Digital Image Processing Week V

transcript

Thurdsak LEAUHATONG

Frequency Domain Processing

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

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.:

2 cos 2 sin 2j uxe ux j ux

Example of 1-D Fourier Transforms

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

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

Fourier Tr. and Frequency Domain (cont.)

1-D, Discrete case

/2)(1)(M

MuxjexfM

uF Fourier Tr.:

Inv. Fourier Tr.:

/2)()(M

MuxjeuFxf

u = 0,…,M-1

x = 0,…,M-1

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

22 )()()( uIuRuF

)()(tan)( 1

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

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

Hz02.05.0100

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

2-Dimensional Discrete Fourier Transform

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

NvyMuxjeyxfMN

2-D IDFT

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

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

),(),(tan),( 1

vuRvuIvu

2-Dimensional Discrete Fourier Transform (cont.)

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

2-D DFT Properties

2-D DFT Properties (cont.)

Relation Between Spatial and Frequency Resolutions

whereDx = spatial resolution in x directionDy = spatial resolution in y direction

Du = frequency resolution in x directionDv = frequency resolution in y directionN,M = image width and height

Dx and Dy are pixel width and height. )

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

f(x,y)

1-D DFT

by row F(u,y)1-D DFT

by column

F(u,v)

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

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

/2)(1)(M

MuxjexfM

uF From DFT:

Conventional Display for 1-D DFT

0 N-1Time Domain Signal

DFTf(x)

Low frequencyarea

High frequencyarea

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

0-N/2 N/2-1

Conventional 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

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

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

NvyMuxjeyxfMN

0 N 2N-N

2-D DFT:g(x,y)

Conventional 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 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

Original displayof 2D DFT

0 N 2N-N

Display of 2D DFTAfter FFT Shift

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

Example of 2-D DFT

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

Example of 2-D DFT

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

Example of 2-D DFT

Original image

2D DFT

2D FFT Shift

F=fft2(f);S=abs(F); imshow(S,[ ]);

Fc=fftshift(F);Sc=abs(Fc); imshow(Sc,[ ]);

Example of 2-D DFT

Original image

2D DFT

2D FFT Shift

F=fft2(f);S=abs(F); imshow(S,[ ]);

Fc=fftshift(F);Sc=abs(Fc); imshow(Sc,[ ]);

Basic Concept of Filtering in the Frequency Domain

From 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 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

Multiplication 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.

“Wrap around” effect

H(u,v)GaussianLowpassFilter

Original image

Filtered image (obtained using circular convolution)

Incorrect areas at image rims

Multiplication in Freq. Domain = Circular Convolution

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)

IDFTZero padding

Zero padding

Concatenation

0 50 100 150 200 2500

Padding zerosBefore DFT

Keep only this part

Linear Convolution by using Circular Convolution and Zero Padding

Filtering in the Frequency Domain : Example

Highpass Filter

Lowpass Filter

Filter Masks and Their Fourier Transforms

Ideal Lowpass Filter

),( 0),( 1

),(DvuDDvuD

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

Ideal LPF Filter Transfer function

Examples of Ideal Lowpass Filters

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

Results of Ideal Lowpass Filters

Ringing effect can be obviously seen!

How ringing effect happens

Ideal Lowpass Filter with D0 = 5

Surface Plot

Abrupt change in the amplitude

),( 0),( 1

),(DvuDDvuD

How ringing effect happens (cont.)

x 10-3

Spatial Response of Ideal Lowpass Filter with D0 = 5

Surface Plot

Ripples that cause ringing effect

Butterworth Lowpass Filter

NDvuDvuH 2

0/),(11),(

Transfer function

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

Results of Butterworth Lowpass Filters

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

Gaussian Lowpass Filter

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.

Results of Gaussian Lowpass Filters

No ringing effect!

Application of Gaussian Lowpass Filters

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

Better LookingOriginal image

Highpass Filters

Hhp = 1 - Hlp

Ideal Highpass Filters

),( 1),( 0

),(DvuDDvuD

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

Ideal LPF Filter Transfer function

Butterworth Highpass Filters

NvuDDvuH 2

0 ),(/11),(

Transfer function

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

Gaussian Highpass Filters

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

Where D0 = spread factor.

Laplacian Filter in the Frequency Domain

)()( uFjudxxfd nn

From Fourier Tr. Property:

Then for Laplacian operator

),(222

22 vuFvu

Surface plot

222 vu

We get

Image of –(u2+v2)

Sharpening 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

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

Original Butterworthhighpass filteredimage

High freq. emphasisfiltered image

AfterHistEq.

a = 0.5, b = 2

Correlation Application: Object Detection

Digital Image Processing Week V

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