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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.

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Definitely not the same - the gradient filter kernels are not equivalent as they are being applied at different sample rates. You would need to re-resample the gradient filter kernel in one of the two cases. –  Paul R Jan 7 '13 at 12:10
    
given that I have a 1080p image, would be easier to resample the signal or the filter? –  subzero Jan 7 '13 at 12:54
    
It really depends on what you're trying to achieve. You haven't explained the motivation for either the down-sampling or the gradient filter. –  Paul R Jan 7 '13 at 13:00
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1 Answer

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.

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