digital image processing-DFT

8/6/2019 digital image processing-DFT

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Part I: Image Transforms

DIGITAL IMAGE PROCESSING


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}10),({ ee Nxxf

!

!

1

0

)(),()(N

x

xfuxTug

1-D SIGNAL TRANSFORM

GENERAL FORM

10 ee Nu

fTg !

Scalar form

Matrix form


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}10),({ ee Nxxf

!

!

1

0

)(),()(N

x

xfuxTug

1-D SIGNAL TRANSFORM cont.

REMEMBER THE 1-D DFT!!!

General form

10 ee Nu

!

!

1

0

2

)(1

)(N

x

N

xuj

xfe

N

ugT

DFT


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1-D INVERSE SIGNAL TRANSFORM

GENERAL FORM

Scalar form

!

!

1

0

)(),()(N

u

uguxIxf

Matrix form

gTf !1


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}10),({ ee Nxxf

1-D INVERSE SIGNAL TRANSFORM cont.

REMEMBER THE 1-D DFT!!!

General form

!

!

1

0

2

)()(N

x

N

xuj

ugexfT

10 ee Nu

DFT

!

!

1

0

)(),()(N

u

uguxIxf


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1-D UNITARY TRANSFORM

fg !

Matrix form

T

TT !1


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SIGNAL RECONSTRUCTION

!

!!

1

0

)(),()(N

u

uguxxfgf

!

1


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2-D IMAGE TRANSFORM

GENERAL FORM

!

!

!1

0

1

0 ),(),,,(),(

N

x

N

y yxfvuyxvug

!

!

!

1

0

1

0 ),(),,,(),(

N

u

N

v vugvuyxIyxf


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2-D IMAGE TRANSFORM

SPECIFIC FORMS

Separable

Symmetric

),(),(),,,( 21 vyTuxTvuyxT !

),(),(),,,( 21 vyTuxTvuyxT !


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Separable and Symmetric

TTfTg 11 !

!! 1111 )( TgTTgTf

TTTT

Separable and Symmetric and Unitary


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Energy Preservation

1-D

2-D

!

!

!

!!

1

0

1

0

21

0

1

0

2

),(),(

N

u

N

v

N

x

N

y vugyxf

22fg !


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Energy Compaction !


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2-D DISCRETE FOURIER TRANSFORM

!

!

!

1

0

1

0

)//(2

),(1

),(

M

x

N

y

NvyMuxj

eyxfMNvuFT

!

!

!1

0

1

0

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

u

N

v

NvyMuxjevuyxf T


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An Atom

Both functions have circular symmetry.

The atom is a sharp feature, whereas its

transform is a broad smooth function. This

illustrates the reciprocal relationship between afunction and its Fourier transform.


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Original Image-FourierAmplitude-Fourier Phase


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A Molecule


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The Fourier Duck


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The Fourier Cat


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Reconstruction from

Phase of Cat and Amplitude of Duck


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Reconstruction from

Phase of Duck and Amplitude of Cat


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Original Image-FourierAmplitude


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Keep Part of the Amplitude Around the Origin and Reconstruct Original

Image (LOW PASS filtering)


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Keep Part of the Amplitude Far from the Origin and Reconstruct

Original Image (HIGH PASS filtering)


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Reconstruction from

phase of one image and amplitude of the other


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Reconstruction from

phase of one image and amplitude of the other


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Amplitude and Log of the Amplitude


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Amplitude and Log of the Amplitude


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Original and Amplitude

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digital image processing-DFT