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7.1.2 Volumetric and mathematical models

Voxel description may be more appropriate if internal object characteristics are not homogenous or if complex object operations must be performed that will significantly alter surface geometry. Voxels are simple building blocks for an object (often cubes), and can possess individual material attributes, allowing nonuniform material descriptions. Thus a pyramid might be described by a sequence of stacked voxels (cubes). Slicing a pyramid with a plane can generate a complex series of geometric shapes; and developing robust algorithms for generating new triangular skin descriptions from an original skin description for a pyramid is a nontrivial problem. Using a voxel−level description, however, the problem is far simpler: if the centroid of a given voxel lies on one side of the slicing plane, assign it to object A, otherwise assign it to object B. Boolean operations, such as calculating the intersection of two arbitrary objects, are also far easier to implement. Applications must typically use a voxel description internally, and generate a skin surface description for rendering. This allows the application to optimize rendering performance (which is required for every frame) while not sacrificing the ability to perform solid modeling operations (which is required occasionally).

Voxel descriptions are often associated with medical visualization applications, particularly for magnetic resonance imaging (MRI) visualization. MRI equipment provides an array of coordinates along with an MRI reading for each coordinate. MRI data can be rendered as an array of cubes with the MRI reading for each cube assigned as a color or transparency for the cube. Finding surfaces across voxels (isosurfaces) may require using the Marching Cubes Algorithm or one of its variants. See W.E. Lorensen, et al., “Marching Cubes: a high resolution 3D surface reconstruction algorithm,” Computer Graphics, vol. 21, no. 4, pp 163–169 (Proc. of SIGGRAPH), 1987, and Alan Watt, et al., Advanced Animation and Rendering Techniques, Addison−Wesley, 1992.

Voxel descriptions also suffer from inherent assumptions regarding the performance of rendering hardware, as the object modeler must decide the voxel resolution required for an object (or part of an object). See figures 7.3, 7.4, and 7.5.


Figure 7.3 Transparent volume rendering of Lucky's (the virtual dog) abdomen. Courtesy Edinburgh Virtual Environment Centre (http://www.edvec.ed.ac.uk)


Figure 7.4 Voxel rendering of human skull from MRI data. Courtesy Ohio State University (


Figure 7.5 Volume rendering of human skull with two sections removed to show internal detail. Courtesy State University of New York at Stony Brook (

Mathematical object descriptions are useful in that they do not contain assumptions about rendering performance—if an object is described as a sphere, with radius 2.5 meters, the rendering system can render the object in a manner appropriate for the underlying hardware. Non−Uniform Relational B−Splines (NURBS) are commonly used in technical modeling applications as they describe an object’s surface mathematically and in a manner that can be rendered (with varying accuracy) compatibly with the hardware available.

It is important to reiterate, however, that current 3D video cards used to accelerate rendering deal exclusively in simple strips of triangles. Thus, no matter what level of description you choose for your application, your objects will need to be decomposed into strips of triangles. The low−level graphics language itself may handle this transformation; OpenGL can render NURBS, for example (figure 7.6). Java 3D does not have the ability to render NURBS directly, the application developer must write code to convert the NURBS into triangles prior to rendering.


Figure 7.6 Rendering of a tube and a deformed sphere, both described using NURBS. Image produced using Amapi 3D (http://www.eovia.com)