Properties of Local Geometric Constraints

Formal analysis shows that the relationship between 2D line segments which is rotationally and translationally invariant can be expressed using four independent variables (five if scale is included). The formalism leads to a language for defining local geometric constraints and describing their properties. This terminology allows us to establish a criterion for evaluating the quality of a constraint set. Finally, a 2D constraint set is described that is deemed useful in light of the criterion proposed. Model-based vision makes extensive use of the local constraint paradigm for matching models to images [1, 2, 3, 4, 5, 6, 7]; in this paradigm unary and binary constraints between linear segments are often used to prune an interpretation tree (describing the correspondence between model and image features) that is potentially combinatorial. The efficiency of this search is largely determined by the constraint set, and how capable it is of meeting the ideal criterion: that it should prune the tree at any node representing an incorrect correspondence, and grow the tree at any node representing a correct one. Various sets of constraints have been described for either matching in 3-space or 2-space. Such constraint sets are largely proposed ad hoc and justified by either practical results or intuitive plausibility (with some exceptions e.g. [8]). In this paper we examine the concept of a local constraint and its properties. We formalise the vocabulary for talking about constraints that is commonly adopted in the literature, and propose qualitative criteria for assessing a constraint set. This vocabulary and criterion allow us to say of a constraint set: "It is bad because . . . " . The paper restricts itself to the relationship between a pair of 2D segments, although much of the content lends itself to 3D constraints also. In as much as the analysis is two dimensional, it has especial relevance to recognition systems that either recognise 2D objects, or recognise 3D objects using 2D features.