Definition (Vector Quantizer)

A $k$-dimensional $N$-point vector quantizer (VQ) is a mapping $Q:\mathbb{R}^{k}\to \mathcal{C}$ where $\mathcal{C}=\{ \mathbf{c}_{1},\dots,\mathbf{c}_{N} \}\subset \mathbb{R}^{k}$

• $\mathcal{C}$ is called the codebook
• $\mathbf{c}_{1},\dots,\mathbf{c}_{N}$ are the reproduction points
• $R_{i}=\{ \mathbf{x}:Q(\mathbf{x})=\mathbf{c}_{i} \},\quad i=1,\dots,N$ are the quantizer cells which form a partition of $\mathbb{R}^{k}$

Definition (VQ Distortion)

For a random vector $\mathbf{X}=(X_{1},\dots,X_{k})^{T}$, the distortion of $Q$ is $$D(Q)=\sum_{j=1}^{N}\int\limits _{R_{j}}d(\mathbf{x},\mathbf{c}_{j})f(\mathbf{x}) \, d\mathbf{x}$$

Definition (Weighted squared error)

The weighted squared error is a distortion measure s.t. $$d(\mathbf{x},\mathbf{y})=(\mathbf{x}-\mathbf{y})^{T}\mathbf{W}(\mathbf{x}-\mathbf{y})$$where $\mathbf{W}$ is symmetric and positive definite $k\times k$ matrix.