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porterbo...@yahoo.com wrote: > Closest approximation to the Karhonen Loeve transform? Same acronym as the Dallas Children's Theater? (www.dct.org) Ciao, Peter K.

p...@yahoo.com wrote: > > Closest approximation to the Karhonen Loeve transform? That is usually mentioned, but I have discovered a more compelling reason. But I would like to know what the original poster has found. Regards Guido

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Neo wrote:
> I was curious to know why is it that DCT is used in image compression.
> why not FFT?
The difference in use between DST, DCT, and DFT are the boundary
conditions. (This is explained in Numerical Recipes, where I looked
when I wondered some time ago.)
DFT has periodic boundary conditions, DST has the function go to
zero at the boundary, and DCT has the derivative go to zero at the
boundary. It seems that DCT has a smaller effect on the image
than the others. That is, the effect of the blocks is less
visible with DCT than DFT or DST. FFT is an algorithm,
or class of algorithms, for evaluating DFT.
-- glen
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glen herrmannsfeldt wrote: > Neo wrote: > > > > I was curious to know why is it that DCT is used in image compression. > > why not FFT? > > > The difference in use between DST, DCT, and DFT are the boundary > conditions. (This is explained in Numerical Recipes, where I looked > when I wondered some time ago.) > > DFT has periodic boundary conditions, DST has the function go to > zero at the boundary, and DCT has the derivative go to zero at the > boundary. It seems that DCT has a smaller effect on the image > than the others. That is, the effect of the blocks is less > visible with DCT than DFT or DST. FFT is an algorithm, > or class of algorithms, for evaluating DFT. > > -- glen Hi Can you explain what is meant by "derivative go to zero" or "function go to zero" ? and when does it go to zero ? Thanks Sandeep

guido, I found out that the DCT distributes most of the energy in the lower order co-efficients compared to the FFT and also gives a purely real output. This energy localisation aids in more efficient encoding later on. Is this reasonable?

Neo wrote: > > guido, > I found out that the DCT distributes most of the energy in the lower > order co-efficients compared to the FFT and also gives a purely real > output. This energy localisation aids in more efficient encoding later > on. > Is this reasonable? No!;-) Your question was why is the DCT used in *image* compression! The general term "energy compaction" does not explain why it is good just for *image* compression, you could say the same thing for any other compression object. The actual reason is much simpler, and therefore apparently very difficult to recognize by complicated-thinking people. Here is the explanation: What are people doing when they have a bunch of images and want a quick preview? They use thumbnails! What are thumbnails? Thumbnails are small downscaled versions of the original image! If you want more details of the image, you can zoom in stepwise by enlarging (upscaling) the image. That is the key to understanding the use of DCT for image compression: The fundamental property of lossy image compression is the similarity of different resolutions of the same image. "Lossy" compression means that we assign *the same* output representation to *multiple*, *similar* input representations. The basic similarity relation for images is resolution, or scale, invariance: If we see the same image in different resolutions (scales, sizes), or the same subject from different distances, we talk about *the same* image (or subject). The DCT provides the best resolution separation property for digital images. The 8-point DCT gives you 8 linearly increasing resolution representations from 8 spatial sample values. You can hardly do better than that. Wavelet transforms, as used in JPEG2000, for example, do *not* provide such optimal resolution separation. See also chapter 4 of my paper at http://jpegclub.org/temp/. Everybody who knows the DCT knows that the DC term represents a 1/8 scale of the input sequence ("thumbnail" version). The DC and first AC together represent a 2/8 or 1/4 scale of the input sequence. The DC and first 2 ACs together represent a 3/8 scale of the input sequence, and so on. Every DCT coefficient adds corresponding resolution detail. This is easy to demonstrate, but was not known before. (See new JPEG scaling features presented at http://jpegclub.org which directly apply this property.) This fundamental DCT property explains why the DCT is the best transform for image compression. Regards Guido