On quantization with the Weaire-Phelan partition

Until recently, the solution to the Kelvin problem of finding a partition of R³ into equal-volume cells with the least surface area was believed to be tessellation by the truncated octahedron. In 1994, D. Weaire and R. Phelan described a partition that outperformed the truncated octahedron partition in this respect. This raises the question of whether the Weaire-Phelan (WP) partition can outperform the truncated octahedron partition in terms of normalized moment of inertia (NMI), thus providing a counterexample to Gersho´s conjecture that the truncated octahedron partition has the least NMI among all partitions of R³. In this correspondence, we show that the effective NMI of the WP partition is larger than that of the truncated octahedron partition. We also show that if the WP partition is used as the partition of a three-dimensional (3-D) vector quantizer (VQ), with the corresponding codebook consisting of the centroids of the cells, then the resulting quantization error is white. We then show that the effective NMI of the WP partition cannot he reduced by passing it through an invertible linear transformation. Another contribution of this correspondence is a proof of the fact that the quantization error corresponding to an optimal periodic partition is white, which generalizes a result of Zamir and Feder (1996)