(IEEE Transactions on Visualization
and Computer Graphics, Vol. 12, No. 5  Special Issue on IEEE Vis'06,
2006, pp. 965972)

Zhanping Liu

Robert J. Moorhead II

Joe Groner

Abstract: This paper presents an advanced evenlyspaced streamline placement algorithm for fast, highquality, and robust layout of flow lines. A fourthorder RungeKutta integrator with adaptive step size and error control is employed for rapid accurate streamline advection. Cubic Hermite polynomial interpolation with large samplespacing is adopted to create fewer evenlyspaced samples along each streamline to reduce the amount of distance checking. We propose two methods to enhance placement quality. Double queues are used to prioritize topological seeding and to favor long streamlines to minimize discontinuities. Adaptive distance control based on the local flow variance is explored to reduce cavities. Furthermore, we propose a universal, effective, fast, and robust loop detection strategy to address closed and spiraling streamlines. Our algorithm is an orderofmagnitude faster than Jobard and Lefer's algorithm with better placement quality and over 5 times faster than Mebarki et al.'s algorithm with comparable placement quality, but with a more robust solution to loop detection. Index Terms: Flow visualization, evenlyspaced streamlines, streamline placement, seeding strategy, closed streamlines. 
One problem with the use of streamlines is that an incomplete view (Fig. 1) or a cluttered display (Fig. 2) of the flow tends to be obtained unless an effective placement mechanism is adopted. A layout of evenly spaced (or uniform) streamlines may provide a visually pleasing as well as informative pattern to facilitate mental reconstruction of the flow. Our ADVESS (ADVanced Evenly Spaced Streamline placement) algorithm allows for fast highquality evenly spaced streamline placement (Fig. 3 and Fig. 4). 
Figure 1. An incomplete view where
some saddles

Figure 2. A heavily cluttered
image where a

Figure 3. Evenly spaced streamlines
constituting an

Figure 4. Evenly
spaced streamlines with a dense

are underrepresented while spirals cause cluttering. 
saddle is still missing in the
middleright part.

elegant pattern to allow for interline
interpolation.

background intended to enhance
"interpolation".

(IEEE Computing in Science and
Engineering, Vol. 9, No. 4, 2007, pp. 86~91)

Zhanping Liu

Robert J. Moorhead II

Without an effective loop detection strategy, a placement of evenly spaced streamlines may be cluttered (Fig. 5) by closed or open but tightly spiraling streamlines, both loosely defined as loops herein. As opposed to some static, global, and computationally expensive loop detectors [1] [2], an ideal one for samplebased evenly spaced streamline placement needs to work in a dynamic, local, and interactive manner. In addition, it should be powerful enough to avoid loopgenic cluttering and artifacts while retaining important flow features such as closed streamlines and spiraling structures. Such a rapid robust loop detector (Fig. 6) is an integral component of our ADVESS algorithm. 
Figure 5. The cluttering problem
caused by some

Figure 6. The evenly spaced streamline
placement


closed and open but tightly spiraling
streamlines.

created by ADVESS with a robust
loop detector.


ADVESS is so far the fastest algorithm for interactive (on a nowadays lowend laptop like HPCompaq Presario / Intel Celeron 1.6GHz / 512MB SDRAM / Windows XP Home) highquality placement of highdensity (1% for 400 x 400 flow fields) evenly spaced streamlines coupled with robust loop detection (Fig. 7). 
0.188 seconds
for 319 streamlines

0.203 seconds
for 388 streamlines

0.234 seconds
for 378 streamlines

0.187 seconds
for 327 streamlines

0.219 seconds
for 387 streamlines

0.190 seconds
for 335 streamlines

Figure 7. Onepercent density evenly
spaced streamline placements generated by using ADVESS with robust loop
detection.

The application of evenly spaced streamlines to 3D exploration of volume flows, curved surface flows, and even planar flows in a perspectiveview setting poses at least three challenges. The first problem is how to create a placement of streamlines such that they are indeed evenly spaced in 2D view space (the output image) because streamlines evenly spaced in 3D physical space (the flow field) may not visually retain the uniformity when projected to view space through a foreshortening transform. Thus viewdependency issues need to be considered to address incompleteness, cluttering, occlusion, and nonuniformity in the resulting image. The second challenge is how to maintain a smooth transition between two consecutive frames of evenly spaced streamlines for coherent flow exploration. This concern is critical for dynamic levelofdetail flow investigation as features may be missing in a static lowdensity placement or indiscernible in a static highdensity layout due to the foreshortening effect. The third issue is how to find a fast universal mechanism, regardless of the data size or the grid type, to enable interactive placement of visually evenlyspaced streamlines for highperformance visualization of large flows defined on complex grids. To tackle the aforementioned problems, we presented IVDESS, an Interactive ViewDriven Evenly Spaced Streamline placement algorithm, for coherent explorative flow visualization. 
REFERENCES 
[1] T. Wischgoll and G. Scheuermann, "Detection and Visualization of Closed Streamlines in Planar Flows," IEEE Transactions on Visualization and Computer Graphics, Vol. 7, No. 2, 2001. pp. 165~172. 
[2] H. Theisel, T. Weinkauf, H.C. Hege, and H.P. Seidel., "GridIndependent Detection of Closed Streamlines in 2D Vector Fields," Proceedings of the 9th International Fall Workshop on Vision, Modeling, and Visualization (VMV'04), 2004. pp. 421~428. 