimensional transfer func-tions combined with photographic data to gener-ate intuitive colors for volume rendering of medicaldata. Hadwiger et al. [11] applied separate trans-fer functions to each material on a per-fragment

basis in order to obtain color values for each mate-rial.

Our main contribution is the combination of vec-tor field properties with MDTFs to provide useful fea-ture extraction. Feature extraction of scalar datasetshas been of interest to the visualization community fora long time. More recently, work has been done on fea-ture extraction of vector field data. The approach ofVerma et al. [12] finds critical points in order to intelli-gently place seeds for streamline generation in 2D vec-tor fields. The method of Mann and Rockwood [13] cal-culates singularities in 3D vector fields. These singular-ities can be useful in limited situations for classifyingflow fields. Wischgoll and Scheuermann [14] describedan algorithm for locating closed streamlines in 3D vec-tor fields, and Mahrous et al. [15] presented a methodfor topological segmentation of 3D steady flow fields.

While these methods describe geometric separa-tions, it is also possible to extract features of certainproperties. Suzuki et al. [16] derived an “S-map” toassign significance to portions of the volume. The S-map is generated by approximating critical points andusing them to assign importance. Post et al. [17] dis-cussed this technique further and mentioned many pos-sible values to extract including vorticity and helicity.A method by Gray et al. [18] maps curl and diver-gence to colors of an isosurface extracted from veloc-ity magnitude. Our technique extracts features in thesame vein, and we use similar quantities to generateMDTFs that allow useful feature extraction.

3. 3D Flow Visualization

Our method permits incorporation of 3D flow visual-ization algorithms in order to enhance visualization ofthe volume. There are many techniques for visualizing3D flow, with the simplest being direct flow visualiza-tion. Some common examples are the hedgehog plot,and mapping RGB colors to vectors in the field. Theseapproaches tend not to be very helpful in 3D becauseof occlusion issues. Another class of algorithms for flowvisualization is made up of the geometric approaches.These are integration approaches such as streamlines,streaklines, timelines, and pathlines. They are inter-esting from the standpoint that long term informationis presented, and that the information can be sparseand directed at areas of importance (such as throughseeding strategies [12]). Recently, texture-based meth-ods have become very popular, especially for 2D fields,mainly because of their ability to show global infor-mation. The most common examples are line integralconvolution (LIC) [19, 20] and texture advection meth-ods, including image based flow visualization (IBFV),

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along with their many variants. Feature-based flow vi-sualization involves indirectly visualizing flow. Usuallythis involves calculating quantities based upon the vec-tor field data. This paper’s foundation is an example offeature-based flow visualization. For more informationon these methods, see the survey paper by Laramee etal. [21].

Our technique makes possible the combination offeature-based flow visualization with the other typesof flow visualization. In Section 6, we discuss the inte-gration of our method with the GPU-based 3D textureadvection method based on the method by Weiskopfand Ertl [22]. By varying different parameters, we show3D LIC-type images of flow volumes. We also combineour methods with the 3D IBFV approach described byTelea and van Wijk [23] to produce 3D IBFV-type im-ages.

4. Vector Field Calculus

A 3D flow field is defined by a trivariate function. LetF : R

3 → R3 be a function such that

F(x, y, z) =

F1(x, y, z)F2(x, y, z)F3(x, y, z)

,

where F1, F2, and F3 represent the three componentsin the directions of the three coordinates x, y, and z.Typically, the coordinates are Cartesian. In many prac-tical settings, the discrete version of a flow field is ob-tained by measuring or simulating the flow componentsat equidistant integer sample points (i, j, k) in each ofthe three coordinate directions, forming a structuredrectilinear grid. In the following, we derive propertiesfrom vector field calculus as described in [2].

4.1. Velocity Magnitude

A straightforward property of a flow field that has beenused by other visualization approaches is velocity mag-nitude. The velocity magnitude is calculated at a dis-crete position (i, j, k) by the Euclidean norm ‖·‖2, i. e.,

|F(i, j, k)| =

∥

∥

∥

∥

∥

∥

F1(i, j, k)F2(i, j, k)F3(i, j, k)

∥

∥

∥

∥

∥

∥

2

. (1)

The velocity magnitude can be used by our MDTFto distinguish between regions of high and lowflow/velocity.

4.2. Gradient

To detect sudden changes in the flow, one is requiredto consider the velocity gradient of the flow. The gra-

dient field of function F is

∇F(x, y, z) =

(

∂F(x, y, z)

∂x,∂F(x, y, z)

∂y,∂F(x, y, z)

∂z

)

,

(2)which defines a tensor represented by a 3 × 3-matrix.

At a discrete position (i, j, k) we can approximatethe nine entries of the matrix by central differencing,for example,

∂F1(i, j, k)

∂x=

1

2(F1(i + 1, j, k) − F1(i − 1, j, k)) . (3)

Gradient Magnitude

The tensor can be investigated by performing an eige-nanalysis of the matrix. Each eigenvalue of the matrixtells us the stretch factor of the gradient tensor in thedirection of its eigenvector, see Figure 1. For the gen-eration of our MDTF, it would be useful to exploitthis property, e. g., to extract features with high gra-dients. A full eigenanalysis for each discrete vertex lo-cation in our grid is not practical however, especiallywhen dealing with unsteady fields, where all data pro-cessing needs to be done in real time as the data isstreaming in. Instead the product of the three eigen-values of a matrix can be computed by taking the deter-minant of the matrix. Thus, |∇F(x, y, z)| can be usedto classify regions due to the dimension of the gradi-ents. Gradient magnitude is given by the determinant

|∇F(x, y, z)| =

∣

∣

∣

∣

∣

∣

∣

∂F1(x,y,z)∂x

∂F1(x,y,z)∂y

∂F1(x,y,z)∂z

∂F2(x,y,z)∂x

∂F2(x,y,z)∂y

∂F2(x,y,z)∂z

∂F3(x,y,z)∂x

∂F3(x,y,z)∂y

∂F3(x,y,z)∂z

∣

∣

∣

∣

∣

∣

∣

.

(4)

x

y y

xe

2

e1

∆

e1

2e

Figure 1. Influence of velocity gradient tensordetermined by eigenanalysis: Unit circle getsstretched by eigenvalues e1 and e2 along the di-rection of the eigenvectors e1 and e2.

Divergence

Another matrix property that is of interest is the trace

3

of the velocity gradient tensor, i. e., its associated ma-trix, since it is the sum of the eigenvalues. Indeed, thetrace of matrix trace(∇F(x, y, z)) turns out to be thedivergence of the field. The divergence of a vector fieldis a scalar value defined by

div F(x, y, z) =∂F1(x, y, z)

∂x+

∂F2(x, y, z)

∂y+

∂F3(x, y, z)

∂z.

(5)The divergence measures the rate of expansion per vol-ume unit, i. e., the difference in inflow and outflow perunit, see Figure 2. Divergence is positive for expand-ing and negative for compressing flow fields.

Figure 2. Divergence measures the difference ininflow and outflow per unit.

Curl Magnitude

The curl of a vector field is a measure for the vectorfield’s vorticity (degree of turbulent behavior). It is de-fined by the vector

curl F(x, y, z) =

∂F3(x,y,z)∂y

− ∂F2(x,y,z)∂z

∂F1(x,y,z)∂z

− ∂F3(x,y,z)∂x

∂F2(x,y,z)∂x

− ∂F1(x,y,z)∂y

. (6)

We can exploit the magnitude of the curl|curl F(i, j, k)| at each discrete grid location forour multi-dimensional transfer function to extractswirling features.

Helicity

Also of interest is the curl in the direction of the veloc-ity of a flow field, which is called the helicity, illustratedin Figure 3. The helicity is a scalar function and canbe computed as the dot product of the curl and the ve-locity:

heli F(x, y, z) = curl F(x, y, z) · F(x, y, z) . (7)

Fcurl

F

Figure 3. Helicity is curl in direction of velocityF, while curl measures vorticity.

5. MDTFs

Transfer functions provide a means to selectively visu-alize different aspects of a volume by defining a func-tion T : R

N → RM , where usually N = 1 (the data is

scalar), and M = 4 (the function is usually RGBA val-ued, where RGB is color and A is opacity). In the vectorfield context, vector/velocity magnitude has been mostcommonly used for the definition of a one-dimensionaltransfer function. To enhance the capability to extractmore subtle features, different properties of a vectorfield can be incorporated generating an N -dimensionaltransfer function to represent each of the N proper-ties. Characteristic properties of a vector field are thosementioned in the previous section, i. e., velocity mag-nitude, gradient tensor determinant, curl magnitude,helicity and divergence. The combination of these fivescalar values leads to a five-dimensional transfer func-tion.

In order to define an MDTF, we map each of theN scalar magnitudes to color channels R, G, and B

and opacity values α by using one-dimensional trans-fer functions Ti : R → R

4, i = 1, . . . , N . We com-bine the one-dimensional transfer functions Ti to anN -dimensional transfer function T : R

N → R4, ac-

cording to the equations

TRGB(x) =1

N

N∑

i=1

Ti,RGB(xi) · Ti,α(xi)

and

Tα(x) =1

N

N∑

i=1

Ti,α(xi) ,

where x = (x1, . . . , xN ) ∈ RN . The color values Ti,RGB

assigned by the individual one-dimensional transferfunctions are averaged in a weighted fashion to de-fine the color TRGB assigned by the MDTF, wherethe weights are given by the individual opacities Ti,α.The opacity values Ti,α assigned by the individual one-dimensional transfer functions are averaged as well, to

4

define the opacity Tα assigned by the MDTF. T al-lows one to visualize each component distinctly, butalso makes possible blending of the values in overlap-ping regions. If extracted features appear to be toodark, one can increase the intensity by multiplying theopacity with a constant factor.

The user interface to manipulate our MDTF isshown in Figure 4(a). For each of the five magnitudesfrom Section 4, we provide the user with a bar that al-lows the user to pick and color the interesting parts ofeach property. For example in the third bar of Figure4(a), we select regions of high curl magnitude and as-sign to them an orange/red color, while in the fifth barof Figure 4(a) we distinguish between regions of highnegative and high positive divergence by assigning tothem the colors blue and white, respectively. The col-ors and opacities selected using the individual bars arecombined to an MDTF as described above.

6. Implementation

We incorporated MDTFs into a GPU-based 3D tex-ture advection application in C++ using NVIDIA’sGeForce FX GPU. The GPU-based advection and ren-dering system we used for steady and unsteady flow issimilar to the one presented by Weiskopf and Ertl [22].

The 3D texture advection visualization goes throughthree basic steps: texture advection, texture blending,and volume rendering.

For texture advection, 3D volume is advected a sliceat a time, using the first-order Eulerian scheme

x(t − ∆t) = x(t) − ∆tF(x(t), t) ,

where F(x(t), t) is our vector field evaluated at posi-tion x(t) at time t. The integration step size is ∆t.

For texture injection and blending, we use the ex-tended blending equation

Tt = Wt · Tt−∆t + Vt · It ,

where the previously advected texture Tt−∆t and theinjection texture It are multiplied by multi-componentweights Wt and Vt to produce the newly advected tex-ture Tt. In our implementation, we tested two differentsettings for noise texture blending. In one case we per-formed an affine combination of values from the previ-ously advected texture to produce a fully opaque LICtype volume. In the other case, we used space-variantscalar injection weights as described in 3D IBFV [23].

For each fragment rendered, properties of the fieldare found using Equation (1) and approximating Equa-tions (4) – (7) by using central differencing as in Equa-tion (3). These properties are then evaluated by theMDTF in Equations (5) and (5).

To render the final image, a direct 3D texture-basedvolume rendering is used, by rendering view-alignedslices.

7. Results

We tested our approach for steady and unsteadydatasets. For steady flow data, we examined a tor-nado dataset [24] of size 1283. For the unsteady case,we examined the CFD simulation of five jets consist-ing of 2000 timesteps of 1283 vector field data1.

Figure 6 shows the components we examined: ve-locity magnitude, determinant of the gradient tensor,magnitude of curl, helicity, and divergence, respec-tively. Solid opaque surfaces are rendered without tex-ture advection to better show the structure of the prop-erties. Figure 6(a) shows regions of constant velocitymagnitude. Figure 6(b) shows high absolute values forthe gradient tensor determinant. Figure 6(c) shows aregion of high curl magnitude, which yields the struc-ture of the vortex core of the tornado. The helical mo-tion of the tornado is shown in Figure 6(d). In Figure6(e), the blue regions are regions of high negative di-vergence, where flow is converging, whereas the whiteregions are regions of high positive divergence, whereflow is diverging. The last image shows the full MDTF,as described in Equations (5) and (5) with N = 5. Theused color and opacity assignments are shown in Fig-ure 4(a).

Figure 5 shows still images from a real-time anima-tion achieved from GPU-based texture advection meth-ods using MDTFs to highlight features of interest. Theparameters used create LIC-type visualizations. Thefirst two images show the tornado data using differ-ent MDTFs. The third image shows the five-jet data ata fixed timestep. When animated with texture advec-tion methods, these surfaces not only show interestingstructures corresponding to the MDTF, but also showthe direction and the movement of the vector field onthe surfaces as well.

In comparison to the LIC-type visualization of vol-umes, we also generated animations combining 3DIBFV with MDTFs. Figure 4(b) shows the results whenapplied to the tornado dataset. We highlight helicity,magnitude, and divergence in the image.

We also show images from three different timestepsof the five-jet dataset with same MDTFs, highlightingproperties of interest in Figure 7. The blue and whiteregions specify areas of high divergence with negativeand positive sign, respectively. Red regions highlights

1 Dataset courtesy of Kwan-Liu Ma, IDAV, University of Cali-fornia, Davis

5

high curl, yellow regions highlight high gradient deter-minant, and green regions show the helical behavior.2

8. Conclusions and Future Work

Vector field visualization continues to be an importantarea of active research. It is important not only to seewhere a particular part of the field advects from one in-stant to the next, but also to visualize globally portionsof the field that have certain behaviors.

We have used various vector calculus quantities thatare important for understanding the flow, and we havediscussed methods for visualizing these behaviors byusing MDTFs. This is beneficial, because it allows usto extract and track features for 3D flow fields. TheMDTFs alleviate part of the occlusion problem, whichhas been the major drawback of 3D flow visualizationtechniques.

In our current work, we have integrated our featureextraction MDTFs with GPU-based 3D texture advec-tion methods to generate LIC-type images of volumesas well as 3D IBFV-type images of volumes. While theysometimes can add clarity to the visualization, they cansometimes distract from the feature extraction visual-ization. This is mainly due to still remaining occlud-ing properties of dense vector field visualization tech-niques.

In future work we may explore integrating geomet-ric techniques with our MDTF method with the hopethat the occlusion problems can be avoided even fur-ther, while gaining valuable insight from them.

Acknowledgments

This work was supported by the National ScienceFoundation under contract ACI 9624034 (CAREERAward), through the Large Scientific and SoftwareData Set Visualization (LSSDSV) program under con-tract ACI 9982251, through the National Partnershipfor Advanced Computational Infrastructure (NPACI)and a large Information Technology Research (ITR)grant; the National Institutes of Health under con-tract P20 MH60975-06A2, funded by the National In-stitute of Mental Health and the National ScienceFoundation; by a United States Department of Educa-tion Government Assistance in Areas of National Need(DOE-GAANN) grant #P200A980307; and through aHewlett-Packard contribution to a Graduate StudentFellowship. We thank the members of the Visualiza-tion and Graphics Research Group at the Institute for

2 The full animation is available as an AVI movie, seehttp://graphics.cs.ucdavis.edu/∼jahsik59/mdtf.avi

Data Analysis and Visualization (IDAV) at the Uni-versity of California, Davis.

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(a) (b)

Figure 4. (a) User interface for an MDTF combining five components: velocity magnitude, determinant ofvelocity gradient tensor, curl magnitude, helicity, and divergence (top to bottom). (b) 3D IBFV methodcombined with MDTF, applied to tornado dataset.

(a) (b) (c)

Figure 5. Texture advectionmethod combinedwithMDTF: (a) highlighting velocitymagnitude, divergence,and curl magnitude, applied to tornado dataset; (b) highlighting helicity, divergence, and curl magnitude,applied to tornado dataset; and (c) highlighting velocitymagnitude, divergence, and curlmagnitude, appliedto timestep 1440 of five-jet dataset.

8

(a) (b) (c)

(d) (e) (f)

Figure 6. Feature extraction from tornado dataset selecting regions of (a) constant velocity, (b) high gra-dient tensor determinant, (c) high curl magnitude, (d) distinct helicity, (e) high negative (blue) and highpositive (white) divergence, and (f) their combination using MDTF.

(a) (b) (c)

Figure 7. Rendering of five-jet dataset at timesteps (a) 1440, (b) 1760, and (c) 1960, with MDTF showinghigh curl magnitude (red), high determinant of velocity gradient tensor (yellow), high positive divergence(white) and high negative divergence (blue), and a feature of helicity (green).

9