Cross-correlation vs. Convolution

In mathematics and physics literature the process we just used, that is placing the kernel (filter) on the top - left of the image and multiply element-wise and then move the kernel one step right for the next element until we find all the values is called as cross-correlation.

In classical Convolution before sliding we flip the kernel both horizontally and vertically. If this is the original kernel :

$$\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$$

then we flip it :

$$\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]$$

and then perform the same operations as cross-correlation

In deep learning the network learns the kernel weights automatically. There is no need to flip anything because if the flipped version were better, gradient descent would simply learn that version.

So in short :

• Cross - correlation uses the kernel as it is
• Convolution flips the kernel and then uses it
• CNNs use the kernel as it is but call it convolution.