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Why is information lost when taking DCT acceptable?

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matrixofdynamism

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The discrete cosine transform (DCT) is apparently a "cut down" version of the discrete fourier transform (DFT). In the DCT, the "imaginary" part of the frequency spectrum is thrown away. I want to know why this is not a problem since DCT is used in image compression to basically get the frequency spectrum of the image.

What can DFT do which DCT cannot do? If DCT really is superior then why not just used DCT and forget about DFT?
 

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