That’s a good point. I didn’t think about the mixing of the interior background color. And yeah, I agree that that’s a weird property, and would guess that depending on how you configure the rest of the parameters, could lead to very strange results.
Actually, I was thinking about this as I went to sleep last night, and I think I was wrong about the opacity being a factor. The convolution kernels will be normalized, which means that their composition will also be normalized. That means that the only parameter to adjust is the radius.
As a side note, it’s kind of interesting that the first property “still a Gaussian blur” is easiest to see in the frequency domain as a product, while the second property “still integrates to 1” is easier to see when thinking in the spatial domain as a convolution.
And one last tidbit my brain wandered to: you might notice that the product of two Gaussian functions results in a new Gaussian narrower than the originals. Yet the blur radius in the spatial domain is obviously wider. This is because the Fourier transform of a wider Gaussian is a narrower Gaussian and vice versa, and that is a special case of the uncertainty principle.
God I miss image processing.