Convolution over Multi-Channel Input

The problem, from first principles

01-single-filter-conv2d works on one 2D grid of numbers. A real image isn't one grid — a color photo is three (red, green, blue), and every layer past the first in a real CNN outputs dozens or hundreds of feature channels stacked on top of each other. A filter that only understands one channel can't use color information at all, let alone combine features from many channels the way a real network needs to.

The fix is to give the kernel its own 2D slice per input channel, run each channel through its own slice, and combine the per-channel results into one number per position — the filter looks at every channel simultaneously and produces a single, richer output that already accounts for all of them.

From theory to code

Theory reduces the multi-channel case to something already built: run 01-single-filter-conv2d's convolution independently on each channel, using that channel's own kernel slice, then add the per-channel outputs together into one combined result.

Implement conv2d_multi_channel(image, kernel) against that reasoning, reusing conv2d_single_filter rather than reimplementing the sliding-window logic per channel.

Constraints

• image: shape (C_in, H, W).
• kernel: shape (C_in, kH, kW) — one 2D filter per input channel.
• Output: shape (H - kH + 1, W - kW + 1), a single 2D result, summed across channels.
• C_in = 1 must reduce exactly to 01-single-filter-conv2d's behavior.
• image and kernel are never modified in place.

Hints

Open one at a time. Each gives away a little more than the last.