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DCT is just the cosine part of the fourier transform
whereas Wavelet is time-frequency resolution of the signal
Ofcourse more computation will be required for wavelet
But Wavelets have many other applications apart from compression.
However I think Wavelet is better for compression.
on applying DCT to an image u will have the DC component accompanying in the analysis side which will create problems.
using DWT u will not get any such thinks and also the subband sampling helps in extraxtion of the features easily.
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