Andrew L. Thall

Document Type



M-reps are a multiscale approach to the modeling and rendering of 30 solid geometry. Traditional geometric models, whether b-reps or CSG, are represented at infinitesimal spatial scale and then require simplification to meet needs requiting coarser scale or smaller data sets. We have developed a model that is designed at successively smaller scales and supports a coarse-to-fine hierarchy in design, rende1ing, physical deformation, and other graphics operations. We base our representation on figural models, defined at coarse scale by a hierarchy of figures - protrusions, indentations, corners, neighboring figures, and included figures - which simultaneously represent solid regions and their boundaries. To capture local shape at scale and thus local zoom-invariance, the figural components imply a fuzzy, i.e., probabilistically described boundary position with a width- and scale-proportional tolerance. At small scale these figures are made precise by displacement maps, geometric textures, or image textures. While these models can exist in 20, we focus on models of 30 objects. We note the needs of a variety of graphics operations and thus motivate the following definition of an m-rep. A model for a single figure is made from a mesh of medial atoms (hence the name m-reps), each atom describing not only a position and width, but also a local figural frame implying figural directions, and an object angle between opposing, corresponding positions on the implied boundary. In addition, width proportionality constants indicate mesh link length, boundary tolerance, boundary curvature limits, and, for 30 image visualization, an interrogation aperture. Thus, a figural mesh defines the figural boundary to within a width-proportional tolerance and provides a width-proportional sampling of the figure's implied medial surface. A figural model, then, consists of a directed acyclic graph of figure meshes, with interfigural, intramesh, and interscale links capturing information about differences in position (and thus figural length and subfigural offset), figural width, narrowing rate, boundary curvature, and figural orientation. The leaves of the figural graph also contain the displacement maps, geometric textures, or image textures that are used to give infinitesimal scale to the model where fine detail is needed. We describe methods for building m-rep models by a design tool, from b-rep or CSG representations, or from a 30-intensity dataset. We present a method for deforming m-reps into image data, allowing model-directed visualization of objects in volume data. We describe algorithms for rendering m-reps, and show two uses in graphics for them - rende1ing for 30 visualization and computer-aided design - with preliminary results. Algorithms for other graphics objectives are sketched.

Publication Date



Link Foundation Fellowship for the years 1998-1999.

FORM Final Report Andrew Thall.pdf (87 kB)
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