How do I prove or disprove:
Smoothing an image with a 3x3 filter using full convolution and then downsampling an image by 2 produces the same result as downsampling an image by 2 and then applying full convolution with a 3x3 smooth filter.

My Thoughts:
It doesn't seem to be true because the information of an image is lost when we downsample the image by 2 first.

Please help me in this regard. Please help me to prove/disprove it mathematically. I really appreciate any help you can provide.


1 Answer 1


Let's rephrase it to a more general question:

Under what assumptions 2 system / operations are Commutative?

We can assure that if 2 system are LTI - Linear Time Invariant they are Commutative.
For more general case, it depends.

In your case, while the linear smoothing is LSI system, the downsampling is not.
It is easy to see even in 1D. Assume the vector: [1, 2, 3, 4, 5, 6, 7, 8]. If we downsample in factor 2 starting from 1 we get: [1, 3, 5, 7].
Yet if we shift the vector and start with 2 we get: [2, 4, 6, 8].
So for sure it is not time invariant.

Now if we apply a blurring of mean of 2 in a running window, we'll get: [1, 3, 5, 7] -> [2, 4, 6] for the first case.

For the 2nd case we'll get [2, 4, 6, 8] -> [3, 5, 7] which is different.

The same will happen in your case, if we blur after the downsampling, we'll be blurring different values in the neighborhood.

Hence you can't just replace them.


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