However block preparation is carried out To compute the transformed value for

However block preparation is carried out to compute

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However, block preparation is carried out. To compute the transformed value for each position in a matrix requires the values in all the locations of the matrix to be processed. It would be too time consuming to compute the DCT of the total matrix in a single step so each matrix is first divided into a set of smaller 8 x 8 sub-matrices. Each is known as a block and these are then fed sequentially to the DCT which transforms each block separately. Image/Block Preparation Multimedia System 030728.ppt 127 Each is known as a block and, as we can see in the figure, these are then fed sequentially to the DCT which transforms each block separately. Image/Block Preparation Forward DCT Blk N Blk 1 Blk 2 Blk 3 Matrix of values to be compressed 8 8 - - - - - - - - - - - - - - Blk N Blk 3 Blk 2 Blk 1 8 8 8 8 Multimedia System 030728.ppt 128 In order to compute the forward DCT, all the values are first centered around zero by subtracting 128 from each intensity/luminance value. The input 2-D matrix is represented by: P[ x , y ] and the transformed matrix by F[ i , j ], the DCT of each 8 x 8 block of values is computed using the expression: Forward DCT ∑∑ = = + + = 7 0 7 0 16 ) 1 2 ( cos 16 ) 1 2 ( cos ] , [ ) ( ) ( 4 1 ) , ( x y j y i x y x P j C i C j i F π π where C(i) and C(j) = for i,j =0 2 1 = 1 for all other values of i and j And x , y , z and j all vary from 0 through 7
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Multimedia System 030728.ppt 129 Each pixel value is quantized using 8 bits which produces a value in the range 0 to 255 for the intensity/luminance values - R, G, B, or Y - and a value in the range -128 to +127 for the two chrominance values - C b , and C r . The follow points are summarized in Figure: Forward DCT 1. All 64 values in the input matrix, P[ x, y ] contribute to each entry in the transformed matrix, F[ i, j ] . 2. For i = j = 0, the two cosine terms (and hence horizontal and vertical frequency coefficients) are both 0. Also, since cos(0)= 1, the value in location F [0,0] of the transformed matrix is simply a function of the summation of all the values in the input matrix. Essentially, it is the mean of all 64 values in the matrix and is known as the DC coefficient. Multimedia System 030728.ppt 130 Forward DCT 3. Since the values in all the other location s of the transformed matrix have a frequency coefficient associated with them – either horizontal ( x = 1 – 7 for y = 0), vertical ( x = 0 for y = 1 – 7) or both ( x = 1 – 7 for y =1 – 7) – they are known as the AC coefficient. 4. For j = 0, only horizontal frequency coefficients are present which increase in frequency for i = 1 – 7. 5. For i = 0, only vertical frequency coefficients are present which increase in frequency for j = 1 – 7. 6. In all other locations in the transformed matrix, both horizontal and vertical frequency coefficients are present to varying degrees. Multimedia System 030728.ppt 131 Forward DCT Figure: DCT computation features P[x, y] =8 x 8 matrix of pixel values F[i, j] = 8 x 8 matrix of transformed values/spatial frequency coefficients In F[i, j]: = DC coefficient = AC coefficients f H = horizontal spatial frequecy coefficient f V =vertical spatial frequency coefficent lncreasing f V and f H coefficients lncreasing f H coefficients F[i, i]: P[x, y]: DCT 1 2 3 4 5 6 7 j =0
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