Quantization of the Image

Suppose we have a binary input image of size $(2N+1)\times (2N+1)$as shown in Figure 1. We call the image ``discrete image,'' since it is the quantized version obtained from the continuous image in which the coordinates are represented by real numbers. For simplicity of the discussion, we put the origin of the coordinate system on the center of the image. Let $\Delta s$ denote the sampling interval of the discrete image. $\Delta s=1$ is convenient, and if so, the discrete image consists of $(2N+1)\times (2N+1)$ pixels.

  

Figure 1: The configuration of the input image

The coordinates (x,y) of each point on a continuous figure are quantized, and the point is represented by the pixel whose center is on the discrete position $(\hat{x},\hat{y})$ given by the following equations.

$$\hat{x}=\left\lfloor\frac{x}{\Delta s}+\frac{1}{2}\right\rflo... ...lfloor\frac{y}{\Delta s}+\frac{1}{2}\right\rfloor\cdot\Delta s$$ (1)

$\lfloor \ \rfloor$ is the floor operator and $\lfloor z\rfloor$denotes the maximum integer which is below or equal to z. Since $\Delta s$ denotes the sampling interval for the image quantization, we call $\delta x=\hat{x}-x$ and $\delta y=\hat{y}-y$ the ``quantization errors.'' Obviously, $\vert\delta x\vert\leq \Delta s/2$ and $\vert\delta y\vert\leq \Delta s/2$ are always satisfied.