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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
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`@.
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" +#
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
<
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 <