2-D DISCRETE FOURIER TRANSFORMdial/ece533/notes9.pdf2-D DISCRETE FOURIER TRANSFORM f 1-D DFT applied...
19
ECE/OPTI533 Digital Image Processing class notes 188 Dr. Robert A. Schowengerdt 2003 2-D DISCRETE FOURIER TRANSFORM DEFINITION forward DFT inverse DFT • The DFT is a transform of a discrete, complex 2-D array of size M x N into another discrete, complex 2-D array of size M x N Approximates the Continuous Fourier Transform (CFT) under certain conditions Both f(m,n) and F(k,l) are 2-D periodic Alternate definitions: • in inverse definition instead, or in forward and inverse definitions (“unitary”) • doesn’t matter as long as consistent Fkl , ( ) 1 MN --------- fmn , ( ) e j –2 π mk M ------ nl N ---- + ⋅ n 0 = N 1 – ∑ m 0 = M 1 – ∑ = fmn , ( ) Fkl , ( ) e + j 2 π mk M ------ nl N ---- + ⋅ l 0 = N 1 – ∑ k 0 = M 1 – ∑ = 1 MN --------- 1 MN -------------
Transcript of 2-D DISCRETE FOURIER TRANSFORMdial/ece533/notes9.pdf2-D DISCRETE FOURIER TRANSFORM f 1-D DFT applied...
EC
E/O
PT
I533 Digital Im
age Processing class notes 188 D
r. Robert A
. Schowengerdt 2003
2-D
DIS
CR
ETE
FO
UR
IER
TRA
NSFO
RM
DEFIN
ITION
forw
ard
DFT
invers
e D
FT
•Th
e D
FT is
a tra
nsfo
rm o
f a d
iscre
te, c
om
ple
x 2
-D a
rray o
f siz
e M
x
N in
to a
noth
er d
iscre
te, c
om
ple
x 2
-D a
rray o
f siz
e M
x N
Ap
pro
xim
ate
s th
e C
on
tinu
ou
s F
ou
rier Tra
nsfo
rm (C
FT) u
nd
er c
erta
in c
on
ditio
ns
Both
f(m,n
) an
d F
(k,l) a
re 2
-D p
erio
dic
Alte
rna
te d
efin
ition
s:
•
in in
vers
e d
efin
ition
inste
ad
, or
in fo
rwa
rd a
nd
invers
e d
efin
ition
s (“
un
itary
”)
• d
oesn
’t ma
tter a
s lo
ng
as c
on
sis
ten
t
Fk
l,(
)1MN
---------f
mn,
()
ej
–2π
mk
M -------n
lN -----+
⋅n
0=
N1
–
∑m
0=
M1
–
∑=
fm
n,(
)F
kl,
()
e+
j2πm
kM -------
nlN -----
+
⋅l
0=
N1
–
∑k
0=
M1
–
∑=
1MN
---------1M
N--------------
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I533 Digital Im
age Processing class notes 189 D
r. Robert A
. Schowengerdt 2003
2-D
DIS
CR
ETE
FO
UR
IER
TRA
NSFO
RM
RELA
TION
OF TH
E D
FT TO
THE C
FT
•O
ne v
iew
of th
e D
FT is
as a
n a
pp
roxim
atio
n to
the C
FT
•“ re
cip
e” to
con
vert C
FT to
DFT:
1. s
am
ple
f(x,y
)
2. tru
nca
te to
MX
x N
Y
3. m
ake p
erio
dic
, i.e. th
e p
erio
dic
exte
nsio
n o
f a 2
-D a
rray f(m
,n) w
ith s
am
ple
inte
rva
ls X
=Y=
1
fx
y,(
)1XY
-------com
bx
Xy
Y⁄,
⁄(
)⋅
fx
y,(
)1XY
-------com
bx
Xy
Y⁄,
⁄(
)rect
xM
X⁄
yN⁄
Y,
()
⋅⋅
fx
y,(
)1XY
-------com
bx
Xy
Y⁄,
⁄(
)rect
xM
X⁄
yN
Y⁄
,(
)⋅
⋅
❉ ❉
1
MX
NY
⋅----------------------co
mb
xM
Xy
NY
⁄,
⁄(
)f
pm
n,(
)=
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I533 Digital Im
age Processing class notes 190 D
r. Robert A
. Schowengerdt 2003
2-D
DIS
CR
ETE
FO
UR
IER
TRA
NSFO
RM
4. ta
ke C
FT
, i.e. th
e p
erio
dic
exte
nsio
n o
f a 2
-D a
rray F
(k,l) w
ith s
am
ple
inte
rva
ls 1
/X=
1/Y
=1
•Th
e a
rrays f a
nd
F a
re b
oth
dis
cre
te a
nd
perio
dic
in s
pa
ce a
nd
sp
atia
l freq
uen
cy, re
sp
ectiv
ely
rep
lica
te (a
liasin
g
occu
rs h
ere
)sm
ooth
(lea
ka
ge
occu
rs h
ere
)sa
mp
le
Fu
v,(
) ❉ ❉
com
bu
XvY
,(
) ❉ ❉
MX
NY
⋅sinc
uM
XvN
Y,
()
[]
com
bu
MX
vNY
,(
)⋅
Fp
kl,
()
=
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I533 Digital Im
age Processing class notes 191 D
r. Robert A
. Schowengerdt 2003
2-D
DIS
CR
ETE
FO
UR
IER
TRA
NSFO
RM
CA
LCU
LA
TION
OF D
FT
•B
oth
arra
ys f(m
,n) a
nd
F(k
,l) are
perio
dic
(perio
d =
M x
N) a
nd
sa
mp
led
(X x
Y in
sp
ace, 1
/MX
x 1
/NY in
freq
uen
cy)
•In
the C
FT, if o
ne fu
nctio
n h
as c
om
pa
ct s
up
port (i.e
. it is s
pa
ce- o
r fre
qu
en
cy-lim
ited
), the o
ther m
ust h
ave
∞
su
pp
ort
•Th
ere
fore
, alia
sin
g w
ill occu
r with
the D
FT, e
ither in
sp
ace o
r fre
qu
en
cy. If w
e w
an
t the D
FT to
clo
sely
ap
pro
xim
ate
the C
FT,
alia
sin
g m
ust b
e m
inim
ized
in b
oth
dom
ain
s
•Th
e F
ast F
ou
rier Tra
nsfo
rm (F
FT) is
an
effic
ien
t alg
orith
m to
ca
lcu
late
the D
FT th
at ta
kes a
dva
nta
ge o
f the p
erio
dic
ities in
the
com
ple
x e
xp
on
en
tial
Ca
n u
se 1
-D F
FT fo
r 2-D
DFT (la
ter)
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I533 Digital Im
age Processing class notes 192 D
r. Robert A
. Schowengerdt 2003
2-D
DIS
CR
ETE
FO
UR
IER
TRA
NSFO
RM
AR
RA
Y C
OO
RD
INA
TES
•Th
e D
C te
rm (u
=v=
0) is
at (0
,0) in
the ra
w o
utp
ut o
f the D
FT (e
.g.
the M
atla
b fu
nctio
n “
fft2”)
•R
eord
erin
g p
uts
the s
pectru
m in
to a
“p
hysic
al”
ord
er (th
e s
am
e
as s
een
in o
ptic
al F
ou
rier tra
nsfo
rms) (e
.g. th
e M
atla
b fu
nctio
n
“ffts
hift”
)
•N
an
d M
are
com
mon
ly p
ow
ers
of 2
for th
e F
FT. Th
ere
fore
, the D
C
term
is a
t (M/2
,N/2
) in th
e re
ord
ere
d fo
rma
t for (0
,0) in
dexin
g
an
d a
t (M/2
+1
,N/2
+1
) for (1
,1) in
dexin
g
Quad I
IIIIIIV
IIIIV
III
raw
ou
tpu
t of D
FT
reord
ere
d o
utp
ut o
f DFT
DC
DC
EC
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I533 Digital Im
age Processing class notes 193 D
r. Robert A
. Schowengerdt 2003
2-D
DIS
CR
ETE
FO
UR
IER
TRA
NSFO
RM
SA
MPLE IN
TER
VA
LS
•Con
stra
ints
pro
du
ct o
f ph
ysic
al s
am
ple
inte
rva
ls in
x a
nd
u, y
an
d v
:X
U =
1/M
, YV
= 1
/N
sa
mp
ling (re
plic
atio
n) fre
qu
en
cy in
u a
nd
v:
u
s
= 1
/X, v
: v
s
= 1
/Y
fold
ing fre
qu
en
cy in
u a
nd
v:
u
f
= 1
/2X
, v
f
= 1
/2Y
•For im
ag
es, a
con
ven
ien
t, norm
aliz
ed
set o
f un
its is
X =
Y =
1 p
ixel
•Th
ere
fore
,
U =
1/M
cycle
s/p
ixel, u
s = 1
cycle
/pix
el, u
f = 1
/2 c
ycle
/pix
el
V =
1/N
cycle
s/p
ixel, v
s = 1
cycle
/pix
el, v
f = 1
/2 c
ycle
/pix
el
•N
ote
, in re
od
ere
d D
FT fo
rma
t, uf a
nd
vf a
re a
lon
g th
e firs
t row
an
d c
olu
mn
s o
f the a
rray
EC
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I533 Digital Im
age Processing class notes 194 D
r. Robert A
. Schowengerdt 2003
2-D
DIS
CR
ETE
FO
UR
IER
TRA
NSFO
RM
Reord
erin
g th
e 2
-D D
FT
•“orig
in-c
en
tere
d” d
isp
lay k
l
123
4 432
010
2030
40
0
10
20
30
40 0 5 10 15 20 25
f(m,n
)
orig
in-c
en
tere
d
|F(k
,l)|
010
2030
40
0
10
20
30
40 0 5 10 15 20 25
010
2030
40
0
10
20
30
40 0
0.2
0.4
0.6
0.8 1
|F(k
,l)|
EC
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I533 Digital Im
age Processing class notes 195 D
r. Robert A
. Schowengerdt 2003
2-D
DIS
CR
ETE
FO
UR
IER
TRA
NSFO
RM
Alia
sin
g in
the fre
qu
en
cy d
om
ain
•D
FT o
f dis
cre
te a
pp
roxim
atio
n to
a re
ct(x
/W,y
/W) fu
nctio
n
010
2030
40
0
10
20
30
40 0
0.2
0.4
0.6
0.8 1
010
2030
40
0
10
20
30
40 0
0.2
0.4
0.6
0.8 1
010
2030
40
0
10
20
30
40 0
0.2
0.4
0.6
0.8 1
010
2030
40
0
10
20
30
40 0 2 4 6 8 10
010
2030
40
0
10
20
30
40 0 20 40 60 80
100
010
2030
40
0
10
20
30
40 0 5 10 15 20 25
5 x
5
9 x
90
1020
3040
0
10
20
30
40 0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
010
2030
40
0
10
20
30
40 0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
010
2030
40
0
10
20
30
40 0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
f(m,n
)|F
(k,l)|
|F(k
,l)| - |F(u
,v)|
3 x
3
EC
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I533 Digital Im
age Processing class notes 196 D
r. Robert A
. Schowengerdt 2003
2-D
DIS
CR
ETE
FO
UR
IER
TRA
NSFO
RM
Dig
ital im
ag
e p
ow
er s
pectru
m (s
qu
are
d a
mp
litud
e o
f F) c
oord
ina
tes
(0,0
)(u
,v)=
(-0.5
,-0.5
)cycle
s/p
ixel
(M/2
,N)
(u,v
) = (+
0.5
-1/N
,0)
cycle
s/p
ixel
(M,N
/2), (u
,v) =
(0,+
0.5
-1/M
) cycle
s/p
ixel
(M,N
)(u
,v)=
(+0
.5-1
/N,
+0
.5-1
/M)
cycle
s/p
ixel
EC
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I533 Digital Im
age Processing class notes 197 D
r. Robert A
. Schowengerdt 2003
2-D
DIS
CR
ETE
FO
UR
IER
TRA
NSFO
RM
EX
AM
PLES O
F IM
AGE P
OW
ER
SPECTR
A
desert
field
s
stre
ets
railro
ad
EC
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PT
I533 Digital Im
age Processing class notes 198 D
r. Robert A
. Schowengerdt 2003
2-D
DIS
CR
ETE
FO
UR
IER
TRA
NSFO
RM
DIS
PLA
Y O
F P
OW
ER
SPECTR
A
•La
rge d
yn
am
ic ra
ng
e
am
plitu
de a
t zero
-freq
uen
cy d
om
ina
tes
Pow
er S
pectra
Dis
pla
y
• M
ask z
ero
-freq
uen
cy te
rm to
zero
• C
on
trast s
tretc
h w
ith s
qu
are
-root tra
nsfo
rm
• R
ep
ea
t con
stra
st s
tretc
h a
s n
eed
ed
EC
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PT
I533 Digital Im
age Processing class notes 199 D
r. Robert A
. Schowengerdt 2003
2-D
DIS
CR
ETE
FO
UR
IER
TRA
NSFO
RM
Exa
mp
le
pow
er s
pectru
mD
C m
asked
22
24
28
du
e to
p
erio
dic
b
ord
er
at n
=0
a
nd
N-1
du
e to
p
erio
dic
b
ord
er
at m
=0
a
nd
M-1
n=
0
m=
0
m=
M-1
n=
N-1
EC
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PT
I533 Digital Im
age Processing class notes 200 D
r. Robert A
. Schowengerdt 2003
2-D
DIS
CR
ETE
FO
UR
IER
TRA
NSFO
RM
MA
TRIX
REPR
ESEN
TATIO
N
This
sectio
n is
from
lectu
re n
ote
s b
y m
y la
te frie
nd
an
d c
olle
ag
ue, P
rofe
ssor S
teve
Pa
rk, o
f the C
olle
ge o
f Willia
m a
nd
Ma
ry, V
irgin
ia
•Com
pa
ct n
ota
tion
•Gen
era
liza
ble
to o
ther tra
nsfo
rms
•D
FT d
efin
ition
let
, wh
ere
WM
is M
x M
, WN is
N x
N
then
,
wh
ich
is th
e fo
rwa
rd tra
nsfo
rm
Fk
l,(
)1MN
---------f
mn,
()
ej
–2π
mk
M -------n
lN -----+
⋅n
0=
N1
–
∑m
0=
M1
–
∑=
WM
mk,
()
ej
–2π
mk
M -------
=
WN
nl,
()
ej
–2π
nlN -----
=
Fk
l,(
)1MN
---------W
Mm
k,(
)f
mn,
()W
Nn
l,(
)n
0=
N1
–
∑m
0=
M1
–
∑1MN
---------WM
fWN
==
EC
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PT
I533 Digital Im
age Processing class notes 201 D
r. Robert A
. Schowengerdt 2003
2-D
DIS
CR
ETE
FO
UR
IER
TRA
NSFO
RM
•N
ote
tha
t
then
,
, wh
ich
is th
e in
vers
e tra
nsfo
rm
WM *
WM
WM
WM *
MI
M (M
x M identity m
atrix)=
=
WN *
WN
WN W
N *N
IN (N
x N identity m
atrix)=
=
WM *
FW
N *1MN
---------W
M *W
M(
)fW
N WN *
()
=
1MN
---------M
IM
()f
NI
N(
)=
f=
EC
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PT
I533 Digital Im
age Processing class notes 202 D
r. Robert A
. Schowengerdt 2003
2-D
DIS
CR
ETE
FO
UR
IER
TRA
NSFO
RM
Ma
trix D
imen
sio
na
lity D
iag
ram
(M > N
)
•D
iag
ram
for in
vers
e tra
nsfo
rm is
sim
ilar, e
xcep
t no 1
/MN
facto
r
•N
ote
, this
rep
resen
tatio
n is
possib
le b
eca
use th
e 2
-D D
FT k
ern
el is
sep
ara
ble
, i.e.
F
M x
N
=W
M
M x
M
f
M x
N
WNN
x N
1MN
---------
F1MN
---------WM
fWN
=
ej
–2π
mk
M -------n
lN -----+
ej2π
mk
M -------
–
ej2π
nlN -----
–
=
EC
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PT
I533 Digital Im
age Processing class notes 203 D
r. Robert A
. Schowengerdt 2003
2-D
DIS
CR
ETE
FO
UR
IER
TRA
NSFO
RM
CA
LCU
LA
TING TH
E 2
-D D
FT
•Ste
p 1
write
ima
ge a
s
wh
ere
f1, f
2, . . . f
N a
re th
e im
ag
e c
olu
mn
s o
f
len
gth
M
then
,
wh
ere
ea
ch
colu
mn
is a
1-D
DFT o
f len
gth
M o
f the im
ag
e c
olu
mn
s
F1MN
---------WM
fWN
=
ff
1f
2 . . .
fN
[]
=
F1N ----
1M -----WM
f1
1M -----WM
f2
. . . 1M -----W
Mf
NW
N=
1N ----F
1F
2 . . .
FN
[]W
N=
EC
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I533 Digital Im
age Processing class notes 204 D
r. Robert A
. Schowengerdt 2003
2-D
DIS
CR
ETE
FO
UR
IER
TRA
NSFO
RM
•Ste
p 2
form
ma
trix tra
nsp
ose
note
, W is
sym
metric
•Ste
p 3
pa
rtition
ima
ge m
atrix
by c
olu
mn
s
, wh
ere
ea
ch
colu
mn
is a
n a
rray o
f len
gth
N
Ft
1N ----WN t
F1 t
F2 t
…FN t
=W
N tW
N=
F1 t
F2 t
…FN t
g1
g2
. . . g
M[
]=
EC
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PT
I533 Digital Im
age Processing class notes 205 D
r. Robert A
. Schowengerdt 2003
2-D
DIS
CR
ETE
FO
UR
IER
TRA
NSFO
RM
then
wh
ere
ea
ch
colu
mn
is a
1-D
DFT o
f len
gth
N
there
fore
•Ste
p 4
tran
sp
ose F
t to g
et F
Ft
1N ----WN g
11N ----W
N g2
. . . 1N ----W
N gN
=
Ft
G1
G2
. . . G
M[
]=
EC
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PT
I533 Digital Im
age Processing class notes 206 D
r. Robert A
. Schowengerdt 2003
2-D
DIS
CR
ETE
FO
UR
IER
TRA
NSFO
RM
Ca
lcu
latin
g th
e 2
-D D
FT - S
um
ma
ry
f
1-D
DFT
ap
plie
d to
ea
ch
colu
mn
ma
trix
tran
sp
ose
1-D
DFT
ap
plie
d to
ea
ch
colu
mn
ma
trix
tran
sp
ose
F
M 1
-D D
FTs
of le
ng
th N
M(N
log
2N
) op
era
tion
sN
1-D
DFTs
of le
ng
th M
N(M
log
2M
) op
era
tion
s
•N
(Mlo
g2M
) + M
(Nlo
g2N
) = M
Nlo
g2(M
N) to
tal o
pera
tion
s
assu
mes 1
-D F
FT is
used
an
d M
,N a
re p
ow
ers
of 2
•Com
pa
res to
M2N
2 to
tal o
pera
tion
s fo
r “b
rute
forc
e” 2
-D D
FT
