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## Geometric Reasoning Using Local Contexts

A major disadvantage of the space filling techniques for object representation discussed in chapter 2 was that the description of an object depends on its orientation with respect to the axes used to partition the space, and its apparent size with respect to the axis units. The axes themselves often have no functional relationship to the task being performed, and the variance in descriptions disguises the similarities between objects that are functionally equivalent, or even identical.

If similarities between objects are to be reflected in their descriptions, the representation must be capable of expressing shape in a way that is independent of surrounding objects, and independent of the viewing position. This can be achieved by describing objects in terms local to the object itself. The head of a bolt, for instance, might appear completely different when seen on the side of a car under assembly, than when seen on a bench with a collection of ball bearings. The functional shape of the bolt head is identical in different situations, however, and it should therefore be described in local terms, relative to the rest of the bolt, rather than relative to the scene in which it appears. If the heads of bolts are always described identically, the invariance of the head's functional shape is easily recognised.

In reasoning about interaction between objects (for instance, analysing fasteners), it is necessary not only for the objects to be expressed in local terms, but also the relationships between the objects. For example, a split pin in a vehicle assembly always works the same way, whether it passes through an axle, or through a steering link. In this case, the overall shape of the axle or link is irrelevant, and it is only a few features near the split pin that are functionally important. It should be possible to represent a fastening situation such as this in terms of the local features, the orientation of those features relative to each other, and the motions required to act on them.

Three aspects of object representation which can be formulated in terms of a local context are size, location, and orientation. Size can be described in local terms by comparing feature sizes either to the overall dimensions of a complete object, or to distinctive features in the vicinity (using a local quantity space in qualitative systems). Location and orientation can be described relative to distinguished points and directions in the remainder of the object shape, rather than with respect to global axes. Using the object itself or an interacting object to provide a reference frame means that descriptions of similar objects or interactions will always be made in consistent terms.

Local orientation must be described by reference to distinctive shape features. Candidates for orientation reference features include object axes, or dominant directions. Dominant directions can be calculated from boundary representations, where dominance is established by the total length of edges and/or surface normals with a particular orientation. Object axes are usually specified in constructive solid geometry representations, where each primitive has intrinsic axes. Both of these directions are functionally significant in many mechanical parts (in fact, the directions often coincide in mechanical parts), and the representations discussed later in this chapter make use of both.

Local context descriptions can be made relative to a reference frame derived from the main object, or derived from neighbours of the object, where interaction between objects is particularly significant. Another possible reference frame for local context is provided in a representation with multiple levels of detail, where a coarsely described shape can be used to provide the local reference frame with respect to which the details that compose it will be defined. One of the representations described later in the chapter does precisely this - in that representation I refer to the coarse feature reference frame as an ``imple context'', because complex concave or convex features of an object's shape can be described at a coarse level as a dimple or pimple on the object. Size, location, and orientation of detailed shape features are then described relative to the imple.

In addition to static descriptions of object shape, size, and orientation, motion of objects can be expressed in terms of local contexts. Where two objects are joined together by small fasteners, the overall motions of the objects for unfastening are best described by reference to the shape of the fasteners. If the fasteners are described using imple contexts, then the motion of the complete object should be described in terms of those contexts. The extensive use of local contexts therefore requires that transformations be available between local and global coordinates, as described by Popplestone, Ambler and Bellos in [PAB80], and by Ballard [Bal84].

This section has described the benefits arising from shape description in terms of local shape context, and has discussed some mechanisms which can provide these abilities. The qualitative representations presented later in the chapter make use of qualitative adaptations of these mechanisms.

Next: Representation of Multiple Features Up: Representation Issues in Qualitative Previous: Representation of Detail
Alan Blackwell
2000-11-17