Two-Dimensional Image Rotation

Abstract

We study the problem of approximating a rotation of the plane, α : R ² → R ² α(x,y) = (x cos θ + y sin θ, y cos θ − x sin θ), by a bijection β: Z ² → Z ². We show by an explicit construction that one may choose β so that $sup_{z \in Z^2 } \left| {\alpha \left( z \right) - \beta \left( z \right)} \right| \leqslant \frac{1} {{\sqrt 2 }}\frac{{1 + r}} {{\sqrt {1 + r^2 } }},$. Where r = tan(θ/2). The scheme is based on those invented and patented by the second author in 1994.