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Lecture 1: Image transformation Image transformation = !"# % & ’())"*#+(, (- +./0"1 (- 1+2" 3 /,4 5/,0" (- +,#",1+#6 7 8 9 : ; & = ->:?@ A B C D E ("-*+ 9FDG : H +< +< F" 3 I B -(5 1+.J)+*+#6 9 /11K." - +1 / 1LK/5" +./0"1 */, M" 0","5/) 3+N 32 +./0" D 8 E % O % B #5/,1-(5. (," +./0" #( /,(#P"5 D Q R+,4 / 1K+#/M)" #5/,1-(5./#+(, SQT %./0+,0 J5(M)".1 U 0 E& VW D M"*(."1 0((4 X Y Z ,(+16 +./0" [ \+]", / ,(+16 U 4+1#(5#"4 +./0" ^ /,4 #5/,1-(5./#+(, V 9 -+,4 (5+0+,/) *)"/, +./0" - ^ & QS B - D _ , K,C,(a, b C,(a, C,#(a, b K,C,(a, B %,]"51" J5(M)". D

Lecture 1: Image transformation

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Page 1: Lecture 1: Image transformation

Lecture 1:Image transformation

Image transformation =

!"# $ % & '())"*#+(, (- +./0"1 (- 1+2" 3 /,4 5/,0" (- $ +,#",1+#6 7 89: ;

<

& = ->:?@ AB C D E ("-*+ 9FDG : H +< +< F" 3 I

B -(5 1+.J)+*+#6 9/11K." - $ +1 $ / 1LK/5" $ +./0"1 */, M" 0","5/) 3+N 32 $ +./0" D

8 E % O % B #5/,1-(5. (," +./0" #( /,(#P"5 D

�Q� R+,4 / 1K+#/M)" #5/,1-(5./#+(, SQT

%./0+,0 J5(M)".1U

0 E& VW D M"*(."1 0((4 X

Y Z,(+16 +./0"

�[� \+]", / ,(+16 U 4+1#(5#"4 +./0" ^/,4 #5/,1-(5./#+(, V

9-+,4 (5+0+,/)

*)"/, $ +./0" -<

^ & QS B - D _ ,`

K,C,(a,

b

C,(a,C,#(a,

bK,C,(a, B %,]"51" J5(M)".D

Page 2: Lecture 1: Image transformation

Definition: (Linear image transformation)c E % O % +1 )+,"/5 B d 8 B /- $ _ 0D & $ / c-- D _ e B 0 D -(5 f< -

9 0 *S % H f g>%h

V/C" - " %<!"#

i >4+#1 <+,#jk<++EE+W&+>b+W(%++%(%%+El!"# 0 & m B - D

<

g11K." e +1 )+,"/59#P", E

0K, E>%>(W>(+%%%+<)+n0(l&#>+>%-/,++N</<<<<"9

oX b

P+N 9/pJ+&7mB q6D ;< 9J

H r & B s0 t<

uE7o HD`@.

Page 3: Lecture 1: Image transformation

Remark:

Definition: (Point spread function)

Definition: (Shift-invariant)

P"N9!

9 6 9 J D & P(a .K*P +,JK# ]/)K" $ /# +N< 6 D +,-)K",*"

#P" (K#JK# ]/)K" /# * [9 J D <

r+N") ' N< 6 D /--"*#+,0 J+N") < ' !9 J D M6 /

a"+0P# P*N<#,9 6 9 J D <

v

'49

P# <

9!

9< J D +1 */))"4 #P" rsR /# '!

9 J D <

V w

R+N [9 J <

!"# N< 6 /1 ]/5+/M)"1 X

3

sP+-# +,]/5+/,# i -5)N 94

9 k 9 -D & P !! S x9 J S k D -(5 fS)1N<6</<J" +#

Page 4: Lecture 1: Image transformation

Definition: (Convolution)

Theorem:

Proof:

Remark:

!"# #<0" %<

B 4"- D , 3

'(,]()K#+(, (- - /K4+# 0 i - $ b 0*/ 9& u

9

-+N 9 6 D 0 B 4S N

9 J S 6 D

B g11K." - /,4 0 /5" J"5+(4+*/))6 "N#",4"4 9 =-+N SQ+,

9 kVy3 D & -+N< 6 D

^C# %39 6#F 3 D&0*@ 9 6 9

P# $ +1F# nDrsR +1 1P+-# S +,]/5+/,# d #P" (J"5/#(5 c +1 /

*(,]()K#+(, a+#P #P" +,JK# +./0" <

!"# 0 E & c)- D < 0*/ 9

& %%9

-+N<6-P*NrD

S PC S N 9 J S 6 D& - $ b P Z

9

< - $ b P<

& PN- B "N"5*+1" D

v '(,]()K#+(, +1 +.J(5#/,# -(5 K,4"51#/,4+,0 +./0" <M)K5

<

Page 5: Lecture 1: Image transformation

Definition: (Separable)

s"J/5/M)"<i P*N<#, 9 6 9 J D & P**N<4)P5)6<J) -(5 f +1 N< T< 4<r>3 <