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How DCT-coeffitients change after shifting the image in the spatial domain?

Suppose we have an image I of dimentions (8*n)x(8*n), and let m=n-1. Consider a cropped image J of dimentions (8*m)x(8*m) obtained from I by reducing t rows form above, 8-t rows from below, k columns from left, and 8-k columns form right.

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If t and k both equal 0, or both equal 8, then DCT matrix of J is a "submatrix" of DCT matrix of I. In other cases the coefficients are very different. Is there any connection between DCT(I) and DCT(J)? I mean, other than DCT(J)=DCT(crop(IDCT(DCT(I)))). Something more direct and useful.

Solution

  • The short answer is no.

    Instead of going into detail here, I refere to a similar question on math.stackexchange: How does a cropping of a 2D matrix/image affect its DCT transform?. The accepted answer also gives an intuitive example of why there is no simple connection between DCT(I) and DCT(J) by considering the one-dimensional case.