The Sliding Template
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Convolution slides a small template of weights across the grid, computing the same weighted sum at every position — reporting, for each neighborhood, how strongly it matches the template's pattern.
One line of numbers proves the idea. Take a row with a hard boundary in it — 0, 0, 0, 9, 9, 9 — and slide the two-cell template (−1, +1), which asks: how much brighter is the right cell than the left? Position by position, the outputs read 0, 0, 9, 0, 0. Uniform stretches score zero; the edge announces itself with a spike. One template, arithmetic only, and the grid's first secret — where things change — has been extracted. Reverse the row to 9, 9, 9, 0, 0, 0 and the spike returns as negative 9: the template reports direction of change, dark-to-bright positive, bright-to-dark negative.
The local constraint is the wisdom — nearby pixels are related, far ones mostly are not, so the machinery is told to look locally.