Downsampling and Gradient filtering

Question

Are the following sequence of operations the same -

• Down-sampling filtering followed by gradient filtering with kernel [-1 0 1]
• Gradient filtering with kernel [-1 0 1] followed by down-sampling

Could you point me to some math which can answer this question.

Intuitively, I am thinking that these operations are the same. That is because down-sampling can be loosely modeled as low-pass filtering and since convolution is associative, I guess they should produce similar results. But I am not able to quantitatively prove this to be a fact.

Answer 1

As the commenters have stated, this is not the same. The relationships between down/up-sampling and filtering are formally described by the noble identities.

In your case down-sampling followed by filtering with [-1 0 1] is equivalent to filtering with [-1 0 0 0 1] first and than downsampling.

In any case you need to make sure your don't alias before down-sampling the picture.