The DCT, just like the DFT, is a discrete version of the (analog) Fourier transform. (FFT is just a clever, fast implementation of the DFT, just like when peopl

The DCT, just like the DFT, is a discrete version of the (analog) Fourier transform. (FFT is just a clever, fast implementation of the DFT, just like when people say “DCT” they usually mean a similarly fast implementation of the DCT. Call it FCT if you wish.) Where they differ is the assumed boundary conditions; DFT works like Fourier-transforming a signal that is repeated and then loops forever, while DCT is like Fourier-transforming a signal that is _reflected_ and then loops forever. (There is also a related transform called DST, where it assumes the signal is _inverted_ and reflected. It's occasionally useful in solving certain kinds of differential equations; only rarely so in compression.) This matches much better with what happens when you cut out small pieces of a signal, so it gives less noise and thus better energy isolation. Say your little pixel block (let's make it 8x1 for simplicity) is a simple gradient, so it goes 1, 2, 3, 4, 5, 6, 7, 8. What do you think has the least amount of high-frequency content you'd need to deal with; 123456788765432112345678… or 123456781234567812345678…? (There's also a DCT version that goes more like 1234567876543212345678 etc., but that's a different story)