形状特征检索：形状不变矩阵法公式详解

形状的不变矩阵法是一种基于形状特征的检索方法，其公式如下：

计算形状的几何中心：

$$\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i,\ \bar{y} = \frac{1}{n}\sum_{i=1}^{n} y_i$$

其中，$n$为形状的点数，$(x_i,y_i)$为形状上的点坐标。

将形状的坐标系平移到几何中心：

$$u_i = x_i - \bar{x},\ v_i = y_i - \bar{y}$$

计算形状的二阶中心距：

$$m_{20} = \frac{1}{n}\sum_{i=1}^{n} u_i^2,\ m_{02} = \frac{1}{n}\sum_{i=1}^{n} v_i^2,\ m_{11} = \frac{1}{n}\sum_{i=1}^{n} u_i v_i$$

计算形状的不变矩：

$$\begin{aligned}\phi_1 &= m_{20} + m_{02} \phi_2 &= (m_{20} - m_{02})^2 + 4m_{11}^2 \phi_3 &= (m_{30} - 3m_{12})^2 + (3m_{21} - m_{03})^2 \phi_4 &= (m_{30} + m_{12})^2 + (m_{21} + m_{03})^2 \phi_5 &= (m_{30} - 3m_{12})(m_{30} + m_{12})[(m_{30} + m_{12})^2 - 3(m_{21} + m_{03})^2] + (3m_{21} - m_{03})(m_{21} + m_{03})[3(m_{30} + m_{12})^2 - (m_{21} + m_{03})^2] \phi_6 &= (m_{20} - m_{02})[(m_{30} + m_{12})^2 - (m_{21} + m_{03})^2] + 4m_{11}(m_{30} + m_{12})(m_{21} + m_{03})\end{aligned}$$

其中，$m_{pq}$表示形状的$p+q$阶中心距。

归一化不变矩：

$$\begin{aligned}\eta_{11} &= \frac{\phi_1}{\phi_1^2 + \phi_2^2} \eta_{12} &= \frac{\phi_2}{\phi_1^2 + \phi_2^2} \eta_{21} &= \frac{\phi_3}{\phi_1^2 + \phi_2^2} \eta_{22} &= \frac{\phi_4}{\phi_1^2 + \phi_2^2} \eta_{31} &= \frac{\phi_5}{(\phi_1^2 + \phi_2^2)^{5/2}} \eta_{32} &= \frac{\phi_6}{(\phi_1^2 + \phi_2^2)^{5/2}}\end{aligned}$$

其中，$\eta_{ij}$为归一化不变矩。