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Standard equation for generating DCT coefficients in JPEG compression process is DCT formula given by:
I have red that this implementation is expensive (slow), and there is much faster way. Is there explicit formula for this faster way of generating DCT coefficients?
Yes, this general version is slow, very slow indeed. There are much faster approximation out there.
Fastest software DCT transformation can be found within the BinDCT family.
They only need some basic additions and shifts, and are therefore very fast, at the expense of some precision.
An excellent presentation of it : On the Process of Realizing the Best BinDCT Configuration for Image Compression (especially slide 12)