r which no fast algorithms exist), and the The original finite sequence f(i) depicted in part (a) is transmission of the eigenvector matrix is required for transformed and reconstructed in (b) using the DFT- every transformed image block. IDFT transform pairs, and in (c) using the DCT-IDCT The DCT is the most widely used block transform for transform pairs. The periodicity of the IDFT in (b) is five digital image and video encoding. It is an orthonormal samples long, and illustrates the discontinuity introduced transform, and has been found to perform close to the by the discrete transform. The periodicity of the IDCT KLT [11] for first-order Markov sources. The DCT is in (c) is 10 samples long, due to the evenness of the DCT defined on an 8 X 8 array of pixels, operation. Discontinuities introduced by the DCT are generally less severe than the DFT. The importance of this property of the DCT is that reconstruction errors and blocking artifacts are less severe in comparison to the DFT. Blocking artifacts are visually striking and occur due xcos(12)+1)vr (7.4o the loss of high frequency components that are either quantized or eliminated from the DCT coefficient and the inverse IDCT is defined as, array. The DCT minimizes blocking artifacts as com- pared to the DFT because it doesn't introduce the same f(i,j) = level of reconstruction discontinuities at the block edges. Figure 7.4-4 depicts blocking artifacts introduced by gross quantization of the DCT coefficients. (7.4.8)This section ends with an example of the energy packing capability of the DCT. Figure 7.4-5 depicts the DCT transform operation. The original 8 X 8 image sub-block from the Lena image is displayed in part (a), Cu - 1/2 for U = 0, Cu = 1 otherwise for V = 0, C = 1 (a) Original Sequence (b) DFT Reconstruction otherwise where i and j are the horizontal and vertical indices of the 8 X 8 spatial array, and U and V are the horizontal and vertical indices of the 8 X 8 coefficient array. The DCT is (c) DCT Reconstruction the chosen method for image transf