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Eurographics Conference on Visualization (EuroVis) 2013B. Preim, P. Rheingans, and H. Theisel(Guest Editors)

Volume 32 (2013), Number 3

Visualization of Input Parameters for Stream and Pathline

Seeding

ID309

Abstract

Uncertainty arises in all stages of the visualization pipeline. However, the majority of flow visualization applica-

tions convey no uncertainty information to the user. In tools where uncertainty is conveyed, the focus is generally on

data, such as error that stems from numerical methods used to generate a simulation or on uncertainty associated

with mapping visualization primitives to data. Our work is aimed at another source of uncertainty - that associ-ated with user-controlled input parameters. The navigation and stability analysis of user-parameters has received

increasing attention recently. This work presents an investigation of this topic for flow visualization, specifically for

three-dimensional streamline and pathline seeding. From a dynamical systems point of view, seeding can be formu-

lated as a predictability problem based on an initial condition. Small perturbations in the initial value may result in

large changes in the streamline in regions of high unpredictability. Analyzing this predictability quantifies the per-

turbation a trajectory is subjugated to by the flow. In other words, some predictions are less certain than others as

a function of initial conditions. We introduce novel techniques to visualize important user input parameters such as

streamline and pathline seeding position in both space and time, seeding rake position and orientation, and inter-seed

spacing. The implementation is based on a metric which quantifies similarity between stream and pathlines. This is

important for Computational Fluid Dynamics (CFD) engineers as, even with the variety of seeding strategies avail-

able, manual seeding using a rake is ubiquitous. We present methods to quantify and visualize the effects that changes

in user-controlled input parameters have on the resulting stream and pathlines. We also present various visualizations

to help CFD scientists to intuitively and effectively navigate this parameter space. The reaction from a domain expert

in fluid dynamics is also reported.

Categories and Subject Descriptors (according to ACM CCS): I.3.3 [Computer Graphics]: Algorithm/TechniqueSeeding Parameter Sensitivity, Uncertainty, Exploratorive Visualization. Flow Visualisation.

1. Introduction

Visualization of uncertainty is identified as one of the topfuture visualization problems [JEH04], [LK07]. Uncertaintymay be introduced in various stages of the visualizationpipeline (Figure 1). Uncertainty may arise from numerical er-ror and visualization parameter instability. The mapping ofdata to visualization primitives may introduce uncertainty. For

example, the use of piecewise linear interpolation to recon-struct data introduces error. Streamline integrators accumulateerror along the length of their curves. However, discrete prim-itives such as streamlines and iso-surfaces imply certainty.

Another source of uncertainty stems from user-input param-eters. Small changes to input parameters can introduce largechanges to the resulting visualization. In the context of dy-namical systems, a seed position represents an initial condi-tion and the trajectory that a stream or pathline takes can bepredicted [AS92]. However, the reliability or certainty of the

prediction fluctuates. Small sub-pixel perturbations in seedingpositions can sometimes result in unpredictable or uncertainbehavior in the forms of very different integral paths. This isclearly demonstrated by Chen et al., [CML07] [CMLZ08].They identify areas in the seeding domain associated with highuncertainty associated with flow trajectories and take this errorinto account in their visualizations based on Morse Decompo-

sition. In this paper, we visualize the predictability uncertaintyassociated with manual seeding directly in the space-time do-main such that there is a direct coupling of the user-input pa-rameter space and the sensitivity (or reliability) of predictinga stream or pathlines trajectory. We use the term sensitivity todescribe the notion that a small change in user input value cancause a large change to visualization output. In other wordsthere are certain regions of the domain where the user is lesscertain what the effects of varying seeding position are. It isnot clear to what extent the output of a flow visualization tech-nique is affected by the combined effect of the uncertainties

c 2013 The Author(s)Computer Graphics Forum c 2013 The Eurographics Association and Blackwell Publish-ing Ltd. Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ,

UK and 350 Main Street, Malden, MA 02148, USA.


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ID309 / Visualization of Input Parameters for Stream and Pathline Seeding

Figure 1: The visualization pipeline. Each stage in the pipeline contains uncertainty. Our goal is to identify, quantify and visualize

the uncertainty associated with user-input (right) typical for flow visualization and to present it to the user in a helpful and

meaningful way.

introduced in the various stages of the visualization pipeline.Without conveying these uncertainties, any flow visualizationmay be misleading. Interestingly, a lot of focus is placed oninteraction in the visualization literature. However very littleattention is paid to the uncertainty introduced by user-inputparameters and their corresponding threshold values - the laststage of the visualization pipeline. Furthermore, interactionmay be iterative, e.g., visualization results are repeatedly re-fined. Conceptually, this paper advocates a visual map of user-input sensitivity.

We focus on uncertainty introduced and associated withuser-controlled input parameters for flow visualization. Whilethere have been a number of papers examining uncertainty inflow visualization due to the data acquisition and visualiza-tion phases of the pipeline, relatively little research has beeninvested into visualizing the effects and uncertainty due to in-teraction or user-input to flow visualization. In many cases theuser may be unaware of effects due to input parameter stabil-ity. Specifically, we analyze the effect that stream and pathlineseed position and seeding rake position, in both space, timeand in orientation, have upon the resulting set of integral lines.Inter-seed spacing along a seeding rake is also investigated.Our investigation includes unsteady flow, highlighting regionswithin the entire spatio-temporal domain that are sensitive to

stream and pathline seeding position and rake orientation. Thegoals of our investigation are to: 1) Investigate the sensitiv-ity of flow visualization with respect to user-input parametersused for stream and pathline seeding position in both space andtime. 2) Evaluate the stability of user-specified values for rakepositioning and orientation. 3) Provide visualization tools toexplore and investigate the behavior and sensitivity of streamand pathline seeding for different user-controlled input param-eters. As part of our contributions we study and visualize thefollowing:

The sensitivity of stream and pathlines to seeding positionin space and time and seeding rake position and orientation.

A streamline similarity metric, based on FTLE, that can beused to quantify flow sensitivity, i.e., reflect areas where a

small change in position results in large changes to stream-line geometry.

Innovative sensitivity visualizations of inter-seed distancebetween adjacent streamline seeds, resulting in a novelmethod for distributing stream and pathline seeds along arake.

Domain expert feedback to our techniques to visualize theseuser-input parameters and their associated sensitivity.

With existing flow visualization tools, discovering how user-input parameters and threshold values affect the visualization

results requires what essentially amounts to a manual searchthrough parameter space. The space may grow exponentiallywith each new user input option. A manual process is verytime consuming and in some cases may even be impossible.Even if parameters can be modified interactively, users may getlost in their search through parameter space and the space-timedomain. To provide insight into a systems behavior requiresspecific visualization techniques for a range of user input.

The rest of the paper is organized as follows: Section 2 de-scribes previous work. Section 3 details a metric for comput-

ing stream and pathline seeding sensitivity. Section 4 demon-strates our method applied to a range of user-input parametersincluding seeding position, rake position and rake orientation.Section 5 discusses the application of our method to adaptivelyspacing seeds along a seeding rake. Section 6 shows how seed-ing position can be visualized in unsteady flow, highlightinginteresting regions in the spatio-temporal domain. An evalu-ation of performance is conducted in Section 7. Finally, thepaper is concluded in Section 8.

2. Related Work

Figure 1 illustrates the visualization pipeline. Uncertainty isaccumulated at each stage. The goal of this work is to identifyuncertainty associated with the parameters controlled by the

user in the interaction stage (right).2.1. Data Uncertainty and Visualization Mapping

Uncertainty

Wittenbrink et al. [WPL96] investigate the use of glyphs to vi-sualize uncertainty. A variety of glyphs are presented and eval-uated, typically mapping uncertainty to glyph attributes suchas length, area, color and/or angle.

Lodha et al. [LPSW96] present a system for visualizinguncertainty called UFLOW. UFLOW is used to analyze thechanges resulting from different integrators and step-sizesused for computing streamlines. Visualization of the differ-ences between the streamlines is achieved using several meth-ods such as glyphs that encode uncertainty through their shape,

size and color. A novel application of mapping sound to un-certainty is presented by Lodha et al. [LWS96]. Cedilnik andRheingans [CR00] simultaneously visualize the data and itsuncertainty while minimizing the distraction that can occurwhen both are visualized together. A procedural texture in-corporates a grid-like structure which is blended with the im-age. To represent the uncertainty the grid is deformed withthe amount of deformation mapped to the uncertainty withina given region. Rhodes et al. [RLBS03] visualize uncertaintyin volume visualization. They extend the Marching Cubes al-gorithm so that an uncertainty value is associated with each

c 2013 The Author(s)c 2013 The Eurographics Association and Blackwell Publishing Ltd.


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sample. Uncertainty is mapped either to hue or to the opacityvalue of a stipple texture pattern on a secondary surface thatenvelops the iso-surface.

Pang et al. [PWL96] and Verma and Pang [VP04], presentcomparative visualization tools to analyze differences between

two datasets. Streamlines and streamribbons are generated ontwo datasets, one being a sub-sampled version of the other.To compare streamlines, Euclidean distance between them isused. Visualization primitives can then be added to aid theuser in seeing how a pair of streamlines differ. Brown [Bro04]demonstrates the use of vibrations to visualize data uncer-tainty. Experiments using oscillations in vertex displacement,and changes in luminance and hue are investigated. Botchenet al. [BW05] present a method of visualizing uncertainty inflow fields. Texture advection is used to provide a dense vi-sualization of the underlying flow field. Error in texture-basedrepresentations of flow is handled by using a convolution fil-ter to smear out particle traces, modifying the spatial frequen-cies perpendicular to the particle traces. Sanyal et al. [SZD10]

depict uncertainty in numerical weather models using glyphs,ribbons and spaghetti plots. Prani et al. [PRH10] highlightareas of ambiguity and allow the user correct potential mis-classification of volume segmentation. Potter et al. [PKRJ10]present an extension to the traditional box plot in order to por-tray uncertainty information.

2.2. User-input Uncertainty

Brecheisen et al. [BVPtHR09] analyze the stability of user-input parameters used for Diffusion Tensor Imaging fibertracking, specifically the terminating criteria for the fibertracking. Often, the same set of fiber tracking parameters isapplied to several different data sets. Thus, the user may beunaware of what effect the current parameter configuration has

on the current data. They visualize the effects of two commontermination criteria for fiber tracking, namely, the anisotropythreshold and the curvature threshold. This is the inspirationbehind the current work. Berger et al. [BPFG11] introducean interactive approach to continuous analysis of a sampledparameter space. Using this approach users are guided to po-tentially interesting parameter regions. Uncertainty in the pre-diction is visualized using 2D scatterplots and parallel coordi-nates.

Hlawatsch et al. [HSW11] augment mouse pointer seed anddirection based on the largest valued regions of a Finite-TimeLypanov Exponent Field. We note that their study is limited to2D fields only and does not focus on visualization of uncer-tainty itself, but rather controlling mouse interaction.

To the authors knowledge the present work is the only workon the visualization of user-input parameter sensitivity specif-ically for stream and pathline seeding and rake positioning. Infact Brecheisen et al. highlight seeding as a direction for futurework. A recent survey covers the topic of streamline seedingand also uncertainty flow visualization [MLP10]. The differ-ence between our work and the above is that we are the first tofocus on the actual seed positioning and rake input parameters,whereas Brecheisen et al. [BVPtHR09] focus on terminatingcriteria.

2.3. Manual Verses Automatic Streamline Seeding

There have been a number of important papers on automaticstreamline seeding for 3D data. See for example Spencer etal. [SLCZ09] and Li et al. [LS07]. See McLoughlin et al.[MLP10] for a complete overview. However the focus of thework we present does not fall into this category. We classifythese into the visualization mapping stage of the pipeline fromFigure 1. In contrast, our work focuses on the last stage - in-teraction.

Another aspect we emphasize is the importance of man-ual seeding. CFD engineers require manual seeding. Interpret-ing the results of fully-automatic streamline seeding strategieswithout a complete understanding of the underlying algorithmis very difficult. In general, automatic seeding strategies do nottake into account the full range of CFD attributes. Not all fea-tures of interest are known a priori. Thus feature-based stream-line seeding has limitations.

2.4. Finite-Time Lyapunov Exponent

The finite-time Lyapunov exponent (FTLE) [Hal01] has

gained increasing attention of late [PPF10]. It quantifies thelocal of separation behavior of the flow. It is used to mea-sure the rate of separation of infinitesimally close flow trajec-tories. It is computed from the set of all trajectories to pro-duce a flow map, . Once the flow map has been computed,(x0, t0, t) maps the position of the trajectory starting at timet0 from position x0 for a finite-time, t. Using the flow map, theCauchy-Green deformation tensor field, Ctt0 , t is obtained byleft-multiplying the flow map with its transpose [Mas99]:

Ctt0 (x) =

(x0, t0, t)

(x0)

T(x0, t0, t)(x0)

(1)

From this, the FTLE is computed by:

FTLEtt0 (x) = d(x) = 1t t0 ln

max(Ctt0 (x)) (2)

where max(M) is the maximum eigenvalue of matrix M[Hal01].

FTLE requires the choice of a temporal window, the effectthat a change in the time-window length has not been studiedsufficiently [PPF10]. A common use ofFTLE isto extractLa-grangian Coherent Structures (LCS). LCS are extracted froman FTLE field by ridge extraction. This is dependent upon thedefinition of ridges themselves as well as upon the quality ofthe ridge extraction technique employed.

From a visualization point of view, FTLE is used to ex-tract Lagrangian Coherent Structures and belongs to the visu-alization mapping stage of the pipeline (Figure 1). Standardfeature-extraction algorithms involve four important stages:Selection, clustering, attribute calculation, and iconic map-ping [PVH02]. Selection generally involves a search processthrough the data space, e.g., searching for extreme minima ormaxima. Then, filtered data is aggregated or clustered together,e.g., ridges or valleys are identified. Finally, icons are mappedto features, e.g., ridge or valley lines.

The work we present does not involve any explicit search-ing, or clustering and thus does not belong to the class of tra-ditional topology or feature extraction algorithms. Our work



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Figure 2: An overview of our workflow used to visualize

stream and pathline seeding and rake parameter sensitivity.

A dense set of streamlines or pathlines are computed on the

simulation data. This set is utilized to construct the sensitiv-

ity field in both space and time. Various visualization tools are

employed to render the parameter-space based on the sensi-

tivity field.

focuses on the interaction stage and is not concerned with ex-plicit extraction of flow features. The work presented here de-velops a visual map of streamline and pathline seeding param-eter space.

3. A Method for Defining Streamline Seeding SensitivityWe examine the effect that seeding position has on a streamlineby analyzing and quantifying the change in curve geometryand direction when compared to another streamline seeded inclose proximity based on FTLE. We describe spatial and tem-poral metrics for comparing the similarity of a pair of stream-lines. We discuss how the metric is used to create both spatialand temporal sensitivity field AS the key behind real-timenavigation, exploration, and visualization of input parameterspace. Figure 2 shows an overview of the workflow.

3.1. A Stream and Pathline Similarity Metric

The Finite Time Lyapunov Exponent (FTLE) [Hal01] is ourchoice of metric. FTLE quantifies divergence.

FTLE is suitable for the extraction of Lagrangian CoherentStructures (LCS) as stretching of the flow (divergence) oftengrows exponentially in time near LCSs. The LCS structuresare derived by applying topological techniques to the FTLEfield e.g., Ridge extraction [PPF10]. In other words Eq. 2.

The sensitivity field however is not only sensitive to di-vergence. Shear, and differences in curvature of neighboringstreamlines are other characteristics which are inherently cap-tured by this technique.

The choice of similarity metric depends on the interests ofthe user. We chose the FTLE-based metric based on feedbackfrom other flow visualisation experts. However, any streamlineor pathline similarity metric can be used in our framework. It

is left as a user option. A full comparison of different metrics isbeyond the scope of this paper. We refer the reader to Mobertset al. [MVvW05] for a full evaluation.

3.2. Candidate Metrics

We have experimented with various metrics for quantifying in-tegral curve similarity. For example, the closest point distance,dc, could be used: dc(sp,sq) = minpksp,qksq ||pk qk||However dc ignores curvature and shear. The meandistance dm(sp,sq) of closest distances is anothercandidate:dM(sp,sq) = mean(dm(sp,sq),dm(sq,sp))

where dM(sp,sq) = meanpksp minqksq ||pk qk|| This doesnot quantify downstream direction or shear. The Hausdorffdistance, dH could be used: dH = max(dh(sp,sq),dh(sq,sp))where dH(sp,sq) = maxpksp minqksq ||pk qk||However, this does not intuitively quantify down-stream direction and is computationally expensive.

Another candidate is the Frchet distance [EM94]:df(sp,sq) = min{||L|| |L is a coupling between P and Q}However, this metric is computationally expensive and doesnot take shear into account. Interpretation by domain experts isalso non-intuitive. Li et al. [LHS08] also describe a similaritymetric:

dl = 1 1

2

v(p) v(p)

|v(p)| |v(p)| + 1

(3)

where v is an approximate vector at p and v(p) is the originalsample vector. This metric compares vectors, not streamlines.

3.3. Computing a Spatial Sensitivity Field

To facilitate interactive analysis, i.e., reporting parameter sta-bility for a given set of seeding points we provide this infor-mation in real time to the user. We pre-compute a spatial sen-sitivity field that utilizes the similarity metric for a dense setof streamlines. We formulate streamline seeding sensitivity interms of the derived sensitivity field defined over the entirespatio-temporal domain.

For each sample position, pi, within the domain, R3, a

streamline, sp, is computed. For every sp within R3, the sim-ilarity is computed by incorporating each of its neighbors(n = 8 in 2D, n = 26 in 3D) (Figure 3). The total spatial simi-larity value is computed using Eq. 4.

D(p0) =

n

i=0

d(s(p0),s(pi))

n(4)

where, n is the number of neighbors and d(s(p0),s(pi)) is thesimilarity distance between a streamline seeded at p0 and an-other seeded at pi. Figure 3 illustrates this for a 2D field.

4. Visualization of Seeding Position and Rake Parameter

Sensitivity

Our first approach to visualizing the sensitivity field directlyis to derive a scalar field D (Eq. 4). This enables fast, intuitiveexploration of data and provides the user a quick overview ofregions with high rates of streamline-based change. With the

Figure 3: A streamline, s, is seeded at every sample point, pi,

in the domain. In order to compute the sensitivity field for a

given pi, the similarity between s(pi) and its neighbors is com-puted. The mean of the similarity computations quantifies pi(Eq. 4). This example is presented for a 2D field for simplicity.



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Figure 4: FTLE computed from the Bernard flow data with a

streamline length of t = 100, rendered using DVR with identi-cal transfer functions.

spatial sensitivity field stored as a scalar field D, it is possi-ble to visualize the sensitivity of the seed positions using anyvolume visualization technique.

Care must be taken by the user when interpreting the re-sults of the sensitivity field. The initial impression is that thesensitivity field directly maps to features within the flow field.This is not necessarily the case. These regions may be dis-

placed far away from any actual flow features. It highlightsregions from where seeded trajectories exhibit more dissimi-lar behavior. In other word, where small perturbations to theinitial seeding value result in large perturbations to the result-ing trajectories. This is an important difference and may pro-vide the user with more information about the flow than justobserving features alone. Features, by definition, have a dis-tinctive characteristic and frequently, flow trajectories passingthrough them exhibit dissimilar behavior. Therefore, these re-gions may be highlighted by the sensitivity field. In addition,regions where the seed of a trajectory does not originate withina feature, but whose trajectory passes through the feature arealso highlighted. Therefore, the similarity field may not onlyhighlight features but regions connected by the flow to these

features. This may aid the user in understanding the nature ofthe interesting behavior. Figure 6 (left) shows this benefit. Theseeding rake is placed outside of the vortical behavior of theflow and the streamlines enter this region as they are traced.Rake placement is guided by the sensitivity field.

4.1. Visualization of Streamline Seed Position Sensitivity

Using a slice probe provides a fast and simple tool for di-rect investigation of streamline seeding position sensitivity.The slice probe may be arbitrarily aligned. Figure 6 (left) de-picts two slices of a steady Arnold-Beltrami-Childress (ABC)

Figure 5: A modified sensitivity field (Eq. 4 ) computed with a

streamline length of t= 100. The image is rendered using DVRwith a transfer function as in Figure 4.

flow, which describes a closed-form solution of Eulers equa-tion [Hal05]. Color is mapped to the similarity distance D(p).This is true for all color-mapping in this paper unless men-tioned otherwise. The color coding conveys positions that arehighly sensitive with respect to seeding position, i.e., where asmall change in seeding position results in a large change to

streamline geometry. The slice probe enables high levels of in-teraction as it is computationally inexpensive. They are partic-ularly useful when swept through the domain. As in this case,the slice provides context information which may allow theuser to navigate to interesting seeding locations more easily.

For DVR we use the Volume Rendering Engine softwaretool (Voreen) [MSRMH09]. Figure 6 (middle) shows the cor-responding DVR of the left image. Rendering D(p) in this wayprovides an overview of the seeding sensitivity and its charac-teristics. Figure 6 (right) shows a direct volume rendering ofthe sensitivity field of the ABC flow. The green and red re-gions in the center of the image highlight an interesting vortexstructure within the flow. The curvature of streamlines change

the most when seeded in or around the vortex core. The trans-parent regions are less sensitive to streamline seeding position.The geometry of streamlines seeded in these regions changesvery little with seed position.

4.2. Visualization of Seeding Rake Sensitivity

This section introduces techniques used in conjunction withthe sensitivity field to provide parameter sensitivity informa-tion for a seeding rake. We experimented with a number ofdifferent approaches before arriving at the one we found mostsuitable.

Figure 7 (left) depicts a naive attempt at an interactive probeconstructed with a dense collection of rakes at various orienta-tions. Streamlines are seeded from all rakes within the probe.This probe attempts to demonstrate the effect that varying the

Figure 7: Previous experiments visualizing seeding rake pa-

rameters. (Left) A seeding probe with a dense collection of

explicit seeding orientations. This probe produces a cluttered

result, especially when streamlines are seeded from all pos-

sible positions. It also provides the user with little intuition

as to how to orient a rake at the given position. (Middle) Aseeding probe with axis-aligned slices. This probe provides in-

sight into the sensitivity information in the vicinity of the seed-

ing rake, through color mapped to sensitivity. However, it suf-

fers from self-occlusion. (Right) A seeding probe depicted as

a semitransparent sphere. However, this approach only shows

the sensitivity information on the surface - which corresponds

to the rake ends. Therefore sensitivity information across the

length of the rake is lost. The orientation of the rake can be

observed through the use of a transparent surface.



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Figure 6: (Left) Visualizing the sensitivity field of Arnold-Beltrami-Childress (ABC) [Hal05] flow using two interactive slice

probes. Sensitive seeding regions within the field are visualized directly. Streamlines seeded in red regions are very sensitive to

seeding position. Using this approach helps the user locate interesting structures more quickly in the domain than a brute force

manual search. In this example, the user quickly navigates the seeding rake to visualize the structure shown using the slice probes

for context information. (Middle-Left) Streamlines in the Bernard flow simulation. (Middle-Right) A direct volume rendering of the

sensitivity field, D(p), from Bernard flow. As well as highlighting the regions to which streamline seeding is sensitive to change, theDVR provides an overview of interesting seed-based regions within the simulation. (Inset) A top view emphasizing the convection

cell structures within the field. (Right) A direct volume rendering of D(p) of the ABC flow and accompanying transfer function.The interesting seeding regions in the flow are highlighted by high sensitivity values mapped to red and green.

orientation of the seeding rake has on the resulting streamlinesby showing many rakes simultaneously. However, this probemay produce a considerable amount of occlusion as can beseen by the dense rakes and streamlines in the image. It alsoprovides the user with little intuition as to how to orientate aseeding rake at a given position.

The probe shown in Figure 7 (middle) uses three axisaligned slices. The seeding rake is interactively rotated aboutthe center point. This approach reduces occlusion and vi-sual complexity from seeding many streamlines, however, theprobe itself is prone to self-occlusion and may occlude partof the seeding rake and resulting streamlines. An ideal toolorientates the rake automatically at an appropriate angle as afunction of local flow properties. A somewhat better method

utilizes a sphere centered at the seeding rake position. See Fig-ure 7 (right). The sphere provides sensitivity information in alldirections around the seeding rake. Occlusion of the seedingrake and the resulting streamlines is reduced through the useof transparency. However, the sphere provides redundant infor-mation. Portions of the surface that are aligned with the flowfail to provide further useful information.

The method that we favor is the use of a planar polygonalways orthogonal to the flow passing through its center, as inFigure 8. We automatically vary the alignment of the probe andthe seeding rake orthogonal to the local flow field in order toguide the user. There is little benefit from seeding a rake thatis tangential to the flow field. In the case of stream surfacesseeded from an interactive rake, a tangential component of the

seeding rake to the flow should be avoided [ PS09]. Our ap-proach provides the user with guidance on how to orient theirseeding rake AS as it automatically remains in the plane lo-cally orthogonal to the flow (at its center). Also redundant vi-sualization information is reduced using a local cross-sectionof the flow. Color is mapped to sensitivity.

The seeding probe is depicted as a planar polygon withregular samples distributed across a central axis. The seed-ing probe sampling can be set to an arbitrary resolution bythe user but the same resolution as the data by default. Fig-

ure 8 illustrates the effect that a region of high sensitivity canhave on streamline geometry with a small change in seedingposition. The seeding probe shows a highly sensitive region -the red vertical band on the probe. Here, the user is uncertainas to which direction a given, individual streamline may fol-low. The streamlines are seeded at equidistant positions alongthe rake. By comparing neighboring streamlines the effect of asmall change in seeding position becomes apparent. Two dis-tinct groups of streamlines can be seen. The sensitive regionhighlighted by the seeding probe corresponds to the separat-ing region in the flow, where the streamlines flow into eithertoroid of the Lorenz attractor.

5. Adaptive Automatic Interseed Spacing

Our spatial similarity metric allowsus to introduce a novel ideaof placing seed points in a distribution that produces a moreillustrative set of streamlines. Choosing the optimal distancebetween seed points along a seeding rake is a topic that hasreceived relatively little attention. Yet this is a virtually univer-sal parameter for flow visualization using 3D streamlines. Ingeneral, the user selects a seeding rake length and the number

Figure 8: A set of streamlines depicting a Lorenz attractor.

Streamlines are seeded with the seeding rake spanning a sen-

sitive region within the domain. The color-mapping shows

streamlines from the center of the rake have the highest sen-

sitivity values. By examining the streamline geometry we ob-

serve that a large change results from a small change in seed-

ing position within this region.



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Figure 9: (Top row) Dense sets of streamlines and pathlines seeded equidistantly for the visualization of Hurricane Isabel. The

first two images are taken from t0 out of 48 time-steps of the simulation. The images are of streamlines and pathlines respectively.The next two images show streamlines and pathlines seeded at t24 of the simulation. In both pathline cases the pathlines are traced

until tmax = 48. All integral curves are colored according to local velocity magnitude. (Bottom row) Spatio-temporal sensitivityvisualizations of the set of corresponding integral curves in the image above.

of streamlines to be seeded from it. Streamline seeds are thendistributed at equidistant intervals along the seeding curve.However, this may not be the optimal distribution. Choosinginter-seed distance along a rake is done using trial-and-errorin practice. Using our similarity metric it is possible to createan automatic distribution. This is based on the idea of streamsurface refinement [Hul92]. In regions of divergence, stream-lines are inserted into a stream surface in order to improve itsapproximation of the underlying velocity field. Similarly, we

Figure 10: (Left) Streamlines seeded at equidistant intervals

along a seeding rake. The bar graphs shows the individual

streamline similarity values. The large values in the bar graph

represent a dissimilarity between pairs of neighboring stream-

lines. (Middle) The previous set of streamline seeds enhanced

by our method. New streamlines are automatically seeded

along the rake between pairs of streamlines with high dissim-

ilarity. These new streamlines aid the user by visualizing theregions with relatively high streamline variation. A threshold

of50% ofthe maximum value ofD(p) is set. (Right) The result-ing streamlines with a smaller threshold of 30% of the maxi-

mum value of D(p). Again, this produces a more dense set ofstreamlines automatically to provide a more detailed visual-

ization. Note that streamlines are only added where needed. If

the similarity of neighboring streamlines are within the spec-

ified threshold, no insertions take place. This simultaneously

helps reduce visual clutter.

introduce extra streamlines along the seeding rake in regionsof high dissimilarity between neighbors.

For a given rake, we begin by distributing the seeds uni-formly along its length. For each pair of neighboring stream-lines, sp and sq, we compute d(sp,sq). We emphasize this simi-larity information in a bar graph. The streamline pairs are plot-ted along the x-axis and their corresponding similarity valuesare plotted along the y-axis (Figure 10). The range along they-axis is [0..Dmax]. This enables the user to observe the cur-

rent streamline sensitivity values in the context ofD(p). Next,we allow the user to select a threshold, d, as a percentage ofDmax. The similarity values of d(sp,sq) are compared to d.Ifdi > d a new streamline is introduced between sp and sq.Figure 10 demonstrates this. The image on the left shows a setof streamlines distributed evenly across the seeding rake. Notethe bar chart visualization shows a large dissimilarity betweenone pair of streamlines. The next images show seeding distri-butions enhanced using our method. These streamlines fill thelarge gaps present using the original seeding pattern.

6. Visualization of Temporal Seeding Position

An extension of our method is used to visualize the seeding

parameter sensitivity of unsteady flow fields. In these casesregions in which seeding at different times results in largechanges to streamlines or pathlines are highlighted. This hasmultiple applications. Searching the entire space-time domainfor optimal seeding positions can be a daunting task for theuser. Therefore, capturing the essential information in a sin-gle static image or volume is valuable. Also, this scheme maybe used to guide down-sampling a data set temporally. Down-sampling is common due to the large output from modern sim-ulations. It is often desirable to use a down-sampled version ofthe simulation that fits into main memory for fast performance



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needed to support interaction. Using our temporal sensitivityfield, the user can measure the uncertainty/change betweentime-steps i.e., D(p,tn) and D(p, tn+1). This can be used asa guide to prevent excessive down-sampling e.g., if the userwants to down-sample the data and encounters a case where||D(p, tn) D(p, tn+1)|| > D, the user knows that temporal

aliasing may reduce visualization result quality. D is a userdefined threshold to control the quality of the down-sampling.

In order to handle unsteady flow data, a modification ismade to how D(p) is computed. Streamlines or pathlines areagain seeded at every sample point, pi R

3. However, theyare seeded at every pi(tn) for every time-step, tn. Thus for Ntime-steps, there are N streamlines or pathlines seeded at pi .The value of d(p,t) is computed starting with the streamlineseeded at time-step t0 and computing the similarity betweenpi(tn) and pi(tn+1). This is repeated for t T with Eq. 5

D(p,t) =N2

n=0

d(S(p, tn),S(p, tn+1)) (5)

where S(pi,tn) is the streamline or pathline seeded at time-step tn and S(pi, tn+1) is the streamline or pathline seeded atthe time-step tn+1.

Figure 11 shows visualizations of the temporal sensitivityfield, D(p, t), from a simulation of Hurricane Isabel. This visu-alization shows that the regions that are most sensitive to seed-ing over the entire 4D space-time domain. These regions arelocated around the path that the eye of the hurricane traversesand the area around that path. The color-coding is mapped toD(p,t). Figure 12 is a direct volume rendering of the temporalsensitivity field from a flow behind a square cylinder simula-tion [CSBI05]. The region most sensitive throughout the entirespatio-temporal domain is located behind the square cylinder.

7. Evaluation

The computation of the spatial and temporal sensitivity fieldsis a one-time computation. After D(p, t) is derived it can besaved to disk for faster loading. If the user requires fast re-sponse times it is possible for the user to take a progressive

Figure 11: A slice visualization of D(p, t) from a simulationof Hurricane Isabel. This shows the regions with the most

variation to streamline seeding positions over all time steps

0 t tmax. The path of the hurricane eye and the regionsclose to it correspond to the most sensitive areas. The left im-

age shows the field constructed using streamlines. The field in

the right image was constructed using pathlines seeded at t0and traced until tmax = 48.

Figure 12:A DVR visualization of the temporal sensitivity (Eq.

5) field from flow behind a square cylinder [CSBI05]. This

visualization shows the region behind the square cylinder is

the most sensitive over time. The left image shows the spatio-

temporal sensitivity field constructed using streamlines. The

right image shows the entire spatio-temporal field constructed

using pathlines seeded at t0 and traced over until the end of

the simulation at tmax = 102.

approach to the sensitivity field construction. Shorter integralcurvescan be used to create a less accurate approximation. Theintegral curves can be lengthened to improve accuracy. SeeFigure 13 for an example. In a multi-threaded environment this

computation can be performed in the background and the re-sults are refined as they become available. This provides userswith immediate feedback and allows them to quickly start ex-ploring the data.

7.1. Domain Expert Review

This work was carried out in close collaboration with a domainexpert in fluid mechanics (Professor Harry Yeh). We report hisfeedback here. The use of sensitivity fields for the visualiza-tion of input parameter space in the context of flow visual-ization is novel. While the initial computation of the sensitiv-ity field may require some seconds or minutes, the benefitsof fast navigation through parameter space outweigh the one-time pre-processing cost. Finding expressive seeding positions

for streamline rakes can be a laborious and time consumingprocess. The sensitivity field facilitates this process. It is typi-cally carried out manually by trial and error. For example, theslice visualization of the sensitivity field in Figure 6 (left) aidsthe navigation of the rake to capture the main vortical struc-ture. This structure is found more quickly than by performinga brute-force search through the domain. And since automaticfeature detection algorithms cannot be implemented for everytype of feature, general strategies are very helpful.

The use of the seeding probe helps provide confidence to

Figure 13: The iterative derivation of the D(p) of the simula-tion of Bernard flow. This field is computed by extending the

length of the streamlines. The streamline lengths used to cre-

ate the field are 10, 50, 100 and 1000 points. The color-coding

is normalized to streamline length in each image.



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the visualization results. In highly sensitive regions extra caremust be taken to ensure that the rake captures the importantflow characteristics. As a small change in rake position maymake the difference if a certain behavior is captured or not.This leads to the topic of automatic, adaptive inter-seed spac-ing which is of particular interest. Having the rake automati-

cally distribute the seeds frees the user from this burden. Asmentioned previously a small change in position may result instreamlines missing a feature which were present previously.Using the adaptive seeding strategy presents several benefits:(1) The adaptive seeding provides a higher sampling frequencyin the required regions and provides a more detailed visualiza-tion result ultimately providing greater insight into the seed-ing behavior and flow structure. (2) Inserting streamlines onlywhere necessary simplifies the visualization by reducing oc-clusion and visual clutter thus making the results easier to in-terpret with less visual overload. (3) Adaptive seeding allevi-ates the problem arising from a feature being lost as the rakeis moved slightly. The accompanying video shows an exam-ple of an interactive session highlighting this phenomena. This

problem could be alleviated by using equidistant seeding andincreasing the sampling resolution. However, this places theresponsibility of detecting this problem and manually updat-ing the seeding resolution on the user.

The sensitivity field provides additional information aboutthe nature of the flow field which are not communicated usingintegral curves alone. Section 7.2 describes a case study basedon the parameter sensitivity data generated from the simulationof Hurricane Isabel.

7.2. Case Study: Hurricane Isabel

Densely distributed streamlines shown in Figure 9 (top) helpobserve seeding position sensitivity and the flow pattern of the

hurricane. First, the tightly organized flow pattern is shownnear the eye of the hurricane. In the area away from the eye, wesee the different flow pattern in the upper and lower altitudes:the lower atmosphere tends to spiral into the hurricane, whilethe flow spreads out in the upper atmosphere. The streamlineseeding pattern in the later time t = t24 is less coherent thanthe previous one t = t0; this could be a sign of deterioration asthe hurricane approaches the landmass. Unlike the streamlines,the pathlines maintain the same pattern as those from the firsttime (top second) including the preferred influence to/from theupper atmosphere in the northeast side of the hurricane. Morepathlines are present there.

As demonstrated, the developed technique is to provide

guidelines to determine seeding rake and orientation to con-struct effective streamline and/or pathline field visualization.Figures 9 (bottom) and 13 however show an additional appli-cation. Visualization of the spatio-sensitivity field, D(p), andthe spatio-temporal field, D(p, t) in the entire flow domain canbe used for the exploratory flow analysis. The high sensitiv-ity means divergence of stream (or path) lines. Figure 9 (bot-tom left) shows the high sensitivity region of the weather frontalong the Eastern Seaboard, which must be strengthened bythe approaching hurricane. The eye of the hurricane has highsensitivity but that is not the case 24 hours later (see bottom

third). This evolution might reflect the growing hurricane att = t0 , and the subsequent weakening at t = t24. Also, see thelarge arc shaped boundary of the high sensitivity ahead of thehurricane. This represents the boundary of air mass influencedby the hurricane; the mass inside of the boundary flows towardthe hurricane, while the air mass outside of the boundary flows

away from the boundary just like the example of Bernard con-vection cell shown in Fig. 7 (middle). Later at time t = t24, theappearance of the sensitivity field is different from that of theearlier time. Note that no weather front line can be seen here. Itmust be pushed further north. It appears that the hurricane hasbeen somewhat diffused and the center of the hurricane (eye)is no longer the location of high sensitivity.

The sensitivity field D(p,t) in terms of pathlines seeded att= t0 (Fig 9 bottom) also shows the weather front strengthenedby the hurricane, but the location is shifted to the north. Thisis because the front is pushed toward north by the hurricaneas it approaches the coast. Just as the sensitivity field of thestreamlines, the arc shaped boundary is observed ahead of the

hurricane: the boundary separates the air mass from the hurri-cane side to the outside. The distinct curly shape pattern of thehurricane is intriguing, which implies that the hurricane mustform the spiral cell toward the eye. The sensitivity field basedon the pathlines at t = t24 is grossly different from that of ear-lier time t = t0. No more spiral pattern is present; instead, asingle diffused ring of high sensitivity has emerged. This mustbe a sign of weakening.

8. Conclusion

We present methods to quantify and visualize parameters as-sociated with seeding streamlines or pathlines for flow visu-alization. We define an FTLE-based similarity metric which

is utilized to derive a sensitivity field. The derived sensitivityfield D(p,t) quantifies the predictability of flow trajectoriesas a function of initial condition, i.e., user-seeding position.This facilitates real-time interaction using our method. Severalvisualization tools are demonstrated to visualize the sensitiv-ity field in both space and time for seeding rake parameters.We provide a tool highlighting the portions of a seeding rakewhich benefit from a denser sampling of seeds, producing amore expressive set of streamlines that would not be achievedby naively increasing the sampling frequency. We present anextension to time-dependent data which highlights the mostinteresting regions over the lifetime of a simulation. We alsoreport the reaction of a fluid dynamics expert in Sections 7.1and 7.2.

Throughout this paper we have focused on presenting theregions that are most sensitive to change. However, the usermay also want to see the opposite. Regions where flow is ad-vected in a steady fashion are also interesting to domain users.These regions correspond to the regions of low sensitivity andthus steady flow advection.

For future research we plan to extend this technique tostreaklines. One of the central challenges of analyzing thismethod in conjunction with streaklines is finding the optimalmoment to seed integral curves.



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