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 . We are going to sweep a window (with displacements in the x direction and in the right direction) and will calculate the variation of intensity.
• 

  where:

  • is the window at position
  • is the intensity at
  • is the intensity at the moved window
• 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:

  

• Using Taylor expansion:

  

• Expanding the equation and cancelling properly:

  

• Which can be expressed in a matrix form as:

  

• Let’s denote:

  

• So, our equation now is:

  

• A score is calculated for each window, to determine if it can possibly contain a corner:

  

  where:

  • det(M) =
  • trace(M) =

  a window with a score greater than a certain value is considered a “corner”

中文说明：

Harris角点算法实现

根据上述讨论，可以将Harris图像角点检测算法归纳如下，共分以下五步：

1. 计算图像I(x,y)I(x,y)在XX和YY两个方向的梯度Ix、IyIx、Iy。

Ix=?I?x=I?(?1 0 1)，Iy=?I?x=I?(?1 0 1)TIx=?I?x=I?(?1 0 1)，Iy=?I?x=I?(?1 0 1)T

2. 计算图像两个方向梯度的乘积。

I2x=Ix?Iy，I2y=Iy?Iy，Ixy=Ix?IyIx2=Ix?Iy，Iy2=Iy?Iy，Ixy=Ix?Iy

3. 使用高斯函数对I2x、I2y和IxyIx2、Iy2和Ixy进行高斯加权（取σ=1σ=1），生成矩阵MM的元素A、BA、B和CC。

A=g(I2x)=I2x?w，C=g(I2y)=I2y?w，B=g(Ix,y)=Ixy?wA=g(Ix2)=Ix2?w，C=g(Iy2)=Iy2?w，B=g(Ix,y)=Ixy?w

4. 计算每个像素的Harris响应值RR，并对小于某一阈值tt的RR置为零。

R={R:detM?α(traceM)2<t}R={R:detM?α(traceM)2<t}

5. 在3×33×3或5×55×5的邻域内进行非最大值抑制，局部最大值点即为图像中的角点。