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lecture04 - ELEC317 Digital Image Processing Image...

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1 ELEC317 Digital Image Processing Lecture 4 Image Transforms In general, transform refers to the representation of a signal as a weighted sum of some basis functions. If we consider a 1-D signal u[n], n=0, 1,….N-1 as a vector = ] 1 [ ] 1 [ ] 0 [ N u u u u M Then transform of u is Au v N v v v = = ) 1 ( ) 1 ( ) 0 ( M where = ) 1 , 1 ( ) 0 , 1 ( ) 2 , 1 ( ) 1 , 1 ( ) 0 , 1 ( ) 2 , 0 ( ) 1 , 0 ( ) 0 , 0 ( N N a N a a a a a a a A M L L where A is the linear transform matrix. Note that = = = 1 0 1 ,... 1 , 0 ], [ ) , ( ] [ N n N k n u n k a k v ( = = N k j i j k B k i A AB 1 , ) , ( ) , ( ) ( ) The most popular 1-D signal transform is the discrete Fourier transform (DFT). In this case, A is the NxN matrix with its (l,m)th entry given by ) 1 )( 1 ( 2 1 ) , ( = m l N j e N m l a π where 1 = j Thus, in DFT = = 1 0 2 ] [ 1 ] [ N n kn n j e n u N k v as we see in ELEC 212. In most transformation, the matrix A is an unitary matrix, i.e.
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2 H T A A A = = * 1 The advantage of using unitary transform is obvious: the inverse transform is easy to find. Thus v A u v A u H = = 1 = = 1 0 * ) , ( ] [ ] [ N k n k a k v n u In the case of DFT, the above expression is = = 1 0 2 1 ] [ ] [ N k kn N j e N k v n u π If we consider = ) , ( ) , 2 ( ) , 1 ( ) 2 , ( ) 2 , 2 ( ) 2 , 1 ( ) 1 , ( ) 1 , 2 ( ) 1 , 1 ( * * * * * * * * * N N a N a N a N a a a N a a a A H M M M K [] Vn V V K 2 1 = Then = + = 1 0 1 ] [ N k k V k v u i.e. u is now written as a sum of basis signals 1 V , 2 V ,… N V . Two Dimensional Transform Similar to the 1-D case, in 2-D, transform refers to representing an image } 1 , 0 ], , [ { = N n m n m u U As a weighted sum of basis signals ] , [ * n m a kl . Hence ∑∑ = = = 1 0 1 0 * ] , [ , ] , [ N k N l kl n m a l k v n m u The coefficients 1 ... 0 , ], , [ = N l k l k v is the transform coefficients, given by = = = 1 0 1 0 , ] , [ ] , [ ] , [ N m N n l k n m a n m u l k v —(*)
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3 In the above, the basis functions 1 ... 1 , 0 , , , ], , [ * = N n m l k n m a kl are chosen carefully to satisfy the following two properties: Orthonormality: ] ' , ' [ ] , [ ] , [ * ' ' 1 0 1 0 l l k k n m a n m a l k N m N n kl = ∑∑ = = δ Completeness: ] ' , ' [ ] ' , ' [ ] , [ * 1 0 1 0 n n m m n m a n m a kl N k N l kl = = = From (*), it can be seen that for a given ( k,l) , computation of v [ k,l ] need approximately 2 N “+” and “x” operations. Î To compute v[ k,l ] for all k, l =0,1,…N-1 needs 4 N additions & multiplications. Separable Unitary Transforms In order to reduce the computations, most practical algorithm uses the so called separable unitary transform . In this case ] , [ n m a kl is separable, ] [ ] [ ] , [ n b m a n m a l k kl = ) , ( ) , ( n l b m k a = Where the transforms (1-D) obtained by a ( k,m ) and b ( l,n ) are both unitary, i.e. = ) 1 , 1 ( ) 0 , 1 ( ) 1 , 1 ( ) 0 , 1 ( ) 1 , 0 ( ) 1 , 0 ( ) 0 , 0 ( N N a N a a a N a a a A L O M L and similarly for B are unitary matrices. In most case, there is no difference between horizontal and vertical directions, so we further choose ] , [ ] , [ m k b m k a = or B A = With the above simplification
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4 ) , ( )] , ( ) , ( [ ] , [ 1 0 1 0 n l a n m u m k a l k v N m N n ∑∑ = = = Î T AUA = Hence * VA A U H = For a M x N rectangular image, We have * N H M T N M VA A U UA A V = = where A M , A N are M x M & N x N unitary matrices respectively.
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This note was uploaded on 04/14/2010 for the course ELEC 317 taught by Professor Nil during the Spring '02 term at HKUST.

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lecture04 - ELEC317 Digital Image Processing Image...

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