Convolution

Only the overlapping spot gets counted

Convolution never looks at the whole photo at once — only as much as the small tile covers. If the tile is three squares by three squares, it's only looking at nine points of the photo at any one time.

The calculation itself is simple, too. Pair up the brightness of each overlapping point with the number written on the tile at that same spot, multiply them, and add up all the results. When a bright point lands where the tile has a big number, the sum grows large; when they don't line up, the sum stays small. The single number that comes out becomes a score for "how clearly that pattern showed up right here."

This calculation is exactly the same at every spot. Top left or bottom right, same tile, same method. Treating every spot identically turns out to be a big advantage later on.

It slides one step at a time, all the way through

Once a spot is scored, the tile slides sideways. It usually moves one square at a time, so the spot it just looked at and the new spot it's about to look at mostly overlap. Sliding this way, with overlap, means a pattern straddling two spots never gets missed.

You can also choose how many squares to slide at a time. Slide one square and you scan densely, but the result table comes out the same size as the original; slide two squares and the number of scans is cut in half, but the result table shrinks to half its size too. When a photo is large and computation needs to be conserved, taking bigger strides is an option.

The wider the tile, the wider the range it covers at once. But make the tile too big without limit and both the values to remember and the computation grow together. That's why a small tile scanned many times is the more common choice — stacking two small tiles ends up covering a wide range anyway, at a fraction of the values.

The edges need special handling

Moving the tile only within the photo's interior creates a problem. The points right at the edge never get a turn at the tile's center, so they end up under-counted, and the result table comes out slightly smaller than the original. Let that loss stack up across many stages and edge information keeps eroding away.

A common fix is to wrap a ring of empty squares around the outside of the photo before scanning — much like adding a mat around a picture in a frame. That lets edge points sit at the tile's center too, and keeps the result table the same size as the original.

Scanning leaves behind a single table

One score from one spot means nothing by itself. It only starts to mean something once scanning finishes and the scores line up in their original arrangement, because exactly where a pattern was caught stays intact as position.

Prepare several tiles and scan with each one separately, and you get that many separate tables. A table made by a tile hunting for vertical lines, a table made by a tile hunting for horizontal lines, stacked up side by side. The next stage receives this bundle of tables as if it were a new photo and scans it again. Stacking scans one on top of another is exactly why this calculation ends up handling bigger and bigger patterns — it's the reason this operation gets piled into layers.