In order to display, transform, and compare volumetric data, it is often convenient or necessary to use different representations derived from the original discrete voxel values. In particular, several methods have been proposed to compute and display an iso-surface defined by some threshold value. In this paper we describe a method to represent the volume enclosed by an iso-surface as the union of simple volume primitives. The needed properties (displayed image, volume, surface, etc.) are derived from this representation. After a survey of properties that might be needed or useful for such representations, we show that some important ones are lacking in the representations used so far. Basic properties include efficiency of computation, storage, and display. Some other properties of interest include stability (the fact that the representation changes little for a small change in the data, such as noise or small distortions), the ability to determine the similarities between two data sets, and the computation of simplified models. We illustrate the concept with two distinct representations, one based on the union of tetrahedra derived from a Delaunay tetrahedralization of boundary points, and an other based on overlapping spheres. The former is simple and efficient in most respects, but is not stable, while the latter needs heuristics to be simplified, but can be made stable and useful for shape comparisons. This approach also helps to develop metrics indispensable to study and compare such representations.
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