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Tutorials 12,13 discrete signals and systems. Technion, CS department, SIPC 236327 Spring 2014. Discrete LSI system. Linear Space invariant. Discrete LSI system. Linear Space invariant. Example. - PowerPoint PPT Presentation
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Technion, CS department, SIPC 236327Spring 2014
Tutorials 12,13discrete signals and systems
1/39
• Linear
• Space invariant
Discrete LSI system
2/39
𝐻1 𝐷 nx ny
• Linear
• Space invariant
Discrete LSI system
3/39
nmg , nmf , 𝐻2 𝐷
• For compression, a rule to predict the pixel value is used:
Is the system linear? Space invariant?
Example
4/39
• System is defined with its impulse response
Discrete LSI system
5/39
nhxknhkxnyk
*
𝛿 [𝑛 ]
𝑛
• Infinite support
• DTFT
Cyclic convolution
6/39
• Finite support
• DFT• Efficient implementation
Convolution
knhkxnhxk
* NknhkxnhxN
k
mod1
0
12
10
01211012
23011210
12
10
NxNx
xx
hhNhNhNhhNhNh
hhhhhhNhh
NyNy
yy
Q: How can we use this system to calculate a linear convolution?A: Zero padding, and truncation of the result.
Exercise
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]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
Q: How can we use this system to calculate a cyclic convolution?A: Duplicate one signal, and truncation of the result.
Exercise
8/39
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
Infinite support Infinite supportContiniuous Continuous
Finite support Finite supportDiscrete Discrete
Discrete Fourier Transform (DFT, FFT)
9/39
)(txFourier)(tx
][nx ][nxDFT
מתבצעות בדרך הרגילה DFT-1 וDFTהתמרות ה•
המקדמים מחזוריים:•[ בד"כ N-1,0לכן במקום להתייחס לתחום ]•
[.N/2,N/2-1מסתכלים על התחום ]-
DFT
10/39
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
11/39
0 50 100-1
-0.5
0
0.5
1
t
x[t]
-50 0 500
5
10
15
20
k|X
[k]|
DFTדוגמאות
12/39
10,2cos
10,
00
00
NknNkDFT
NnnnDFT
• Fourier transform– Time domain – non-periodic infinite signals– Continuous time (t)– Continuous frequency (f)– Formulas
Summary – Fourier Transforms
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
Summary – Fourier Transforms
14/39
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
Summary – Fourier Transforms
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
Summary – Fourier Transforms
16/39
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
17/39
nhxny nx nh kX kH
kHkXkY
• 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
Exercise
18/39
][][)1(][ nNhnhnf n
]}[{]}[)1{(
nNhDFTnhDFT n
• 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.
Example – discrete frequency filtration
19/39
mm fnAnmnoise 2cos],[
Example – added noise in line 100
20/39
radA
325.137.22
100
100
Example – discrete frequency filtration
21/39
Example – discrete frequency filtration - smoothing
22/39
Example – discrete frequency filtration – smoothing vs median (8 pixels)
23/39
No noise but image is blurred
• DFT of the noise in line i
Example – discrete frequency filtration
24/39
elsekeA
elsekeAniNoiseDFT
N
nN
AfnAniNoise
ii
i iiN
ikN
i
iiii
032
032),(
256
322cos2cos),(
3222
• 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
Example – discrete frequency filtration
25/39
Example – discrete frequency filtration
26/39
Original signal in frequency domain
Filtered signal in frequency domain
• Noise removed completely
• Original image not fully restored– We cannot restore the
attenuated frequencies
Example – discrete frequency filtration
27/39
Example – discrete frequency filtration
28/39
Smoothing filter of 8 pixels
Notch filter
• 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
Example –frequency filtration - implementation
29/39
Notch filter in freq. domain
Technion, CS department, SIPC 236327Spring 2014
Tutorials 12,13discrete signals and systemsPart II: 2D
30/39
2D convolution:
2D - definitions
31/39
mnhxmny ,*, mnx , mnh ,
k l
lnkmhlkxnmhx ,,,*
• Cyclic 2D-convolution:
• 2D DFT:
32/39
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:
33/39
nXDnXDFT
TDnmDXnmXDFT ,,
lk nXDFTmXDFTnmXDFT
nXmXnmX
21
21
,,
2D - notes
• Noisy image 512X512
• The noise:Add 100 gray levels for all 16i lines
Example
34/39
mean 4X4
Example
35/39
Noisy image Average filteroriginal + noise
mean 16X16
Example
36/39
Noisy image Average filteroriginal + noise
• How does the noise look like in the frequency domain?
Example
37/39
else
kkmknmnr
0,16161
,
• 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 );
Example
38/39
After freq. filtration
לפני סינון תדרDFT of image + noise
39/39
לפני סינון תדר (הגדלה של מרכז)
40/39
אחרי סינון תדר (הגדלה של מרכז)
41/39
Image filtration
42/39
000011000
h
000110000
h
000101000
21h
• Roberts
• Prewitt
• Sobel
Edge detection of Image A
43/39
AGAG yx *0110
*10
01
AGAG yx *111
000111
*101101101
AGAG yx *121
000121
*101202101
22 ),(),(),( nmGnmGnmG yx
Edge detection of Image A
44/44
Original Roberts
SobelPrewitt
Unsharp masking – edge enhancement
45/44
LInkInOut
L
*
010141
010