td Zip J if
Transcript of td Zip J if
Representation of by a matrix
Lecture 2:! " #
$% &'( )*+,% #
- . / 0 1 / / 2 / / .3 . / 4 / / / 2 / / / / / 4/ 5 /
4 1 /// / 2 / / / / / 6 /
7
/ 89: ,3
;< =+, >?;2;;
>@
>?> A B , 5
C C C
4 =D E+ ?; "FG .3 > H+A B I ' D+(%'*
C CC %JK',+L(M C
;< =+, > N B2'*3
> E+A B , 5 < < C 5 4 = OEE B2P E+3>E
> A B
Q**'(R% #
=K 9 B S / . / F( T',*+U =L*T>D%,
R&VC(D# #
= #/
+M=< OE > +, SWW; B +=#& +M.,#+#X#B C +M#Y##@#
#C
C
CZ2%( # R<
.
0 "==D 9/ . R'+( B Z
# -@[EB\ =, / / B F R+( >E > E?UE? T',*+U
By careful examination, we see that:
Definition:
]<
,C
F F ? F E F F
,2%#Y#:T+M=+,,C=+C #XC +++C++++C+++C2'+,+#L= &L, C C
^DL&_ ^DL&_ L= ^DL&_ # #
=,AC]`aab(^[L+',T',+U2'
>/
>+M F . 2%>E
>+C 7F <
< <
2+T>(>+C a B
O< GEcE
+M &'DD%3 ,2% ,*'(M=L*T',+L( T',*+U L= !C
dLDK,+L( # =V+ 1e? 1ee
= > e = > >=+V 1;e =++
efe +T'R%0=VgP=Rah=Rah8B 1? .
1ee 1e;
=+ > =+V 1;e=+V 4 =V+ 5 4 =V g 4 =MV =V g
4 =VV 4 =e >4 = > i
S?? 0
jF See 0
W%,&
C C C
F k k F
F k k FX##XXXXX#// l%&'DD #
MLDK,+L( # ,*'KT'M C
#P > %T C C
" +M M%A'*'^D% m 2 @ c>?
> [ > A `0 RC P U >
@ B RV O[> A B C
+M##+(M'(% #### ## ##X#n # ', #### ###
m+(M+M,M
##### +( +M#+XX+oR
V F ? > F B O= F
F( LK* %'M%> "0P ?R'D
>
2'3C
B h -C 0P Fp
>B
++C S >P ;>; B 0 ; S?W>;;0?
S > P ?>; B0p RV P ?>; B 0 ;
S; P ;>?q0? RV P ;>?q0e
S > P ?>?;0; S? P ?>W0Wj/
GL*% '^LK, &L(rLDK,+L(
-%LT%,*+&/ +(, 5 *%AK,',+L( L= 3+M&*%,% &L(rLDK,+L(
@%, =U<R_CAD 0 FF> >
=+U > ] > R &'< UC A < ] B
OL(M+3%* ' M+TAD% &'M% )2%*% L(D] M%r%*'D %(,*+%M L= R '*% (L( < V%*LC
E'T%D] >0 F
-DL >L B 0 -OE > E B R%,
M+&#X+X##s+XXXMM++X+XXXX+X+# #XXX+X#+, C
C
0 R P E < F>E < , B 0 F
]0]]0L5 ] << F
D D D
F D F.
F D
U <
<< F C
< < <FZL=< <
c < L < < <F=_<t<< 5 C
U <
< F < < < Y:t< <
F D FD FD D D
bUA'(3 ,2% MKTT',+L( #
= QR2 > A B0 =D'A B R%L >
L B , =K( >A,uR&L ><; B ,= O3
> A < ;B R%L > F B ,
=D',> R%, >
L B 4 = O"F> 6,+ B R,D >
<; B , =&,,D >A < ;B R%, >F B ,
=< ", > =B R%, >L B , =D',
> A,+ B RK >< 2, =D3 <;
> A< F B RK( B
C
!r%*D'] ' ]0 >RL
v0<;
Y+UC. / .w. / F(+#u /
F;
9 >x @
p
y =&?CA,/ ,2% ; C
V z='K,
/ +T'R% CF
y
=&?CAD y
z % &L
p
{R]AM]
yC =D3,2=/ /
]'DL
RF
y
R]T >y z F '
p
y ,L y
R P _%*(%D Bd%*+=
/
= f >< F F
D
Properties of shift-invariant/separable image transformationDefinition: (Circulant matrix)
(
tQ &+*&K+, '(, T',*+U o # 0 &+*&D% B 'MML&+',%3 ,L ' r%&,L* +, <
< OrL>r
>>
C C CC+(,
+M ' (U( T',*+U )2LM% &LDKT(M '*% R+r%( ^] +,%*',+L(M L= M2+=,
LA%*',L* Z '&,+(R L( F.
C"%*%
>Z # '
/| b
/
^]}( < F
]X#"X C
C
C
C2,2 &LDKT( +M R+r%( ^] Z2 <5 .
P b B
+(, #XX++XX+#
/ / / / 5 //
Definition:(Block circulant)
Theorem:
+M ^DL&_ C &+*)+'(, 5 n o <
<
C C
'
%'&2 "+ +M ' &+*&KD'(, T',*+UC
F= " 0 ,*'(M=C
T',*+U L= M2+=, <+(r'*+'(, LA%*',L* >
,2%( " <
= Q''9# )2%*% %'&2 Q+* +M
' &+*&)D'(, T',*+U C
O QMMKT+(R 2&U >?
> ]> A B 0 R O@ < U
> A< ] B '(3 R
+M
A%*+L3+& +( ;M, '(3 ?(3 '*RKT%(, B
Proof: U |
OL(M+3%* Q+a 50
P' P [A B B, +
a + a + a >+
+C Q+MC 5 <
C =,##XX++2+#X.MC+M.
2',+++CX /C
hB/
>
/
'
2&D >E >a> +,2DVCECaC+,<
< C 2, E>E>a>+Bd2+=, < +(r'*+'(, 5 m 2DUC,( > ] > A B 0 R O 3 < U > A < ] B =L* MLT% R C
R&L >+ < a B R& < F >
+ < a B CC 5 C R OD < E
>+ < a B
# C Q+a 50
P RK> >+< a+ R 9+ < F B < < < R >
+ <~;B x&+*&KD'*
# # #RK]C F>+ < 7 B -OE <?
>+ < 7 B <
< < R P L>+ < a BP QMMKT% A%*+L3+& A*LA%*,] B
Properties of separable image transformation
Recall: d%A'*'^D% 2 n 2&U >@
> ] > A B 0 2&&U >@ B 2* D ] > A B C
@%, M FFF='*+('C)M T',*+U
n S_ CAD 2&&UCTR`F=&UC]+2*&] >
<OL(M+3%* 2* 0 P 2* D [> 6 B B > M] > A%(
b GEcE G',*+U 5 TKD,+AD+&',+L(
2% 0 @2& OU>T B > > 5 f >
' %TbG >rfE @%, d 0 =2*
C
= 5 0 P = OU > ] B B > m f> > %(
b G / / .
b'M] ,L M%% #
RK( > A > 0 '2&&f>' > d+U>
0 2+T >U B M+U
>
+ R 0 2+M 0 "Fd 2* P G',*+U =L*T B
Image decomposition
dKAALM% 2&U>'
> ] > A B 0 2% O c+( 2 * D] > A B P d%A'*'^D%BC
Z2%( # R 0 2+, 2* n = 5 0 F 2+ B< .
R P 2* B/
)*+,% #
&2%M, C t C C C 9C B h 2+.
0
Z2%( # = 5 0 +v+aZ+7 GTT
<
O2%&_ ,2', # O2+,< .
P 9 2+.
0 +, +ra.
O+>a F < %(,*]
C
.
C = 5 0 D+(%'* <^+(',+L( L= � +, +ra, g +C a
bU'TAD% # $% M%% ,2', =< 5 0
>
R+aK+Ka,
QMMKT% R+M ' 3+'RL('D T',*+U P ,2', +M
> R+a0L +=+,a B
'(3 MKAALM% L(D] * 3+'RL('D %(,*+%M '*% (L( < H%*L C
Z2%( # = 5 0 b*>
R++K+r+,
Z2% M,L*'R% *%JK+*%T%(, +M 0 P E, E , ;B U *
Z Z ,.
<
Z+o+
-++
Similarity between images
Definition:
Example:
E%%3 ,L 3%=+(% T',*+U (L*T ;;C;; MK&2 ,2', # =L* o => R % F
>)% &'(
3%=+(% M+T+D'*+,] ^%,)%%( 1 '(3 R 'M "= C RDD C
Q r%&,L* z T',*+U (L*T 5 +M ' =K(&,+L( ;;C;; # _T P L* ;lTf/ B | Fl ML ,2',
=L* '(] FC w% FlG P L*;lTf
/ B '(3 @bFl>
)% 2'r% #
;C
;;~;;q�>;;fW;0� +== F0L
C
?C
"FZw"b ;;~.DD 4 "F" ; ,*+'(RD% 5 +(%JK'D+,] B
e C
;; @F .DD 0 ;?;;;~.DD
C;;f4;; >0 F( DU+D F0 P f > >fV >
CCC>fTBZ
y "+DDC 0=bU+[_
y / Y / CFb`TDUCC >"F