Filtering in the Frequency Domain

Image Processing

CSE 166

Lecture 7

Announcements

• Assignment 3 will be released today

– Due Apr 27, 11:59 PM

• Reading

– Chapter 4: Filtering in the Frequency Domain

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Overview: Image processing in the frequency domain

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Image in spatial domain

f(x,y)

Image in spatial domain

g(x,y)

Fouriertransform

Image in frequency domain

F(u,v)

Inverse Fourier

transform

Image in frequency domain

G(u,v)

Frequency domain processing

Jean-Baptiste Joseph Fourier1768-1830

2D continuous Fourier transform

• (Forward) Fourier transform

• Inverse Fourier transform

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2D continuous Fourier transform

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1D

2D

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Example: box function

Unit discrete impulse

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1D

2D

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Impulse train

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1D

2D

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Fourier transform of sampled functionand extracting one period

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1D

2D

Over-sampled Under-sampled

Recovered Imperfect recoverydue to

interference

Aliasing

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1D

2D

Aliasing

Original

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Aliasing in real images

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AliasingOriginal No aliasing

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2D Discrete Fourier Transform

2D discrete Fourier transform (DFT)

• (Forward) Fourier transform

• Inverse Fourier transform

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Centering the DFT

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1D

2D

In MATLAB, use fftshift and ifftshift

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Centering the DFT

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Original

DFT(look at corners)

Shifted DFTLog of

shifted DFT

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DFT magnitude of geometrically transformed images

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Translated

Rotatedabout center

Same magnitude as original

(invariant to translation)

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DFT phase of geometrically transformed images

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TranslatedRotated

about centerOriginal

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Contributions of magnitude and phaseto image formation

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Phase

IDFT: Phase only

(zero magnitude)

IDFT: Magnitude

only (zero phase)

IDFT: Boy magnitude

and rectangle phase

IDFT: Rectangle magnitude

and boy phase 17

2D convolution theorem

• 2D discrete (circular) convolution

• 2D convolution theorem

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Filtering using convolution theorem

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Filtering in spatial domain

using convolution

expectedresult

Filtering in frequencydomain

using productwithout

zero-padding

wraparounderror

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Filtering using convolution theorem

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Filtering in frequencydomain

using productwith

zero-padding

no wraparounderror

Gaussian lowpass filter in

frequency domain

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Fourier transform

Product

Inverse Fourier transform

Zero padding

Filtering using convolution theorem

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Filtering in spatial

domain using

convolution

Filtering in frequencydomain

usingproduct

Identical results

DFT

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Filtering in the frequency domain

• Ideal lowpass filter (LPF)

– Frequency domain

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Filtering in the frequency domain

• Ideal lowpass filter (LPF)

– Spatial domain

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H(u,v) h(x,y)

Filtering in the frequency domain

• Gaussian lowpass filter (LPF)

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Filtering in the frequency domain

• Butterworth lowpass filter (LPF)

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Gaussian LPF

Filtering in the frequency domain

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Ideal LPF Butterworth LPF

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Example: character recognition

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Gaussian LPFjoins broken characters

Highpass filter (HPF)Frequency domain

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Ideal HPF

Gaussian HPF

Butterworth HPF

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Butterworth HPF

Highpass filter (HPF)Spatial domain

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Ideal HPF Gaussian HPF

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Filtering in the frequency domain

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Ideal HPF Gaussian HPF Butterworth HPF

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H (u)

u u

x x

H (u)

h (x)

1

16–– 3

h (x)

2 12 12 1

2 1 8 2 1

2 12 12 1

0 2 1 0

2 1 4 2 1

0 2 1 0

1 2 1

2

1

9–– 3

4 2

1 2 1

1 1 1

1 1 1

1 1 1

Filtering in the frequency domain

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1D

Lowpass filter Sharpening filter31

Frequencydomain

Spatialdomain

Filtering in the frequency domain

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2D

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Lowpass filter Highpass filter Offset highpass filter

Bandreject filters

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ButterworthIdeal Gaussian

Filtering in the frequency domain

• Sharpening filter

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Overview: Image processing in the frequency domain

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Image in spatial domain

f(x,y)

Image in spatial domain

g(x,y)

Fouriertransform

Image in frequency domain

F(u,v)

Inverse Fourier

transform

Image in frequency domain

G(u,v)

Frequency domain processing

Jean-Baptiste Joseph Fourier1768-1830

Next Lecture

• Image restoration

• Reading

– Chapter 5: Image Restoration and Reconstruction

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