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In this DCT video isn't $X^c$ the input image and what does the superscript $c$ signify? Why do they call $X^c$ "DCT coefficients " although it's the input image?

jojeck
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Suvi
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2 Answers2

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That's the inverse DCT. $x[m,n]$ is the image, $X[k,l]$ are the coefficients.

Jdip
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  • I'm surprised. The video doesn't say it's the inverse. But it makes sense... – Suvi Aug 18 '22 at 15:39
  • Actually, I'm not perfectly happy with the answer. Look at the scanned page from Koppel, Eisen and Ribeiro posted by @Laurent Duval. There the inverse transform uses $cos$ of $[m∗pi((2k+1)/2M]$ whereas in the video it is $[k∗pi((2m+1)/2M]$. So $k$ and $m$ change places! Which one is the REAL formula of inverse DCT? – Suvi Aug 21 '22 at 16:16
  • @Suvi I agree, it’s confusing: Looks like $m$ And $k$ Move around from definition to definition: matlab inverse 2D DCT – Jdip Aug 21 '22 at 17:48
  • I do think my answer is valid: Matlab has the same equation as in the video, so I’d say that’s the correct expression. Not sure why Laurent’s paper shows different indices. – Jdip Aug 21 '22 at 18:05
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    Also this link and this link seem to support your position. In this video they call the inverse transformation DCT-III and the first link provides the same formula for DCT-III as what was seen in the original video. But still don't understand why the book tells otherwise. Maybe you can use either one for direct transformation as long as you use the opposite formula for the inverse? – Suvi Aug 22 '22 at 13:13
  • @Suvi that's an interesting thought. Again, not well-versed enough in DCT to answer without spending a little time on it, which I don't have too much of at the moment! – Jdip Aug 22 '22 at 13:26
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The answer given by Jdip can be accepted. As a complement, such a notation is fairly common, though I am more used to a subscript ($X_C$ instead of $X^c$). A classical convention is:

  • $x$ (lowercase) denotes input signals/images
  • $X$ (capital) denotes transformed signals/images, also called coefficients.

To be more explicit regarding the action on $x$ of a transformation or basis $C$ (here DCT-II), the transformed coefficients are denoted $X_C$ as "$x$ transformed with respect to $C$". You can find more details in: The Discrete Cosine Transform and JPEG, Alec Koppel, Mark Eisen, Alejandro Ribeiro, 2021. Excerpt:

The Discrete Cosine Transform and JPEG

Jdip
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Laurent Duval
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  • The lower formula for $\vec{x_C}$ is different than in the video. In the video we take the cosine of $[k*pi((2m+1)/2M]$, in the book it's $[m*pi((2k+1)/2M]$. So k and m change places, why? – Suvi Aug 19 '22 at 11:24
  • In the video, I am not sure of which DCT the speaker is talking about. He does not talk about an "inverse DCT", maybe he is using a direct one. – Laurent Duval Aug 19 '22 at 12:16
  • matlab inverse DCT Looks like $m$ and $k$ are inter-changeable? Not really versed in 2D DCT so can't really provide any explanation... – Jdip Aug 19 '22 at 12:27