harris推导，参见opencv文档

Let’s look for corners. Since corners represents a variation in the gradient in the image, we will look for this “variation”.
Consider a grayscale image $I$ . We are going to sweep a window $w(x,y)$ (with displacements $u$ in the x direction and $v$ in the right direction) $I$ and will calculate the variation of intensity.
$E(u,v) = \sum _{x,y} w(x,y)[ I(x+u,y+v) - I(x,y)]^{2}$

where:
- $w(x,y)$ is the window at position $(x,y)$
- $I(x,y)$ is the intensity at $(x,y)$
- $I(x+u,y+v)$ is the intensity at the moved window $(x+u,y+v)$
Since we are looking for windows with corners, we are looking for windows with a large variation in intensity. Hence, we have to maximize the equation above, specifically the term:

$\sum _{x,y}[ I(x+u,y+v) - I(x,y)]^{2}$
Using Taylor expansion:

$E(u,v) \approx \sum _{x,y}[ I(x,y) + u I_{x} + vI_{y} - I(x,y)]^{2}$
Expanding the equation and cancelling properly:

$E(u,v) \approx \sum _{x,y} u^{2}I_{x}^{2} + 2uvI_{x}I_{y} + v^{2}I_{y}^{2}$
Which can be expressed in a matrix form as:

$E(u,v) \approx \begin{bmatrix} u & v \end{bmatrix} \left ( \displaystyle \sum_{x,y} w(x,y) \begin{bmatrix} I_x^{2} & I_{x}I_{y} \\ I_xI_{y} & I_{y}^{2} \end{bmatrix} \right ) \begin{bmatrix} u \\ v \end{bmatrix}$
Let’s denote:

$M = \displaystyle \sum_{x,y} w(x,y) \begin{bmatrix} I_x^{2} & I_{x}I_{y} \\ I_xI_{y} & I_{y}^{2} \end{bmatrix}$
So, our equation now is:

$E(u,v) \approx \begin{bmatrix} u & v \end{bmatrix} M \begin{bmatrix} u \\ v \end{bmatrix}$
A score is calculated for each window, to determine if it can possibly contain a corner:

$R = det(M) - k(trace(M))^{2}$

where:
- det(M) = $\lambda_{1}\lambda_{2}$
- trace(M) = $\lambda_{1}+\lambda_{2}$
a window with a score $R$ greater than a certain value is considered a “corner”