Shape representation and recognition based on invariant unary and binary relations

The problem of transformation invariant object recognition is considered. We develop a projective transformation invariant representation for both scene and model which facilitates an attributed relational graph object matching based only on unary and binary relations. The unary and binary measurements used for matching are derived from sets of reference points such as corners and bi-tangent points which are stable under the various transformations considered. Each set of reference points is used to generate a distinct barycentric coordinate basis system associated with one node of the object graph representation. We show that barycentric coordinates of the reference image points can be made invariant under any arbitrary projective transformation. The conditions that must hold for a basis to be valid are stated. We illustrate the construction of the barycentric coordinate systems for the affine and perspective transformations. For the object and scene representation we use the barycentric coordinates of the reference points generating the barycentric coordinate system, together with auxiliary measurements such as colour and texture as the node's unary measurements. For binary measurements we use the product of the barycentric coordinate system for one node with the inverse of the barycentric coordinate system associated with another node. The unary and binary relations provide an orthogonal decomposition of the shape being matched. They are used in a relaxation process to detect instances of objects consistent with a given model. We demonstrate the proposed methodology of projective transformation invariant object representation on several examples. First we illustrate the stability of the shape representation in terms of unary relations both visually and numerically. We then experimentally demonstrated the invariance of binary relations on a star-like object. We show experimentally that the binary relations derived are invariant. The final example demonstrates the proposed approach as a tool for 3D object recognition. The aim is to recognize 3D objects in terms of planar faces. A hexagonal model shape is hypothesized in the image. The only instance of the hypothesized model is successfully recovered.