DCT vs. DFT

On Tuesday, November 4, 2003 at 1:29:40 PM UTC-6, Matthias Alles wrote:
> Hi!
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> Can anyone explain me, why you use DCT instead of DFT for transformation=
=20
> of image-data (like it is done for JPEG). I know that the DCT can be=20
> performed by a DFT with a post-rotation. But what are the advantages of=
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> the DCT compared to the DFT? Are there any benefits from the post-rotatio=
n?
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> Regards,
> Matthias

JPEG isn't really using the DCT in its full generality but rather a highly =
specialized 8x8 table which can just as well be expressed as without the ne=
ed or use of trig (and tabulated efficiently without trig). It's just a way=
 to get LOCALIZED transforms, but without employing the full machinery of t=
ime-frequency analysis (or the 2D equivalent of spectrographs).

The 8x8 table is simply this:
	DDDDDDDD
	GECAaceg
	FBbffbBF
	EagcCGAe
	DddDDddD
	CgAEeaGc
	BfFbbFbB
	AcEgGeCa
where
	-a =3D A =3D 1/4 root(2 - root(2 + root(2)))
	-b =3D B =3D 1/4 root(2 - root(2))
	-c =3D C =3D 1/4 root(2 - root(2 - root(2)))
	-d =3D D =3D 1/4 root(2)
	-e =3D E =3D 1/4 root(2 + root(2 - root(2)))
	-f =3D F =3D 1/4 root(2 + root(2))
	-g =3D G =3D 1/4 root(2 + root(2 + root(2))).
The coefficients satisfy simple algebraic relations with one another which =
allows for efficient methods that factor the matrix or use other tricks sim=
ilar to those used for the Fourier Transform.