i. 2d dft and applications

Digital Image Processing, 3rd ed. & 2nd ed.

www.ImageProcessingPlace.com

© 1992–2008 R. C. Gonzalez & R. E. Woods

Gonzalez & Woods

DIP Week 3 –part I (largely based on Chapter 4)

I. 2D DFT and Applications

I.1 2D DFT Definitions

I.2 Properties

I.3 Filtering in Frequency Domain

Appendixes (Fourier series, Fourier

transform, FT of sampled signal and

discrete Fourier transform)




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1 12 ( / / )

0 0

[ , ] [ , ]M N

j ux M vy N

x y

F u v f x y e

1 12 ( / / )

0 0

1[ , ] [ , ]

M Nj ux M vy N

x y

f x y F u v eMN

I.1 2D DFT Definition:

Forward & Inverse DFT




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% create a 512x512 image of black with centre 20x40 white

x=zeros(512,512);

for i=1:20

for j=1:40

x(256-10+i,256-20+j)=255;

end

end

% display the image

figure(1); imshow(x);

% simple 2D DFT

figure(2); y=fft2(x);

% display the DFT magnitude image

imshow(abs(y)/max(max(abs(y))));

% display shifted DFT magnitude image

figure(3); y=fftshift(fft2(x));

imshow(abs(y)/max(max(abs(y))));

% display log magnitude (shifted) DFT magnitude iamge

figure(4); imshow(log(abs(y)),[]); colormap(gray);

Recreate this 2D DFT example in Matlab:




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I.2 2D DFT Properties




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2-Dimensional DFT

& Inverse




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Result of a notch filter that set to 0

the F(0,0) term in the Fourier

transform




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Result of lowpass

filter (a) and

highpass filters (b)

and (c)





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Lowpass filter:

Frequency

domain response

and

Spatial domain

masks

Laplacian of

Gaussian (LoG)

edge masks

2f(x,y)

Highpass filter:

Frequency

domain response

and

Spatial domain

masks




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Laplacian filter:

G(x,y)=f(x,y)-2f(x,y)




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Lowpass Filters

2 2 1/ 2( , ) [( / 2) ( / 2) ]D u v u M v N

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




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Highpass Filters




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Homomorphic Filtering:

f(x,y)=i(x,y)r(x,y), product of illumination and reflectance




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Homomorphic Filtering:

Relations between Fourier Series, Fourier Transform and Discrete Fourier Transform Their relationships can be summarised in the following table, together with examples with rectangular waves in both continuous- and discrete –time, periodic and aperiodic forms.

Time Domain Frequency Domain

Fourier Series

∫+

Ω−=Tt

t

tjkk dtetx

TC

0

0

)(1

tjk

kk eCtx Ω

∞

−∞=∑=)(

Continuous x(t)

tTτ

A

Tktj

k

eT

kcTAtx /2)(sin)( ππττ∑

∞

−∞=

=

Discrete

-5 0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25

)/(sin TkcTACk πττ

=

Fourier Transform

∫∞

∞−

−≡ dtetxjX tjωω )()(

∫∞

∞−

= ωωπ

ω dejXtx tj)(21)(

Continuous x(t)

tτ

A

Continuous

-40 -30 -20 -10 0 10 20 30 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ω

|X(jw

)|

6π/τ −2π/τ −6π/τ −4π/τ 4π/τ 2π/τ )2/(sin)( ωττω cAjX =

Fourier Transform

kj

k

j ekxeX ωω −∞

−∞=∑= ][)(

ωπ

ωπ

π

ω deeXnx njj∫−

= )(21][

Discrete

n

1

-3 -2 -1 0 1 2 3 4 5 6 7 8 9

⎩⎨⎧ −≤≤

=otherwise

Mnnx

,010 ,1

][

Continuous

-8 -6 -4 -2 0 2 4 6 80

1

2

3

4

5

6

7

|X (e )|jw

2π −2π π −π 2π/5 −2π/5

2/)1(

)2/sin()2/sin()( ωω

ωω −−= Mjj eMeX

Discrete Fourier Transform

NknjN

n

enxkX /21

0

][][ π−−

=∑=

∑−

=

=1

0

/2][1][N

k

NknjekXN

nx π

Discrete

n

1

-3 -2 -1 0 1 2 3 4 5 6 7 8 9

⎩⎨⎧

−≤≤−≤≤

=1 ,0

10 ,1][

NnMMn

nx

Discrete

0 1 2 3 4 5 6 7 8 90

1

2

3

4

5

6

7

k

|X(k

)|

NMkje

NkNkMkX /)1(

)/sin()/sin(][ −−= π

ππ

i. 2d dft and applications

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