It has been shown in this dissertation that principal curvature texture combined with either shad- ows or correspondence provides the best intersecting-surface visualization (of the techniques eval- uated). This dissertation contains the most comprehensive comparitive evaluation yet performed of techniques for display of intersecting surfaces. As such, these two techniques (texture plus shadows or texture plus correspondence) are thede factorecommended techniques for visualization of layered surfaces.
I do not believe that the use of texture that indicates principal curvature information on the surface is of critical importance to the effectiveness of the techniques. The important aspects of the exterior surface texture are that it contains both translucent and opaque regions and that the opaque regions are dense enough to reveal exterior surface shape while simultaneously sparse enough to reveal interior surface shape. Indeed, Interrante found that a regular grid texture in some cases outperformed princi- pal curvature texture for a shape task identifying closest approach between nested surfaces [IFP97].
In general, either of these two techniques can be recommended for visualizing nested or inter- secting surfaces. Specifically, when point correspondence information is important or available it is recommended that the texture with correspondence technique be employed. If the correspondence in- formation is not available and not important to the exploration of the surfaces, it is recommended that the texture with shadows technique be employed. The computation of a suitable point correspondence mapping from two surfaces with noa prioricorrespondence relationship is to expensive to justify (re- call that no statistical difference exists between the two techniques for performance of the evaluated tasks).
Neither of these two techniques can be said to solve all visualization problems for intersecting surfaces. The following sections will lay out the limitations of the two techniques and, where possible, recommendations of the best way to handle facing those limitations.
Figure 6.1: The image on the left is of a tumor surface. Note the protuberance on the left side of the tumor which folds back toward the right occluding a significant portion of the tumor. The image on the right is a 2D sketch of an oppositely folded surface.
6.3.1 Self-occluding surfaces
Many real world objects (such as the examples in Figure 6.1) have complex geometries that fold over such as to occlude other regions of the object when viewed from certain directions. For instance, the fingers of the relaxed human hand curl so as to “fold” over the palm. Fully exploring such surfaces usually relies on interactive visualization tools where the user controls the position and orientation of the view.
One should expect the same – interactive control – in a tool for exploring nested or intersecting surfaces. Indeed, because the most effective techniques for displaying nested or intersecting surfaces display portions of the occluding surface opaquely (via opacity-modulating texture), interactive con- trol is necessary to explore the interior surface to the fullest. Also, interactive control greatly increases the understanding of the readily viewable shapes because of the kinetic depth effect, so such control is beneficial even when objects are not self-occluding.
6.3.2 Shape at scale
The techniques shown through human-subjects experiments (in this dissertation and other works) to effectively convey shape for nested or intersecting surfaces use opacity-modulating texture. Opacity-
modulating texture uses surface texture to alternate the occluding surface between opaque and translu- cent, enabling portions of both surfaces to be visible. The density of the opaque regions (equivalently, the frequency of the modulation) is critical to understanding both surface shapes simultaneously. The frequency of opaque regions in the texture pattern determines the texture’s sampling of the occlud- ing surface, but the surface may contain features at higher frequency than the texture’s modulation frequency.
A common solution to sampling domains with varying frequency content is adaptive sampling. In adaptive sampling, some local metric on the domain is used to determine the local sampling rate within the domain. Adaptive sampling can be applied to adjust the sampling rate of the texture in an effort to capture significant high-frequency features. One method for accomplishing this is to vary the sampling frequency proportional to the magnitude of the local surface curvature. The resulting texture will be irregular unless the underlying surface has uniformly distributed frequency characteristics, so it generally will not be possible to produce regular texture as in Rheingans work [Rhe96]. However, localized irregularities in an otherwise regular texture will draw attention to higher-curvature regions, which may itself be a desirable effect of the visualization.
6.3.3 Small regions of intersection
Often, two intersecting objects may share a very small volume of overlap relative to their own volumes. Such cases are very different from those studied in this dissertation, and the techniques developed herein assume a large amount of object overlap. The two recommended techniques are developed around cases where the two objects are relatively close in volume as well as close in “reg- istration” (i.e. their common features are relatively close in space).
When the two intersecting surfaces are not close in volume (i.e. one is significantly smaller than the other), neither of the two recommended techniques may provide a better understanding of the two surfaces than texture alone. The addition of shadows in such a case is unlikely to hinder understanding but correspondence glyphs may. When the two intersecting surfaces a not close in registration, it may be that it is sufficient simply to highlight the intersection without applying any of the techniques in
this dissertation. If the penetration depth between the two surfaces is important, one solution might be to use texture on both surfaces in the vicinity of their intersection, and use the point correspondence glyphs within the penetration.
6.3.4 Obvious versus non-obvious correspondences
The bump surfaces used in the evaluation studies in this dissertation were generally in close reg- istration with each other – meaning their obvious features were in close proximity to each other. The data collected from domain scientists for this work also have the property that pairs of surfaces are in close registration. However, real data from other application areas may not have this property. In particular, objects that have obvious correspondences that are not closely registered with each other may pose special problems for visualizing with these techniques.
As an example of objects with obvious correspondence, suppose we had two statistical models of some recognizable object, say a human hand in a relaxed pose. Assume the difference between the two models has to do with some condition that effects the curl of the fingers in a relaxed pose. So the features of each hand are the same, but they are not all in close registration. Assuming the correspondence mapping is the expected or obvious one, it is helpful to display in the sense that it reinforces the expected correspondence.
Examples of non-obvious correspondences include shape features of one surface (i.e. bumps, dim- ples, vortices, folds, cusps) that correspond to very different features of the other surface. Assuming the correspondence mapping is not the expected or obvious one, it is helpful to show the true corre- spondence if the correspondence information itself is of importance. I claim it is also helpful to show the non-obvious correspondences when domain specific questions pertain to the evolution of one sur- face into the other, such changes in an object over time or changes in a model due to modification of model parameters.
Many of these shape features of interest may have a critical or extremal point that is especially useful to denote. Finding these points to best approximation and including them in the sampling of both the exterior surface (through use of opaque texture) and as an end point for correspondence
glyphs would also be of significant value in depicting the surfaces and their correspondences.
6.3.5 More than two surfaces
The evaluations in this dissertation focus on a pair of intersecting surfaces. Real science often requires the exploration of more than a pair of surfaces. For example, radiation treatment planning can involve the three common tumor volumes (Gross Tumor Volume, Clinical Target Volume, Planning Target Volume), level sets of radiation dosage, and surrounding healthy tissues. Displaying all these potential surfaces (or more usefully, enabling the clinician to select arbitrary combinations of these) can result in complex surface layerings.
Color and texture can be used to label object surfaces in the scene. Color alone may be used to distinctly label seven or more surfaces, and texture may be used to extend or reinforce the color labelings. Different texture shapes (i.e. the cross used throughout this work) could be used on dif- ferent surface layers to disambiguate the layers (i.e. interior, middle, exterior), while color is used to disambiguate the objects.
It is an open question how effective the shadows would be at enhancing the perception of depth between more than two surfaces. Though not physically accurate, it may be that restricting shadows to fall only on the “next” innermost surface would yield the strongest benefit to pair-wise depth per- ception. For example, assuming for simplicity nested surfacesAcontainingBcontainingC; shadows from the texture on surfaceAfall ontoBbut not ontoC, and shadows from the texture on surfaceB fall ontoC. Thus, the shadows fromAontoBwould enhance the depth perception between surfaces AandBwhile minimally interfering with the shadows cast byBontoC.
Labeling the different layers of the surfaces now becomes more complex. Obviously interior/exterior alone is not generally sufficient. Further, there is a combinatorial explosion of possible regions to la- bel. Two closed, intersecting surfacesA andBpartition space into four (22) regions,A∩B, A∩B¯,
¯
A∩B, and ¯A∩B¯. Similarly, three closed, intersecting surfaces potentially partition space into eight (23) regions, andnclosed, intersecting surfaces potentially partition space into 2n regions. Finding and labeling the layers of surface separating space interior tomsurfaces from space interior tom−1
surfaces is complex computationally and perceptually.