Affine invariants of generalized polygons and matching under affine transformations

A generalized polygon is an ordered set of vertices. This notion generalizes the concept of the boundary of a polygonal shape because self-intersections are allowed. In this paper we study the problem of matching generalized polygons under affine transformations. Our approach is based on invariants. Firstly we associate an ordered set of complex numbers with each polygon and construct a collection of complex scalar functions on the space of plane polygons. These invariant functions are defined as quotients of the so-called Fourier descriptors, also known as discrete Fourier transforms.Each one of these functions is invariant under similarity transformations; that is, the function associates the same complex number to similar polygons. Moreover, if two polygons are affine related (one of them is the image of the other under an affine transformation), the pseudo-hyperbolic distance between their associated values is a constant that depends only on the affine transformation involved, but independent of the polygons.More formally, given a collection {Z1,Z2,…,Zm}{Z1,Z2,…,Zm} of n-sided polygons in the plane and a query polygon W  , we give algorithms to find all ZℓZℓ such that f(Zℓ)=W+ΔWf(Zℓ)=W+ΔW, where f   is an unknown affine transformation and ΔW=(Δw1,…,Δwn)ΔW=(Δw1,…,Δwn) with |Δwk|≤ρ|Δwk|≤ρ, where ρ is certain tolerance.