Digital Image Procesing

Unitary Transforms

Discrete Fourier Trasform (DFT) in Image Processing

DR TANIA STATHAKI READER (ASSOCIATE PROFFESOR) IN SIGNAL PROCESSING IMPERIAL COLLEGE LONDON

}10 ),({ Nxxf

1

0

)(),()(N

x

xfxuTug

1-D Signal Transforms

10 Nu

fTg

Scalar form

Matrix form

}10 ),({ Nxxf

1

0

)(),()(N

x

xfxuTug

1-D Signal Transforms-Remember the 1-D DFT

General form

DFT

10 Nu

1

0

2

)(1

)(N

x

N

xuj

xfeN

ug

1-D Inverse Signal Transforms-General Form

Scalar form

Matrix form

1

0

)(),()(N

u

uguxIxf

gTf 1

}10 ),({ Nxxf

1-D Inverse Signal Transforms-Remember the 1D DFT

General form

Inverse DFT

1

0

2

)()(N

x

N

xuj

ugexf

10 Nu

1

0

)(),()(N

u

uguxIxf

1-D Unitary Transforms

fTg

Matrix form

T

TT

1

Signal Reconstruction

1

0

)(),()(N

u

T

ugxuTxfgTf

T

TT

1

Why do we use Image Transforms?

Often, image processing tasks are best performed in a

domain other than the spatial domain.

Key steps:

• Transform the image

• Carry the task(s) of interest in the transformed domain.

• Apply inverse transform to return to the spatial domain.

2-D (Image) Transforms-General Form

1

0

1

0

),(),,,(),(N

x

N

y

yxfyxvuTvug

1

0

1

0

),(),,,(),(N

u

N

v

vugvuyxIyxf

2-D (Image) Transforms-Special Cases

Separable

Symmetric

),(),(),,,( 21 yvTxuTyxvuT

),(),(),,,( 11 yvTxuTyxvuT

Separable and Symmetric

Separable, Symmetric and Unitary

TTfTg 11

11 TgTf

T

2-D (Image) Transforms-Special Cases (cont.)

Energy Preservation

1-D

2-D

1

0

1

0

21

0

1

0

2),(),(

M

u

N

v

M

x

N

y

vugyxf

22fg

Energy Compaction

• Most of the energy of the original data is

concentrated in only a few transform

coefficients, which are placed close to the

origin; remaining coefficients have small

values.

• This property facilitate compression of the

original image.

Let’s talk about DFT in images: Why is it useful?

• It is easier for removing undesirable

frequencies.

• It is faster to perform certain operations in

the frequency domain than in the spatial

domain.

• The transform is independent of the signal.

Example: Removing undesirable frequencies

Result after removing high frequencies

frequencies noisy signal Example of

removing a high

frequency using

the transform

domain

How do frequencies show up in an image?

• Low frequencies correspond to slowly

varying information (e.g., continuous

surface).

• High frequencies correspond to quickly

varying information (e.g., edges)

Original Image Low-passed version

2-D Discrete Fourier Transform

1

0

1

0

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

x

N

y

NvyMuxjeyxfvuF

1

0

1

0

)//(2),(1

),(M

u

N

v

NvyMuxjevuFMN

yxf

• It is separable, symmetric and unitary

• It results in a sequence of two 1-D DFT

operations (prove this)

Visualizing DFT

• The dynamic range of F(u,v) is typically very large

• Small values are not distinguishable

• We apply a logarithmic transformation to enhance

small values.

original image before transformation after transformation

constant a is ],),(1log[),( cvuFcvuD

Amplitude and Log of the Amplitude

Amplitude and Log of the Amplitude

Original Image and Log of the Amplitude

1

0

/2

1

0

/2

1

0

1

0

/2/2

),(),(

),(),(

),(),(

M

x

Muxj

N

y

Nvyj

M

x

N

y

NvyjMuxj

vxGevuF

eyxfvxG

eyxfevuF

DFT properties: Separability

).( of rows of DFT

),(1

),(),(

1

0

/2

1

0

/2

yxfN

eyxfN

N

eyxfvxG

N

y

Nvyj

N

y

Nvyj

),( of columns of DFT

),(1

),(1

0

/2

vxGM

evxGM

MvuFM

x

Muxj

DFT properties: Separability

DFT properties: Separability

The DFT and its inverse are periodic.

),(),(),(),( NvMuFNvuFvMuFvuF

DFT properties: Separability

odd andimaginary ),( odd and real ),(

even and real ),( even and real ),(

),(),(

integersany , with ),,(),( *

vuFyxf

vuFyxf

vuFvuF

qpqNvpMuFvuF

DFT properties: Conjugate Symmetry

Translation in spatial domain:

Translation in frequency domain:

),(),( 00)//(2 00 vvuuFeyxf

NyvMxuj

)//(200

00),(),(NyvMxuj

evuFyyxxf

DFT properties: Translation

Warning: to show a full period, we need to translate

the origin of the transform at )2/,2/(),( NMvu

)(uF

)2

(N

uF

DFT properties: Translation

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

),(),(

Using

)1(

caseIn that

2/ and 2/ replace

)2/,2/(at ),( move To

)(

00)//(2

)()()//(2

00

00

00

nvMuFyxf

vvuuFeyxf

ee

NvMu

NMvuF

yx

NyvMxuj

yxyxjNyvMxuj

DFT properties: Translation

without translation after translation

)2/,2/()1)(,( )( nvMuFyxf yx

DFT properties: Translation

by rotates by rotating ),,(),( vuFyxf

DFT properties: Rotation

transform Fourier the is ][ where

)],([)],([)],(),([

)],([)],([)],(),([

yxgyxfyxgyxf

yxgyxfyxgyxf

DFT properties: Addition and Multiplication

)0,0(1

),(

),()0,0(

),(1

),(

1

0

1

0

1

0

1

0

FMN

yxf

yxfF

yxfMN

yxf

M

x

N

y

M

x

N

y

:image the of value Average

DFT properties: Average value of the signal

Original Image Fourier Amplitude Fourier Phase

Magnitude and Phase of DFT

What is more important?

Hint: use inverse DFT to reconstruct the image using magnitude or phase only information

magnitude phase

Reconstructed image using

magnitude only

(i.e., magnitude determines the

contribution of each component!)

Reconstructed image using

phase only

(i.e., phase determines

which components are present!)

Magnitude and Phase of DFT

Magnitude and Phase of DFT

Low pass filtering using DFT

High pass filtering using DFT

Experiment: Verify the importance of phase in images

Reconstruction from

phase of one image and amplitude of the other

phase of cameraman phase of grasshopper

amplitude of grasshopper amplitude of cameraman

Experiment: Verify the importance of phase in images

Reconstruction from

phase of one image and amplitude of the other

phase of buffalo phase of rocks

amplitude of rocks amplitude of buffalo

DFT of a single edge

• Consider DFT of image with single edge.

• For display, DC component is shifted to the centre.

• Log of magnitude of Fourier Transform is displayed

DFT of a box

DFT of rotated box

DFT computation: Extended image

• DFT computation assumes image repeated

horizontally and vertically.

• Large intensity transition at edges = vertical and

horizontal line in middle of spectrum after shifting.

Windowing

• Can multiply image by windowing function before

DFT to reduce sharp transitions between borders of

repeated images.

• Ideally, causes image to drop off towards ends,

reducing transitions