matrix multiplication - How is a convolution calculated on an image with three (RGB) channels?

Question

Say we have a single channel image (5x5)

A = [ 1 2 3 4 5
      6 7 8 9 2
      1 4 5 6 3
      4 5 6 7 4
      3 4 5 6 2 ]

And a filter K (2x2)

K = [ 1 1
      1 1 ]

An example of applying convolution (let us take the first 2x2 from A) would be

1*1 + 2*1 + 6*1 + 7*1 = 16

This is very straightforward. But let us introduce a depth factor to matrix A i.e., RGB image with 3 channels or even conv layers in a deep network (with depth = 512 maybe). How would the convolution operation be done with the same filter ? A similiar work out will be really helpful for an RGB case.

Answer 1

Lets say we have a 3 Channel (RGB) image given by some matrix A

    A = [[[198 218 227]
          [196 216 225]
          [196 214 224]
          ...
          ...
          [185 201 217]
          [176 192 208]
          [162 178 194]]

and a blur kernal as

    K = [[0.1111, 0.1111, 0.1111],
         [0.1111, 0.1111, 0.1111],
         [0.1111, 0.1111, 0.1111]]

    #which is actually 0.111 ~= 1/9

The convolution can be represented as shown in the image below

As you can see in the image, each channel is individually convoluted and then combined to form a pixel.