Shape-Gain Wrapped Spherical Vector Quantizer

The proposed gain-shape vector quantizer for Gaussian sources uses a wrapped spherical code for the shape quantizer codebook. We impose the constraint that the quantized gain $\hat{g}$ depends only on the true gain $g$ and the quantized shape vector $\hat{S}$ depends only on the true shape vector $S$. This allows the gain and shape quantizers to operate in parallel and independently of each other, and it simplifies the analysis of the distortion. A small performance improvement can be realized by allowing $\hat{g}$ and $\hat{S}$ to each depend on both $g$ and $S$, which is discussed in Section V. The rates $R_s$ and $R_g$ of the shape and gain codebooks, respectively, are defined as the number of bits used to quantize the shape and gain per scalar component of $X\in{\mathbb{R}}^k$. Thus the number of bits used to quantize each $(k-1)$-dimensional shape vector is $kR_s$ and the number of bits used to quantize each scalar gain is $kR_g$. The choice of rates $R_s$ and $R_g$ is discussed in Sections III-B and III-C.

We optimize the gain codebook with the Lloyd-Max algorithm [33,18] using the gain pdf $f_g(r)$ from (1) (no training vectors are needed). Since $f_g(r)$ is strictly log-concave the Lloyd-Max algorithm converges to a globally optimum gain codebook [34,35]. The centroid condition implies that $E[\hat{g}] = E[g]$ and the MSE is $E[g^2] - E[\hat{g}^2]$.

The shape codebook is generated by a wrapped spherical code whose construction is reviewed here (for more details see [5]). Let $\Lambda$ denote a sphere packing in ${\mathbb{R}}^{k-1}$ which has minimum distance $d_\Lambda$ and density $\Delta_\Lambda$. The latitude of a point $X = (x_1, \ldots, x_k) \in \Omega _k$ is defined as $\sin^{-1}(x_k)$, i.e., the angle subtended from the ``equator'' to $X$. Let $-\pi/2 = \alpha_0 < \cdots < \alpha_N = \pi/2$ be a sequence of latitudes, where $N= \left\lceil \pi/\sqrt{d_\Lambda}\right\rceil$ and $\alpha_i = \pi (\frac{i}{N}-\frac{1}{2})$. The $i$th annulus is defined as the set

$$A_i = \{ (x_1,\ldots,x_k) \in \Omega _k ~:~ \alpha_i \le \sin^{-1} x_k < \alpha_{i+1}\}$$

i.e., points between consecutive latitudes (see Figure 4).

Figure 4: $\Omega _k$ is partitioned into annuli.

Let $(x)_+ \equiv \max(0,x)$, and for each $X = (x_1, \ldots, x_k) \in A_i$, let

$$X_L = \arg\min_Z \{\Vert X-Z\Vert : Z = (z_1, \ldots, z_{k-1},\sin\alpha_i) \in \Omega _k \}.$$

i.e., the closest point to $X$ that lies on the border between $A_{i-1}$ and $A_i$ (see Figure 5). Let prime notation denote the mapping from ${\mathbb{R}}^k$ to ${\mathbb{R}}^{k-1}$ obtained by deletion of the last coordinate, so that for example, $X' = (x_1, \ldots, x_{k-1})$. For each $i$, define a one-to-one mapping $h_i$ from $A_i$ to a subset of ${\mathbb{R}}^{k-1}$ by

$$h_i(X) \equiv {X' \over \Vert X'\Vert} \cdot \left(\Vert(X_L)'\Vert - \Vert X_L-X\Vert\right)_+ .$$ (6)

Figure 5: (a) The mapping of a point $X$ under $h_i$. (b) The mapping of many points under $h_i$. The lattice in the plane has similar structure on $\Omega _3$.

The wrapped spherical vector quantizer codebook $\mbox{${\rm W}_\Lambda$}$$$ with respect to a packing $\Lambda$ is defined as

$$\mbox{${\rm W}_\Lambda$} \equiv \bigcup_i h_i^{-1}(\Lambda\setminus \{0\}).$$ (7)

An example of a wrapped spherical vector quantizer in ${\mathbb{R}}^3$ is shown in Figure 6, where the codevectors are the centers of the spherical caps. Table III describes the procedure for using ${\rm W}_\Lambda$ .

Figure 6: A wrapped spherical vector quantizer.

Table I: Algorithmic description of the quantizer ${\rm W}_\Lambda$ .