eee508_Restoration_Part2

eee508_Restoration_Part2 - Wiener filter e e te Non-zero...

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Wiener filter Non-zero mean signal and noise ¾ In our derivation of the optimal LS solution, we assumed that signal and noise are zero-mean. What about non-zero mean signals and/or noise? ¾ Can use same derived filter but subtract mean before wiener filtering zero-mean signal + + G ( Ȧ 1 , Ȧ 2 ) + H (0,0) m x + m K ± ² 2 1 , ˆ n n x + m x m xd x d ( n 1 , n 2 ) + ¾ Note : Wiener filter significantly blurs the image ³ use adaptive approach where m x , m , S xx and S KK are estimated locally (pixel-wise or block-wise) instead of estimating them for entire image EEE 508 or block wise) instead of estimating them for entire image. 1
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Variation of LS approach: Constrained Least Squares Define: 2 2 1 2 1 ) , ( ˆ ) , ( Z x Q J where Q is a chosen suitable “penalty” function 1 > @ 2 1 2 1 1 2 2 2 , ) 2 , ( ˆ ) 2 , ( 4 S d d x Q ³³ or, equivalently (from Parseval’s Theorem) 2 2 1 2 1 ) , ( ˆ * * ) , ( n n x n n q J > @ 2 2 1 2 1 ) , ( ˆ * * ) , ( 2 1 n n x n n q n n ¦ ¦ EEE 508 2
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Variation of LS approach ¾ We want to minimize J subject to the constraint 9 This constraint forces a good appro imation since 2 2 2 1 2 1 2 1 ) , ( ˆ ) , ( ) , ( H Z d ± X H X d for some 0 ! X X ˆ This constraint forces a good approximation since X d = H ( Ȧ 1 , Ȧ 2 ) X ( Ȧ 1 , Ȧ 2 ) ¾ What should Q Ȧ Ȧ ) be like? | What should ( Ȧ 1 , Ȧ 2 ) ² ³ Q X ˆ Since should be low pass, Q penalizes us for having a lot of Ȧ ² ³ X ˆ high frequency energy ´ Q used to smooth out the noise ´ Q tends to impose smoothness EEE 508 3
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Variation of LS approach ¾ Recall ) , ( ) , ( ) , ( ˆ 1 1 1 1 1 1 Z d X G X restoration filter ¾ We want to find such that J is minimized subject to constraint. ) , ( 1 1 G ¾ How do we do the minimization? 9 Use Lagrange multipliers ¿ ¾ ½ ¯ ® ­ ± ± ± 2 2 2 ˆ ˆ H O X H X X Q obj d constrain J where d GX X ˆ EEE 508 constraint 4
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Variation of LS approach ¾ Set 2 2 1 2 2 1 2 1 * 2 1 ) , ( ) , ( ) , ( ) , ( 0 Z O Q H H G G obj ± ² w w where Ȝ is a design parameter which is determined such that 0 2 2 ³ ³ H d d HGX X X ˆ (Equality constraint satisfied) where 2 2 * Q H H G ± EEE 508 5
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Implementation of LS solution Note that the obtained Wiener filter and constrained filter are IIR (since G has a denominator) ¾ Consider Wiener filter H S * ± we need to know or estimate S xx and S ȘȘ KK S H S G xx xx ² 2 9 Noise power spectrum easily estimated using spectral estimation methods.
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eee508_Restoration_Part2 - Wiener filter e e te Non-zero...

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