A polygon matching problem in computer vision (abstract only)

• Authors: L. W. Brinn
• Venue: CSC '87 Proceedings of the 15th annual conference on Computer Science
• Year: 1987

A central problem in computer vision is that of determining whether a given object O appears in a scene S. We outline a general method of solution applicable to r dimensional location spaces, where rWe assume that the scene S is a bounded region in r dimensional Euclidean space (Rr), and we consider the object O as represented by the values of a given feature function f at certain distinguished points p1, p2,…,pn on O. If Q is the polyhedron in Rr with vertices p1,p2,…,pn, we refer to Q as the query polyhedron. We suppose that a metric d is defined for feature values and that a tolerance t = (t1,t2,…,tn), has been given.For each i, l ≤ i ≤ n, let Ri be the set of points p in S for which d(f(p),f(pi)) ≤ ti. We refer to the Ri as the object polyhedra and say that the object O has been found in S to within the tolerance t iff there exists a translation T of the query polyhedron Q with T(pi) &egr; Ri for each i. The original problem is now a problem in computational geometry. This approach is due to W. Grosky.We consider the restricted problem in which each Ri is a closed nonempty convex polyhedron. The Ri need not be disjoint, nor is there any restriction (beyond convexity) on their shape. We assume that each Ri is given as the solution set of a system Li of ki linear inequalities (that is, as an intersection of ki closed halfspaces). The representation is convenient for higher dimensional spaces. We can determine whether the translation T exists in time O(k), where k = k1 + k2 + … + kn.The approach is to translate not Q, but the object polyhedra Ri. For each i, 2 ≤ i ≤ n, let Ti be the unique translation in Rr with Ti(pi) = p1, and let Ri′ = Ti(Ri). There is a translation T with T(pi)&egr; Ri for each i iff there is a translation T with T(p1) &egr; R = R1∩ R2'∩ R3'∩…∩Rn'. Thus, the required translation T exists iff R is nonempty. For 2 ≤ i ≤ n, the linear system Li′ which defines Ri′ can be found from the system Li in time O(ki) by a simple change of variable applied to each inequality in Li separately. The linear system L consisting of the conjunction L1⋏L2′⋏ L3′⋏…⋏Ln′ to be satisfied by the coordinates of p1 can thus be constructed in time O(k). This system consists of k linear constraints and defines the feasible region of a r variable linear programming problem. The method of N. Megiddo [1] will find a point p in the feasible set or characterize the set as empty in time O(k). The translation T is then defined by the requirement that T(p1) = p.By an optimal translation T of Q we mean one which minimizes (or maximizes) some linear function h of r variables at T(p1). If r = 4, for example, we may wish to locate object O when it first appears in S (that is, to minimize h(x,y,z,t) = t = 0x+0y+0z+1t). By the method outlined above, an optimal translation T may be found in time 0(k) if the feasible set is nonempty.Additional results have been obtained concerning nontranslational motion of the query polyhedron Q and more general matching conditions.