Backpropagation on a convolutional layer

Online tutorials describe in depth the convolution of an image with a filter, etc; However, I have not seen one that describes the backpropagation on the filter (at least visually).

First let me try to explain how I understand backpropagation on a fully connected network.

For example, the derivative of the $Error$ with respect to $W_1$ is the following:

$$
frac{partial Error}{partial W_1} = frac{partial Error}{partial HA_1} frac{partial HA_1}{partial H_1} frac{partial H_1}{partial W_1}
$$

The last partial derivative is the most interesting one in this case … and it is equal to the value of the first input (Single value).

$$
frac{partial H_1}{partial W_1} = I_1
$$

EDITED

The original question was how does one perform backpropagation on a convolutional layer – for example $$frac{partial Error}{partial W_1} = ?$$

The convolutional layer as described online.

$$
G_1 = V_1W_1 + V_2W_2 + V_4W_3 + V_5W_4 \
G_2 = V_2W_1 + V_3W_2 + V_5W_3 + V_6W_4 \
G_3 = V_4W_1 + V_5W_2 + V_7W_3 + V_8W_4 \
G_4 = V_5W_1 + V_6W_2 + V_8W_3 + V_9W_4
$$

Notice that there are groups of pixels that share the same weights ($W$s), so I can picture the equations above as follows:

So, applying the chain rule as in the first example, we get the following:

$$
frac{partial Error}{partial W_1} = frac{partial Error}{partial G_1}frac{partial G_1}{partial W_1} + frac{partial Error}{partial G_2}frac{partial G_2}{partial W_1} + frac{partial Error}{partial G_3}frac{partial G_3}{partial W_1} + frac{partial Error}{partial G_4}frac{partial G_4}{partial W_1}
$$

And the derivatives of interest …
$$
frac{partial G_1}{partial W_1} = V_1 \
frac{partial G_2}{partial W_1} = V_2 \
frac{partial G_3}{partial W_1} = V_4 \
frac{partial G_4}{partial W_1} = V_5
$$

That’s it! Chain rule all the way.

I don’t like to answer my own question – so if you leave some feedback or tell me I am wrong – I ‘ll give you the credit.