Indexing Point Triples Via Triangle Geometry

Database search for images containing icons with specific mutual spatial relationships can be facilitated by an appropriately structured index. For the case of images containing subsets each of which consist of three icons, the one-to-one correspondence between (distinct) point triples and triangles allows the use of such triangle attributes as position, size, orientation, and "shape" in constructing a point-based index, in which each triangle maps to a single point in a resulting hyperdimensional index space. Size (based on the triangle perimeter) can be represented by a single linear dimension. The abstract "shape" of a triangle induces a space that is inherently two-dimensional, and a number of alternative definitions of a basis for this space are examined. Within a plane, orientation reduces to rotation, and (after assignment of a reference direction for the triangle) can be represented by a single, spatially closed dimension. However, assignment of a reference direction for triangles possessing a k-fold rotational symmetry presents a significant challenge. Methods are described for characterizing shape and orientation of triangles, and for mapping these attributes onto a set of linear axes to form a combined index. The shape attribute is independent of size, orientation, and position, and the characterization of shape and orientation is stable with respect to small variations in the indexed triangles.