 o2; is the variance of each subband i, for CHAPTER 7.4 Video compression OKi<N- - GTC also measures the gain of block implementations using real calculations. It is even sim- transform coding over PCM coding. The coding gain pler to compute than the DFT because it does not re- generated by a block transform is realized by packing quire the use of complex numbers. most of the original signal energy content into a small The second reason for its success is that the recon- number of transform coefficients. This results in a loss- structed input of the inverse DCT (IDCT) tends not to less representation of the original signal that is more produce any significant discontinuities at the block edges. suitable for quantization. That is, there may be many Finite discrete transforms create a reconstructed signal transform coefficients containing little or no energy that that is periodic. Periodicity in the reconstructed signal can be completely eliminated. Spatial transforms can produce discontinuities at the periodic edges of the should also be orthonormal, i.e., generate uncorrelated signal or pixel block. The DCT is not as susceptible to coefficients, SO that simple scalar quantization can be this behavior as the discrete Fourier transform (DFT). used to quantize the coefficients independently. Since the cosine function is real and even, i.e., cos(x) The Karhunen-Loève transform (KLT) creates =cos(-x), and the input F(u, v) is real, the IDCT gen- uncorrelated coefficients, and is optimal in the energy erates a function that is even and periodic in 2n, where n packing sense. But the KLT is not widely used in practice. is the length of the original sequence. On the other hand, It requires the calculation of the image block covariance the IDFT produces a reconstruction that is periodic in n, matrix SO that its unitary orthonormal eigenvector matrix but not necessarily even. This phenomenon is illustrated can be used to generate the KLT coefficients. This in Figure 7.4-3 for the one dimensional signal f(i). calculation (fo