Can I reduce the "number of weights" in CNN to 1/3 by restricting the input as greyscale image? In a CNN, does each new filter have different weights for each input channel, or are the same weights of each filter used across input channels?

This question helps me a lot.

Let, I have RGB input image. (3 channels)
Then each filter has n×n weights for one channel.
It means, actually the filter has totally 3×n×n weights.

For channel R, it has own n×n filter.

For channel G, it has own n×n filter.

For channel B, it has own n×n filter.

After inner product, add them all to make one feature map.
Am I right?

And then, my question starts here.
For some purpose, I will only use greyscale images as input.
So the input images always have the same values for each RGB channel.

Then, can I reduce the number of weights in the filters?
Because in this case, using three different n×n filters and adding them is same with using one n×n filter that is the summation of three filters.

Does this logic hold on a trained network?
I have a trained network for RGB image input, but it is too heavy to run in real time.
But I only use the greyscale images as input, so it seems I can make the network less heavy (theoretically, almost 1/3 of original).

I'm quite new in this field, so detailed explanations will be really appreciated.
Thank you.
After inner product, add them all to make one feature map. Am I right?


yes, you are right.


  Then, can I reduce the number of weights in the filters? Because in this case, using three different n×n filters and adding them is same with using one n×n filter that is the summation of three filters.


If you have transformed the image into greyscale then you no longer need 3 filters. You should retrain your model on greyscale images. In a greyscale image the value of each pixel is a single sample representing only an amount of light (the light intensity).

The network will run faster if that is the only architectural change you make, but keep in mind that by converting the image to greyscale you will lose information and probably some of the predictive power of your network.