Optimum mapping between tesselated parallelograms and tesselated rectangles?

Question

I have a lattice whose points are the vertices of many tessellated parallelograms. Each point is located at $\mathbf{x}=\alpha \mathbf a + \beta \mathbf b$ where $\alpha$ and $\beta$ are integers and $\mathbf {a,b}$ are linearly independent. I would like to (bijectively) map this on to a rectangular lattice $\mathbf{x}'=\lambda \mathbf x + \mu \mathbf y$, where $\lambda,\mu$ are again integers but $\mathbf{x,y}$ are not only linearly independent but also are orthogonal. Except that they are orthogonal, I have freedom to choose any $\mathbf x$ and $\mathbf y$ that I want. Each point should move as little as possible when moved from its place on the parallel lattice to its new place on the rectangular lattice. How can this be done?