The display of four-dimensional data is usually accomplished by assigning three dimensions to location in three-space, and the remaining dimension to some scalar property at each three-dimensional location. This assignment is quite apt for a variety of four-dimensional data, such as tissue density in a region of a human body, pressure values in a volume of air, or temperature distribution throughout a mechanical object.
While there exist a number of methods to approach the visualization of three-dimensional scalar fields ([Chen 85], [Drebin 88], [Kajiya 84], [Lorensen 87], and [Sabella 88] are good examples), there are few methods that are effective on true four-dimensional data, where the data do not represent a three-dimensional scalar field.
This paper approaches the problem of displaying 4D objects as physical models through two main approaches: wireframe methods and raytracing. Both of these methods employ true four-dimensional viewpoints and viewing parameters, and light (or depthcue) the rendered objects from four-space.
The tremendous difficulty of visualizing true four-space data lies in the fact that there are no solid paradigms for three-space creatures such as ourselves. This difficulty is best understood in imagining the plights of two-space creatures who try to comprehend our three-space world (see [Abbott 52] or [Dewdney 84] for explorations of this idea).
The method we use to comprehend three-dimensional scenes is quite complex, since each eye is presented with only the two-dimensional projection of the three-dimensional scene. There are several methods we employ to convert these 2D projections to an imaginary 3D model. These methods include focusing (of limited help but useful when viewing with a single eye), parallax (deriving depth through binocular vision), application of object knowledge to understand different views, and motion to derive depth information.
These methods of visualizing 3D objects are very strong, and are usually quite accurate. When dealing with a single rendered 2D projection, however, the image becomes a bit more difficult to comprehend, because the viewer can no longer use focusing or parallax to extract information from the scene.
Usually, however, rendered 2D projections are quite intelligible because these projections involve objects or structures familiar to the viewer, and because shadows and highlights also help the viewer to extract depth information. The main thing that aids our understanding of a scene when given only 2D projections is our experience and intuitive understanding of the 3D world. This additional understanding helps us to intuitively reconstruct the original 3D scene, and to accurately guess the portions for which we have no visual information.
If we are deprived of this intuition and experience, then reconstructing the original scene from a projection is very difficult. This is what makes visualizing 4D data such a complex task. In fact, when rendering a 4D scene, not one but two dimensions of the original data are lost; one can think of a screen image of 4D data as a projection of the projection of the scene. This further loss of information, coupled with our lack of intuitive understanding, demands that the 4D visualization methods provide additional information, usually in the form of motion, multiple views or other visual cues. These techniques will be presented in more detail later in this paper.
The idea of understanding four-dimensional space is at least as old as the nineteenth century, and has been studied mathematically at least as early as the 1920's. Understanding four-space has also been explored in texts that attempt to model two-dimensional creatures and their perceptions of three-space (see [Abbott 52] and [Dewdney 84]). Studying the difficulties of two-space creatures who attempt to understand three-space often yields insight into the problem of understanding four-space from our three-dimensional world.
The task of viewing four-space structure has been explored as early as [Noll 67], who rendered four-dimensional wireframes with 4D perspective projection. Noll was limited in his exploration of four-dimensional structures by the technology of that time; his method consisted of generating pictures via plotter and then transferring each drawing onto film. The movies he produced yielded a great amount of insight into the structure of various four-dimensional objects. However, the lack of interaction was certainly a significant hindrance.
In 1978, David Brisson ([Brisson 78]) presented hyperstereograms of 4D wireframes. These hyperstereograms are unconventional in the sense that the viewer must rotate the hyperstereograms in order to resolve the second degree of parallax of the 4D view. These hyperstereograms are very difficult to view, but do provide another method of understanding the four-dimensional structures.
In the early 1980's, Thomas Banchoff (who is heavily involved in the visualization of four-space) rendered hypersphere ``peels'' ([Dewdney 86] and [Banchoff 90]) which resulted in some beautiful images of their rotation in four-space.
Several people have rendered four-dimensional objects by producing the three-dimensional slices of the object; this is presented in [Banchoff 90].
Rendering solid four-dimension objects yields a three-dimensional ``image'', rather than the two-dimensional image of a three-dimensional object. [Steiner 87] and [Carey 87] use scanplane conversion to render four-dimensional objects into three-dimensional voxel fields. Both approaches also tackle the difficult problem of hidden volume removal in four-dimensional views. The resulting fields are then displayed with semi-transparent voxels in order to view the internal structure of the three-dimensional projections.
This research builds on the existing wireframe display techniques and tackles the visualization of solid four-dimensional objects via raytracing techniques.
The wireframe viewer takes as input a list of four-space vertices and a list of edges between pairs of these vertices. It produces a single image which is the 2D projection of the 4D scene, and which may be interactively rotated, depthcued, and parallel- or perspective-projected.
The limitations of the 4D wireframe viewer are pretty much the same as those for a 3D wireframe viewer: the lack of surface information, the multiple ambiguities of a wireframe view, and the fact that all objects must be decomposed to vertices and edges.
The significant advantage of the wireframe viewer is that image display is quite fast, especially when compared to raytracing methods. This speed of display allows for interactive rotation of the 4D object, which greatly aids in an understanding of the object and its relationship to the rest of the scene. Another advantage is that the wireframe viewer is able to display curves in four-space.
The 4D raytracer solves the hidden volume and lighting problem in a very straight-forward manner (as for 3D). It takes an input file of 4D object information and probes the scene that contains these objects. The implemented 4D raytracer handles three primitives: hyperspheres, tetrahedra, and parallelepipeds, although the methods presented in this paper apply to a much broader class of objects.
The output of the 4D raytracer is a gridded 3D volume of RGB triples, analogous to the gridded 2D volume of RGB data produced by a 3D raytracer. The disadvantages of the raytracer include the increased rendering time and the resultant 3D volume, which must be further rendered with other methods.
The main advantages of the raytracer include the fact that the rendered image has hidden volumes, shadows, highlights, reflections and other artifacts that aid the understanding of the scene.
Chapter 2 covers the four-dimensional geometry that is used in this research, and focuses on vector operations and 4D rotations. The 3D and 4D viewing models are presented on a functional level in Chapter 3 and implemented in detail in Chapters 4 and 5. Chapter 4 covers the rendering of 4D data via wireframe methods. It describes the method of projecting the image from four-space to three-space, and then from three-space to the 2D viewport. It also covers the implementation of 4D object rotation and other wireframe visual aids. The four-dimensional raytracing method is presented in Chapter 5; this includes the generation of the ray target grid, the intersection algorithms used for the fundamental 4D objects, and the methods of visualizing the resulting 3D ``image'' of RGB data. Finally, Chapter 6 provides a conclusion to this research. It includes notes on the research in general, suggestions for further research and exploration, and provides a few comments on the implementation of the programs discussed in this paper.
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