Multi-Channel Convolution

The real convolution layer: many input channels in, many output channels out.

Input x has shape (C_in, H, W). The kernel bank has shape (C_out, C_in, kH, kW) and the bias is (C_out,).

For each output channel o and each spatial position, sum the elementwise product over every input channel and the whole window, then add the bias:

out[o, i, j] = bias[o] + sum over c, u, v of x[c, i*s+u, j*s+v] * kernel[o, c, u, v]

Zero pad the spatial dimensions by padding on all four sides, never the channel axis. Output shape is

(C_out, (H + 2p - kH) // s + 1, (W + 2p - kW) // s + 1)

Two things to get right:

• Sum over input channels, do not keep them. Each output channel is a single 2-D map, not a stack of C_in maps. Each of the C_out filters sees the whole depth of the input at once, which is how a layer combines edges from different colour channels into one feature.
• One bias per output channel, added once per position, not per input channel.

This is still cross correlation, so the kernel is not flipped. If the padded input is smaller than the kernel, return an array of shape (C_out, 0, 0).

Function signature

def conv2d_multichannel(x, kernels, bias, stride=1, padding=0):

Example

Input

x = np.array([[[1, 2], [3, 4]],
              [[10, 20], [30, 40]]])     # (C_in=2, 2, 2)
kernels = np.ones((1, 2, 2, 2))             # (C_out=1, C_in=2, 2, 2)
bias = np.array([0.5])
conv2d_multichannel(x, kernels, bias)

Output

array([[[110.5]]])

Reasoning

There is one window position. The kernel of ones sums both channels: 1 + 2 + 3 + 4 = 10 and 10 + 20 + 30 + 40 = 100. Add the bias once to get 110.5.

Constraints and notes

• Pad the spatial axes only, never the channel axis.
• Add one bias per output channel, once per position.
• If the padded input is smaller than the kernel, return shape (C_out, 0, 0).