7transform - 7. Image Transform 2-D Discrete Cosine...

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7. Image Transform • 2-D Discrete Cosine Transform (Section 8.5.2)
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Discrete Fourier Transform (DFT) • 1-D DFT pair for a signal sequence x(n) of length N: 1 0 2 1 0 2 . 1 ,..., 0 , ) ( ) ( 1 ,..., 0 , ) ( 1 ) ( N k N nk j N n N nk j N n k X n x N k n x N k X e e  1 0 2 1 1 0 ) ( 2 2 1 2 1 1 0 2 1 1 0 ) ( 2 2 1 2 2 1 12 2 2 1 1 2 2 1 1 1 ,..., 0 , , ) , ( ) , ( , 1 ,..., 0 , , ) , ( 1 ) , ( NN N j N j k N n n k k n k n k k X n n x n N k k n k n k n n n x k k X e e N • 2-D DFT pair for a signal sequence x(n 1 ,n 2 ) of area NxN:
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1-D General Transform • A more general transform would be: where g(n,k) is a “ forward transform kernel •A n inverse transform is: 1 0 1 ,..., 0 ), , ( ) ( ) ( ~ N n N k k n g n x k X . 1 ,..., 0 ), , ( ) ( ~ ) ( 1 0 N k N n k n h k X n x * In the discussions, we often neglect that there may be a scaling constant in the forward and/or inverse transform so that a function going through a forward transform followed by an inverse transform will get back to the same function.
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Examples of 1-D g(n,k) k n cosine sine Hadamard Harr Slant KLT
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Example: 1-D DCT (Discrete Cosine Transform) 1 0 1 ,..., 0 , 2 ) 1 2 ( cos ) ( ) ( ) ( ~ N n N k N k n n x k k X N k for N k where N n N k n k X k n x N k / 2 0 / 1 ) ( , 1 ,..., 0 , 2 ) 1 2 ( cos ) ( ~ ) ( ) ( 1 0 Ex. N=8: ] 16 7 15 cos ) 7 ( ~ ... 16 1 15 cos ) 1 ( ~ )[ 2 / 1 ( 16 0 15 cos ) 0 ( ~ ) 8 / 1 ( ) 7 ( ... ] 16 7 3 cos ) 7 ( ~ ... 16 1 3 cos ) 1 ( ~ )[ 2 / 1 ( 16 0 3 cos ) 0 ( ~ ) 8 / 1 ( ) 1 ( ] 16 7 1 cos ) 7 ( ~ ...
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7transform - 7. Image Transform 2-D Discrete Cosine...

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