16_DCT and KL

16_DCT and KL - EE 211A Fall Quarter 2011 Instructor John...

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1 EE 211A Digital Image Processing I Fall Quarter, 2011 Handout 16 Instructor: John Villasenor DCT and KL Transform Comparison Approach: 1. Define a class of matrices 123 (, , ) Jk k k . 2. Show that for a Markov sequence, 21 (,,0 ) Jpp R β = , eigenvectors of (,, 0 ) give KL transform. 3. Show that (1,1, 0) J gives eigenvectors of DCT when 1 p , DCT KL. 13 32 10 0 0 01 0 00 0 1 000 1 kk αα α −− ΡΤ ΢Υ = ΣΦ L L L MM O O O M O We want to show that [ ] ) / ) R Jpp R I ββ = = where 2 1 p p = + , 2 2 2 1 1 p p = + 2 2 0 0 1 0 1 0 ( , ,0) 0 1 1 0 0 1 N N NN N p pp p p p ppp p −− − = L L L L L OO O M O O O M MO M O L 22 3 2 23 2 1 1 n p p p p p p JR m p p p + + = + + L L In general [ ] 2 2 2 222 2 1 12 1 111 mn p p JR p αβ = + = = = = = +++ .

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2 [ ] ( ) 1 2 ,, 3 1 22 33 1 2 11 0. 1 mn mnm n JR p p p pp pp p p p p αα −− = ΛΞ = ΜΟ ++ ΝΠ + == + Result: 1 (,,0 ) Jpp R I J R ββ = = Recall that KL transform matrix Φ consists of eigenvectors k φ of R , i.e. 1 , , 1 . kk k k k R RR R R φλ φφ λ = = = eigenvectors of R are also eigenvectors of 1 R .
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16_DCT and KL - EE 211A Fall Quarter 2011 Instructor John...

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