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Digital Image Processing
Chapter 4: Image Enhancement in the Frequency Domain
Background
The French mathematiian Jean Baptiste Joseph Fourier Born in 1768 Published Fourier series in 1822 Fourier’s ideas were met with
skepticism Fourier series
Any periodical function can be expressed as the sum of sines and/or cosines of different frequencies, each multiplied by a different coefficient
Fourier transform Functions can be expressed as the
integral of sines and/or cosines multiplied by a weighting function
Functions expressed in either a Fourier series or transform can be reconstructed completely via an inverse process with no loss of information
1),( yx2222222 2/)()(222 vuyx AeeA
Applications Heat diffusion Fast Fourier transform (FFT) developed
in the late 1950s
Introduction to the Fourier Transform and the Frequency Domain
The one-dimensional Fourier transform and its inverse Fourier transform
Inverse Fourier transform
dxexfuF uxj 2)()(
dueuFxf uxj 2)()(
Two variables
dxdyeyxfvuF vyuxj )( 2),(),(
dudvevuFyxf vyuxj )( 2),(),(
Fourier transform
Inverse Fourier transform
Discrete Fourier transform (DFT) Original variable
Transformed variable
1,...,2,1,0),( Mxxf
1,...,2,1,0),( MuuF
1,...,2,1,0
,)(1
)(1
0
/ 2
Mu
exfM
uFM
x
Muxj
1,...,2,1,0
,)()(1
0
/ 2
Mx
euFxfM
u
Muxj
DFT The discrete Fourier transform and its
inverse always exist f(x) is finite in the book
Sines and cosines
sincos je j
1
0
]/ 2sin/ 2)[cos(1
)(M
x
MuxjMuxxfM
uF
Time domain
Time components
Frequency domain
Frequency components
x
)(xf
u
)(uF
Fourier transform and a glass prism Prism
Separates light into various color components, each depending on its wavelength (or frequency) content
Fourier transform Separates a function into various
components, also based on frequency content
Mathematical prism
Polar coordinates
Real part
Imaginary part
)()()( ujeuFuF
)(uR
)(uI
Magnitude or spectrum
Phase angle or phase spectrum
Power spectrum or spectral density
2122 )()()( uIuRuF
)(
)(tan)( 1
uR
uIu
)()()()( 222uIuRuFuP
Samples
)()( 0 xxxfxf
)()( uuFuF
xMu
1
Some references http://local.wasp.uwa.edu.au/~pbourke/
other/dft/ http://homepages.inf.ed.ac.uk/rbf/HIPR2
/fourier.htm
Examples test_fft.c fft.h fft.c Fig4.03(a).bmp test_fig2.bmp
The two-dimensional DFT and its inverse
1,...,2,1,0
1,...,2,1,0
,),(1
),(1
0
1
0
)// (2
Nv
Mu
eyxfMN
vuFM
x
N
y
NvyMuxj
Spatial, or image variables: x, y Transform, or frequency variables:
u, v
1,...,2,1,0y
1,...,2,1,0 x
,),(),(1
0
1
0
)// (2
N
M
evuFyxfM
u
N
v
NvyMuxj
Magnitude or spectrum
Phase angle or phase spectrum
Power spectrum or spectral density
2122 ),(),(),( vuIvuRvuF
),(
),(tan),( 1
vuR
vuIvu
),(),(),(),( 222vuIvuRvuFvuP
Centering
Average gray level F(0,0) is called the dc component of the
spectrum
)2/,2/()1)(,( NvMuFyxf yx
1
0
1
0
),(1
)0,0(M
x
N
y
yxfMN
F
Conjugate symmetric If f(x,y) is real
Relationships between samples in the spatial and frequency domains
),(*),( vuFvuF
),(),( vuFvuF
xMu
1
yNv
1
The separation of spectrum zeros in the u-direction is exactly twice the separation of zeros in the v direction
Filtering in the frequency domain
Strong edges that run approximately at +45 degree, and -45 degree
The inclination off horizontal of the long white element is related to a vertical component that is off-axis slightly to the left
The zeros in the vertical frequency component correspond to the narrow vertical span of the oxide protrusions
Basics of filtering in the frequency domain 1. Multiply the input image by
to center the transform 2. Compute F(u,v) 3. Multiply F(u,v) by a filter function
H(u,v) 4. Compute the inverse DFT 5. Obtain the real part 6. Multiply the result by
yx )1(
yx )1(
Fourier transform of the output image
zero-phase-shift filter Real H(u,v)
),(),(),( vuFvuHvuG
),(
),(tan),( 1
vuR
vuIvu
Inverse Fourier transform of G(u,v)
The imaginary components of the inverse transform should all be zero When the input image and the filter
function are real
),(1 vuG
Set F(0,0) to be zero, a notch filter
otherwise1
)2/,2/(),( if0),(
NMvuvuH
Lowpass filter Pass low frequencies, attenuate high
frequencies Blurring
Highpass filter Pass high frequencies, attenuate low
frequencies Edges, noise
Convolution theorem
1
0
1
0
),(),(1
),(*),(M
m
N
n
nymxhnmfMN
yxhyxf
),(),(),(*),( vuHvuFyxhyxf
),(*),(),(),( vuHvuFyxhyxf
Impulse function of strength A
),( 00 yyxxA
),(),(),( 00
1
0
1
000 yxAsyyxxAyxs
M
x
N
y
)0,0(),(),(1
0
1
0
AsyxAyxsM
x
N
y
MN
eyxMN
vuFM
x
N
y
NyvMxuj
1
),(1
),(1
0
1
0
)/ / (2
),(1
),(),(1
),(*),(1
0
1
0
yxhMN
nymxhnmMN
yxhyxfM
m
N
n
),(),(),(*),( vuHvuFyxhyxf
),(),(),(*),( vuHyxyxhyx
),(),( vuHyxh
Gaussian filter
22 2/)( uAeuH 22222)( xAexh
Highpass filter
22
221
2 2/2/)( uu BeAeuH 22
2222
12 2
22
1 22)( xx BeAexh
21 and BA
Smoothing Frequency-Domain Filterers
Ideal lowpass filters
),(),(),( vuFvuHvuG
0
0
Dv)D(u, if0
Dv)D(u, if1),( vuH
2/1 22 )2/()2/(),( NvMuvuD
Cutoff frequency
Total image power
Portion of the total power
0D
1
0
1
0
),(M
u
N
vT vuPP
u vTPvuP /),(100
Blurring and ringing properties Filter
Convolution
: Spatial filter was multiplied by Then the inverse DFT The real part of the inverse DFT was
multiplied by
),(),(),( vuFvuHvuG
),(*),(),( yxfyxhyxg ),( yxh
),( vuH vu )1(
yx )1(
The filter A dominant component at the origin Concentric, circular components about
the center component --- ringing The radius of the center component
and the number of circles per unit distance from the origin are inversely proportional to the value of the cutoff frequency of the ideal filter.
),( yxh
Butterworth lowpass filters
when
nDvuDvuH 2
0/),(1
1),(
2/1 22 )2/()2/(),( NvMuvuD
5.0),( vuH 0),( DvuD
Butterworth lowpass filters Order 1: No ringing Order 2: Imperceptible ringing Higher order: Ringing becomes a
significant factor
Gaussain lowpass filters
When
No ringing
22 2/),(),( vuDevuH
2/1 22 )2/()2/(),( NvMuvuD
20
2 2/),(),( DvuDevuH
0D
Additional examples of lowpass filters Machine perception Printing and publishing Satellite and aerial images
Sharpening Frequency Domain Filters
Highpass filter
Spatial filter: was multiplied by Then the inverse DFT The real part of the inverse DFT was
multiplied by
),(1),( vuHvuH lphp
),( yxh),( vuH vu )1(
yx )1(
Ideal highpass filters
0
0
Dv)D(u, if1
Dv)D(u, if0),( vuH
2/1 22 )2/()2/(),( NvMuvuD
Butterworth highpass filters
nvuDDvuH 2
0 ),(/1
1),(
2/1 22 )2/()2/(),( NvMuvuD
Gaussian highpass filters
2/1 22 )2/()2/(),( NvMuvuD
20
2 2/),(1),( DvuDevuH
The Laplacian in the frequency domain
)()()(
uFjudx
xfd nn
n
),()(
),()(),()(
),(),(
22
22
2
2
2
2
vuFvu
vuFjvvuFju
y
yxf
x
yxf
)(),( 22 vuvuH
),()(),( 222 vuFvuyxf
22 )2/()2/(),( NvMuvuH
After centering
Inverse Fourier transform
Fourier-transform pair
),()2/()2/(
),(221
2
vuFNvMu
yxf
),()2/()2/(
),(22
2
vuFNvMu
yxf
Subtracting the Laplacian from the original image
),( )2/()2/(1
),(221 vuFNvMu
yxg
),(),(),( 2 yxfyxfyxg
Unsharp masking, high-boost filtering, and high-frequency emphasis filtering Highpass filtering
High-boost filtering
),(),(),( yxfyxfyxf lphp
),(),(),( yxfyxAfyxf lphb
Frequency domain
),(),(),()1(),( yxfyxfyxfAyxf lphb
),(),()1(),( yxfyxfAyxf hphb
),(1),( vuHvuH lphb
),()1(),( vuHAvuH hphb
High-frequency emphasis
where and
),(),( vubHavuH hphfe
0a ab
Homomorphic Filtering
Illumination and reflectance components
Derivations
),(),(),( yxryxiyxf
),(),(),( yxryxiyxf
),(ln),(ln
),(ln),(
yxryxi
yxfyxz
Or
),(ln),(ln
),(ln),(
yxryxi
yxfyxz
),(),(),( vuFvuFvuZ ri
Frequency domain
Spatial domain
),(),(),(),(
),(),(),(
vuFvuHvuFvuH
vuZvuHvuS
ri
),(),(
),(),(
),(),(
1
1
1
vuFvuH
vuFvuH
vuSyxs
r
i
),(),(),(' 1 vuFvuHyxi i
),(),(),(' 1 vuFvuHyxr r
),('),('),( yxryxiyxs
),(),(
),(
00
),('),('),(
yxryxi
eeeyxg yxryxiyxs
Decrease the contribution made by the low frequencies (illumination)
Amplify the contribution made by high frequencies (reflectance)
Simultaneous dynamic range compression and contrast enhancement
LDvuDc
LH evuH )/),(( 20
2
1)(),(
Implementation
Translation
When and
),(),( 00)//(2 00 vvuuFeyxf NyvMxuj
)//(200
00),(),( NvyMuxjevuFyyxxf
yx
yxjNyvMxuj ee
)1(
)()//(2 00
2/0 Mu 2/0 Nv
)2/,2/()1)(,( NvMuFyxf yx
vuvuFNyMxf )1)(,()2/,2/(
Distributivity and scaling
),(),(),(),( 2121 yxfyxfyxfyxf
),(),( vuaFyxaf
)/,/(1
),( bvauFab
byaxf
Rotation Polar coordinates
Rotating by an angle rotates by the same angle
cosrx sinry coswu sinwv
),(),( wFrf ),(),( 00 wFrf
),( yxf 0),( vuF
Periodicity and conjugate symmetry Periodicity property
),(),(
),(),(
NvMuFNvuF
vMuFvuF
),(),(
),(),(
NyMxfNyxf
yMxfyxf
Conjugate symmetry
Symmetry of the spectrum
),(),( * vuFvuF
),(),( vuFvuF
),(),( vuFvuF
),(),( vuFvuF
Separability
1
0
)/ (2
1
0
)/ v(21
0
)/ (2
1
0
1
0
)// (2
),(1
),(11
),(1
),(
M
x
Muxj
N
y
NyjM
x
Muxj
M
x
N
y
NvyMuxj
evxfM
eyxfN
eM
eyxfMN
vuF
where
We can compute the 2-D transform by first computing a 1-D transform along each row of the input image, and then computing a 1-D transform along each column of this intermediate result
1
0
)/ v(2),(1
),(N
y
NyjeyxfN
vxF
Computing the inverse Fourier transform using a forward transform algorithm
1,...,2,1,0
,)(1
)(1
0
/ 2
Mu
exfM
uFM
x
Muxj
Calculate
1,...,2,1,0
,)()(1
0
/ 2
Mx
euFxfM
u
Muxj
1
0
/ 2** )(1
)(1 M
u
MuxjeuFM
xfM
Inputting into an algorithm designed to compute the forward transform gives the quantity
)(* uF
)(1 * xfM
2-D
1
0
1
0
)// (2*
*
),(1
),(1
M
u
N
v
NvyMuxjevuFMN
yxfMN
More on periodicity: the need for padding Convolution: Flip one of the functions
and slide it pass the other
1
0
)()(1
)()(M
m
mxhmfM
xhxf
