lecture 13 - math.cuhk.edu.hk
TRANSCRIPT
Image deblurring in the frequency domain:Lecture 13:
! "##$%& ' (# )*+,* -
.&/0+1 2 3 4(5&6/ 7 (*8&5#& 9(:/&5(*;
< &/ = ! >?8 - @
ABB
C&/D - EF 7 G*1 H;*:D - @ B 2 +/0&5,(#& -' ! $% - / I #;* ! 'G% ?J
! #;*6 I- @ 2 (9
B
"8+(1 7 #(*;$:G5(/KL+%M$/& 2 ! $
?8 - @ N6$ ?8 ? =6$ ?
8 -O
P(*1 (*8&5#& 4P= +9 I6$ ?$ - /+ ;&/ 7 G* 7 (%G;& P(Q
? ? ?O
N++1 3 H(%M:&
RG1 3 R+G#/ $M *+(#&
P:$?8 - @ N6KK5:=6$
?8 - @ P:$
?8 - ST:$BB'($ ?
6$? 7 0/
T&$%
'%% - / I #;*<'6$O$#-
T+/& 'G(5 - (# U(; 9+5 L$ ?8 - 6:+#& /+ ! + ?+ - ! )&&M :+, 95&V$&*6(&# -
(# H%G:: 9+5 ! $ ?8 - 9G5 7 G,GK 95+% ! +
? 7 + -
(O '/W2XY (# U(; 9+5 6G* ? 9G* G,GK 95+% 6+ ?O ?
B 0(3<Z&*/(/(&#R+G#/ $M *+(#&# Z Z
[5(;(*G: R:$55&1 (%G;& 4(5&6/ (*8&5#& 9(:/&5(*;
.&/0+1 \ 3 .+1(9(&1 (*8&5#& 9(:/&5(*;
<&/R:$O82B;]K*G*1=6$O8- @ /O&$^_9<#;*`= O
=0&* 3 7 1&9(*& 3 7 P:$ ?8 - @ =6$ ?8 - N6$8 - aa P:$ ?8 - R:$ ?8 -
/O.2__OR_;O8*`$$/__O(_S2_;2GG/_6_O(_PG? 9+5 ! $% @(b?b -
T!$?8-R6$@ R:$% (# #%G:: - 9+5 ! $ ?8 - 9G5 G,GK'L G(5 -/&#;*! '&
(# H%G:: ! G#
95+% ! F?F -?
AcA0< #$MM5&##&# /0& 0(;0 B 95&V$&*6K ;G(* O
':$ ?8-/I#;*! '$% -
RG1 3 'G# /+ 60++#& 4 G*1 * 6G5&9$::K O
,(/0 G U&#/ 4 G*1 *O
.&/0+1 c 3 d(&*&5 9(:/&5
<&/ =6$ ?8 - @
'e ,0&5& H*6$ ?8 - @f T6$ O*:
g
2' :$8 g/ g/ h2_? H; (* ?8 ? @ 2 P(*(#0
g
29 H$:$ ?8 - G*1 H9 :$8 - G5& 7 *+/ )*+,*?/0&* ,& :&/ i@ h2_$? U ;&/ 3
=6$ ?8 - @
'2 '&G8K g/ i
<&/ 7 I6$?8 - @ =6$
? 78#N6$
?$ -
O L+%M$/& 2* ? K - @ 7 (*8&5#& 4P= +9 7 26$?8 ? O
2* 9G6/?
/0& d(&*&5 9(:/&5 6G* U& 7 1A(U&1 G# G* 7 (*8&5#& 9(:/&5(*; G# 9+::+,# 3
I%%(//? 9%/2 'G%%
R&0G8& :()&j
.+1(9(&1 (*8&5#& 2*+ (9 ' <$ ?
$ - *&+ ! (9 2 > ?8E 9G5 7 G,GK9(:/&5(*; j
95+% 7 + -@< (9 '%% (# :G5;& ! (9 <$*; @ 6+
?+* Z
What does Wiener filtering do mathematically?d& g:: #0+, 3 d(&*&5 9(:/&5 7 %(*(%(D&# /0& 7 %&G* #V$G5& 7 &55+5 7 3
I:9?I- @ kk 29 (QO K ? B I$QK((1Q1K
+5(/(*1 C&#/+5&1
! ,& G##$%& /0& 6+*/(*$+$# 7 &G#& /+ G8+(1 6+%M:(6G/&11&;5G1G/(+* (*1(6&# g1
[U#&58&1 2
<&/ j
; @ 0l9 7 S * m *+(#&
n +5(;(*G:
"##$%& /0G/ 9 7 G*1 * 7 G5& #MG/(G::K $*6+55&:G/&1 3
h P+Q? K - * ! o/5
? K/# - 1Q 1K 9+5 7 G:: 5?#
O
4&9(*& 3 2 ! Q ? _ - @ dLo ? K - l ;6Q ? _ - 9+5 H+%& B ,B LQ ? K -
C&6G:: 3 2 (# 7 +U/G(*&1 G# 9+::+,# O
p ((((((((((((((((((((((6((3 (*33XXXqq;&/ 9j
6
g
O 2 @ dg l ; 9+5 H+%& dO
(Sketch of proof)'& *&&1 /+ $#& 3 rG5#&̀ G: =0&+5&% 3
IB ! 9?2 - 3 @
O
kk 2 9(Q? 7 ? ? B
P(Q ?K(:g
B
B L 2 P:$ ?8 ? B
I6$ O*:g
9+5 #+%& 6+*#/G*/ L
@ 4P= +99 G*1G
,0&5& P:$ ? 8A I6$ ?8 ? @ 4P= +9 9[U#&58&1 (%G;&1
<&/ N6$ ?8# @ 4P= +9 ; G*1 T6$ ?8 ? @ 4P= +9 *
=0&* O
O P:$ ?8 - @ d0&* - N6$ ?
8 - ! G# 2jg/ 7 QO K ? @ 7 2P= ! d6G ?8 - N&$% - -
H+?
! $?8 - @ d6, ?
8 - N6$ ?8 - @ d0(5: ! '%% '$% / T6$
?8 ? -
2* 7 +/0&5 ,+51#? 9 B 2 @ !2 B d' - P B dT
G*1 Ig ! 9?
I- @
Lk-? ( X d' - P BdT=
H(*6& 9 7 G*1 * 7 G5& #MG/(G::K $*6+55&:G/&1?
,& 6G* #0+, /0G/ (
Ig
L9?
2/ @ 2:: O ,(/0 P:g
/ 2dT2g
;+! ! 2 B ,(/ g1 P '= T/ @ -
H(*6& ,& :++) 9+5 7 ,6Q ? K - ! 0&*6& d0&* ? - #$60 /0G/ Ij
(# 7 %(*(%(D&1 ? ,& 6G* 7 5&;G51
s (# 1&M&*1&*/ 7 +* dO
=+ %(*(%(D& It ! d -?
,& 6+*#(1&5 3
11/9/+IB ! d//8 - @ [ 9+5 7 G:: u
O
d& ;&/ 3 k9 B L( B d=22'2P2tu B ! 2 B d' - 'g 2P2 g= / d=T2u / d2TIg= B
B + 9+5 G5& 8O
r$/ u@ B (/ B d' - 2P 2P2 g/ 7 d2T=O
=0&* 3 7 ,& 0G8& 3 9- 2 B L2 B d'2== 2P2 g/ d2 T(:g
@+ O
O
g
O
B ! /,'-'/I/g7g
S d2T2g
B +
'
d B
IB2'2 g/ 2._2P=
O