therefore, the properties of quantization error are first investigated analytically. The result will be used to develop the binary hashing algorithm we propose.

Most of linear binary hashing methods consist of translation operation and linear transformation. $$\begin{aligned}
 \label{eq:binhash}
&\bm{b}(\bm{x}) \equiv \text{sgn} (A (\bm{x} - \bm{t})), \\
\bm{x}, \bm{t} \in &\mathbb{R}^n, \bm{b}(\bm{x}) \in \{-1,1\}^m, A \in \mathbb{R}^{m \times n}. \nonumber\end{aligned}$$ It is assumed that a translation is mean centering in the following discussion.

Quantization error is defined as the sum of squared Euclidean distance between an original feature vector and its binarized vector. $$\begin{aligned}
 \label{eq:e_q_emp}
E_{q} \equiv \frac{1}{N} \sum_i |\bm{x}_i - \bm{b}(\bm{x}_i)|^2,\end{aligned}$$ where $N$ is the number of data points. When the data is distributed as arbitrary distribution function $p(\bm{x})$, it is possible to write down mean quantization error. If two-dimensional data are assumed, $$\begin{aligned}
E_{q}^{N\to \infty} &= \int_0^{\infty} \!\!\! dx_1 \int_0^{\infty}\!\!\! dx_2 \left[(x_1 - 1)^2 + (x_2 - 1)^2\right] p(\bm{x}) \nonumber \\
&+ \int_{-\infty}^{0} \!\!\! dx_1 \int_0^{\infty}\!\!\! dx_2 \left[(x_1 + 1)^2 + (x_2 - 1)^2\right]p(\bm{x}) \nonumber \\
&+ \int_{-\infty}^{0}\!\!\! dx_1 \int_{-\infty}^{0}\!\!\! dx_2 \left[(x_1 + 1)^2 + (x_2 + 1)^2\right]p(\bm{x}) \nonumber \\
&+ \int_{0}^{\infty}\!\!\! dx_1 \int_{-\infty}^{0}\!\!\! dx_2 \left[(x_1 - 1)^2 + (x_2 + 1)^2\right]p(\bm{x}).\end{aligned}$$ Then it is generally calculated as follows; 