Importance in a visualization is controlled by the importance field. The importance field is a scalar field imposed over a vector field that is used by the visualization methods to determine which regions of the visualization require more focus than others. The importance field is normalized with a value of 0 being completely unimportant and 1 being the highest importance.
The importance field is stored as a 2D texture. This texture is populated with the superposition of all the focus points’ wave equations (see Section 3.1.1) sampled at the center of each texture point. The importance field at any continuous point is then acquired by sampling this texture using bi-linear interpolation. Figure 3.2. shows the importance field resulting from two focus points.
A more accurate approach would be to sample each focus point’s wave equation every time the importance field is sampled, but this would be slow since the field is constantly re-sampled and some of the wave equations can be computationally costly. This texture method is used as an optimization to improve performance by reducing the number of wave equation evaluations required.
Figure 3.2: Shows the importance field resulting from multiple focus points. The importance field varies from 0 (blue) to 1 (orange). Focus points are visualized by the circles with width representing the radius of the focus. In this figure both focus points use an inverse square wave function.
CHAPTER 4
GLYPHS
Glyphs are chosen because they offer the most control over information density. By varying the number of glyphs placed in a particular region, the information density can be either increased or decreased. Glyphs are also computationally inexpensive compared to other visualization methods and often give a clearer indication of direc- tion than streamline based methods.
4.1
Glyph Pool
The glyph pool is a high-density collection of potential glyphs created before the visualization. Each glyph in the glyph pool has its position chosen and fixed before the visualization. During visualization glyphs are chosen from the pool based on the importance field and rendered. A higher importance results in more glyphs being chosen from the pool and thus a higher density. A lower importance results in fewer glyphs being chosen from the pool and thus a lower density.
A glyph pool is chosen for two reasons. The first is for efficiency (to avoid unnecessary addition and deletion of glyphs during visualization). The second is to maintain temporal coherence. By fixing the positions of all potential glyphs before visualization, temporal disruption is limited to glyphs fading in and out rather than moving around.
cussed. Each attempting to address the shortcomings of the method before it. • Random Glyphs
• Grid-Based Glyphs • Mipmap Glyphs
4.2
Random Glyphs
The first method for glyphs involves randomly selecting the positions and importance values. The desired result is to create a uniform distribution of glyphs throughout the visualization. The problem with this method is that some glyphs may be placed too close together and result in the glyphs occluding one another.
Attempts to resolve these occlusion problems included attempting to get a more uniform distribution of glyphs by further jittering the glyph positions after generation, but this did not appear to have a significant impact and made little difference visually. Even with an ideal uniform distribution of glyphs and with occlusion avoided; another problem with this method exists. The set of all glyphs may have a uniform spatial distribution, but the subset of all visible glyphs can still have pockets of unintended density.
Figure 4.1: Shows a vector field visualization using randomly placed glyphs. Note how some glyphs overlap. Also note that there are small regions of higher density that are far away from the focus point.
Figure 4.1. shows a visualization using random glyphs. Note how some glyphs are occluding one another. This figure also shows small regions of high density even in regions which should clearly be low density since they are far from the focus point.
4.3
Grid-Based Glyphs
The next method attempts to avoid occlusion by placing the glyphs on a uniform or jittered grid and choosing the importance thresholds randomly. Glyphs placed on
(a) Uniform 32x32 grid (b) Jittered 32x32 grid
Figure 4.2: Shows the results of placing glyphs on a uniform and jittered grid while choosing importance threshold randomly. Notice that occlusion is suppressed, but pockets of uneven density still exist.
4.4
Mipmap Glyphs
The reason pockets of high density appear in low importance regions is because two or more glyphs near each other are all assigned low importance thresholds. This causes a small region of increased density. The reason pockets of low density appear is that too many glyphs in a small region are all assigned high importance thresholds.
Randomly choosing the importance threshold may result in a uniform distribution of importance threshold values, but it is also essential that these importance values be uniformly distributed in space.
To avoid these pockets of high or low density and create a uniform spacial distribu- tion of importance thresholds, the importance threshold is calculated using a mipmap approach. For a grid of dimension 2n × 2n, n sub-grids of decreasing dimension are generated. For each value i from 1 to n a grid of size 2i × 2i is generated and each element is assigned a random value between i−n1 and ni. These sub-grids represent multiple levels-of-detail with each sub-grid having importance threshold values randomly chosen around a value proportional to the number of elements in the sub-grid. For example a 23 ×23 sub-grid will have higher importance threshold values than a 22×22 sub-grid.
The final threshold value for each glyph is selected by sampling all valid sub-grids and choosing the lowest threshold value among them. A sub-gridiis considered valid for some index (u, v) if and only if u mod 2n−i = 0 and v mod 2n−i = 0. Figure 4.3 shows an example of a 8x8 grid and its sub-grids being used to chose importance thresholds.
This mipmap approach ensures that no two neighbouring glyphs both have low importance values and thus avoids pockets of high density. It also ensures that there exists at least one glyph with a low importance threshold in each region thereby avoiding pockets of low density.
Figure 4.4. shows uniform and jittered grid glyph placement using the mipmap approach to assign importance thresholds. Notice there are no longer localized regions of higher density and occlusion is mostly avoided.
(d) n= 3
Figure 4.3: The sub-grids for a 23×23 grid are shown above and the final sampled grid is shown below. Importance thresholds are shown with blue being the highest, then orange, and green being the lowest.
(a) Uniform 32x32 grid. (b) Jittered 32x32 grid.
Figure 4.4: Shows uniform and jittered grids using mipmap importance thresholds.
some range is so the transition between LODs is smoother. If each sub-grid was filled with a constant value proportional to its depth, then artifacts are visible when transitioning between the LODs as seen in Figure 4.5.