For some might wondering why cosine rather than sine, here is a piece of interesting information I found:
"The use of cosine rather than sine functions is critical for compression since it turns out (as described below) that fewer cosine functions are needed to approximate a typical signal, whereas for differential equations the cosines express a particular choice of boundary conditions. In particular, a DCT is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using only real numbers. The DCTs are generally related to Fourier Series coefficients of a periodically and symmetrically extended sequence whereas DFTs are related to Fourier Series coefficients of only periodically extended sequences. DCTs are equivalent to DFTs of roughly twice the length, operating on real data with even symmetry (since the Fourier transform of a real and even function is real and even), whereas in some variants the input and/or output data are shifted by half a sample. There are eight standard DCT variants, of which four are common."
melodysifry
What exactly does it mean here to transform onto DCT space? is each pixel then represented as a linear combination of these 64 bases shown in the diagram? Conceptually, how does this end up pushing error into higher frequency?
kevintli
^ I believe what's happening is that each 8x8 block of image values (rather than each pixel) gets converted into DCT space, which as you said is a linear combination of the 64 bases in this diagram. This by itself doesn't result in any compression or error; that part happens in the quantization step on the next slide, which explicitly zeros out higher frequencies. The reason for the DCT step is so that we can represent our image in terms of its frequencies rather than individual pixel values, which makes it much more convenient for the quantization step.
For some might wondering why cosine rather than sine, here is a piece of interesting information I found: "The use of cosine rather than sine functions is critical for compression since it turns out (as described below) that fewer cosine functions are needed to approximate a typical signal, whereas for differential equations the cosines express a particular choice of boundary conditions. In particular, a DCT is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using only real numbers. The DCTs are generally related to Fourier Series coefficients of a periodically and symmetrically extended sequence whereas DFTs are related to Fourier Series coefficients of only periodically extended sequences. DCTs are equivalent to DFTs of roughly twice the length, operating on real data with even symmetry (since the Fourier transform of a real and even function is real and even), whereas in some variants the input and/or output data are shifted by half a sample. There are eight standard DCT variants, of which four are common."
What exactly does it mean here to transform onto DCT space? is each pixel then represented as a linear combination of these 64 bases shown in the diagram? Conceptually, how does this end up pushing error into higher frequency?
^ I believe what's happening is that each 8x8 block of image values (rather than each pixel) gets converted into DCT space, which as you said is a linear combination of the 64 bases in this diagram. This by itself doesn't result in any compression or error; that part happens in the quantization step on the next slide, which explicitly zeros out higher frequencies. The reason for the DCT step is so that we can represent our image in terms of its frequencies rather than individual pixel values, which makes it much more convenient for the quantization step.