Derive the filter mask that corresponds to the double filtering process

Question

There are two mask filters,

f1 = np.array([[1, 1, 1],
               [1, 1, 1],
               [1, 1, 1]]) / 9


f2 = np.array([[1, 1, 1],
               [1, -8, 1],
               [1, 1, 1]])

I filter an input image using f2 and then filter the result with f1, how can I derive the filter mask that corresponds to the double filtering process?

I want to get a combine filter mask.

Answer 1

The convolution is associative: $(I * f_1) * f_2 = I * (f_1 * f_2)$.

In other words, applying the two filters in sequence to an image is equivalent to applying the convolution of the two filter kernels as a single filter to the image.

Simply convolve f1 with f2, making sure to extend the domain with zeros so that the output is not cropped.