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Discrete Fourier TransformDiscrete Fourier Transform
Nuno VasconcelosUCSD
The Discrete-Space Fourier Transform• as in 1D, an important concept in linear system analysis
is that of the Fourier transform• the Discrete-Space Fourier Transform is the 2D
extension of the Discrete-Time Fourier Transform2211][)( ωω ωω eennxX njnj∑∑ −−
2121
2211
2211
1 2
)(1][
],[),(
ωωωω
ωω
ωω ddeeXnnx
eennxX
njnj
jj
n n
∫∫
∑∑
=
=
• note that this is a continuous function of frequency
2121221 ),()2(
],[ ωωωωπ
ddeeXnnx ∫∫=
q y– inconvenient to evaluate numerically in DSP hardware– we need a discrete version
this is the 2D Discrete Fourier Transform (2D DFT)
2
– this is the 2D Discrete Fourier Transform (2D-DFT)
• before that we consider the sampling problem
Sampling in 2D
• consider an analog signal xc(t1,t2) and let its analog Fourier transform be X (Ω1 Ω2)Fourier transform be Xc(Ω1,Ω2)– we use capital Ω to emphasize that this is analog frequency
• sample with period (T1,T2) to obtain a discrete-space signal
222111 ;2121 ),(],[TntTntc ttxnnx
===
3
Sampling in 2D
• relationship between the Discrete-Space FT of x[n1,n2]and the FT of x (t1 t2) is simple extension of 1D resultand the FT of xc(t1,t2) is simple extension of 1D result
∑ ∑∞ ∞
⎟⎠
⎞⎜⎜⎝
⎛ −−= 2211
212,21),( c T
rT
rXTT
X πωπωωω
DSFT of x[n1,n2] FT of xc(ω1,ω2)
−∞= −∞= ⎠⎝1 2 2121 r r TTTT
“discrete spectrum” “analog spectrum”
• Discrete Space spectrum is sum of replicas of analog• Discrete Space spectrum is sum of replicas of analog spectrum– in the “base replica” the analog frequency Ω1 (Ω2) is mapped
i t th di it l f T ( T )4
into the digital frequency Ω1T1 (Ω2T2)– discrete spectrum has periodicity (2π,2π)
For exampleΩ2
Ω’
Ω”ω2
Ω’
2π
Ω1
Ω”T2
ω1
2π−Ω”T2
2π
2
1
''''''
TT
Ω=→ΩΩ=→Ω
βα
ω1
Ω’T1
2π2π
• no aliasing if
2π−Ω’T1
⎧ Ω≤Ω '2' TT π
ω1
2π⇔⎩⎨⎧
Ω−≤ΩΩ−≤Ω
22
11
'2''2
TTTT
ππ
⎨⎧ Ω≤
⇔'/1 πT
5⎩⎨ Ω≤
⇔''/2 πT
Aliasing• the frequency (Ω’/π,Ω’’/π)
is the critical sampling frequency
• below it we havealiasing
ω2
aliasing
ω1
• this is just like the 1D case, but now there are morenow there are more possibilities for overlap
6
Reconstruction• if there is no aliasing we can recover the signal in a way
similar to the 1D case
( ) ( )∑ ∑∞ ∞
−−=
2222
1111
2121
sinsin],[),(c
TntT
TntTnnxtty π
π
π
π
t i 2D th ibiliti th i 1D
( ) ( )∑ ∑−∞= −∞= −−1 2
2222
1111
n n TntT
TntT
ππ
• note: in 2D there are many more possibilities than in 1D– e.g. the sampling grid does not have to be rectangular, e.g.
hexagonal sampling when T2 = T1/sqrt(3) and
⎩⎨⎧
= ==
otherwise0odd or even both21;21
21,),(
],[ 222111nnttx
nnx TntTntc
7– in practice, however, one usually adopts the rectangular grid
• a sequence of images obtained by down-sampling without any filteringg
• aliasing: the low-frequency parts are
li t d th h t th8
replicated throughout the low-res image
The role of smoothing
none
some
l ta lot
9• too little leads to aliasing• too much leads to loss of information
Aliasing in video• video frames are the result of temporal sampling
– fast moving objects are above the critical frequency– above a certain speed they are aliased and appear to move
backwards– this was common in old western movies and become known as
the “wagon wheel” effect– here is an example: super-resolution increases the frame rate
and eliminates aliasing
from “Space-TimeResolution in Video” by yE. Shechtman, Y. Caspi and M. Irani
(PAMI 2005).
10
2D-DFT• the 2D-DFT is obtained by sampling the DSFT at regular
frequency intervals
22
211
12,22121 ),(],[
kN
kN
XkkX πωπωωω
===
• this turns out to make the 2D-DFT somewhat harder to work with than the DSFT
it i th i 1D– it is the same as in 1D– you might remember that the inverse transform of the product of
two DFTs is not the convolution of the associated signals– but, instead, the “circular convolution”– where does this come from?
• it is better understood by first considering the 2D
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it is better understood by first considering the 2D Discrete Fourier Series (2D-DFS)
2D-DFS• it is the natural representation for a periodic sequence• a sequence x[n1,n2] is periodic of period N1xN2 if
[ ] [ ][ ]
21121 ,,nnNnnx
nNnxnnx∀+=
+=
• note that
[ ] 21221 , ,, nnNnnx ∀+=
∑ ∑∞
−∞=
∞
−∞=
−−=1 2
21212121 ],[),(
n n
jnjn rrnnxrrX
• makes no sense for a periodic signal– the sum will be infinite for any pair r1,r2
ith th 2D DSFT th Z t f ill k h
12
– neither the 2D DSFT or the Z-transform will work here
2D-DFS• the 2D-DFS solves this problem• it is based on the observation that
– any periodic sequence can be represented as a weighted sum of complex exponentials of the form
222211 nkjnkj ππ
( )10 ,10 ,,
22
1121
222
111
−≤≤−≤≤××
NkNkeekkX
nkN
jnkN
j
– this is a simple consequence of the fact that
10 22 ≤≤ Nk
22 kk ππ
is an orthonormal basis of the space of periodic sequences
10 ,10 , 2211
2222
211
1 −≤≤−≤≤× NkNkeenk
Njnk
Nj ππ
13
– is an orthonormal basis of the space of periodic sequences
2D-DFS• the 2D-DFS relates x[n1,n2] and X[k1,k2]
( ) [ ] 22111 2221 1 nkjnkjN N ππ
−−− −
( ) [ ] 222
111
1 2
221 1
0 02121 ,,
N N
Nj
Nj
n neennxkkX
ππ
∑∑= =
=
[ ] [ ] 222
111
1
1
2
2
221
0
1
021
2121 ,1,
nkN
jnkN
jN
k
N
keekkX
NNnnx
ππ
∑∑−
=
−
=
=
• note that X[k1,k2] is also periodic outside
1010 ≤≤≤≤ NkNk• like the DSFT,
properties of the 2D DFS are identical to those of the 1D DFS
10 ,10 2211 −≤≤−≤≤ NkNk
14
– properties of the 2D-DFS are identical to those of the 1D-DFS– with the straightforward extension of separability
Periodic convolution• like the Fourier transform,
– the inverse transform of multiplication is convolution
[ ] [ ] ( ) ( )2121
2121 ,,,, kkYkkXnnxnnxDFS
×↔∗
– however, we have to be careful about how we define convolution– since the sequences have no end, the standard definition
],[],[],[ 221121211 2
knknhkkxnnyk k
−−= ∑ ∑∞
−∞=
∞
−∞=
makes no sense– e.g. if x and h are both positive sequences, this will allways be
infinite
15
infinite
Periodic convolution• to deal with this, we introduce the idea of periodic
convolution• instead of the regular definition
],[],[* 221121 knknhkkxyx −−= ∑ ∑∞ ∞
• which, from now on, we refer to as linear convolution
],[],[ 2211211 2
yk k∑ ∑
−∞= −∞=
• periodic convolution only considers one period of our sequences
1 11 2N N− −
h l diff i i h i li i
],[ ],[ 2211
1
0
1
021
1
1
2
2
knknhkkxhxN
k
N
k
−−= ∑ ∑= =
o
16
• the only difference is in the summation limits
Periodic convolution• this is simple, but produces a convolution which is
substantially different• let’s go back to our example, now assuming that the
sequences have period (N1=3,N2=2)n2 n2
(1) (2)*(3) (4)
(1) (2)
(3) (4)
n1 n1
(1) (2)*(1) (2)
(3) (4)
(1) (2)
(1) (2)
(3) (4)
• as before we need four steps],[ 21 nnx ],[ 21 nnh
17
• as before, we need four steps
Periodic convolution• step 1): express sequences in terms of (k1, k2), and
consider one period only
],[ ],[ 2211
1
0
1
021
1
1
2
2
knknhkkxhxN
k
N
k−−= ∑ ∑
−
=
−
=
o
k2 k2
(3) (4)
k1 k1(1) (2)
we next proceed exactly as before
],[ 21 kkx ],[ 21 kkh
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we next proceed exactly as before
Periodic convolution• step 2): invert h(k1, k2)
1 11 2N N− −
2
],[ ],[ 22110 0
21
1
1
2
2
knknhkkxhxk k
−−= ∑ ∑= =
o
k2
(1)(2)
k2
(3) (4)
k2
(1) (2)
k1
( )( )
(3)(4)
k1(1) (2) k1
(1) (2)
(3) (4)1
],[],[ 2121 kkhkkg −−=],[ 21 kkh ],[ 21 kkh −
19
Periodic convolution• step 3): shift g(k1, k2) by (n1, n2)
1 11 2N N− −
],[ ],[ 22110 0
21
1
1
2
2
knknhkkxhxk k
−−= ∑ ∑= =
o
k2
(1)(2)
k2
(1)(2)
(3)(4)
n2
this sendswhatever is at (0,0) to (n1,n2)
k1
( )( )
(3)(4)
k1n1
],[],[ 2121 kkhkkg −−=][
],[ 2211
knknhnknkg =−−
20
],[ 2211 knknh −−
Periodic convolution• e.g. for (n1,n2) = (1,0)
k2k2
k
(1)(2)n2
k k1
(3)(4)
n1
k1
th t t ld b t i t i lti l th t
],[ 21 kkx],[
],[
2211
2211
knknhnknkg−−
=−−
the next step would be to point-wise multiply the two signals and sum
][][1 11 2
kkhkkhN N
∑ ∑− −
21
],[],[ 22110 0
211 2
knknhkkxhxk k
−−= ∑ ∑= =
o
Periodic convolution• this is where we depart from linear convolution• remember that the sequences are periodic, and we really
only care about what happens in the fundamental periodonly care about what happens in the fundamental period• we use the periodicity to fill the values missing in the
flipped sequencek2
(4)(3)
k2
n
k1(1) (2)
(4)( )
k1
(1)(2)
(3)(4)
n2
=
],[ 2211 knknh −−
n1
],[ 2211 knknh −−
22
Periodic convolution• step 4): we can finally point-wise multiply the two signals
and sum
],[ ],[ 2211
1
0
1
021
1
1
2
2
knknhkkxhxN
k
N
k−−= ∑ ∑
−
=
−
=
o
– e.g. for (n1,n2) = (1,0)k2 k2k2
3x1+1x1=4
k1
x =k1k1
(1) (2)
(4)(3)3x1+1x1=4
][ kkx ][ nny],[ 2211 knknh −−
23
],[ 21 kkx ],[ 21 nny],[ 2211
Periodic convolution• note: the sequence that results from the convolution is
also periodic• it is important to keep in mind what we have done• it is important to keep in mind what we have done
– we work with a single period (the fundamental period) to make things manageableb t b th t h i di– but remember that we have periodic sequences
– it is like if we were peeking through a window– if we shift, or flip the sequence we need to remember that – the sequence does not simply move out of the window, but the
next period walks in!!!– note, that this
k2 k2
,can make thefundamentalperiod change k1 k1shift
24
considerably by(1,1)
Discrete Fourier Transform• all of this is interesting,
– but why do I care about periodic sequences?– all images are finite, I could never have such a sequence
• while this is true– the DFS is the easiest route to learn about the discrete Fourierthe DFS is the easiest route to learn about the discrete Fourier
transform (DFT)
• recall that the DFT is obtained by sampling the DSFT
22
211
12,22121 ),(],[
kN
kN
XkkX πωπωωω
===
– we know that when we sample in space we have aliasing in frequency
– well, the same happens when we sample in frequency: we get
25
pp p q y galiasing in time
Discrete Fourier Transform• this means that
– even if we have a finite sequence– when we compute the DFT we are effectively working with a
periodic sequence
• you may recall from 1D signal processing thaty y g p g– when you multiply two DFTs, you do not get convolution in time– but instead something called circular convolution
this is strange: when I flip a signal (e g convolution) it wraps– this is strange: when I flip a signal (e.g. convolution) it wraps around the sequence borders
– well, in 2D it gets much stranger
th l I k h t d t d thi i t• the only way I know how to understand this is to– think about the underlying periodic sequence– establish a connection between the DFT and the DFS of that
26
sequence– use what we have seen for the DFS to help me out with the DFT
Discrete Fourier Transform• let’s start by the relation between DFT and DFS• the DFT is defined as
22
211
12,22121 ),(],[
kN
kN
XkkX πωπωωω
===
(here X(ω1,ω2) is the DSFT) which can be written as
[ ]⎪⎧ <≤∑∑
−−− − NknkN
jnkN
jN N 0 11221 1
22111 2ππ
[ ] [ ]⎪⎩
⎪⎨
⎧
<≤<≤
= ∑∑= =
otherwiseNkNk
eennxkkXN
jN
j
n n
000
,,,22
11
0 021
21
222
111
1 2
[ ] [ ]⎪
⎪⎨
⎧
<≤<≤
= ∑∑−
=
−
= NnNn
eekkXNNnnx
nkN
jnkN
jN
k
N
k 00
,1,
22
11221
0
1
021
2121
222
111
1
1
2
2
ππ
27
⎪⎩ otherwise0
Discrete Fourier Transform• comparing this
[ ]⎪⎧ <≤∑∑
−−− − NknkN
jnkN
jN N 0 11221 1
22111 2ππ
[ ] [ ]⎪⎩
⎪⎨ <≤
<≤= ∑∑
= =
otherwiseNkNk
eennxkkXNN
n n
000
,,,22
11
0 021
21
21
1 2
[ ] [ ]⎪⎩
⎪⎨
⎧
<≤<≤
= ∑∑−
=
−
=
hNnNn
eekkXNNnnx
nkN
jnkN
jN
k
N
k 00
,1,
22
11221
0
1
021
2121
222
111
1
1
2
2
ππ
with the DFS
⎪⎩ otherwise0
[ ] [ ] 222
111
1 2221 1 nkN
jnkN
jN N
kkXππ
∑∑−−− −
[ ] [ ]
[ ] [ ] 222
111
1 2
21
1 2
221 1
0 02121
1
,,
nkN
jnkN
jN N
NN
n n
eekkXnnx
eennxkkX
ππ
∑∑
∑∑− −
= =
=
=
28
[ ] [ ]1 20 0
2121
21 ,,k k
eekkXNN
nnx ∑∑= =
=
Discrete Fourier Transform• we see that inside the boxes
110 Nk <≤ 110 Nn <≤
the two transforms are exactly the same22
11
0 Nk <≤ 22
11
0 Nn <≤
• if we define the indicator function of the box
⎪⎧ <≤ Nn01 11
[ ]⎪⎩
⎪⎨ <≤=×
otherwiseNnnnR NN
00
,1,222121
• we can write
[ ] [ ] [ ]212121 ,,, nnRnnxnnx NN ×= [ ] [ ] [ ]212121 ,,, kkRkkXkkX NN ×=
29
[ ] [ ] [ ]212121 ,,,21 NN × [ ] [ ] [ ]212121 ,,,
21 NN ×
Discrete Fourier Transform• note from
[ ] [ ] [ ]212121 ,,,21
nnRnnxnnx NN ×= [ ] [ ] [ ]212121 ,,,21
kkRkkXkkX NN ×=
that working in the DFT domain is equivalent to– working in the DFS domain
[ ] [ ] [ ]212121 21 NN [ ] [ ] [ ]212121 21 NN
– extracting the fundamental period at the end
• we can summarize this as
[ ] [ ] [ ] [ ]2121
2121 ,
,,
, kkXkkXnnxnnx
truncate
eperiodiciz
DFS
eperiodiciz
t t←→
↔←→
• in this way, I can work with the DFT without having to worry about aliasing
eperiodiciztruncate
30
worry about aliasing
Discrete Fourier Transform
[ ] [ ] [ ] [ ]2121
2121 ,
,,
, kkXkkXnnxnnx
truncate
DFS
eperiodiciz→
↔→
• this trick can be used to derive all the DFT properties
[ ] [ ] [ ] [ ]21212121 ,,,, kkXkkXnnxnnxeperiodiciztruncate
←↔←
• e.g. what is the inverse transform of a phase shift?– let’s follow the steps
22 kk ππ
– 1) periodicize: this causes the same phase shift in the DFS
[ ] [ ] 222
111
22
2121 ,,mk
Njmk
Nj
eekkXkkYππ
−−
=
[ ] [ ] 222
111
22
2121 ,,mk
Njmk
Nj
eekkXkkYππ
−−
=
31
Discrete Fourier Transform
[ ] [ ] [ ] [ ]2121
2121 ,
,,
, kkXkkXnnxnnx
truncate
DFS
eperiodiciz
←→
↔←→
– 2) compute the inverse DFS: it follows from the properties of the DFS (page 142 on Lim) that we get a shift in space
eperiodiciztruncate←←
DFS (page 142 on Lim) that we get a shift in space
[ ] [ ]221121 ,, mnmnxnny −−=
– 3) truncate: the inverse DFT is equal to one period of the shifted periodic extension of the sequence
[ ] [ ] [ ]
– in summary, the new sequence is obtained by making the
[ ] [ ] [ ]21221121 ,,,21
nnRmnmnxnny NN ×−−=
32
y q y goriginal periodic, shifting, and taking the fundamental period
Examplek2 k2
periodicize
k1 k1
shift by (1,1)
k2k2
2
k
truncate
k1k1
33
• note that what leaves on one end, enters on the other
Example• for this reason it is called a circular shift
k2
i l hift
k2
k1
circular shift by (1,1)
k1
• note that this is way more complicated than in 1D• to get it right we really have to think in terms of the
i di t i f thperiodic extension of the sequence• we will see that this shows up in the properties of the
DFT,
34
,• namely that convolution becomes circular convolution
35