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EC
E/O
PT
I533 Digital Im
age Processing class notes 10 D
r. Robert A
. Schowengerdt 2003
1
-D M
ATH
REV
IEW
CO
NTIN
UO
US 1
-D F
UN
CTIO
NS
•K
ron
ecker d
elta
fun
ctio
n a
nd
its re
lativ
es
delta
fun
ctio
n
NO
TE: Th
e d
elta
fun
ctio
n’s
am
plitu
de is
infin
ite a
nd
its a
rea
is 1
. The a
mp
litud
e is
sh
ow
n a
s 1
for c
on
ven
ien
ce in
plo
ts.
Write
the e
qu
atio
n th
at d
efin
es th
e a
rea
of a
delta
fu
nctio
n a
s 1
.
Revie
w th
e d
efin
ition
of d
elta
fun
ctio
n in
term
s o
f the lim
it of c
on
ven
tion
al fu
nctio
ns, s
uch
as th
e re
cta
ng
le fu
nctio
n
δx
x0
–(
)
x
1
x0
x
1x
0 = 0
EC
E/O
PT
I533 Digital Im
age Processing class notes 11 D
r. Robert A
. Schowengerdt 2003
1
-D M
ATH
REV
IEW
even
delta
pa
ir
od
d d
elta
pa
ir
δδx
x0
–b-------------
bδ
xx
0–
b+
()
δx
x0
–b
–(
)+
[]
=
x
|b|
bx
- b
|b|
x0 +
bx
0 - b
x0 =
0x
0 ≠
0
δδx
x0
–b-------------
bδ
xx
0–
b+
()
δx
x0
–b
–(
)–
[]
=
x
|b|
bx
- b
x0 +
b
x0 - b
x0 =
0
-|b|
|b|
-|b|
x0 ≠
0
EC
E/O
PT
I533 Digital Im
age Processing class notes 12 D
r. Robert A
. Schowengerdt 2003
1
-D M
ATH
REV
IEW
com
b (s
ha
h) co
mb
xx
0–b
-------------
b
δx
x0
–n
b–
()
n∞–
= ∞∑=
x
|b|
b- b
x0 =
0
02b
- 2bx
|b|
x0 +
bx
0 -b0
x0 +
2bx
0 -2bx
0
. . .. . .
. . .. . .x
0 ≠
0
Even
delta
pa
ir, od
d d
elta
pa
ir a
nd
com
b h
ave
am
plitu
de o
f b
EC
E/O
PT
I533 Digital Im
age Processing class notes 13 D
r. Robert A
. Schowengerdt 2003
1
-D M
ATH
REV
IEW
•U
se o
f the
δ
fun
ctio
n
siftin
g
• N
OTE
: Siftin
g is
a c
on
volu
tion
, eva
lua
ted
for a
pa
rticu
lar s
hift
• F
ind
s th
e v
alu
e o
f a fu
nctio
n a
t a s
pecific
va
lue o
f the in
dep
en
den
t va
riab
le (s
imila
r to a
look-
up
tab
le) f
α()δ
αx
0–
()
αd∞– ∞∫
fx
0(
)constant
==
EC
E/O
PT
I533 Digital Im
age Processing class notes 14 D
r. Robert A
. Schowengerdt 2003
1
-D M
ATH
REV
IEW
sa
mp
ling
• N
OTE
: Sa
mp
ling
is a
mu
litplic
atio
n
• O
utp
ut is
a d
elta
fun
ctio
n, w
ith a
rea
dete
rmin
ed
by th
e v
alu
e o
f the fu
nctio
n a
t the s
pecifie
d
va
lue o
f the in
dep
en
den
t va
riab
le.
un
iform
sa
mp
ling
• N
OTE
: Mu
st d
ivid
e c
om
b fu
nctio
n b
y |b
| to re
tain
am
plitu
de o
f f(x).
• N
OTE
: f(x) m
od
ula
tes th
e c
om
b fu
nctio
n.
fx()δ
xx
0–
()
fx
0(
)δx
x0
–(
)=x
x0
xx
0x
x=
f(x)f(x0 )
x0
1b -----f
x()com
bx
x0
–b-------------
fx
0n
b+
()δ
xx
0–
nb
–(
)n
∞–= ∞∑
=
xx
0 +b
x0 -b
0x
0 +2b
x0 -2b
x0
. . .. . .
xx
0 +b
x0 -b
0x
0 +2b
x0 -2b
x0
. . .. . .
1
EC
E/O
PT
I533 Digital Im
age Processing class notes 15 D
r. Robert A
. Schowengerdt 2003
1-D
MA
TH R
EV
IEW
sh
ifting
rep
lica
ting
• N
OTE
: Mu
st d
ivid
e b
y |b
| to re
tain
am
plitu
de o
f f(x)
gx()
fx() ❉
δx
x0
–(
)f
α()δ
xx
0–
α–
()
αd∞– ∞∫
fx
x0
–(
)=
==
xx
0x
❉=
f(x)
xx
0
g(x)1
gx()
1b -----f
x() ❉ co
mb
xx
0–b
-------------
=
xx
0 +b
x0 -b
0x
0 -2bx
0
. . .. . .
1
x
❉
f(x)
xx
0 +b
x0 -b
0x
0 -2bx
0
. . .. . .
g(x)
=
EC
E/O
PT
I533 Digital Im
age Processing class notes 16 D
r. Robert A
. Schowengerdt 2003
1-D
MA
TH R
EV
IEW
recta
ng
le (s
qu
are
pu
lse)
trian
gle
rectxb ---
0
xb⁄
12⁄
>
12
x
b⁄⁄
12⁄
=
1
xb⁄
12⁄
<
=
trixb ---
0
xb⁄
1≥
1x
b⁄
xb⁄
1<
–=
Wh
at is
the v
alu
e o
f b in
the
ab
ove g
rap
h?
-0.2 0
0.2
0.4
0.6
0.8 1
1.2-60-40
-200
2040
60
recttri
f(x)
x
For a
giv
en
b,
the tri fu
nctio
n
is tw
ice a
s w
ide
as th
e re
ct
fun
ctio
n
EC
E/O
PT
I533 Digital Im
age Processing class notes 17 D
r. Robert A
. Schowengerdt 2003
1-D
MA
TH R
EV
IEW
sin
c
sin
c-s
qu
are
d
sinc(xb
)⁄
πx
b⁄[
]sinπ
xb⁄
--------------------------=
sinc2
xb⁄
()
Wh
at is
the v
alu
e o
f b in
th
e a
bove g
rap
h?
-0.4
-0.2 0
0.2
0.4
0.6
0.8 1
1.2-60-40
-200
2040
60
sincsinc squared
f(x)
x
EC
E/O
PT
I533 Digital Im
age Processing class notes 18 D
r. Robert A
. Schowengerdt 2003
1-D
MA
TH R
EV
IEW
ga
us(s
ian
) ga
us
xb⁄
()
eπ
xb⁄
() 2
–=
Wh
at is
the v
alu
e o
f b in
the
ab
ove g
rap
h?
-0.2 0
0.2
0.4
0.6
0.8 1
1.2-60-40
-200
2040
60
gaus
f(x)
x
EC
E/O
PT
I533 Digital Im
age Processing class notes 19 D
r. Robert A
. Schowengerdt 2003
1-D
MA
TH R
EV
IEW
Wh
at is
the v
alu
e o
f b in
th
e a
bove g
rap
h?
-1.2
-0.8
-0.4 0
0.4
0.8
1.2-60-40
-200
2040
60
cossin
f(x)x
cosin
e
sin
e
2πx
b⁄(
)cos
ej2π
xb⁄
()
ej
–2π
xb⁄
()
+2----------------------------------------------
=
2πx
b⁄(
)sin
ej2π
xb⁄
()
ej
–2π
xb⁄
()
–2j
----------------------------------------------=
EC
E/O
PT
I533 Digital Im
age Processing class notes 20 D
r. Robert A
. Schowengerdt 2003
1-D
MA
TH R
EV
IEW
CO
NV
OLU
TION
(1-D
)
Wh
y is
it imp
orta
nt?
•D
escrib
es th
e e
ffect o
f a L
inea
r Sh
ift Inva
rian
t (LSI) s
yste
m o
n
inp
ut s
ign
als
• L
is th
e s
yste
m o
pera
tor
•D
escrip
tion
of g
en
era
l syste
m
•For a
n L
SI s
yste
m, L
is a
con
volu
tion
system
(operator L)
inputsignal f(x)
outputsignal g(x)
gx()
Lf
x()[
]=
gx()
fx() ❉
hx()
fα(
)hx
α–
()
αd∞– ∞∫
==
EC
E/O
PT
I533 Digital Im
age Processing class notes 21 D
r. Robert A
. Schowengerdt 2003
1-D
MA
TH R
EV
IEW
Exa
mp
le
2
03
x or α
f(x) or f(α)
03
x or α
h(x) or h(α)
-1
1
EC
E/O
PT
I533 Digital Im
age Processing class notes 22 D
r. Robert A
. Schowengerdt 2003
1-D
MA
TH R
EV
IEW
α
h(- α)
1
x = 0
α
f(α)h(0 - α
)
area = g(0)
α
f(α)h(3 - α
)
area = g(3)
α
f(α)h(2 - α
)
area = g(2)
α
f(α)h(1 - α
)
area = g(1)
α
f(α)h(4 - α
)
area = g(4)
α
h(1 - α)
1x = 1
α
h(2 - α)
1
x = 2
α
h(3 - α)
1
x = 3
α
h(4 - α)
1
x = 4
shiftm
ultiplyintegrate
EC
E/O
PT
I533 Digital Im
age Processing class notes 23 D
r. Robert A
. Schowengerdt 2003
1-D
MA
TH R
EV
IEW
• P
lot g
(x)
2
0
31
g(x)
x
The s
hifts
in
this
exa
mp
le
are
by in
teg
er
ste
ps, fo
r illu
stra
tion
con
ven
ien
ce
EC
E/O
PT
I533 Digital Im
age Processing class notes 24 D
r. Robert A
. Schowengerdt 2003
1-D
MA
TH R
EV
IEW
Recip
e
Con
volu
tion
• 1
. write
both
as a
fun
ctio
n o
f αf(α
) an
d h
(α)
• 2
. flip h
(or f) a
bou
t α =
0h
(-α)
• 3
. sh
ift h (o
r f) by a
n a
mou
nt x
h(x
- α)
• 4
. mu
ltiply
the tw
o fu
nctio
ns
f(α)h
(x -α
)
• 5
. inte
gra
te th
e p
rod
uct fu
nctio
n o
ver a
ll αg
(x)
• 6
. rep
ea
t ste
ps 3
thro
ug
h 5
un
til don
e
EC
E/O
PT
I533 Digital Im
age Processing class notes 25 D
r. Robert A
. Schowengerdt 2003
1-D
MA
TH R
EV
IEW
Con
volu
tion
Pro
pertie
s
1-D C
ON
VO
LU
TIO
N P
RO
PE
RT
IES
property
comm
utative
distributive
associative
fx() ❉
hx()
hx() ❉
fx()
=
fx() ❉
h
1x()
h2
x()+
[]
fx() ❉
h1
x()f
x() ❉ h
2x()
+=
fx() ❉
h
1x() ❉
h2
x()[
]
fx() ❉
h1
x()[
] ❉ h
2x()
=
EC
E/O
PT
I533 Digital Im
age Processing class notes 26 D
r. Robert A
. Schowengerdt 2003
1-D
MA
TH R
EV
IEW
Con
volu
tion
Exa
mp
les
1-D C
ON
VO
LU
TIO
N E
XA
MP
LE
S
f(x)h(x)
g(x)
f(x)δ(x)
f(x)
f(x-x0 )
h(x)g(x-x
0 )
f(x)h(x-x
0 )g(x-x
0 )
rect(x)rect(x)
tri(x)
sinc(x)sinc(x)
sinc(x)
gaus(x)gaus(x)
12-------g
au
sx2
-------
EC
E/O
PT
I533 Digital Im
age Processing class notes 27 D
r. Robert A
. Schowengerdt 2003
1-D
MA
TH R
EV
IEW
FO
UR
IER
TRA
NSFO
RM
S (1
-D)
Wh
y is
it imp
orta
nt?
•For a
n L
SI s
yste
m, th
e c
on
volu
tion
op
era
tor b
ecom
es a
mu
ltiplic
atio
n
op
era
tor in
the F
ou
rier d
om
ain
•Ta
kin
g th
e F
ou
rier tra
nsfo
m o
f the s
yste
m e
qu
atio
n,
wh
ere
G(u
) is th
e s
pectru
m o
f the o
utp
ut s
ign
al, F
(u) is
the s
pectru
m o
f th
e in
pu
t sig
na
l, an
d H
(u) is
the s
yste
m tra
nsfe
r fun
ctio
n
Gu(
)F
u()H
u()
=
EC
E/O
PT
I533 Digital Im
age Processing class notes 28 D
r. Robert A
. Schowengerdt 2003
1-D
MA
TH R
EV
IEW
• In
ma
ny c
ases, it is
ea
sie
r to a
na
lyze a
n L
SI s
yste
m in
the F
ou
rier
dom
ain
• F
orw
ard
tran
sfo
rm
• In
vers
e tra
nsfo
rm F
u()
fx()e
j2πxu
–xd
∞– ∞∫=
fx()
Fu(
)ej2π
xuud
∞– ∞∫=
EC
E/O
PT
I533 Digital Im
age Processing class notes 29 D
r. Robert A
. Schowengerdt 2003
1-D
MA
TH R
EV
IEW
Fou
rier Tra
nsfo
rm P
rop
ertie
s
•f(x
) an
d F
(u) a
re, in
gen
era
l, com
ple
x fu
nctio
ns
•f(x
) rea
l ➞ F
(u) =
F*(-u
)
• F
is H
erm
itian
: Re[F
(u)] e
ven
, Im[F
(u)] o
dd
•f(x
) rea
l an
d e
ven ➞
Im[F
(u)] =
0 , i.e
. F(u
) is re
al
•Forw
ard
tran
sfo
rm is
the a
na
lysis
of f(x
) into
its s
pectru
m F
(u)
•In
vers
e tra
nsfo
rm is
the s
yn
thesis
of f(x
) from
F(u
)
EC
E/O
PT
I533 Digital Im
age Processing class notes 30 D
r. Robert A
. Schowengerdt 2003
1-D
MA
TH R
EV
IEW
Fou
rier Tra
nsfo
rm P
airs
1-D F
OU
RIE
R T
RA
NSF
OR
M PA
IRS
f(x)F
(u)
1δ(u)
δ(x)1
rect(x)sinc(u)
sinc(x)rect(u)
comb(x)
comb(u)
gaus(x)gaus(u)
tri(x)sinc 2(u)
2πu
0 x(
)cos
12
u0
-----------δδuu
0-----
12
x0
-----------δδxx0
-----
2π
ux
0(
)cos
2πu
0 x(
)sin
j2
u0
----------- δδuu
0-----
EC
E/O
PT
I533 Digital Im
age Processing class notes 31 D
r. Robert A
. Schowengerdt 2003
1-D
MA
TH R
EV
IEW
Fou
rier Tra
nsfo
rm P
rop
ertie
s
1-D F
OU
RIE
R T
RA
NSF
OR
M P
RO
PE
RT
IES
name
f(x)F
(u)f(x)
F(u)
f(±x)F
(±u)
F(±x)
scalingf(x/b)
|b|F(bu)
shifting
f(x ± x0 )
derivative
linearity
f1
x() ❉ f
2x()
F1
u()F
2u(
)
fu
+ −()
f1
x()f2
x()F
1u(
) ❉ F
2u(
)
f1
x() ★ f
2x()
F1
u()F
2u–
()
ej2π
x0 u
±F
u()
f1
x()f2
x–(
)F
1u(
) ★ F
2u(
)
ej2π
xu0
±f
x()F
uu
0+ −
()
fk()
x()j2π
u(
)kF
u()
j2πx
–)
kfx()
Fk()
u()
a1 f
1x()
a2 f
2x()
+a
1 F1
u()
a2 F
2u(
)+
EC
E/O
PT
I533 Digital Im
age Processing class notes 32 D
r. Robert A
. Schowengerdt 2003
1-D
MA
TH R
EV
IEW
LIN
EA
R, S
HIF
T-INV
AR
IAN
T (LSI) S
YSTE
MS (1
-D)
•Lin
ea
r: ou
tpu
t of a
su
m o
f inp
uts
is e
qu
al to
the s
um
of th
e in
div
idu
al
ou
tpu
ts
•sh
ift-inva
rian
t: syste
m re
sp
on
se d
oes n
ot c
ha
ng
e o
ver s
pa
ce
Syste
m e
qu
atio
n
• w
here
h(x
) is th
e s
yste
m im
pu
lse re
sp
on
se
g1
x()a
f1
x() ❉ h
x()=
g2
x()b
f2
x() ❉ h
x()=
gx()
af
1x()
bf
2x()
+[
] ❉ h
x()=
af
1x() ❉
hx()
bf
2x() ❉
hx()
+(
)=
g1
x()g
2x()
+=
gx()
fx() ❉
hx()
=
EC
E/O
PT
I533 Digital Im
age Processing class notes 33 D
r. Robert A
. Schowengerdt 2003
1-D
MA
TH R
EV
IEW
•Fou
rier tra
nsfo
rm o
f syste
m e
qu
atio
n
• w
here
H(u
) is th
e s
yste
m tra
nsfe
r fun
ctio
n
• H
(u) is
a c
om
ple
x filte
r tha
t mod
ifies th
e s
pectru
m F
(u) o
f f(x)
•For c
om
ple
x fu
nctio
ns G
, F a
nd
H
Gu(
)F
u()H
u()
=
am
pl
Gu(
)[
]a
mp
lF
u()
[]a
mp
lH
u()
[]
=
ph
ase
Gu(
)[
]p
ha
seF
u()
[]
ph
ase
Hu(
)[
]+
=
EC
E/O
PT
I533 Digital Im
age Processing class notes 34 D
r. Robert A
. Schowengerdt 2003
1-D
MA
TH R
EV
IEW
1-D
CA
SCA
DED
SYSTE
MS
N c
asca
ded
LSI s
yste
ms
•Sin
gle
syste
m e
qu
iva
len
t
wh
ere
hn
et is
the n
et s
yste
m im
pu
lse re
sp
on
se
. . .
f(x)g(x)
h1 (x)
hN
(x)
gx()
fx() ❉
h1
x()[
] ❉ h
2x()
{}…
❉ …
hN
x()=
f(x)
hnet (x)
g(x)
hn
etx()
h1
x() ❉ h
2x()…
❉ …
hN
x()=
EC
E/O
PT
I533 Digital Im
age Processing class notes 35 D
r. Robert A
. Schowengerdt 2003
1-D
MA
TH R
EV
IEW
FO
UR
IER
TRA
NSFO
RM
EX
AM
PLES
Ex 1
. sin
c(x
/2) ❉
sin
c(x
/3)
•Con
volu
tion
in th
is c
ase is
very
diffic
ult!
•Ta
ke th
e F
ou
rier tra
nsfo
rm
•Ta
ke th
e in
vers
e F
ou
rier tra
nsfo
rm
2rect
2u
()
3rect
3u
()
⋅6
rect3
u(
)=
21/4-1/4
31/6-1/6
x
uu
61/6-1/6
u
=
63 ---sinc(x/3)2
sinc(x/3)=
EC
E/O
PT
I533 Digital Im
age Processing class notes 36 D
r. Robert A
. Schowengerdt 2003
1-D
MA
TH R
EV
IEW
Ex 2
. Fin
d s
pectru
m o
f sq
ua
re w
ave w
ith a
DC b
ias
•W
rite s
qu
are
wa
ve a
s c
on
volu
tion
•Ta
ke th
e F
ou
rier tra
nsfo
rm
• s
pectru
m is
com
b fu
nctio
n, m
od
ula
ted
by s
inc fu
nctio
n, s
am
ple
d a
t freq
uen
cy in
terv
al ∆
u =
1/5
, i.e
1/p
erio
d. z
ero
s a
t u =
n/2
, n =
± 1
, ±2
, . . .
+1
-15
10-5
1 f(x)
x
. . .. . .
fx()
15 ---rectx2 ---
❉
com
bx5 ---
=
Fu(
)15 ---
25
sinc2
u(
)co
mb
5u
()
⋅⋅
⋅⋅
=
2sinc
2u
()co
mb
5u
()
=
EC
E/O
PT
I533 Digital Im
age Processing class notes 37 D
r. Robert A
. Schowengerdt 2003
1-D
MA
TH R
EV
IEW
• if s
qu
are
wa
ve p
erio
d P
= 2
x p
uls
e w
idth
, we h
ave th
e c
lassic
sq
ua
re w
ave s
pectru
m a
t
• u
= 0
, ±1
/P, ±
3/P
, ±5
/P, . . .
? Verify
the a
bove s
tate
men
t for a
n a
rbitra
ry
perio
d P
• s
inc(2
u) a
nd
com
b(5
u)
-0.4
-0.2 0
0.2
0.4
0.6
0.8 1
1.2
01/2
13/2
-1/2-1
-3/2
1/52/5
3/54/5
5/56/5
7/58/5
-8/5-7/5
-6/5-5/5
-4/5-3/5
-2/5-1/5
21/5u
EC
E/O
PT
I533 Digital Im
age Processing class notes 38 D
r. Robert A
. Schowengerdt 2003
1-D
MA
TH R
EV
IEW
• s
inc(2
u) tim
es c
om
b(5
u)
-0.4
-0.2 0
0.2
0.4
0.6
0.8 1
1.2
01/2
13/2
-1/2-1
-3/2
1/52/5
3/54/5
5/56/5
7/58/5
-8/5-7/5
-6/5-5/5
-4/5-3/5
-2/5-1/5
2/5
com
b,
mod
ula
ted
by
sin
c fu
nctio
n
EC
E/O
PT
I533 Digital Im
age Processing class notes 39 D
r. Robert A
. Schowengerdt 2003
1-D
MA
TH R
EV
IEW
•Sca
ling
Pro
perty
•W
idth
of F
(u) is
invers
ely
pro
portio
na
l to w
idth
of f(x
)
rect(x)
x
rect(x/2)
x
uu
sinc(u)
2sinc(2u)
rect(x/3)
x
3sinc(3u)
u
F123
1/2-1/2
1-1
3/2-3/2
EC
E/O
PT
I533 Digital Im
age Processing class notes 40 D
r. Robert A
. Schowengerdt 2003
1-D
MA
TH R
EV
IEW
δδ(2x)
x
δδ(x)
x
δδ(2x/3)
x
uuu
cos(πu)
2cos(2πu)
3cos(3πu)
F
1/2-1/2
1-1
3/2-3/2
1/2
13/2
EC
E/O
PT
I533 Digital Im
age Processing class notes 41 D
r. Robert A
. Schowengerdt 2003
1-D
MA
TH R
EV
IEW
δδ(u)/2
u
δδ(u/2)/4
u
δδ(u/3)/6
u
Fcos(2πx)
xxx
cos(4πx)
cos(6πx)
1/2
1/2
1/2
1
2
3
-1
-2
-3
EC
E/O
PT
I533 Digital Im
age Processing class notes 42 D
r. Robert A
. Schowengerdt 2003
1-D
MA
TH R
EV
IEW
Su
perp
ositio
n P
rop
erty
•Fou
rier tra
nsfo
rm o
f su
m o
f fun
ctio
ns e
qu
als
su
m o
f their in
div
idu
al
Fou
rier tra
nsfo
rms
1 + cos(2πx)
x
δ(u)+δδ(u)/2
u
F1/2
1-1
1
cos(2πx)+cos(4πx)
δδ(u)/2+δδ(u/2)/4
u
1/2
1-1
2-2
x
EC
E/O
PT
I533 Digital Im
age Processing class notes 43 D
r. Robert A
. Schowengerdt 2003
1-D
MA
TH R
EV
IEW
SYSTE
M A
NA
LYSIS
WITH
THE F
OU
RIE
R TR
AN
SFO
RM
• Th
ree a
pp
lica
tion
s
Application
Given
Find
Spatial Dom
ainF
ourier Dom
ain
system output
f(x), h(x)
g(x)
system
identificationf(x), g(x)
h(x)N
Aill-conditioned
inversionh(x), g(x)
f(x)N
Aill-conditioned
gx()
fx() ❉
hx()
=G
u()
Fu(
)Hu(
)=
gx()
F1–
Gu(
)[
]=
Hu(
)G
u()
Fu(
)⁄
=
hx()
F1–
Hu(
)[
]=
Fu(
)G
u()
Hu(
)⁄
=
fx()
F1–
Fu(
)[
]=