Convolution & 2D spatial filtering

This image is just a dark-to-light edge. Slide the window across it. The response is near zero on either flat side, but spikes right where the edge actually is — the same small filter, reused at every position, looking for the exact same pattern everywhere.

A convolution slides a small filter (the kernel) across an image and, at every position, computes one number — the sum of the kernel multiplied elementwise with the patch underneath it:

• $\text{feature}(r,c)$ — the output feature-map value at row $r$, column $c$.
• $\text{kernel}(i,j)$ — the learned filter weight at offset $(i,j)$ within the small window.
• $\text{image}(r+i,\, c+j)$ — the input image's pixel value at the position the kernel currently overlaps.
• $r, c$ — the row and column of the output position being computed.
• $i, j$ — the row and column offsets that range over the kernel's own window.

1. One kernel, reused everywhere

   The same 9 kernel numbers are reused at all 16 positions here — the identical filter weights get multiplied against every patch of the image as the window slides across it.

2. Why sharing makes sense

   That's the whole idea: a pattern worth detecting in one corner of an image is worth detecting anywhere else in it too, so the filter doesn't need its own separate weights for every location.

Drag the row slider alone — the response doesn't change at all. This particular filter only cares about columns, because the edge it's built to find runs vertically. Drag the column slider and the response swings from $0$ to $3$ and back to $0$, tracing the edge exactly.

1. Window position (0, 1) — on the edge

   The patch under the kernel is columns 1-3 of the image: two dark pixels ($0$) and one light pixel ($1$), in every one of the 3 rows. The kernel row $(-1, 0, 1)$ gives $(-1)(0) + (0)(0) + (1)(1) = 1$ per row, times 3 rows $= 3$ — the maximum possible response, because the patch is a perfect textbook edge.

2. Window position (0, 3) — off the edge

   One column over, the patch is three light pixels: $(-1)(1)+(0)(1)+(1)(1) = 0$ per row — no edge in that patch, no response.