Cross correlation¶
There is a close relationship between cross correlation and convolution.
The convolution operator fo a 2D image H and a 2D kernel F is defined as:
\[\begin{split}
G = H * F \\
G[i,j]= \sum_{u=-k}^k \sum_{v=-k}^k H[u,v]F[i-u, j-v]
\end{split}\]
The cross correlation is defined as:
\[\begin{split}
G = H \bigotimes F \\
G[i,j]= \sum_{u=-k}^k \sum_{v=-k}^k H[u,v]F[i+u, j+v]
\end{split}\]
The difference between them is that:
convolution is equivalent to flipping the filter in booth dimensions (bottom to top, right to left) and apply cross correlation.
if the kernel is symmetric the result is the same
in general most ml library use cross correlation and call it convolution.
