ch04-frequency domain processing

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Digital Image Processing, 2nd ed.www.imageprocessingbook.com

2002 R. C. Gonzalez & R. E. Woods

1

Frequency

DomainProcessing

Chapter 4

www.ImageProcessingPlace.com

www.csie.ntnu.edu.tw/~violet/IP93/Chapter04.ppt


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2

Complex Numbers

http://en.wikipedia.org/wiki/Complex_number


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3

Polar Coordinates

http://en.wikipedia.org/wiki/Polar_coordinates


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4

Periodic Function

per iod p

x x+px

A functionfis periodic with period pgreater than zero if

f(x +p) =f(x)

forallvalues ofx in the domain off.

If a functionfis periodic with periodp, then for allx in the domain offand all integers n,

f(x + np) =f(x)


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5

If a function f is periodic with period p > 0

then thefundamental frequency off is 1/p

p has units of length 1/p has units of cycles/length

Periodic Function


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6

p is large (large period) 1/p is small (low frequency)

p is small (small period) 1/p is large (high frequency)

p

p

Periodic Function


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7

Fourier Series & Fourier Transform

Any function thatperiodically repeats itself can

be expressed as the sum of sines and/or cosines of

different frequencies, each multiplied by a

different coefficient (Fourier series).

Even functions that are not periodic (but whose

area under the curve is finite) can be expressed as

the integral of sines and/or cosines multiplied bya weighting function (Fourier transform).


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8

Fourier Transform & Glass Prism

The prism is a physical device that separates lightinto various color components, each dependingon its wavelength (or frequency) content.

The Fourier transform may be viewed as amathematical prism that separates(characterizes) a function into variouscomponents, also based on frequency content.

This is a powerful concept that lies at the heart oflinear filtering.


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9

Frequency Domain

The frequency domain

refers to the plane of the

two dimensional discrete

Fourier transform of animage.

The purpose of the

Fourier transform is to

represent a signal as a

linear combination of

sinusoidal signals of

various frequencies.


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10

Fourier TransformationAnd Its Inverse

The one-dimensional Fourier transform and its inverse

Fourier transform (continuous case)

Inverse Fourier transform:

The two-dimensional Fourier transform and its inverse

Fourier transform (continuous case)


dueuFxf uxj 2)()(

dydxeyxfvuF vyuxj )(2),(),(

1where)()(2

jdxexfuF

uxj

dvduevuFyxf vyuxj )(2),(),(


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11

The one-dimensional Fourier transform and its inverse

Fourier transform (discrete case)


1,...,2,1,0for)(

1

)(

1

0

/2

MuexfMuF

M

x

Muxj

1,...,2,1,0for)()(1

0

/2

MxeuFxfM

u

Muxj

Discrete Fourier Transformation

One Dimensional


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12

Since (Eulers formula)

then discrete Fourier transform can be redefined

Frequency (time) domain: the domain (values ofu) over

which the values ofF(u) range; because u determines thefrequency of the components of the transform.

Frequency (time) component: each of theMterms ofF(u).

1,...,2,1,0for Mu

1

0]/2sin/2)[cos(

1

)(

M

xMuxjMuxxfMuF

sincos je j

Discrete Fourier Transformation

One Dimensional


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13

F(u) can be expressed in polar coordinates:

R(u): the real part ofF(u)

I(u): the imaginary part ofF(u)

Power spectrum:

)()()()( 222

uIuRuFuP

spectrum)phaseorangle(phase)(

)(tan)(

spectrum)or(magnitude)()()(where

)()(

1

22

)(

uR

uIu

uIuRuF

euFuF uj

Discrete Fourier Transformation (DFT)

One Dimensional


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14

f(0) = 2

f(1) = 3

f(2) = 4

f(3) = 4

The One-Dimensional DFT

Examples

1,...,2,1,0for Mu

1

0

]/2sin/2)[cos(1

)(M

x

MuxjMuxxfM

uF

)()()( 22 uIuRuF

F(0) = 3.25

F(1) = -0.5 +j 0.25

F(2) = 0.25

F(3) = -0.5 -j 0.25

|F(0) | = 3.25

|F(1) | = 0.75

|F(2) | = 0.25

|F(3) | = 0.75


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15


Examples


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16


Examples

M= 1024 points, A = 1, K= 8 points

Spectrum is centered at u = 0 (discussed in the following slides)

1,...,2,1,0for Mu

1

0

]/2sin/2)[cos(1

)(M

x

MuxjMuxxfM

uF

)()()( 22 uIuRuF

1


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17


Examples

Important Features:

The height of the spectrum doubled as the area

under the curve in thex-domain doubled. The number of zeros in the spectrum in the same

interval doubled as the length of the function

doubled. This reciprocal nature of the Fourier transform

pair is most useful in interpreting results of image

processing in the frequency domain.

l 2 d d 18


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The One-Dimensional Fourier TransformSome Examples

The transform of a constant function is a DC value only.

The transform of a delta function is a constant.

Di i l I P i 2 d d 19


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The transform of an infinite train of delta functions spaced by

T is an infinite train of delta functions spaced by 1/T.

The transform of a cosine function is a positive delta at the

appropriate positive and negative frequency.

Di i l I P i 2 d d 20


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The transform of a sin function is a negative complex delta

function at the appropriate positive frequency and a negative

complex delta at the appropriate negative frequency.

The transform of a square pulse is a sinc function.

Di it l I P i 2 d d 21


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21

The Two Dimensional DFT

The two-dimensional Fourier transform and its inverse

Fourier transform (discrete case) DFT


u, v : the transform or frequency variables

x, y : the spatial or image variables

1,...,2,1,0,1,...,2,1,0for

),(1

),(1

0

1

0

)//(2

NvMu

eyxf

MN

vuFM

x

N

y

NvyMuxj

1,...,2,1,0,1,...,2,1,0for

),(),(

1

0

1

0

)//(2

NyMx

evuFyxf

M

u

N

v

NvyMuxj



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22


Properties

(shift))2

,2

()1)(,()1(N

vM

uFyxf yx



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We define the Fourier spectrum, phase angle, and power

spectrum as follows:

R(u,v): the real part ofF(u,v)

I(u,v): the imaginary part ofF(u,v)

spectrum)(power),(),(),()(

angle)(phase),(

),(tan),(

spectrum)(),(),(),(

222

1

2

122

vuIvuRvuFu,vP

vuR

vuIvu

vuIvuRvuF




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24

Magnitude and Phase

Of sinusoidal wave

Digital Image Processing 2nd ed 25


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25

Complex Numbers



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26

A Memory Aid for Evaluating j



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DFT Translation Property



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DFT Rotation Property



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DFT Periodicity



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spectrum)theofcomponentor DC(average

),(1

)0,0(1

0

1

0

M

x

N

y

yxfMN

F


Properties

F(u, v) =F(u+M, v) =F(u, v+N) =F(u+M, v+N)

(infinitely periodic in both u & v directions)

Same is for inverse DFT:

f(x,y) =f(x+M,y) =f(x,y+N) =f(x+M,y+N)



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(a)f(x,y) (b)F(u,y) (c)F(u, v)

The 2D DFTF(u, v) can be obtained by:

1. taking the 1D DFT of every row of imagef(x,y),F(u,y),

2. taking the 1D DFT of every column ofF(u,y)




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shift




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Digital Image Processing, 2nd ed. 35


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35

Computing & Visualizing

2-D DFT>> f = imread('rectangle.tif');

>> F = fft2(f);

>> Fc = fftshift(F);

>> imshow(abs(Fc), [])>> S2 = log(1+abs(Fc));

>> imshow(S2, [])

Note:[(-1)x+yf(x, y)]

= fftshift(fft2(f))



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36

The Property of Two-Dimensional DFT

Rotation

DFT

DFT



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Linear Combination

DFT

DFT

DFT

A

B

0.25 * A

+ 0.75 * B



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g g g,www.imageprocessingbook.com


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Expansion

DFT

DFT

A

Expanding the original image by a factor of n (n=2), filling

the empty new values with zeros, results in the same DFT.

B

Digital Image Processing, 2nd ed.i i b k

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39

Two-Dimensional DFT with Different Functions

Sine wave Its DFT


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Two-Dimensional DFT with Different Functions

2D Gaussian

function

Impulses

Its DFT

Its DFT


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Filtering in the Frequency Domain

Digital Image Processing, 2nd ed.www imageprocessingbook com

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g g gwww.imageprocessingbook.com


1. Since frequency is directly related

to rate of change it can be

associated with frequencies in the

Fourier transform.

2. Slowest varying frequencycomponent (u = v = 0)

corresponds to the average gray

level of an image.

3. As we move further away from

the origin, the higher frequencies

begin to correspond to faster and

faster gray level changes in an

image, which are the edges of

objects or noise etc.

Filtering in the Frequency DomainSome Basic Properties


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g g gwww.imageprocessingbook.com


Filtering in the Frequency DomainBasics of Filtering in the Frequency Domain


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www.imageprocessingbook.com


1) Multiply the input image by (-1)x+y to center the

transform.

2) ComputeF(u,v), the DFT of the image from (1).

3) MultiplyF(u,v) by a filter functionH(u,v).

4) Compute the inverse DFT of the result in (3).

5) Obtain the real part of the result in (4).6) Multiply the result in (5) by (-1)x+y.

Filtering in the Frequency DomainBasics of Filtering in the Frequency Domain


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1. Obtain the padding parameters:

PQ = paddedsize(size(f));

2. Obtain the Fourier transform with padding:

F = fft2(f, PQ(1), PQ(2));

3. Generate a filter function, H, of size PQ using any of the methods

discussed in the later slides. Apply: H = fftshift(H) if necessary.

4. Multiply the transform by the filter:

G = H.*F;5. Obtain the real part of the inverse FFT of G:

g = real(ifft2(G));

6. Crop the top, left rectangle to the original size:

g = g(1:size(f, 1), 1:size(f, 2));

Filtering in the Frequency DomainImplementation Through MATLAB


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Filtering in the spatial domain is described by the following

equation:

g(x,y) = h(x,y)*f(x,y)

wheref(x,y) is the original image, h(x,y) is the filter and * denotes theconvolution operation.

Filtering in the frequency domain is described by the following

equation:

G(u, v) =H(u, v)F(u, v)where

F(u, v) is the Fourier transform of an image

H(u, v) is the frequency domain filter

G(u, v) is the Fourier transform of the filtered image

Filtering in the Spatial and Frequency Domain


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Correspondence between Filtering in

the Spatial and Frequency Domain

Convolution Theorem:

The foundation for linear filtering in both the spatial and

frequency domains is the convolution theorem, which may

be written as:

where

F(u,v) andH(u,v) denote the Fourier transforms off(x,y) and h(x,y). * Indicates the convolution of the two functions.

The discrete convolution of two functionsf(x,y) and h(x,y) of sizeMN

is defined as:

1

0

1

0

),(),(1

),(),(M

m

N

n

nymxhnmf

MN

yxhyxf


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g p g




The third equivalence relation says that given a filter in the

frequency domain, we can obtain the corresponding filter in the

spatial domain by taking the inverse Fourier transform of the

former. The reverse is also true.

The convolution theorem links filters in both domains (spatial and

frequency). Filtering is often more intuitive in the frequency

domain, since one can easily foresee the results, given a filter.

However, filters in the spatial domain have smaller masks than

those in the frequency domain. That is, one can first specify a

filter in the frequency domain, take its inverse Fourier transform,

and then use the resulting filter in the spatial domain as a guide for

constructing smaller spatial filter masks!


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g p g


Spatial domain Frequency domain

Average filter Gaussian lowpass filter

Laplacian Gaussian highpass filter

The convolution theorem establishes a correspondence

between filtering in the spatial and frequency domains as

shown in the following table.




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g p g


Letf(x,y) and h(x,y) are of sizeA B and C D, resp., then wraparound

error can be avoided by choosing:

P A + C -1

Q B + D -1

Filtering in the Frequency DomainPadding


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Filtering in the Frequency DomainPadding


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Filtering in the Frequency DomainPadding Function

Exercises:

(1) [1, 4.3.1]

Write an M-function:

function PQ = paddedsize(AB, CD, PARAM)

and use it on figure squareBW.tif to visualize the

difference between images with padding and without

padding.

(2) [1, 4.3.3]

Write an M-function for filtering in the frequency

domain.


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Notch Filter: Force the average value of an image to zero.

Multiply all values ofF(u,v) by the filter function:

All this filter would do is setF(0,0) to zero and leave all

frequency components of the Fourier transform untouched.

otherwise.1

)2/,2/(),(if0),(

NMvuvuH

Filtering in the Frequency DomainSome Basic Filters & Their Properties


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Lowpass filter is a filter that attenuates high frequencies

while passing' low frequencies. Low frequencies

correspond to the slowly varying components of an image

(e.g., uniform background).

Highpass filter is a filter that attenuates low frequencies

while passing high frequencies. High frequencies

correspond to the sharply varying components of an image(e.g., edges).


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Lowpass Filter

Highpass Filter


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A 2-D ideal low-pass filter is one whose transfer

function satisfies the relation:

1 ifD(u, v) Do

whereDo is a non-negative quantity and D(u,v) is the distance from

the point (u,v) to the origin of the frequency plane:

Filtering in the Spatial and Frequency DomainIdeal Filters

22

22

),(

Nv

MuvuD


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H(u, v)

D(u, v)0

1

Do

Do

Remove (multiply by zero) all frequencies >Do

Ideal Low Pass Filter



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Ideal low pass filter

Spatial domain image Filtered frequency domain imageFrequency domain image

ForwardFFT

Filtered spatial domain image

InverseFFT

Ideal Frequency

Domain Filtering

causes ringing



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Filtering in the Spatial and Frequency DomainIdeal Lowpass Filters (ILPF)


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Ideal Lowpass Filters


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Forward

FFT


Inverse

FFT

Ideal Frequency

Domain Filtering

causes ringing

Ideal high pass filter



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Butterworth Lowpass Filters (BLPFs)With order n

nDvuDvuH

2

0/),(1

1),(

Digital Image Processing, 2nd ed.www.imageprocessingbook.com 66


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Butterworth Lowpass

Filters (BLPFs)

n=2

D0=5,15,30,80,and 230



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Butterworth Lowpass Filters (BLPFs)

Spatial Representation

n=1 n=2 n=5 n=20



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Gaussian Lowpass Filters (GLPFs)

20

2 2/),(),( DvuDevuH



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Gaussian Lowpass

Filters (GLPFs)

D0=5,15,30,80,and 230



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Additional Examples of Lowpass Filtering



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Additional Examples of Lowpass Filtering



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Forward

FFT


Inverse

FFT

Smoother Gaussian

Filtering in the

Frequency

Domain shows no

ringing


Gaussian high pass filter



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Sharpening Frequency Domain Filter

),(),( vuHvuH lphp Ideal highpass filter

Butterworth highpass filter

Gaussian highpass filter

),(if1

),(if0),(

0

0

DvuD

DvuDvuH

nvuDDvuH

2

0 ),(/1

1),(

20

2 2/),(1),( DvuDevuH



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Highpass Filters

Spatial Representations



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Ideal Highpass Filters

),(if1

),(if0),(

0

0

DvuD

DvuDvuH



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Butterworth Highpass Filters

nvuDDvuH

2

0 ),(/1

1),(



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Gaussian Highpass Filters

20

2 2/),(1),( DvuDevuH

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