Digital Image Processing Chapter 4: Image Enhancement in the Frequency Domain 22 June 2005

Digital Image ProcessingChapter 4:

Image Enhancement in the Frequency Domain

22 June 2005

Background: Fourier Series Background: Fourier Series

Any periodic signals can beviewed as weighted sumof sinusoidal signals with different frequencies

Fourier series:

Frequency Domain: view frequency as an independent variable

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Fourier Tr. and Frequency Domain Fourier Tr. and Frequency Domain

Time, spatial Domain Signals

Frequency Domain Signals

Fourier Tr.

Inv Fourier Tr.

1-D, Continuous case

dxexfuF uxj 2)()(Fourier Tr.:

dueuFxf uxj 2)()(Inv. Fourier Tr.:

Fourier Tr. and Frequency Domain (cont.) Fourier Tr. and Frequency Domain (cont.) 1-D, Discrete case

1

0

/2)(1)(M

x

MuxjexfM

uF Fourier Tr.:

Inv. Fourier Tr.:

1

0

/2)()(M

u

MuxjeuFxf

u = 0,…,M-1

x = 0,…,M-1

F(u) can be written as)()()( ujeuFuF or)()()( ujIuRuF

22 )()()( uIuRuF

where

)()(tan)( 1

uRuIu

Example of 1-D Fourier Transforms Example of 1-D Fourier Transforms

Notice that the longerthe time domain signal,The shorter its Fouriertransform


Relation Between Relation Between x and x and u u For a signal f(x) with M points, let spatial resolution x be space between samples in f(x) and let frequency resolution u be space between frequencies components in F(u), we have

xMu

1

Example: for a signal f(x) with sampling period 0.5 sec, 100 point, we will get frequency resolution equal to

Hz02.05.0100

1

u

This means that in F(u) we can distinguish 2 frequencies that are apart by 0.02 Hertz or more.

2-Dimensional Discrete Fourier Transform2-Dimensional Discrete Fourier Transform

1

0

1

0

)//(2),(1),(M

x

N

y

NvyMuxjeyxfMN

vuF

2-D IDFT

1

0

1

0

)//(2),(),(M

u

N

v

NvyMuxjevuFyxf

2-D DFT

u = frequency in x direction, u = 0 ,…, M-1v = frequency in y direction, v = 0 ,…, N-1

x = 0 ,…, M-1y = 0 ,…, N-1

For an image of size MxN pixels

F(u,v) can be written as),(),(),( vujevuFvuF or),(),(),( vujIvuRvuF

22 ),(),(),( vuIvuRvuF

where

),(),(tan),( 1

vuRvuIvu

2-Dimensional Discrete Fourier Transform (cont.)2-Dimensional Discrete Fourier Transform (cont.)

For the purpose of viewing, we usually display only theMagnitude part of F(u,v)


2-D DFT Properties2-D DFT Properties


2-D DFT Properties (cont.)2-D DFT Properties (cont.)






Computational Advantage of FFT Compared to DFTComputational Advantage of FFT Compared to DFT

Relation Between Spatial and Frequency ResolutionsRelation Between Spatial and Frequency Resolutions

xMu

1yN

v

1

wherex = spatial resolution in x directiony = spatial resolution in y direction

u = frequency resolution in x directionv = frequency resolution in y directionN,M = image width and height

x and y are pixel width and height. )

How to Perform 2-D DFT by Using 1-D DFTHow to Perform 2-D DFT by Using 1-D DFT

f(x,y)

1-D DFT

by row F(u,y)

1-D DFTby column

F(u,v)

How to Perform 2-D DFT by Using 1-D DFT (cont.)How to Perform 2-D DFT by Using 1-D DFT (cont.)

f(x,y)

1-D DFT

by rowF(x,v)

1-D DFTby column

F(u,v)

Alternative method

Periodicity of 1-D DFTPeriodicity of 1-D DFT

0 N 2N-N

DFT repeats itself every N points (Period = N) but we usually display it for n = 0 ,…, N-1

We display only in this range

1

0

/2)(1)(M

x

MuxjexfM

uF From DFT:

Conventional Display for 1-D DFTConventional Display for 1-D DFT

0 N-1

Time Domain Signal

DFTf(x)

)(uF

0 N-1

Low frequencyarea

High frequencyarea

The graph F(u) is not easy to understand !

)(uF

0-N/2 N/2-1

)(uF

0 N-1

Conventional Display for DFT : FFT ShiftConventional Display for DFT : FFT Shift

FFT Shift: Shift center of thegraph F(u) to 0 to get betterDisplay which is easier to understand.

High frequency area

Low frequency area

Periodicity of 2-D DFTPeriodicity of 2-D DFT

For an image of size NxM pixels, its 2-D DFT repeats itself every N points in x-direction and every M points in y-direction.

We display only in this range

1

0

1

0

)//(2),(1),(M

x

N

y

NvyMuxjeyxfMN

vuF

0 N 2N-N

0

M

2M

-M

2-D DFT:

g(x,y)


Conventional Display for 2-D DFTConventional Display for 2-D DFT

High frequency area

Low frequency area

F(u,v) has low frequency areasat corners of the image while highfrequency areas are at the centerof the image which is inconvenientto interpret.


2-D FFT Shift : Better Display of 2-D DFT2-D FFT Shift : Better Display of 2-D DFT

2D FFTSHIFT

2-D FFT Shift is a MATLAB function: Shift the zero frequency of F(u,v) to the center of an image.

High frequency area Low frequency area(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Original displayof 2D DFT

0 N 2N-N

0

M

2M

-M


Display of 2D DFTAfter FFT Shift

2-D FFT Shift (cont.) : How it works2-D FFT Shift (cont.) : How it works

Example of 2-D DFTExample of 2-D DFT

Notice that the longer the time domain signal,The shorter its Fourier transform



Notice that direction of an object in spatial image andIts Fourier transform are orthogonal to each other.



Original image

2D DFT

2D FFT Shift


Original image

2D DFT

2D FFT Shift

Basic Concept of Filtering in the Frequency DomainBasic Concept of Filtering in the Frequency DomainFrom Fourier Transform Property:

),(),(),(),(),(),( vuGvuHvuFyxhyxfyxg

We cam perform filtering process by using


Multiplication in the frequency domainis easier than convolution in the spatialDomain.

Filtering in the Frequency Domain with FFT shiftFiltering in the Frequency Domain with FFT shift

f(x,y)

2D FFT

FFT shift

F(u,v)

FFT shift

2D IFFTX

H(u,v)(User defined)

G(u,v)

g(x,y)

In this case, F(u,v) and H(u,v) must have the same size andhave the zero frequency at the center.

0 20 40 60 80 100 1200

0.5

1

0 20 40 60 80 100 1200

0.5

1

0 20 40 60 80 100 1200

20

40

Multiplication in Freq. Domain = Circular ConvolutionMultiplication in Freq. Domain = Circular Convolution

f(x) DFT F(u)G(u) = F(u)H(u)

h(x) DFT H(u)g(x)IDFT

Multiplication of DFTs of 2 signalsis equivalent toperform circular convolutionin the spatial domain.

f(x)

h(x)

g(x)

“Wrap around” effect

H(u,v)GaussianLowpassFilter with D0 = 5

Original image

Filtered image (obtained using circular convolution)

Incorrect areas at image rims

Multiplication in Freq. Domain = Circular ConvolutionMultiplication in Freq. Domain = Circular Convolution

Linear Convolution by using Circular Convolution and Zero Padding Linear Convolution by using Circular Convolution and Zero Padding

f(x) DFT F(u)G(u) = F(u)H(u)

h(x) DFT H(u)

g(x)

IDFTZero padding

Zero padding

Concatenation

0 50 100 150 200 2500

0.5

1

0 50 100 150 200 2500

0.5

1

0 50 100 150 200 2500

20

40

Padding zerosBefore DFT

Keep only this part





Zero padding area in the spatialDomain of the mask image(the ideal lowpass filter)

Filtered image

Only this area is kept.

Filtering in the Frequency Domain : ExampleFiltering in the Frequency Domain : Example

In this example, we set F(0,0) to zerowhich means that the zero frequencycomponent is removed.

Note: Zero frequency = average intensity of an image


Filtering in the Frequency Domain : ExampleFiltering in the Frequency Domain : Example


Highpass Filter

Lowpass Filter

Filtering in the Frequency Domain : Example (cont.)Filtering in the Frequency Domain : Example (cont.)


Result of Sharpening Filter

Filter Masks and Their Fourier TransformsFilter Masks and Their Fourier Transforms


Ideal Lowpass FilterIdeal Lowpass Filter

0

0

),( 0),( 1

),(DvuDDvuD

vuH

where D(u,v) = Distance from (u,v) to the center of the mask.

Ideal LPF Filter Transfer function


Examples of Ideal Lowpass FiltersExamples of Ideal Lowpass Filters


The smaller D0, the more high frequency components are removed.

Results of Ideal Lowpass Filters Results of Ideal Lowpass Filters

Ringing effect can be obviously seen!


How ringing effect happens How ringing effect happens

Ideal Lowpass Filter with D0 = 5

-200

20

-20

0

20

0

0.2

0.4

0.6

0.8

1

Surface Plot

Abrupt change in the amplitude

0

0

),( 0),( 1

),(DvuDDvuD

vuH

How ringing effect happens (cont.) How ringing effect happens (cont.)

-200

20

-20

0

20

0

5

10

15

x 10-3

Spatial Response of Ideal Lowpass Filter with D0 = 5

Surface Plot

Ripples that cause ringing effect

How ringing effect happens (cont.) How ringing effect happens (cont.)


Butterworth Lowpass Filter Butterworth Lowpass Filter

NDvuDvuH 2

0/),(11),(

Transfer function

Where D0 = Cut off frequency, N = filter order.


Results of Butterworth Lowpass Filters Results of Butterworth Lowpass Filters

There is less ringing effect compared to those of ideal lowpassfilters!


Spatial Masks of the Butterworth Lowpass Filters Spatial Masks of the Butterworth Lowpass Filters

Some ripples can be seen.(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Gaussian Lowpass Filter Gaussian Lowpass Filter

20

2 2/),(),( DvuDevuH Transfer function

Where D0 = spread factor.

Note: the Gaussian filter is the only filter that has no ripple and hence no ringing effect.


Gaussian Lowpass Filter (cont.) Gaussian Lowpass Filter (cont.)

-200

20

-20

0

20

0.2

0.4

0.6

0.8

1

Gaussian lowpass filter with D0 = 5

-200

20

-20

0

20

0

0.01

0.02

0.03

Spatial respones of the Gaussian lowpass filter with D0 = 5

Gaussian shape

20

2 2/),(),( DvuDevuH

Results of Gaussian Lowpass Filters Results of Gaussian Lowpass Filters

No ringing effect!


Application of Gaussian Lowpass FiltersApplication of Gaussian Lowpass Filters

The GLPF can be used to remove jagged edges and “repair” broken characters.

Better LookingOriginal image


Application of Gaussian Lowpass Filters (cont.) Application of Gaussian Lowpass Filters (cont.)

Remove wrinkles

Softer-Looking

Original image


Application of Gaussian Lowpass Filters (cont.) Application of Gaussian Lowpass Filters (cont.)

Remove artifact lines: this is a simple but crude way to do it!

Filtered imageOriginal image : The gulf of Mexico andFlorida from NOAA satellite. (Images from Rafael C. Gonzalez and Richard E.

Wood, Digital Image Processing, 2nd Edition.


Highpass Filters Highpass Filters

Hhp = 1 - Hlp


Ideal Highpass Filters Ideal Highpass Filters

0

0

),( 1),( 0

),(DvuDDvuD

vuH

where D(u,v) = Distance from (u,v) to the center of the mask.

Ideal LPF Filter Transfer function

Butterworth Highpass Filters Butterworth Highpass Filters


NvuDDvuH 2

0 ),(/11),(

Transfer function

Where D0 = Cut off frequency, N = filter order.

Gaussian Highpass Filters Gaussian Highpass Filters


20

2 2/),(1),( DvuDevuH Transfer function

Where D0 = spread factor.

Gaussian Highpass Filters (cont.) Gaussian Highpass Filters (cont.)

20

2 2/),(1),( DvuDevuH

1020

3040

5060

20

40

600

0.2

0.4

0.6

0.8

1

1020

3040

5060

20

40

60

0

1000

2000

3000

Gaussian highpass filter with D0 = 5

Spatial respones of the Gaussian highpass filter with D0 = 5


Spatial Responses of Highpass Filters Spatial Responses of Highpass Filters

Ripples


Results of Ideal Highpass Filters Results of Ideal Highpass Filters

Ringing effect can be obviously seen!


Results of Butterworth Highpass Filters Results of Butterworth Highpass Filters


Results of Gaussian Highpass Filters Results of Gaussian Highpass Filters


Laplacian Filter in the Frequency DomainLaplacian Filter in the Frequency Domain

)()( uFjudx

xfd nn

n

From Fourier Tr. Property:

Then for Laplacian operator

),(222

2

2

22 vuFvu

yf

xff

Surface plot

222 vu We get

Image of –(u2+v2)

Laplacian Filter in the Frequency Domain (cont.)Laplacian Filter in the Frequency Domain (cont.)

Spatial response of –(u2+v2) Cross section

Laplacian mask in Chapter 3


Sharpening Filtering in the Frequency DomainSharpening Filtering in the Frequency Domain

),(),(),( yxfyxfyxf lphp

),(),(),( yxfyxAfyxf lphb

Spatial Domain

Frequency Domain Filter

),(),(),()1(),( yxfyxfyxfAyxf lphb

),(),()1(),( yxfyxfAyxf hphb

),(1),( vuHvuH lphp

),()1(),( vuHAvuH hphb


Sharpening Filtering in the Frequency Domain (cont.)Sharpening Filtering in the Frequency Domain (cont.)

p P2

P2 PP 2


Sharpening Filtering in the Frequency Domain (cont.)Sharpening Filtering in the Frequency Domain (cont.)),(),()1(),( yxfyxfAyxf hphb

Pfhp2f

A = 2 A = 2.7


High Frequency Emphasis FilteringHigh Frequency Emphasis Filtering),(),( vubHavuH hphfe

Original Butterworthhighpass filteredimage

High freq. emphasisfiltered image

AfterHistEq.

a = 0.5, b = 2

Homomorphic FilteringHomomorphic FilteringAn image can be expressed as

),(),(),( yxryxiyxf

i(x,y) = illumination componentr(x,y) = reflectance component

We need to suppress effect of illumination that cause image Intensity changed slowly.


Homomorphic FilteringHomomorphic Filtering


Homomorphic FilteringHomomorphic Filtering

More details in the room can be seen!


Correlation Application: Object DetectionCorrelation Application: Object Detection

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Digital Image Processing Chapter 4: Image Enhancement in the Frequency Domain 22 June 2005