Processing for compression - The MPEG - page 84

Figure 3.44 (b) shows that the phase of all the components of one block are in the opposite sense to those in the
other. This means that when the components are added to give the transform of the double length block all the sine
components cancel out, leaving only the cosine coefficients, hence the name of the transform. [ 15 ] In practice the
sine component calculation is eliminated. Another advantage is that doubling the block length by mirroring doubles
the frequency resolution, so that twice as many useful coefficients are produced. In fact a DCT produces as many
useful coefficients as input samples.
Figure 3.44: The DCT is obtained by mirroring the input block as shown in (a) prior to an FFT. The mirroring
cancels out the sine components as in (b), leaving only cosine coefficients.
For image processing two-dimensional transforms are needed. In this case for every horizontal frequency, a search
is made for all possible vertical frequencies. A two-dimensional DCT is shown in Figure 3.45 . The DCT is separable
in that the two-dimensional DCT can be obtained by computing in each dimension separately. Fast DCT algorithms
are available. [ 16 ]
Figure 3.45 shows how a two-dimensional DCT is calculated by multiplying each pixel in the input block by terms
which represent sampled cosine waves of various spatial frequencies. A given DCT coefficient is obtained when
the result of multiplying every input pixel in the block is summed. Although most compression systems, including
JPEG and MPEG, use square DCT blocks, this is not a necessity and rectangular DCT blocks are possible and are
used in, for example, Digital Betacam and DVC.
Figure 3.45: A two-dimensional DCT is calculated as shown here. Starting with an input pixel block one calculation
is necessary to find a value for each coefficient. After 64 calculations using different basis functions the coefficient
block is complete.
The DCT is primarily used in MPEG because it converts the input waveform into a form where redundancy can be
easily detected and removed. More details of the DCT can be found in Chapter 5 .
[ 15 ] Ahmed, N., Natarajan, T. and Rao, K., Discrete Cosine Transform. IEEE Trans. Computers , C-23 , 90-93 (1974)