HARRIS CORNER DETECTOR

This detector relies on a continuous reformulation of the Moravec neighbor differences. A generic infinitesimal shift from the point location can be described as

$$E(\Delta x,\Delta y)=\sum _{x,y}w(x,y)(I(x+\Delta x,y+\Delta y)-I(x,y){)}^2$$

With $w(x,y)$ as a weight function (gaussian function)

Due to the fact that it’s a continuous formulation intensity function can be approximated with it’s Taylor polynomial form

$$I(x+\Delta x,y+\Delta y)-I(x,y)\simeq \frac{\delta I(x,y)}{\delta x}\Delta x+\frac{\delta I(x,y)}{\delta y}\Delta y=I_x(x,y)\Delta x+I_y(x,y)\Delta y$$

So the edge function can be described as

$$E(\Delta x,\Delta y)=\left[\begin{array}{cc}\Delta x& \Delta y\end{array}\right]\left[\begin{array}{cc}{\sum }_{x,y}w(x,y)I_x(x,y{)}^2& {\sum }_{x,y}w(x,y)I_x(x,y)I_y(x,y)\\ {\sum }_{x,y}w(x,y)I_x(x,y)I_y(x,y)& {\sum }_{x,y}w(x,y)I_y(x,y{)}^2\end{array}\right]\left[\begin{array}{c}\Delta x\\ \Delta y\end{array}\right]=$$ $$\left[\begin{array}{c}\Delta x\\ \Delta y\end{array}\right]M\left[\begin{array}{cc}\Delta x& \Delta y\end{array}\right]$$

Where the $M$ matrix encode the characteristics of the structure around the pixel, also $M$ is real and symmetric so the correspondent diagonal matrix can always be computed from the eigenvectors

$$M=R\left[\begin{array}{cc}{\lambda }_1& 0\\ 0& {\lambda }_2\end{array}\right]R^T$$

the coefficients ${\lambda }_1,{\lambda }_2$ can be used to distinguish corners

But Computation of the eigenvector is expensive so a better approximation can be deployed

$$C=\det (M)-ktr(M{)}^2={\lambda }_1{\lambda }_2-k({\lambda }_1+{\lambda }_2{)}^2$$

And with this corner can be detected as

So in conclusion the corner detection algorithm can be described as follows

flowchart TD
A[compute C at each point of the image]
B[filter points based on threshold T]
C[detect points that are local maxima NMS]
A --> B
B --> C

In order to weights better the points in the neighborhood the weight function $w(x,y)$ used is a Gaussian function

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