形状特征检索：不变矩阵法及 Hu 不变矩公式

形状的不变矩阵法是一种用于描述和比较形状特征的方法，它基于形状的几何特征，通过计算一组不变矩阵来描述形状的特征。其中，最常用的不变矩阵是 Hu 不变矩，其公式如下：

$H_u = \begin{bmatrix} \eta_{20}+\eta_{02} & (\eta_{20}-\eta_{02})^2+4\eta_{11}^2 \ 3\eta_{21}-3\eta_{03} & (\eta_{30}-3\eta_{12})^2+(3\eta_{21}-\eta_{03})^2 \ (\eta_{30}+\eta_{12})^2+(\eta_{21}+\eta_{03})^2 & (\eta_{30}-3\eta_{12})(\eta_{30}+\eta_{12})[(\eta_{30}+\eta_{12})^2-3(\eta_{21}+\eta_{03})^2]+(3\eta_{21}-\eta_{03})(\eta_{21}+\eta_{03})[3(\eta_{30}+\eta_{12})^2-(\eta_{21}+\eta_{03})^2] \ (\eta_{20}-\eta_{02})[(\eta_{30}+\eta_{12})^2-(\eta_{21}+\eta_{03})^2]+4\eta_{11}(\eta_{30}+\eta_{12})(\eta_{21}+\eta_{03}) & (\eta_{30}-3\eta_{12})(\eta_{30}+\eta_{12})[(\eta_{30}+\eta_{12})^2-3(\eta_{21}+\eta_{03})^2]+(3\eta_{21}-\eta_{03})(\eta_{21}+\eta_{03})[3(\eta_{30}+\eta_{12})^2-(\eta_{21}+\eta_{03})^2] \ \end{bmatrix}$

其中，$\eta_{pq}$ 表示形状的归一化矩，计算公式如下：

$\eta_{pq} = \frac{\mu_{pq}}{\mu_{00}^{1+\frac{p+q}{2}}}$

其中，$\mu_{pq}$ 表示形状的矩，计算公式如下：

$\mu_{pq} = \sum_{i=1}^{n}\sum_{j=1}^{m}x_{ij}i^pj^q$

其中，$x_{ij}$ 表示形状在像素 $(i,j)$ 处的灰度值，$n$ 和 $m$ 分别表示形状的宽度和高度。