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Chapter 5 Chapter 5 Image Image
TransformsTransforms
Chapter 5 Chapter 5 Image Image
TransformsTransforms
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Image Image Processing for Processing for Pattern Pattern RecognitionRecognition
Feature Extraction
Acquisition
Preprocessing
Classification
Post Processing
ScalingCenteringEnhancementFiltering (Transform) Binarization (Thresholding)Edge detectionThinning
Pixel Feature (Histogram)Boundary ProjectionMomentsTransformation
MatchingTree Classification Neural Network
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Why need transformation?Why need transformation?
• By image transformation with different basis functions (kernels), image f(x,y) is decomposed into a series expansion of basis functions, which are used as the featuresfeatures for further recognition.
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Image TransformsImage TransformsImage TransformsImage Transforms
• Fourier transform
• Discrete Fourier transform
• Discrete Cosine transform
• Hough transform
• Wavelet transform
Transform
t
f(t)
F()
TransformTransform Input function
Basis function
Basis function g(t)
Operation: Inner Product
),(),( ),()()( wtgtfdtwtgtfwF
Wave transforms
• Wave transforms use the waves as their basis functions• Fourier transform uses sinusoidal waves as its orthogonal basis functions
dttjttf
dtetfF tj
)sin)(cos(
)()(
Transform
t
f(t)
0 t
0 t
0 t
F()
Fourier Transform
f0(x) = 1;
f1(x) = sin(x);
f2(x) = cos(2x);
f3(x) = cos(3x);
f4(x) = sin(18x);
f(x) = f0(x) +
f1(x) +
2f2(x) -
4f3(x) +
f4(x)
f1
2f2
- 4f3
f4
f0
f0(x) = 1;
f1(x) = sin(x);
f2(x) = cos(2x);
f3(x) = cos(3x);
f4(x) = sin(18x);
f0(x) = 1;
f1(x) = sin(x);
f2(x) = cos(2x);
f3(x) = cos(3x);
f4(x) = sin(18x);
f = f0f = f0 + f1 + f1 +2f2 +2f2 - 4f3- 4f3 + f4
-6
-4
-2
0
2
4
6
8
0 1 2 3
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Fourier TransformsFourier Transforms
• Fourier integral transform• Discrete Fourier transform (DFT)• Fast Fourier Transform (FFT)
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• Let f (x) be a continuous function of a real variable x. The Fourier transform of f (x) is
dxuxjxfuF ]2exp[)()(
Input signal Basic function
• F(u) is complex: )()()( ujIuRuF Real component Imaginary component
• Fourier spectrum: |)()(||)(| 22 uIuRuF
• Phase angle:
)()(
tan)( 1
uRuI
u
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• Example:
uXj
X
euXuA
dxuxjA
dxuxjxfuF
)sin(
]2exp[
]2exp[)()(
0
)()sin(
|||)sin(||)(|
uXuX
AX
euXuA
uF uXj
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• The 2-D Fourier transform of f (x,y) is
( , ) ( , ) exp[ 2 ( )] F u v f x y j u ux vy dxdy
• Fourier spectrum:
|),(),(||),(| 22 vuIvuRvuF
• Phase angle:
),(),(
tan),( 1
vuRvuI
vu
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• Example:
Input function
)()sin(
)()sin(
),(),(
22
0
2
0
2
)(2
uYeuY
uXeuX
AXY
dyedxeA
dxdyeyxfvuF
uYjuXj
Yvyj
Xuxj
vyuxj
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Input function
Spectrum displayedas an intensity function
Fourier spectrum
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Discrete Fourier Transform• 1D:
• 2D: (N=M)
1
0
/2
1
0
/21
N
u
Nuxj
N
x
Nuxj
euFxf
exfN
uF
1
0
1
0
/2
1
0
1
0
/2
,1
,
,1
,
N
u
N
v
Nvyuxj
N
x
N
y
Nvyuxj
evufN
yxf
eyxfN
vuF
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Discrete Fourier Transform (cont’)
• The Fourier spectrum, phase, and energy spectrum of 1D and 2D discrete functions are the same as the continuous case. But unlike the continuous case, both F(u) and F(u,v) always exist in the discrete case.
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Some Properties of the 2D Fourier Transform
• Separability:
– The principle advantage of the separability property is that F(u, v) or f(x, y) can be obtained in two steps by successive applications of the 1D FT or its inverse.
1
0
/21
0
/2
1
0
/21
0
/2
,1
,
,1
,
N
v
NvyjN
u
Nuxj
N
y
NvyjN
x
Nuxj
evuFeN
yxf
eyxfeN
vuF
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Some Properties of the 2D Fourier Transform (cont’)
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Some Properties of the 2D Fourier Transform (cont’)
• Periodicity and Conjugate Symmetry:
– If f(x, y) is real, the FT also exhibits conjugate symmetry:
NvNuFNvuFvNuFvuF ,,,,
vuFvuF
vuFvuF
,,
,, *
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Some Properties of the 2D Fourier Transform (cont’)
• Translation:
where the double arrow is used to indicate the correspondence between a function and its FT (and vice versa).
00
/2
00/2
,,
,,00
00
yyxxfevuF
vvuuFeyxfNvyuxj
Nyvxuj
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Some Properties of the 2D Fourier Transform (cont’)
• Scaling and Distributivity
– FT and its inverse are distributive over addition, but not over multiplication.
1 2 1 2
/ , / ,
, , , ,
f x a y b F au bv
F f x y f x y F f x y F f x y
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Some Properties of the 2D Fourier Transform (cont’)
• Average Value:
Substituting u=v=0 into F(u, v) yields
Giving
1 1
20 0
1, ,
N N
x y
f x y f x yN
1
0
1
0
,1
0,0N
x
N
y
yxfN
F
1, 0,0f x y F
N
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Fast Fourier Transform (FFT)Fast Fourier Transform (FFT)• The number of complex multiplications and
additions required to implement a 1D discrete Fourier Transform is proportional to N2. The FFT computation of this is Nlog2N.
• In the 2D case, the number of direct operations is N4 and the FFT operation is 2N2log2N.
• FFT offers considerable computation advantage over direct implementation when N is relatively large (>256).
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Fast Fourier Transform (cont’)
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Fourier Transform (FFT) and Fourier Inverse Transform (FFT)Fourier Transform (FFT) and Fourier Inverse Transform (FFT)
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Fourier High Pass FilteringFourier High Pass Filtering
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Fourier Low Pass FilteringFourier Low Pass Filtering
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Discrete Cosine TransformDiscrete Cosine Transform• The 1-D DCT of a function f (x) is C(u), u = 0,
1, 2, …, N-1
1
0
1
0
2
)12(cos)(
2)(
)(1
)0(
N
x
N
x
N
uxxf
NuC
xfN
C
• By the DCT, a function f(x) is decomposed into a series expansion of basis functions, which are used as the features
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• The 2-D DCT of an image f (x,y) is C(u,v), u,v = 0, 1, 2, …, N-1
])12[cos(])12)[cos(,(2
1),(
),(1
)0,0(
1
0
1
03
1
0
1
0
vyuxyxfN
vuC
yxfN
C
N
y
N
x
N
y
N
x
• By the DCT, image f(x,y) is decomposed into a series expansion of basis functions, which are used as the features
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Hough TransformHough Transform
Consider a point (xi, yi) and the general equation of a straight line in slope-intercept form,
yi=axi+b.
There is an infinite number of lines that pass through (xi, yi), but they all satisfy the above equation for varying values of a and b.
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Hough Transform (cont’)Hough Transform (cont’)
• Consider b=-xia+yi, and the ab plane (parameter space), then we have the equation of a single line for a fixed pair (xi, yi).
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Find the locations of strong peaks in the Hough transform matrix. The locations of these peaks correspond to the location of straight lines in the original image.
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In this example, the strongest peak in R corresponds to and , . The line perpendicular to that angle and located at x’ is shown below, superimposed in red on the original image. The Radon transform geometry is shown in black.
94q
101x
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Waves, Wavelets, and Transforms
Waves & Wavelets
Book and booklet
A new word in English - Wavelets
Waves & WaveletsWaves & WaveletsWavesWaves
• Waves are non-compact support functions• Non-compact support function The functions extend to infinity in both directions They are non-zero over their entire domain
f(x), x = - , …, 0, …, f(-) 0, f() 0
WaveletsWavelets
Wavelets are compact support functions
Compact support functions:
The functions are in a limited duration
f(x) 0, for x = (a, b)
•These basis functions vary in position as well as frequency
WavesWaves
WaveletsWavelets
Low-frequency High-frequency
Position
0 dttR
a is a scale parametera scale parameter, b is a translation parametera translation parameter.
,, 1 dttfbafW abt
Ra
Wavelet Transform
For any f(t) L2(R), the wavelet transform is
A function (t) R is called a wavelet, if it satisfies
where
dtetfF tj
)()(Wave Transform
An example of Wavelet Transform• Haar function (mother)
• Haar baby wavelets
(t)
0
1
-1
10.5
t
otherwiswfor,0
121for,1
21t0for,1
)( tt
10.5
t
0
)12(2 t
0 42 t
)12
(2
1 t
Transform
t
f(t)
0 t
ba,
0 t
ba,
0 t
ba,
t
f(t)
Signal
WaveletWavelettransformtransform
Inverse waveletInverse wavelettransformtransform
Time
Frequency
Musical notationWavelet components
t
f(t)
Signal
FourierFouriertransformtransform
Inverse FourierInverse Fouriertransformtransform
Time
Frequency
Musical notationFourier components
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