Chapter 4 Filtering in the Frequency Domain - BGUdip111/wiki.files/DIP3E_Chapter04_Art.pdf ·...
Transcript of Chapter 4 Filtering in the Frequency Domain - BGUdip111/wiki.files/DIP3E_Chapter04_Art.pdf ·...
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
+)sin(tA +++
=
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
Step FunctionStep FunctionThe unit step function, is a discontinuous function whose value is zero for negative argument and one for positive argument.
A-A
0000
)(tt
tH
Rectangle FunctionThe unit step function, is a discontinuous function whose value is zero for negative argument and one for positive argument.
AtA
Attct 1
0)(Re
A-A
At0
Impulse FunctionThe Dirac delta function, or δ function, is zero for all values of the parameter except when the parameter is zero, and its integral
th t f t i l tover the parameter from −∞ to ∞ is equal to one
000
)(tt
t
1)( dtt
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
)()()(
)0()()(
tfdttttf
fdtttf
)()()( 00 tfdttttf
In discrete variable
0001
)(tx
x
)0()()( fxxfx
© 1992–2008 R. C. Gonzalez & R. E. Woods
)()()( 00 tfxxxfx
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
)()( Tntts T
n
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
Complex NumberComplex Number
12
jIRCj
jIRC
jIRC
)t (
|| 22
IIRC
))sin()(cos(||
)tan(
jCCR
© 1992–2008 R. C. Gonzalez & R. E. Woods
)sin()cos( je j
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
Function Representationu c o ep ese o• Taylor Series
nn xxxfxf )()(1)( 00)(
...!
...!4!3!2!1
1432
xxxxxe
nx
n
xxxfn
xf )()(!
)( 000
• Fourier Series !!4!3!2!1 n
a
)sin()cos((2
)(1
0 ntbntaatf nn
n
ktktktktktf 0];)/2sin()/2sin()/[sin(2)( 11
© 1992–2008 R. C. Gonzalez & R. E. Woods
ktktktkttf 0..];)/2sin()/2sin()/[sin()( 31
21
3
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
xe
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
Fourier TransformFor a give function g(x) of a real variable x, the F i t f ti f ( ) hi h i d t dFourier transformation of g(x) which is denoted as
2)(G(u){g(x)} dxexg uxj -
)(G(u){g(x)} dxexg j
-
21 )(G(u){g(x)} dxexg uxj
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
SinceSince
F th t th F i t f F( ))2cos()2sin(2 tjte tj
For that reason the Fourier transform F(u)F(u) = R(u) + jI(u)
Which often appear as
|)(|)( )(euFuF uj
)())(t (
)()(|)(| 22
uIuIuRuF
© 1992–2008 R. C. Gonzalez & R. E. Woods
)()())(tan(
uRu
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
Spatial Domain Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
Spatial and FrequencySpatial and Frequency
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
Spatial and Frequency RepresentationsSpatial and Frequency Representations
mai
n )()()()(' 321 xbxbxbxb
atia
lDom
)12(2sin(12
10
3
1xfn
nn
Spa
mai
n
)32sin(1)(
)2sin()( 01
Xfxb
Xfxb
uenc
y D
om
)52sin(1)(
)32sin(3
)(
03
02
Xfxb
Xfxb
© 1992–2008 R. C. Gonzalez & R. E. Woods
Freq
u )52sin(5
)( 03 Xfxb
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
2/2/ WW A
2/
2/22/
2/
22
2)()( W
WutjW
W
utjutj euj
AdtAedtetfuF
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
2D Fourier Transform2D Fourier TransformFor a give function g(x) of a real variable x, the Fourier transformation of g(x) which is denoted as
dxdyeyxgvuGyxg vyuxj )(2),(),()},({
and
dudvevuGyxgvuG vyuxj )(21 ),(),()},({
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
AAeeA
dyedxeAdxdyeyxfvuF
vYjuXjYvyjXuxj
Y uyjX uxjvyuxj
]1[]1[
),(),(
)(2)(2)(2)(2
0
)(2
0
)(2)_(2
vYuXAXY
evj
eujvjuj
A vYjuXj
)sin()sin(
]1[2
]1[222
)(2)(2
00
© 1992–2008 R. C. Gonzalez & R. E. Woods
vYuX
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
Discrete Fourier TransformDiscrete Fourier Transform• Let us discretize a continuous function f(x) into the N uniform
samples that generate the sequence f(x0) f(x0+Δx) f(x0+2Δx) f(x0+3Δx) f(x0+[N-1]Δx)f(x0), f(x0+Δx), f(x0+2Δx), f(x0+3Δx), …, f(x0+[N-1]Δx)
• Hence f(x) = f(x0+i Δx)• We could denote the samples as
f(0) f(1) f(2) f(N 1)f(0), f(1), f(2), …,f(N-1).• The Fourier Transform is
1
/2 1210;)(1)(N
Nuxj NuexfuF
1
0
1,...,2,1,0;)()(
N
x
j
and
NuexfN
uF
© 1992–2008 R. C. Gonzalez & R. E. Woods
1
0
/2)()(N
u
NuxjeuFxf
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
2D Discrete Fourier Transform
eyxfvuFM
NvyMuxjN
)(1)(1
)//(21
2D Discrete Fourier Transform
NvMu
eyxfNM
vuFx y
1,...,2,1,0;1,...,2,1,0
),(),(0 0
Ff
andM
NvyMuxjN
)()(1
)//(21
evuFyxfu
NvyMuxj
v
11
),(),(0
)//(2
0
© 1992–2008 R. C. Gonzalez & R. E. Woods
yNv
xMu
;
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
Display F(u v)Display F(u,v)The Dynamic rang of Fourier spectra usually is much higher than the typical display device is able to reproduce fathfuly. Therefore, often use the logarithm function to perform the appropriate compression of the rang.
|)),(|1log(),( vuFcvuD
Where c is a scale factor
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
Examples p
© 1992–2008 R. C. Gonzalez & R. E. Woods
http://www.cs.unm.edu/~brayer/vision/fourier.html
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
Properties of 2D Fourier TransformProperties of 2D Fourier Transform• Spatial and Frequency Domain
f(t, z) sampled from f(x, y) using the separation between samples as T and Z1
ZNv
TMu
1
1
• Translation and RotationMultiplying f(x,y) by the exponential shifts the original of DFT to (u0,v0).Multiplying F(u v) by the exponential shifts the original of f(x y) to (x0 y0);Multiplying F(u,v) by the exponential shifts the original of f(x, y) to (x0,y0);
)//(200
00),(),( NvyMuxjevuFyyxxf
© 1992–2008 R. C. Gonzalez & R. E. Woods
)//(200
00),(),( NyvMxujeyxfvvuuF
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
Properties of 2D Fourier Transform pPeriodicity
The Fourier transform and inverse are infinitely periodic on the u and vinfinitely periodic on the u and v directions. (k1 and k2 are integers).
),(),(),( 21 NkvuFvMkuFvuF
)(),(),(),(
),(
21
21
NkyMkxfNkyxfyMkxfyxf
NkvMkuF
),( 21 NkyMkxf To show the origin of F(u,v) at the center we shift the data by M/2 and N/2
© 1992–2008 R. C. Gonzalez & R. E. Woods
)2/,2/()1)(,( NvMuFyxf yx
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
Properties of 2D Fourier Transform pSymmetryAny real or complex function can be expressed as the sum of even and odd part
)()()( yxwyxwyxw
2),(),(),(
),(),(),(yxwyxwyxw
yxwyxwyxw
e
oe
2),(),(),( yxwyxwyxwe
Whi h h th t f ti t i d dd f tiWhich shows that even functions are symmetric and odd functions are antisymmetric
),(),( yxwyxw ee
© 1992–2008 R. C. Gonzalez & R. E. Woods
),(),( yxwyxw oo
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
Properties of 2D Fourier Transform pSymmetryThe Fourier transform of a real function f(x,y) is conjugate symmetric
)()(* vuFvuF ),(),( vuFvuF The Fourier transform of a imaginary function f(x,y) is conjugate anti-symmetric
),(),(* vuFvuF *
1
),(1),(
1 1
*1
0
)//(21
0
* eyxfNM
vuF
M N
M
x
NvyMuxjN
y
Proof
)(1
),(1
1)/][/]([2
1
0
)//(2
0
*
eyxf
eyxfNM
MNyvMxuj
N
x
NvyMuxj
y
© 1992–2008 R. C. Gonzalez & R. E. Woods
),(
),(0 0
vuF
eyxfNM x y
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
Sampling and Fourier TransformSampling and Fourier Transform1. Converting continuous function/signal into
a discrete one.2. The sampling is uniform at T intervalsTh l d f ti d th l f hThe sampled function and the value of each Sample are :
)()()()()(~ Tnttftstftf T
)()()(
)()()()()(
TkfdtTkttff
Tnttftstftf
k
nT
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
The Fourier of Sampled functionThe Fourier of Sampled functionLet F() and be the Fourier transform of the continuous function f(t) and its equivalent Sampled function
)(~ F
)(~ tf
)()()}()({)}(~{)(~ SFtstftfF T
)(1)(Tn
TS
TT n
n
dSFSFF
1
)()()()()(~
dTnF
T
dTnF
T
n
n
)()(1
)()(1
© 1992–2008 R. C. Gonzalez & R. E. Woods
TnF
T n
n
1
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
Sampling TheoremSampling TheoremBand limited-A function f(t) whose Fourier transform is zero out of the interval [-max , max] is called band limited
We can recover a function f(t) from its sampled representation if we can isolate a copy of F() from the periodic sequence of copies.
Extracting from a single period that represents F() is possible in the separation between copies is sufficient , which is guaranteed if ½T > max
)(~ F
max21 T
© 1992–2008 R. C. Gonzalez & R. E. Woods
Which is call the Nyquist Rate
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
To extract a single copy we multiply )(~ Fg py p yby H().
)(F
0)( maxmax
otherwise
TH
)(~)()(
0
FHF
otherwise
Once we have F() we can recover f(t)
deFtftj2
)()(
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
Aliasing gAppears when the sampling rate is less than the Nyquist rate – under-sampling. The inverse Fourier transform would then yield a corrupted f nction of t hich is kno n as freq encfunction of t, which is known as frequency aliasing or just aliasing.
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
Sampling in the 2D Domainp g
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
Examples of Moire peffect
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 4Filtering in the Frequency Domain
Chapter 4Filtering in the Frequency Domain
© 1992–2008 R. C. Gonzalez & R. E. Woods