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+(%'* &LT^+(',+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