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 $ k/2 $-dimensional lattices $ D_{k/2} $ and $ A^{*}_{k/2} $ 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 $ D_{k/2} $ lattice and quadratic for the $ A^{*}_{k/2} $ 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 $ D_4 $ 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 $ A^{*} $ 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.

中文翻译：

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基于扁平花托的形状增益矢量量化方法

在本文中，我们提出了一种用于高斯源的矢量量化框架，该框架结合了平面花托层上的球形代码以及形状和增益技术。回顾了托里层球形编码的基本概念，并通过利用$ k / 2 $维格$ D_ {k / 2} $和$ A ^ {*} _ {k / 2 } $作为其原像。通过使用Lloyd-Max算法针对给定的速率，对增益进行标量量化器优化。量化过程的计算复杂度由晶格解码过程决定，该过程对于$ D_ {k / 2} $晶格是线性的，而对于$ A ^ {** __ {k / 2} $晶格则是平方的。详细介绍了所提出的量化器，并以SNR为量化速率的函数表示了一些数值结果，单位维数。结果表明，当速率等于每维1位时，由$ D_4 $晶格设计的量化器的性能优于以前的记录。对于所有测试的速率，这些量化器的性能也优于由双点阵A ^ {*} $设计的量化器。通常，两个提议的框架在速率失真函数的2 dB之内执行，考虑到它们的低计算复杂性，这可能是一个很好的权衡。