correlation and convolution

Sliding a small weight grid over an image and summing at every position; convolution flips the grid first.

2D correlation slides a small weight grid (a kernel) $H$ over an image $I$ and computes $(I\star H)(x,y) = \sum_{i,j} I(x+i,y+j)\,H(i,j)$ at every position. Convolution, $(I*H)(x,y)$ , computes the same kind of sum but reads the kernel flipped $180°$ first. Most deep-learning frameworks call their layers "convolutional" while actually running correlation -- a naming mismatch between vocabulary and the mathematical operation, not a property of either operation itself.

Examples

A $3\times3$ box kernel (every entry $1/9$ ) produces the identical output under correlation and convolution, since it looks the same after a $180°$ flip.