Revisiting Lattice Tiling Decomposition and Dithered Quantisation

A lattice tiling decomposition induces dual operations: quantisation and wrapping, which map the Euclidean space to the lattice and to one of its fundamental domains, respectively. Applying such decomposition to random variables over the Euclidean space produces quantised and wrapped random variables. In studying the characteristic function of those, we show a ‘frequency domain’ characterisation for deterministic quantisation, which is dual to the known ‘frequency domain’ characterisation of uniform wrapping. In a second part, we apply the tiling decomposition to describe dithered quantisation, which consists in adding noise during quantisation to improve its perceived quality. We propose a non-collaborative type of dithering and show that, in this case, a wrapped dither minimises the Kullback-Leibler divergence to the original distribution. Numerical experiments illustrate this result.