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3/20/2012
1
Digital Image Processing
Filtering in the Frequency Domain(Fundamentals)
Christophoros Niko
University of Ioannina - Department of Computer Science
Christophoros [email protected]
2 Filtering in the Frequency Domain
Filter: A device or material for suppressing or minimizing waves or oscillations of certain frequencies.
Frequency: The number of times that a periodic function repeats the same sequence of values during a unit variation of the independent variable.
C. Nikou – Digital Image Processing (E12)
Webster’s New Collegiate Dictionary
3/20/2012
2
3 Jean Baptiste Joseph Fourier
Fourier was born in Auxerre, France in 1768.
– Most famous for his work “La Théorie Analitique de la Chaleur” published in 1822.
– Translated into English in 1878: “The Analytic Theory of Heat”.
C. Nikou – Digital Image Processing (E12)
Nobody paid much attention when the work was first published.One of the most important mathematical theories in modern engineering.
4 The Big Idea
g (2
002)
=
& W
oods
, Dig
ital I
mag
e P
roce
ssin
C. Nikou – Digital Image Processing (E12)
Any function that periodically repeats itself can be expressed as a sum of sines and cosines of different frequencies each multiplied by a different coefficient – a Fourier series
Imag
es ta
ken
from
Gon
zale
z &
3/20/2012
3
5 1D continuous signalsg
(200
2) • It may be considered both as
& W
oods
, Dig
ital I
mag
e P
roce
ssin
0, x x+∞ =⎧⎨
considered both as continuous and discrete.
• Useful for the representation of
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z & 0
0
,( )
0 otherwisex x
x xδ+⎧
− = ⎨⎩
discrete signals through sampling of continuous signals. 0 0( ) ( ) ( )f x x x dx f xδ
+∞
−∞
− =∫
6 1D continuous signals (cont.)
g (2
002) Impulse train function
& W
oods
, Dig
ital I
mag
e P
roce
ssin
( ) ( )Tn
S t t n Tδ+∞
Δ=−∞
= − Δ∑
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z &
[ ] ( ) ( ) ( ) ( ) ( ) ( )Tn n
x n x t S t x t t n T x n T t n Tδ δ+∞ +∞
Δ=−∞ =−∞
= = − Δ = Δ − Δ∑ ∑
3/20/2012
4
7 1D continuous signals (cont.)g
(200
2)
[ ] ( ) ( )x n x t S t
& W
oods
, Dig
ital I
mag
e P
roce
ssin [ ] ( ) ( )Tx n x t S tΔ=
( ) ( )n
x t t n Tδ+∞
=−∞
= − Δ∑
+∞
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z &
( ) ( )n
x n T t n Tδ=−∞
= Δ − Δ∑
8 1D continuous signals (cont.)
g (2
002) • The Fourier series expansion of a periodic
signal f (t)
& W
oods
, Dig
ital I
mag
e P
roce
ssin signal f (t).
2
( )j nt
Tn
nf t c e
π+∞
=−∞
= ∑
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z &
/ 2 2
/ 2
1 ( )T
j ntT
nT
c f t e dtT
π−
−
= ∫
3/20/2012
5
9 1D continuous signals (cont.)g
(200
2) • The Fourier transform of a continuous signal f (t)
& W
oods
, Dig
ital I
mag
e P
roce
ssin signal f (t).
2( ) ( ) j tF f t e dtπμμ+∞
−
−∞
= ∫
2( ) ( ) j tf t F e dπμμ μ+∞
= ∫
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z & ( ) ( )f μ μ
−∞∫
• Attention: the variable is the frequency (Hz) and not the radial frequency (Ω=2πμ) as in the Signals and Systems course.
10 1D continuous signals (cont.)
g (2
002)
& W
oods
, Dig
ital I
mag
e P
roce
ssin
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z &
/ 2sin( )( ) ( ) ( )
( )WWf t P t F W
Wπμμ
πμ= Α ↔ = Α
3/20/2012
6
11 1D continuous signals (cont.)g
(200
2) • Convolution property of the FT.
& W
oods
, Dig
ital I
mag
e P
roce
ssin
( )* ( ) ( ) ( )
( )* ( ) ( ) ( )
f t h t f h t d
f t h t F H
τ τ τ
μ μ
+∞
−∞
= −
↔
∫
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z &
( ) ( ) ( )* ( )f t h t F Hμ μ↔
12 1D continuous signals (cont.)
g (2
002) • Intermediate result
− The Fourier transform of the impulse train
& W
oods
, Dig
ital I
mag
e P
roce
ssin − The Fourier transform of the impulse train.
1( )n n
nt n TT T
δ δ μ+∞ +∞
=−∞ =−∞
⎛ ⎞− Δ ↔ −⎜ ⎟Δ Δ⎝ ⎠∑ ∑
• It is also an impulse train in the frequency
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z & • It is also an impulse train in the frequency
domain.• Impulses are equally spaced every 1/ΔΤ.
3/20/2012
7
13 1D continuous signals (cont.)g
(200
2) Sampling
& W
oods
, Dig
ital I
mag
e P
roce
ssin
[ ] ( ) ( )Tx n x t S tΔ=
( ) ( )n
x t t n Tδ+∞
=−∞
= − Δ∑
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z &
( ) ( )n
x n T t n Tδ+∞
=−∞
= Δ − Δ∑
14 1D continuous signals (cont.)
g (2
002) • Sampling
− The spectrum of the discrete signal consists of
& W
oods
, Dig
ital I
mag
e P
roce
ssin − The spectrum of the discrete signal consists of
repetitions of the spectrum of the continuous signal every 1/ΔΤ.
− The Nyquist criterion should be satisfied.
( ) ( )f t F μ↔
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z & ( ) ( )
1( ) [ ] ( )n
f t Fnf n T f n F F
T T
μ
μ μ+∞
=−∞
↔
⎛ ⎞Δ = ↔ = −⎜ ⎟Δ Δ⎝ ⎠∑
3/20/2012
8
15 1D continuous signals (cont.)g
(200
2)&
Woo
ds, D
igita
l Im
age
Pro
cess
in
Nyquist theorem
max1 2T
μ≥Δ
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z & TΔ
16 1D continuous signals (cont.)
g (2
002)
FT of a continuous signal
& W
oods
, Dig
ital I
mag
e P
roce
ssin
Oversampling
Critical sampling with
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z &
Undersampling Aliasing appears
C ca sa p gthe Nyquist frequency
3/20/2012
9
17 1D continuous signals (cont.)g
(200
2) • Reconstruction (under correct sampling).
& W
oods
, Dig
ital I
mag
e P
roce
ssin
( ) ( ) ( )F F Hμ μ μ=
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z &
( ) ( )* sin tf t f t T cT
⎛ ⎞= ⎜ ⎟Δ⎝ ⎠
18 1D continuous signals (cont.)
g (2
002) • Reconstruction
−Provided a correct sampling the continuous
& W
oods
, Dig
ital I
mag
e P
roce
ssin −Provided a correct sampling, the continuous
signal may be perfectly reconstructed by its samples.
( )T+∞ Δ⎡ ⎤
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z & ( )( ) ( ) sinc
n
t n Tf t f n Tn T
+∞
=−∞
− Δ⎡ ⎤= Δ ⎢ ⎥Δ⎣ ⎦∑
3/20/2012
10
19 1D continuous signals (cont.)g
(200
2)
• Under aliasing, the
& W
oods
, Dig
ital I
mag
e P
roce
ssin reconstruction of
the continuous signal not correct.
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z &
20 1D continuous signals (cont.)
g (2
002) Aliased signal
& W
oods
, Dig
ital I
mag
e P
roce
ssin
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z &
3/20/2012
11
21 The Discrete Fourier Transformg
(200
2) • The Fourier transform of a sampled (discrete) signal is a continuous function of the frequency.
& W
oods
, Dig
ital I
mag
e P
roce
ssin
g q y
1( )n
nF FT T
μ μ+∞
=−∞
⎛ ⎞= −⎜ ⎟Δ Δ⎝ ⎠∑
• For a N-length discrete signal, taking N samples of its Fourier transform at frequencies:
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z & q
provides the discrete Fourier transform (DFT) of the signal.
0,1, .., 1,k k Nk
N Tμ = −=
Δ
22The Discrete Fourier Transform
(cont.)
g (2
002) • DFT pair of signal f [n] of length N.
& W
oods
, Dig
ital I
mag
e P
roce
ssin
21
021
0 1
0 1
[ ] [ ] ,
1[ ] [ ] ,
nkN jN
nnkN j
N
k N
n N
F k f n e
f n F k e
π
π
− −
=
−
≤ ≤ −
≤ ≤ −
=
=
∑
∑
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z &
0
[ ] [ ] ,n
fN =∑
3/20/2012
12
23The Discrete Fourier Transform
(cont.)g
(200
2) • PropertyThe DFT of a N length f [n] signal is periodic
& W
oods
, Dig
ital I
mag
e P
roce
ssin – The DFT of a N-length f [n] signal is periodic
with period N.
[ ] [ ]F k N F k+ =
– This is due to the periodicity of the complex exponential:
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z & exponential:
( )2 2n nj jn nN N
N Nw e w eπ π
− −⎛ ⎞= ⇔ =⎜ ⎟⎝ ⎠
24The Discrete Fourier Transform
(cont.)
g (2
002) • Property: sum of complex exponentials
& W
oods
, Dig
ital I
mag
e P
roce
ssin
0
1, ,10, otherwise
NknN
n
k rN rw
N =
= ∈⎧= ⎨⎩
∑
The proof is left as an exercise
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z & The proof is left as an exercise.
2jN
Nw eπ
−=
3/20/2012
13
25The Discrete Fourier Transform
(cont.)g
(200
2) • DFT pair of signal f [n] of length N may be expressed in matrix-vector form
& W
oods
, Dig
ital I
mag
e P
roce
ssin expressed in matrix-vector form.
1
0
1
0 1
0 1
[ ] [ ] ,
1[ ] [ ] ,
NnkN
n
Nnk
N
k N
n N
F k f n w
f n F k wN
−
=
−−
≤ ≤ −
≤ ≤ −
=
=
∑
∑
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z &
0[ ] [ ] ,N
nf
N =∑
2jN
Nw eπ
−=
26The Discrete Fourier Transform
(cont.)F = Af
( ) ( ) ( ) ( )( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
0 1 2 10 0 0 0
0 1 2 11 1 1 1
0 1 2 1
N
N N N N
N
N N N N
N
w w w w
w w w w
−
−
−
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥
A
…
…
C. Nikou – Digital Image Processing (E12)
[ ] [ ][0], [1],..., [ 1] , [0], [1],..., [ 1]T Tf f f N F F F N− −f = F =
( ) ( ) ( ) ( )0 1 2 11 1 1 1 NN N N NN N N Nw w w w− − − −⎢ ⎥
⎣ ⎦…
3/20/2012
14
27The Discrete Fourier Transform
(cont.)
Example for N=4
1 1 1 11 11 1 1 11 1
j j
j j
⎡ ⎤⎢ ⎥− −⎢ ⎥=⎢ ⎥− −⎢ ⎥− −⎣ ⎦
A
C. Nikou – Digital Image Processing (E12)
1 1j j− −⎣ ⎦
28The Discrete Fourier Transform
(cont.)The inverse DFT is then expressed by:
-1f = A Ff = A F
( )
( ) ( ) ( ) ( )( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
*0 1 2 10 0 0 0
0 1 2 11 1 1 11 *
0 1 2 1
1 1
TN
N N N N
NT N N N N
N
w w w w
w w w wN N
−
−
−
−
⎛ ⎞⎡ ⎤⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥= = ⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥
A A
…
…
C. Nikou – Digital Image Processing (E12)
( ) ( ) ( ) ( )0 1 2 11 1 1 1 NN N N NN N N Nw w w w− − − −⎜ ⎟⎢ ⎥⎜ ⎟⎣ ⎦⎝ ⎠
…
This is derived by the complex exponential sum property.
3/20/2012
15
29 Linear convolution
1 2[ ] 1, 2, 2, [ ] 1, 1, 3, 2f n h n N N= = − = =
[ ] [ ]* [ ] [ ] [ ]m
g n f n h n f m h n m+∞
=−∞
= = −∑
C. Nikou – Digital Image Processing (E12)
is of length N=N1+N2-1=4
30 Linear convolution (cont.)
1 2[ ] 1, 2, 2, [ ] 1, 1, 3, 2f n h n N N= = − = =+∞
[ ] 1 2 2 [ ]0 [0 ] 1 1 11 [1 ] 1 1 1
f m g nn h mn h m= − − →= − − →
[ ] [ ]* [ ] [ ] [ ]m
g n f n h n f m h n m=−∞
= = −∑
C. Nikou – Digital Image Processing (E12)
1 [1 ] 1 1 12 [2 ] 1 1 03 [3 ] 1 1 2
n h mn h mn h m
→= − − →= − − → −
[ ] 1, 1, 0, 2g n = −
3/20/2012
16
31 Circular shift
• Signal x[n] of length N.• A circular shift ensures that the resulting
signal will keep its length N.• It is a shift modulo N denoted by
[( ) ] [( ) mod ]Nx n m x n m N− = −
C. Nikou – Digital Image Processing (E12)
• Example: x[n] is of length N=8.
8[( 2) ] [( 2) ] [6]Nx x x− = − =
8[(10) ] [(10) ] [2]Nx x x= =
32 Circular convolution
1 2[ ] 1, 2, 2, [ ] 1, 1, 3, 2f n h n N N= = − = =
[ ] [( ) ]Nm
f m h n m+∞
=−∞
= −∑g[n]=f [n] h[n]
Circ lar shift mod lo N
C. Nikou – Digital Image Processing (E12)
The result is of length 1 2max , 3N N N= =
Circular shift modulo N
3/20/2012
17
33 Circular convolution (cont.)
1 2[ ] 1, 2, 2, [ ] 1, 1, 3, 2f n h n N N= = − = =
[ ] 1 2 2 [ ]0 [(0 ) ] 1 1 1 1N
f m g nn h m= − − − −
[ ] [( ) ]Nm
f m h n m+∞
=−∞
= −∑g[n]=f [n] h[n]
C. Nikou – Digital Image Processing (E12)
1 [(1 ) ] 1 1 12 [(2 ) ] 1 1 0
N
N
n h mn h m= − −= − −
[ ] 1, 1, 0,g n = −
34 DFT and convolution
[ ] [ ] [ ]G k F k H k↔ =g[n]=f [n] h[n]
• The property holds for the circular convolution.
• In signal processing we are interested in linear convolution
C. Nikou – Digital Image Processing (E12)
linear convolution.• Is there a similar property for the linear
convolution?
3/20/2012
18
35 DFT and convolution (cont.)
[ ] [ ] [ ]G k F k H k↔ =g[n]=f [n] h[n]
Let f [ ] be of length N and h[ ] be of length N• Let f [n] be of length N1 and h[n] be of length N2.
• Then g[n]=f [n]*h[n] is of length N1+N2-1.
• If the signals are zero-padded to length N=N1+N2-1then their circular convolution will be the same as their linear convolution:
C. Nikou – Digital Image Processing (E12)
[ ] [ ]* [ ] [ ] [ ] [ ]g n f n h n G k F k H k= ↔ =
Zero-padded signals
36 DFT and convolution (cont.)
1 2[ ] 1, 2, 2, [ ] 1, 1, 3, 2f n h n N N= = − = =
Zero padding to length N=N +N 1 =4[ ] 1, 2, 2, 0, [ ] 1, 1, 0, 0f n h n= = −
Zero-padding to length N=N1+N2-1 =4
4
[ ] 1 2 2 0 [ ][( 0) ] 0 0 1 1 0 0 1 1[( 1) ] 0 0 1 1 0 0 1
f m g nh nh n
− − −
C. Nikou – Digital Image Processing (E12)
4
4
4
[( 1) ] 0 0 1 1 0 0 1[( 2) ] 0 0 1 1 0 0[( 3) ] 0 0 1 1 2
h nh nh n
− −− −− − −
The result is the same as the linear convolution.
3/20/2012
19
37 DFT and convolution (cont.)
Verification using DFT1 1 1 1 1 5⎡ ⎤ ⎡ ⎤ ⎡ ⎤1 1 1 1 1 51 1 2 1 21 1 1 1 2 11 1 0 1 2
j j j
j j j
⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥− − − −⎢ ⎥ ⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥ ⎢ ⎥− −⎢ ⎥ ⎢ ⎥ ⎢ ⎥− − − +⎣ ⎦ ⎣ ⎦ ⎣ ⎦
F = Af
1 1 1 1 1 0⎡ ⎤ ⎡ ⎤ ⎡ ⎤
C. Nikou – Digital Image Processing (E12)
1 1 1 1 1 01 1 1 11 1 1 1 0 21 1 0 1
j j j
j j j
⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥− − − +⎢ ⎥ ⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥ ⎢ ⎥− −⎢ ⎥ ⎢ ⎥ ⎢ ⎥− − −⎣ ⎦ ⎣ ⎦ ⎣ ⎦
H = Ah
38 DFT and convolution (cont.)
[ ] [ ] [ ]G k F k H k=
Element-wise multiplication
5 0 0( 1 2) (1 ) 1 3j j j
×⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥− − × + −⎢ ⎥ ⎢ ⎥G F H
C. Nikou – Digital Image Processing (E12)
( ) ( )1 2 2
( 1 2) (1 ) 1 3
j j j
j j j
⎢ ⎥ ⎢ ⎥× = =⎢ ⎥ ⎢ ⎥×⎢ ⎥ ⎢ ⎥− + × − +⎣ ⎦ ⎣ ⎦
G = F H
3/20/2012
20
39 DFT and convolution (cont.)
Inverse DFT of the result
( )1 *
1 1 1 1 0 11 1 1 3 111 1 1 1 2 041 1 1 3 2
T j j j
j j j
−
⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥− − −⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥− −⎢ ⎥ ⎢ ⎥ ⎢ ⎥− − + −⎣ ⎦ ⎣ ⎦ ⎣ ⎦
g = A G = A
C. Nikou – Digital Image Processing (E12)
j j j⎣ ⎦ ⎣ ⎦ ⎣ ⎦
The same result as their linear convolution.
40 2D continuous signals
g (2
002) 0 0
0 0
, ,( , )
0 otherwisex x y y
x x y yδ+∞ = =⎧
− − = ⎨⎩
& W
oods
, Dig
ital I
mag
e P
roce
ssin 0 otherwise⎩
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z &
0 0 0 0( , ) ( ) ( )x x y y x x y yδ δ δ− − = − −Separable:
0 0 0 0( , ) ( , ) ( , )f x y x x y y dydx f x yδ+∞ +∞
−∞ −∞
− − =∫ ∫
3/20/2012
21
41 2D continuous signals (cont.)g
(200
2)&
Woo
ds, D
igita
l Im
age
Pro
cess
in
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z &
The 2D impulse train is also separable:
( , ) ( ) ( ) ( , )X Y X Yn m
S x y S x S y x n X y n Yδ+∞ +∞
Δ Δ Δ Δ=−∞ =−∞
= = − Δ − Δ∑ ∑
42 2D continuous signals (cont.)
g (2
002) • The Fourier transform of a continuous 2D
signal f (x y)
& W
oods
, Dig
ital I
mag
e P
roce
ssin signal f (x,y).
2 ( )( , ) ( , ) j x vyF f x y e dydxπ μμ ν+∞ +∞
− +
−∞ −∞
= ∫ ∫
+∞ +∞
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z &
2 ( )( , ) ( , ) j x vyf x y F e d dπ μμ ν ν μ+∞ +∞
+
−∞ −∞
= ∫ ∫
3/20/2012
22
43 2D continuous signals (cont.)g
(200
2) • Example: FT of f (x,y)=δ(x)y f (x,y)=δ(x)
& W
oods
, Dig
ital I
mag
e P
roce
ssin
2 ( )( , ) ( ) j x vyF x e dydxπ μμ ν δ+∞ +∞
− +
−∞ −∞
= ∫ ∫
2 2( ) j x j vyx e dx e dyπμ πδ+∞ +∞
− −
−∞ −∞
= ∫ ∫
x
ν F(μ,ν)=δ(ν)
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z &
2j vye dyπ+∞
−
−∞
= ∫ ( )δ ν= μ
44 2D continuous signals (cont.)
g (2
002) • Example: FT of f (x,y)=δ(x-y)
yf (x,y)=δ(x-y)
& W
oods
, Dig
ital I
mag
e P
roce
ssin
2 ( )( , ) ( ) j x vyF x y e dydxπ μμ ν δ+∞ +∞
− +
−∞ −∞
= −∫ ∫
2 2( ) j x j vyx y e dx e dyπμ πδ+∞ +∞
− −
−∞ −∞
⎡ ⎤= −⎢ ⎥
⎣ ⎦∫ ∫
x
ν F(μ,ν)=δ(μ+ν)
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z & ⎣ ⎦
2 2j y j ye e dyπμ πν+∞
− −
−∞
= ∫( )δ μ ν= +
μ2 ( )j ye dyπ μ ν+∞
− +
−∞
= ∫
3/20/2012
23
45 2D continuous signals (cont.)g
(200
2)&
Woo
ds, D
igita
l Im
age
Pro
cess
in
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z &
2/ 2, / 2
sin( ) sin( )( , ) ( , ) ( , )( ) ( )W W
W Wf x y P x y F WW Wπμ πνμ ν
πμ πν= Α ↔ = Α
46 2D continuous signals (cont.)
g (2
002) • 2D continuous convolution
+∞ +∞
& W
oods
, Dig
ital I
mag
e P
roce
ssin
( , )* ( , ) ( , ) ( , )f x y h x y f x y h d dα β α β α β+∞ +∞
−∞ −∞
= − −∫ ∫
• We will examine the discrete convolution in more detail.C l i
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z &
( , )* ( , ) ( , ) ( , )f x y h x y F Hμ ν μ ν↔
• Convolution property
3/20/2012
24
47 2D continuous signals (cont.)g
(200
2) • 2D sampling is accomplished by+∞ +∞
& W
oods
, Dig
ital I
mag
e P
roce
ssin ( , ) ( , )X Y
n m
S x y x n X y n YδΔ Δ=−∞ =−∞
= − Δ − Δ∑ ∑
• The FT of the sampled 2D signal consists of repetitions of the spectrum of the 1D continuous signal.
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z & g
1 1( , ) ,n
m nF FX Y X Y
μ ν μ ν+∞
=−∞
⎛ ⎞= − −⎜ ⎟Δ Δ Δ Δ⎝ ⎠∑
48 2D continuous signals (cont.)
g (2
002) • The Nyquist theorem involves both the
horizontal and vertical frequencies.
& W
oods
, Dig
ital I
mag
e P
roce
ssin
q
max max1 12 , 2vX Y
μ≥ ≥Δ Δ
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z &
Over-sampled Under-sampled
3/20/2012
25
49 Aliasingg
(200
2)&
Woo
ds, D
igita
l Im
age
Pro
cess
in
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z &
50 Aliasing
g (2
002)
& W
oods
, Dig
ital I
mag
e P
roce
ssin
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z &
The difference is more pronounced in the area inside the bottle
3/20/2012
26
51 Aliasing - Moiré Patterns
• Effect of sampling a scene with periodic or nearly periodic components (e.g. y p p ( goverlapping grids, TV raster lines and stripped materials).
• In image processing the problem arises when scanning media prints (e.g.
C. Nikou – Digital Image Processing (E12)
magazines, newspapers).• The problem is more general than
sampling artifacts.
52 Aliasing - Moiré Patterns (cont.)
• Superimposed grid drawings (not digitized) produce the effect of new frequencies not existing in the original components.
C. Nikou – Digital Image Processing (E12)
3/20/2012
27
53 Aliasing - Moiré Patterns (cont.)
• In printing industry the problem comes when scanning photographs from the g p g psuperposition of:
• The sampling lattice (usually horizontal and vertical).
• Dot patterns on the newspaper image.
C. Nikou – Digital Image Processing (E12)
54 Aliasing - Moiré Patterns (cont.)
C. Nikou – Digital Image Processing (E12)
3/20/2012
28
55 Aliasing - Moiré Patterns (cont.)
• The printing industry uses halftoning to cope with the problem.
• The dot size is inversely
ti l t
C. Nikou – Digital Image Processing (E12)
proportional to image intensity.
56 2D discrete convolution
g (2
002) m f [m,n] m h [m,n]
& W
oods
, Dig
ital I
mag
e P
roce
ssin
n3 2 n
1 11 -1
[ , ] [ , ]* [ , ] [ , ] [ , ]k l
g m n f m n h m n f k l h m k n l+∞ +∞
=−∞ =−∞
= = − −∑ ∑
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z & k l=−∞ =−∞
• Take the symmetric of one of the signals with respect to the origin.
• Shift it and compute the sum at every position [m,n].
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29
57 2D discrete convolution (cont.)g
(200
2) n f [m,n] n h [m,n]
& W
oods
, Dig
ital I
mag
e P
roce
ssin
m3 2 m
1 11 -1
llg [0 0]=0 g [1 1]=0
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z &
k
h [1-k,1-l]
-1 11 1
3 2kh [-k,-l]
-1 11 1
3 2
g [0,0]=0 g [1,1]=0
58 2D discrete convolution (cont.)
g (2
002) n f [m,n] n h [m,n]
& W
oods
, Dig
ital I
mag
e P
roce
ssin
m3 2 m
1 11 -1
llg [2 2]=3 g [3 2]=-1
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z &
k
h [3-k,2-l]-1 11 13 2k
h [2-k,2-l]-1 11 1
3 2
g [2,2]=3 g [3,2] -1
3/20/2012
30
59 2D discrete convolution (cont.)g
(200
2) n f [m,n] n h [m,n]
& W
oods
, Dig
ital I
mag
e P
roce
ssin
m3 2 m
1 11 -1
n g[m,n]3 5 2
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z &
m3 -1 -3
M1+M2-1=3N1+N2-1=2
60 The 2D DFT
g (2
002) • 2D DFT pair of image f [m,n] of size MxN.
& W
oods
, Dig
ital I
mag
e P
roce
ssin
1 1 2
0 0
1 1 2
[ , ] [ , ]
1[ , ] [ , ]
km lnM N jM N
m n
km lnM N jM N
F k l f m n e
f m n F k l eMN
π
π
⎛ ⎞− − − +⎜ ⎟⎝ ⎠
= =
⎛ ⎞− − +⎜ ⎟⎝ ⎠
=
=
∑∑
∑∑
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z & 0 0m nMN = =
0 1 0 1
0 1 0 1,
k M m M
l N n N
≤ ≤ − ≤ ≤ −
≤ ≤ − ≤ ≤ −
⎧ ⎧⎨ ⎨⎩ ⎩
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31
61 The 2D DFT (cont.)g
(200
2) • All of the properties of 1D DFT hold.• Particularly:
& W
oods
, Dig
ital I
mag
e P
roce
ssin • Particularly:
– Let f [m,n] be of size M1xN1 and h[m,n] of size M2xN2 .
– If the signals are zero-padded to size (M1+M2-1)x(N1+N2-1) then their circular convolution will be the same as their linear convolution and:
C. Nikou – Digital Image Processing (E12)
Imag
es ta
ken
from
Gon
zale
z & be the same as their linear convolution and:
[ , ] [ , ]* [ , ] [ , ] [ , ] [ , ]g m n f m n h m n G k l F k l H k l= ↔ =