4.1.1. Harris Corner Detection

Harris Corner Detection

Harris corner detection is a corner detection algorithm used to identify corner points in an image where the intensity changes significantly in both directions (i.e., where edges intersect or where there is strong local variation). These points are often good features to track or match between images.

For each pixel $(x,y)$, consider a small window $W$ and compute the sum of squared differences (SSD) between the window and its shifted version:
$E(u,v)={\sum }_{(x,y)\in W}w(x,y)[I(x+u,y+v)-I(x,y){]}^2,$
where $w(x,y)$ is a Gaussian weighting function.

Approximating with a Taylor expansion leads to
$E(u,v)\approx \left[\begin{array}{cc}u& v\end{array}\right]M\left[\begin{array}{c}u\\ v\end{array}\right],$
where $M={\sum }_{(x,y)\in W}w(x,y)\left[\begin{array}{cc}{I}_x^2& I_x{I}_y\\ I_x{I}_y& I_y^2\end{array}\right]$ is the second-moment (structure) matrix.

The eigenvalues ${\lambda }_1,{\lambda }_2$ of $M$ describe intensity variation, such that a point can be a corner (both $\lambda$ large), an edge (one $\lambda$ large), or a flat region (both $\lambda$ small).

To avoid explicit eigenvalue computation, Harris proposed a corner response function $R$,
$R=\det (M)-k(\text{trace}(M){)}^2,$
such that pixels with large $R$ (positive and above threshold) are identified as corners. Here, $k$ is typically between 0.04–0.06.