Filtering in the Frequency Domaincseweb.ucsd.edu/classes/sp20/cse166-a/lec7.pdfFiltering in spatial...

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

CSE 166, Spring 2020 2

Overview: Image processing in the frequency domain


Image in spatial domain

f(x,y)


g(x,y)

Fouriertransform

Image in frequency domain

F(u,v)

Inverse Fourier

transform


G(u,v)

Frequency domain processing

Jean-Baptiste Joseph Fourier1768-1830

2D continuous Fourier transform

• (Forward) Fourier transform

• Inverse Fourier transform


2D continuous Fourier transform

CSE 166, Spring 2020

1D

2D

5

Example: box function

Unit discrete impulse


1D

2D

6

Impulse train


1D

2D

7

Fourier transform of sampled functionand extracting one period


1D

2D

Over-sampled Under-sampled

Recovered Imperfect recoverydue to

interference

Aliasing


1D

2D

Aliasing

Original

9

Aliasing in real images


AliasingOriginal No aliasing

10

2D Discrete Fourier Transform

2D discrete Fourier transform (DFT)

• (Forward) Fourier transform

• Inverse Fourier transform

CSE 166, Spring 2020 12

Centering the DFT


1D

2D

In MATLAB, use fftshift and ifftshift

13

Centering the DFT


Original

DFT(look at corners)

Shifted DFTLog of

shifted DFT

14

DFT magnitude of geometrically transformed images


Translated

Rotatedabout center

Same magnitude as original

(invariant to translation)

15

DFT phase of geometrically transformed images


TranslatedRotated

about centerOriginal

16

Contributions of magnitude and phaseto image formation


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

CSE 166, Spring 2020 18

Filtering using convolution theorem


Filtering in spatial domain

using convolution

expectedresult

Filtering in frequencydomain

using productwithout

zero-padding

wraparounderror

19




using productwith

zero-padding

no wraparounderror

Gaussian lowpass filter in

frequency domain

20

Fourier transform

Product

Inverse Fourier transform

Zero padding



Filtering in spatial

domain using

convolution


usingproduct

Identical results

DFT

21

Filtering in the frequency domain

• Ideal lowpass filter (LPF)

– Frequency domain

CSE 166, Spring 2020 22


• Ideal lowpass filter (LPF)

– Spatial domain

CSE 166, Spring 2020 23

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


• Gaussian lowpass filter (LPF)

CSE 166, Spring 2020 24


• Butterworth lowpass filter (LPF)

CSE 166, Spring 2020 25

Gaussian LPF



Ideal LPF Butterworth LPF

26

Example: character recognition

CSE 166, Spring 2020 27

Gaussian LPFjoins broken characters

Highpass filter (HPF)Frequency domain


Ideal HPF

Gaussian HPF

Butterworth HPF

28

Butterworth HPF

Highpass filter (HPF)Spatial domain


Ideal HPF Gaussian HPF

29



Ideal HPF Gaussian HPF Butterworth HPF

30

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



1D

Lowpass filter Sharpening filter31

Frequencydomain

Spatialdomain



2D

32

Lowpass filter Highpass filter Offset highpass filter

Bandreject filters

CSE 166, Spring 2020 33

ButterworthIdeal Gaussian


• Sharpening filter

CSE 166, Spring 2020 34

Overview: Image processing in the frequency domain

CSE 166, Spring 2020 35


f(x,y)


g(x,y)

Fouriertransform


F(u,v)

Inverse Fourier

transform


G(u,v)

Frequency domain processing

Jean-Baptiste Joseph Fourier1768-1830

Next Lecture

• Image restoration

• Reading

– Chapter 5: Image Restoration and Reconstruction

CSE 166, Spring 2020 36

Filtering in the Frequency Domaincseweb.ucsd.edu/classes/sp20/cse166-a/lec7.pdfFiltering in spatial...

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