A shape-gain approach for vector quantization based on flat tori

In this paper we present a vector quantization framework for Gaussian sources which combines a spherical code on layers of flat tori and the shape and gain technique. The basic concepts of spherical codes in tori layers are reviewed and two constructions are presented for the shape by exploiting the \begin{document}$ k/2 $\end{document} -dimensional lattices \begin{document}$ D_{k/2} $\end{document} and \begin{document}$ A^{*}_{k/2} $\end{document} as its pre-image. A scalar quantizer is optimized for the gain by using the Lloyd-Max algorithm for a given rate. The computational complexity of the quantization process is dominated by the lattice decoding process, which is linear for the \begin{document}$ D_{k/2} $\end{document} lattice and quadratic for the \begin{document}$ A^{*}_{k/2} $\end{document} lattice. The proposed quantizer is described in details and some numerical results are presented in terms of the SNR as a function of the quantization rate, in bits per dimension. The results show that the quantizer designed from the \begin{document}$ D_4 $\end{document} lattice outperform previous records when the rate is equal to 1 bit per dimension. These quantizer also outperform the quantizers designed from the dual lattice \begin{document}$ A^{*} $\end{document} for all rates tested. In general the two proposed frameworks perform within 2 dB of the rate distortion function, which may be a good trade-off considering their low computational complexity.