Existing techniques

This section describes existing techniques used for comparison with the novel techniques in this study; pseudo-color mapping and line glyphs. These techniques are quite appropriate for displaying scalar information on a surface. These techniques are often used for layered-surface display because more effective techniques are relatively new and not generally available in visualization libraries. These frequently-used techniques serve as baselines for comparison.

Pseudo-color mapping

Pseudo-color mapping recolors a surface, usually at the vertices of the geometric representation, to display metric information through mapping to a color scale. In general, the metric may be any scalar field sampled on the surface. For layered surfaces the metric is typically distance between surfaces. This distance is often Euclidean shortest distance, which is asymmetric (see Figure 3.3).

Choosing the correct perceptual color scale is a critical design step when using color maps. Any color scale used to convey secondary metric information on a curved surface should be isoluminant1so that it does not interfere with the perception of the surface’s shape [War88]. The remaining perceptual color channels for encoding metric information on a surface are the red-green and blue-yellow color channels. They may be combined to produce color scales varying in hue, saturation, or both, but care must be taken that the color scale remains isoluminant.

Because the surfaces of interest to this work intersect each other, the metric data to be encoded by color has positive, negative, and zero values (often referred to asratio data). The positive and negative values should be clearly distinguishable. This rules out broad spectrum color scales such as the hot-body spectrum or those employed by Ware for metric encoding [War88].

In the first set of experiments (see Section 5.3), the color scale used was red-grey-blue. This color scale encodes sign with hue and magnitude with saturation. Red denotes a positive distance (the displayed geometry is on the exterior) and blue a negative distance (the displayed geometry is on the interior). Unsaturated grey denotes the intersection itself. See Figure 4.3 for an example. This scale

is perceptible to those with red-green colorblindness (about 10% of the male population, and about 1% of the female population [War04]). It also uses a natural hot-cold encoding of the two ends of the color scale, making interpretation of the color scale easily remembered, if not intuitive.

Figure 4.3: Two surfaces visualized using the red-grey-blue version of the color mapping technique. Blue indicates that the visible geometry is inside the second surface, red that the geometry is outside, and grey that the two surfaces are in contact.

In the third set of experiments (see Section 5.5), the color scale used was red-yellow-green. This color scale encodes both sign and magnitude with hue. Red denotes a positive distance (the displayed geometry is on the exterior) and green a negative distance (the displayed geometry is on the interior). Yellow denotes the intersection itself (see Figure 4.4). This scale uses only the most sensitive of the chromatic perceptual color channels (red-green).

Why the switch in color scales? When publishing the results of the first experiments (those using red-grey-blue), reviewers repeatedly commented that the red-grey-blue color scale probably underper- formed relative to a red-yellow-green scale due to limited sensitivity for saturation and the sensitivity mismatch between the red and blue ends of the scale. The third set of user study experiments showed that the reviewers predictions were true for one task but not for the other (see Chapter 5, Section 5.5).

Figure 4.4: The hill and tumor data sets visualized using the red-yellow-green version of the color mapping technique. Red indicates that the visible geometry is inside the second surface, green that the geometry is outside, and yellow that the two surfaces are in contact.

Line glyphs

Line glyphs2 are frequently used to display scalar data on a surface. In some cases, such as uncertainty magnitude in uncertainty visualization, the line glyphs may carry geometric information as well; if the “free” end of the glyphs rests on the error boundary, the glyphs effectively sample the

2_{Line glyphsare line segments attached at one end to the surface and extending for a distance related to the magnitude of}

the scalar field at the point of attachment. When used for scalar display, they typically extend in the direction of the surface normal at the point of attachment.

surface of the error.

Line glyphs may be used to show correspondence between two surfaces, and they are used to do so in this work (see Figure 4.5). Line glyphs used to show point correspondence will be referred to as point-correspondence glyphs throughout this work. To show correspondence between surfaces, the ends of each point-correspondence glyph rest on one of the two different surfaces. Although the dis- tance between the two endpoints can be inferred, it is not necessarily computed when determining cor- respondence. Nor is distance necessarily directly displayed. For instance, the point-correspondence glyphs studied here do not follow the shortest paths between corresponding points, but instead fol- low smoothly curved paths. These curved paths are the trajectories of the corresponding points for a distortion-minimizing animation from one surface to the other.

For two arbitrary surfaces there may be noa prioricorrespondence information between the sur- faces. If the surfaces are the result of a statistical model, there may exist a closed-form solution to a point-correspondence map between surfaces. If the two surfaces are described only by their geome- try, the correspondence must be approximated. The desired property for a correspondence mapping between two surfaces is that the mapping be a bijection (both one-to-one and onto). One common method for computing a bijective map between surfaces is to solve Laplace’s equation using the two surfaces as boundaries. Following an integral path of the gradient of the heat diffusion from one sur- face to another then yields the geometry of the point-correspondence glyph between those two points. Computing correspondence between arbitrary surfaces is discussed in more detail in Section 4.4.3.

Point-correspondence glyphs appear in two techniques in this work. The first of these is described here. This technique displays the interior of a pair of layered surfaces, as computed by the refactoring algorithm. The interior pieces are colored to show to which of the potentially intersecting layered surfaces each piece originally belonged. Point-correspondence glyphs are computed between the interior and exterior surfaces. The correspondence-glyph endpoints on the exterior surface are spread over the surface such that the geodesic distance between endpoints follows a Poisson law distribution. Where the point-correspondence glyphs terminate on the interior surface, an additional proximity glyph is placed. The proximity glyph approximates shadowing on the interior surface by the point- correspondence glyph due to global illumination. This enhances the perception that the interior surface

and point-correspondence glyphs do indeed contact [TSS+98]. See Figure 4.5.

Figure 4.5: The hill and tumor data sets visualized using the point-correspondence technique. The correspondence glyphs connect points on the (hidden) exterior surface to their corresponding points on the (visible) interior surface.