The auto correlation function for Harris Corner Detector is defined as

$E(u, v) = \Sigma_{x, y} w(x, y) [I(x + u, y + v) - I(x, y)]^ 2$

The idea here is that we will consider a patch around a point in the image and we will displace this patch to a small extent to see the variation in the $E$. If the variation is large in all directions, the point/ patch can be considered as a corner. We also use a window or weight function, for smoothing or averaging. Here $(u, v)$ are displacement and $(x, y)$ are pixel coordinates/ indices. I am trying to understand what range of values can these variables have and how this auto correlation function works. The $I$ indicates image and $w$ indicates window/ weight function.

Let's consider an image of size 10 x 10 and patches of size 3 x 3. The value of both $u$ and $v$ can be in the range {-1, 0, 1}. Let us say we are trying to see if a point (3, 3) and associated patch is a corner or not. In this case, the $x$ and $y$ will have values in the range {2, 3, 4}. I am making an assumption that $x$ and $y$ will always belong to a set of original image pixel coordinates. However, for a particular patch, this set will have only 3 values since the patch size is 3.

Q1) Does $u$ and $v$ have the right range of values?

Q2) Is the assumption regarding values of $x$ and $y$ correct?

Q3) Are we evaluating a point or a patch to be called a corner?

I believe that my story regarding the values acquired by $x$, $y$, $u$ and $v$ are consistent upto this point. However, the values of $x$ and $y$ does not make sense when we consider the window or weight function.

Q4) Shouldn't weight function have another set of variables to indicate its indices or am I selecting the values for $x$ and $y$ wrong?


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