Polygon Matching and Indexing Under Affine Transformations

Given a collection {Z1, Z2, . . . , Zm} of n-sided polygons in the plane and a query polygon W we give algorithms to find all Zl such that W = f(Zl) with f an unknown similarity transformation in time independent of the size of the collection. If f is a known affine transformation, we show how to find all Zl such that W = f(Zl) in O(n+ log(m)) time. For a pair W,W ′ of polygons we can find all the pairs Zl, Zl′ such that W = f(Zl) and W ′ = f(Zl′) for an unknown affine transformation f in O(m+ n) time. For the case of triangles we also give bounds for the problem of matching triangles with variable vertices, which is equivalent to affine matching triangles in noisy conditions.