Chapter14

240-373: Chapter 14: The Frequency Domain

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Montri [email protected]://fivedots.coe.psu.ac.th/~montri

240-373 Image Processing


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Chapter 14

The Frequency Domain


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The Frequency Domain

• Any wave shape can be approximated by a sum of periodic (such as sine and cosine) functions.

• a--amplitude of waveform• f-- frequency (number of times the wave

repeats itself in a given length)• p--phase (position that the wave starts)• Usually phase is ignored in image

processing


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5


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The Hartley Transform

• Discrete Hartley Transform (DHT)– The M x N image is converted into a second i

mage (also M x N)– M and N should be power of 2 (e.g. .., 128, 25

6, 512, etc.)– The basic transform depends on calculating t

hh hhhhhh hhh hhh hhhh hhhhh hh hhh hhh M x N hhhhh

1

0

1

0

)2sin()2cos(),(1

),(M

x

N

y N

vy

M

ux

N

vy

M

uxyxf

MNvuH


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where f(x,y) is the intensity of the pixel at posit ion (x,y)

H(u,v) is the value of element in frequency domain

– The results are periodic– The cosine+sine (CAS) term is call “ the kern

el of the transformation ” (or ”hhhhh hhhhhhhh”)


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• Fast Hartley Transform (FHT)– M hhh N must be power of 2– Much faster than DHT– hhhhhhhhh

2/),(),(),(),(),( vNuMTvNuTvuMTvuTvuH


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The Fourier Transform

• The Fourier transform– Each element has real and imaginary values– hhhh hhhh

– f(x,y) is point (x,y) in the original image andF(u,v) is the point (u,v) in the frequency imagh

dxdyeyxfvuF vyuxi )(2),(),(


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The Fourier Transform

• Discrete Fourier Transform (DFT)

– Imaginary part

– Real part

– The actual complex result is Fi(u,v) + Fr(u,v)

1

0

1

0

2sin),(1

),(M

x

N

yi N

vy

M

uxyxf

MNvuF

1

0

1

0

2cos),(1

),(M

x

N

yr N

vy

M

uxyxf

MNvuF

1

0

1

0

2

),(1

),(M

x

N

y

N

vy

M

uxi

eyxfMN

vuF


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Fourier Power Spectrum and Inverse Fourier Transform

• Fourier power spectrum

• Inverse Fourier Transform

22 ),(),(),( vuFvuFvuF ir

1

0

1

0

2

),(1

),(M

x

N

y

N

vy

M

uxi

evuFMN

yxf


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• Fast Fourier Transform (FFT)– Much faster than DFT– M hhh N must be power of 2– Computation is reduced from M2N2 to

MN log2

M . log2

N (~1 /1 0 0 0 times)


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• Optical transformation– A common approach to view image in

frequency domain

Original image Transformed image

A BD C

C D

B A


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Power and Autocorrelation Functions

• Power function:

• Autocorrelation function

– hhhhhhh hhhhhhh hhhhhhhhh hhhh– hhhhhhh hhhhhhhhh hh

22 ),(),(2

1),(),( vuHvuHvuFvuP

2),( vuF

22 ),(),(2

1vuHvuH


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Hartley vs Fourier Transform


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Interpretation of the power function


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Applications of Frequency Domain Processing

• Convolution in the frequency domain


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Applications of Frequency Domain Processing

– 102useful when the image is larger than 41024x and the template size is greater than

16x16– Template and image must be the same size

0000

0000

0011

0011

11

11


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– Use FHT or FFT instead of DHT or DFT– Number of points should be kept small– The transform is periodic

• zeros must be padded to the image and the template• minimum image size must be (N+n-1) x (M+m-1)

– Convolution in frequency domain is “real convolution” Normal convolution

Real convolution


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– Convolution in frequency domain is “real convolution”

Normal convolution

Real convolution

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Convolution using the Fourier transform

1Technique : Convolution using the Fourier transform

USE: To perform a convolution OPERATION:

– - zero padding both the image (MxN) and the temp - -late (m x n) to the size (N+n 1 ) x (M+m 1 )

– hhhhhhhh hhh hh hhh hhhhhhhh hhhhh hhh hhhhhat e

– hhhhhhhhhhh hhhhhhh hh hhhhhhh hh hhh hhhhhh or med i mage agai nst t he t r ansf or med t empl at

e


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Convolution using the Fourier transform

OPERATION: (cont’d)– Multiplication is done as follows: F (image) F (template) F(result)

(r1

hi1

) (r2

h i2

) (r1

r2

- i1

i2

h r1

i2

+r2

i1

)

4 2i.e. real multiplications and additions– Per f or mi ng I nver se Four i er t r ansf or m


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Hartley convolution

2Technique : Hartley convolution USE: To perform a convolution

OPERATION:– - zero padding both the image (MxN) and the t

emplate ( mx n) to the size - (N+n 1 ) x (M+m-1)

image template

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Hartley convolution

– hhhhhhhh h hhhhhh hhhhhhhhh hh hhh h hhhhhhh h mage and template

image template

2.75-1.25-0.25-1.75-

1.25-1.25-1.75-1.75-

2.250.751.753.25

0.750.753.253.25

25.125.325.075.9

75.125.175.075.3

75.275.075.325.2

25.575.325.225.11


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Hartley convolution

– Multiplying them by evaluating:

),(),(

),(),(

),(),(),(),(

2),(

vMuNTvMuNI

vuTvMuNI

vMuNTvuIvuTvuINM

vuR


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Hartley convolution: Cont’d

h hhhhhh

– Performing Inverse Hartley transform, gives:

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Hartley convolution: Cont’d


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Deconvolution

• Convolution R = I * T

• Deconvolution I = R *-1 T

• Deconvolution of R by T = convolution ofR

• by some ‘inverse’ of the template T (T’)


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Deconvolution

• Consider periodic convolution as a matrix operation. For example

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011

011

*

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Deconvolution

is equivalent to A B C

AB = C

ABB-1 = CB-1

A = CB-1

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Chapter14

Technology

x n image

vt n u

n mn x

x n array

size n

log2 n

point u

fourier power spectrumf