Tutorials 11,12,13 discrete signals and systems

עיבוד תמונות ואותות במחשבDFT: 10תרגול

Tutorials 11,12,13discrete signals and systems

Technion, CS department, SIPC 236327Winter 2012-2013

1


Discrete LSI system

Linear

Space invariant

2

𝐻1 𝐷 nx ny


Discrete LSI system

Linear

Space invariant

3

nmg , nmf , 𝐻2 𝐷


Example

For compression, a rule to predict the pixel value is used:

Is the system linear? Space invariant?

4


Discrete LSI system

System is defined with its impulse response

5

nhxknhkxnyk

*

𝛿 [𝑛 ]

𝑛


Cyclic convolution

• Infinite support

• DTFT

6

• Finite support

• DFT• Efficient implementation

Convolution

knhkxnhxk

* NknhkxnhxN

k

mod1

0

12

10

01211012

23011210

12

10

NxNx

xx

hhNhNhNhhNhNh

hhhhhhNhh

NyNy

yy


Exercise

Q: How can we use this system to calculate a linear convolution?A: Zero padding, and truncation of the result.

7

]0,0,0,3,2,1,0[]0,0,0,4,3,2,1[]3,2,1,0[*]4,3,2,1[

H nx nyx )( ny

Q: If both signals are of length N, how many zeros will we add?

A: N-1 zeros


Exercise

Q: How can we use this system to calculate a cyclic convolution?A: Duplicate one signal, and truncation of the result.

8

H nx nyx )*( ny

]3,2,1,0[*]3,2,1,4,3,2,1[]3,2,1,0[]4,3,2,1[ Q: If both signals are of length N, how much should we

duplicateA: N-1 cells


Discrete Fourier Transform (DFT, FFT)

Infinite support infinite supportContiniuous continuous

Finite support Finite supportDiscrete Discrete

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)(txFourier)(tx

][nx ][nxDFT


DFT

מתבצעות בדרך הרגילה DFT-1 וDFTהתמרות ה•

המקדמים מחזוריים:•[ בד"כ מסתכלים על N-1,0לכן במקום להתייחס לתחום ]•

[.N/2,N/2-1התחום ]-

10

1

0

/2

1

0

/2/2

1

0

*

1][

11,][

,

N

k

Nkni

N

n

NkniNkni

N

n

ekXN

nx

enxN

eN

nxkX

ngnfngnf

...2,1,0, mmNkXkX


DFTהפעלת

11

0 50 100-1

-0.5

0

0.5

1

t

x]t[

-50 0 500

5

10

15

20

k|X

]k[|


DFTדוגמאות

12

10,2cos

10,

00

00

NknNkDFT

NnnnDFT


Summary – Fourier Transforms

Fourier transform– Time domain – non-periodic infinite signals– Continuous time (t)– Continuous frequency (f)– Formulas

13/39

TransformFourier )( )(

TransformFourier Inverse )()(

2

2

dtetxfX

dkefXtx

fti

fti



DTFT: Discrete Time Fourier Transform– Time domain – non-periodic infinite signals– Discrete time (n)– Continuous frequency (f)– Formulas

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ansformFourier tr DT ][ )(

ansformFourier tr DT inverse )(21][

2

-n

2

fni

fti

etxfX

dfekXnx

מד נל

לא

קורסב



Fourier series– Time domain – periodic infinite signals– Continuous time (t)– Discrete frequency (f)– Formulas

15/39

fixed is , 1 Lperiod has

)(T1 ),(][

][)(

)2()2(

)2(

ff

x(t)

dtetxetxkX

ekXtx

T

ktfiktfi

k

ktfi



DFT or Discrete Time Fourier Series– Time domain – periodic infinite signals– Discrete time (n)– Discrete frequency (f)– Formulas

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N period a have X[k] and x[n]

1][

11,][

1

0

/2

1

0

/2/2

N

k

Nkni

N

n

NkniNkxi

ekXN

nx

enxN

eN

nxkX


DFT ומערכת LSI

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nhxny nx nh kX kH

kHkXkY


Exercise

We have an N-length filter with impulse response h]n[.We create a new filter as follows:

Express F]k[ with H]k[, where H]k[=DFT{h]n[},F]k[=DFT{f]n[}

Instructions: calculate

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][][)1(][ nNhnhnf n

]}[{]}[)1{(

nNhDFTnhDFT n


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Example – discrete frequency filtration

Noisy image of size 256X256Im_out]m,n[=Im_in]m,n[+noise]m,n[

•Harmonic noise:

•f = 1/(8 pixels)•Amplitude A and phase φ are random and

independent for each line.

mm fnAnmnoise 2cos],[


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Example – added noise in line 100

radA

325.137.22

100

100


21/39



22/39

Example – discrete frequency filtration - smoothing


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Example – discrete frequency filtration – smoothing vs median (8 pixels)

No noise but image is blurred


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DFT of the noise in line i

elsekA

niNoiseDFT

N

nN

AfnAfnAniNoise

i

iiii

032

),(

256

322cos2cos2cos),(


25/39


Design an LSI filter–Such filter multiplies each frequency with a complex

number–Can handle each frequency separately

In this example, we want to handle frequencies 32 and -32.

–Notch filter – attenuates specific frequency


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Original signal in frequency domain

Filtered signal in frequency domain


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Noise removed completely

Original image not fully restored

–We cannot restore the attenuated frequencies


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Smoothing filter of 8 pixels

Notch filter


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Example –frequency filtration - implementation

Filter in freq. domain:Filter=ones(1,256);Filter(32+1)=0;Filter(224+1)=0;

Filtration:For k=1:size(I,1),

Y=fft(I(k,:)).*Filter;I(k,:)=ifft(Y);

end

Notch filter in freq. domain


Tutorials 11,12,13discrete signals and systemsPart II: 2D

Technion, CS department, SIPC 236327Winter 2012-2013

30


2D - definitions

2D convolution:

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mnhxmny ,*, mnx , mnh ,

k l

lnkmhlkxnmhx ,,,*


Cyclic 2D-convolution:

2D DFT:

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nmhxnmy ,, nmx , nmh ,

lkX , lkH ,

lkHlkXlkY ,,,

1

0

1

0modmod ,,,

M

k

N

lNM lnkmhlkxnmhx

1

0

1

0

22

, ,1,M

m

N

n

Nnli

Mmki

lk eenmxMN

nmxDFT

2D - definitions


DFT is linear, we have an operation matrix:

2D-DFT can be implemented as:

If the input is separable:

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nXDnXDFT

TDnmDXnmXDFT ,,

lk nXDFTmXDFTnmXDFT

nXmXnmX

21

21

,,

2D - notes


Example

Noisy image 512X512

The noise:Add 100 gray levels for all 16i lines

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Example

35

Noisy image Average filteroriginal + noise mean 4X4


Example

36

Noisy image Average filteroriginal + noise mean 16X16


Example

How does the noise look like in the frequency domain?

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else

kkmknmnr

0,16161

,


Example

38

After freq. filtration• Filter implementation in the freq.

domain:

H=ones(512,512);for n=1:32:512

H(n,1) = H(1,n) = 0;endH(1,1) = 1;

• Image filtration:out = ifft( fft(img).*H );

עיבוד תמונות ואותות במחשבDFT: 10תרגול 39

לפני סינון תדרDFT of image + noise


לפני סינון תדר (הגדלה של מרכז)


אחרי סינון תדר (הגדלה של מרכז)

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Tutorials 11,12,13 discrete signals and systems