Re: {JPEG}Discrete Cosine Transformation

From: Dave Martindale (davem_at_cs.ubc.ca)
Date: 07/13/04 

Date: Tue, 13 Jul 2004 06:22:54 +0000 (UTC)

hoffmann@fho-emden.de (Gernot Hoffmann) writes:

>in at least one of the URLs in the other posts the DCT is considered
>as the cosine part of the discrete Fourier series. Thatīs wrong.
>The cosines in the DCT have a phase shift.
>Some illustrations are here:
>http://www.fho-emden.de/~hoffmann/jpeg131200.pdf
>The graphics may help to understand the SHAPES of the functions in the
>function system.

In addition, the discrete Fourier transform assumes that the input
function is periodic in a very simple way: copies of the input image are
effectively tiled out to infinity in all directions, but each tile has
the same orientation. This means there are often abrupt discontinuities
at the tile boundaries, where left edge of one copy meets right edge of
the adjacent tile, and so on. This in turn means there is sometimes
lots of high-frequency content, which is difficult to compress.

The discrete cosine transform assumes a more complex tiling, where the
original image is mirror-imaged across each tile boundary. This means
the input is mathematically an even function, so if you did a Fourier
transform of the double-sized mirrored and reflected image it would have
zero imaginary component to every coefficient. This also makes the DCT
coefficients compress better than FFT coefficients.

        Dave