现代数字信号处理 英文版课件

90
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1D signal
2D signal=image
3D signal
A. Mohammad-Djafari, Advanced Signal and Image Processing
Huazhong & Wuhan Universities, September 2012,
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Linear Transformations
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A. Mohammad-Djafari, Advanced Signal and Image Processing
Representation of signals
Biblioteka Baidu
g(t) 2.5
2
1.5
1
0.5
Amplitude
0
−0.5
−1
−1.5
−2
−2.5
0
10
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50 time
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D
Huazhong & Wuhan Universities, September 2012, 6/59
g (r , φ) =
A. Mohammad-Djafari, Advanced Signal and Image Processing
Linear and Invariant systems: convolution
h(r , r ′ ) = h(r − r ′ ) f (r ) −→ h(r ) −→ g (r ) = h(r ) ∗ f (r )
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1–D : g (t ) =
D
f (t ′ ) h(t − t ′ ) dt ′ f (x ′ ) h(x − x ′ ) dx ′
g (x ) =
D ◮
2–D : g (x , y ) =
Huazhong & Wuhan Universities, September 2012,
1D Fourier Transform F1
g (ω ) = f (t ) = f (t ) exp {−j ω t } dt
1 2π
g (ω ) exp {+j ω t } dω
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2D Fourier: F2 g (ωx , ωy ) = f (x , y )
f (t )
=
1 2π
g (ω ) exp {+j ω t } dω
f (x , y ) exp {−j (ωx x + ωy y } dx dy g (ωx , ωy ) exp {+j (ωx x + ωy y } dωx dωy
D
f (x , y ) h(x − x ′ , y − y ′ ) dx ′ dy ′
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h(t ) impulse response h(x , y ) Point Spread Function
Huazhong & Wuhan Universities, September 2012, 7/59
Wuhan University, September 2012
September 2012
A. Mohammad-Djafari, Advanced Signal and Image Processing Huazhong & Wuhan Universities, September 2012, 1/59
. Advanced Signal and Image processing
Ali Mohammad-Djafari
Groupe Probl` emes Inverses Laboratoire des signaux et syst` emes (L2S) UMR 8506 CNRS - SUPELEC - UNIV PARIS SUD 11 Sup´ elec, Plateau de Moulon, 91192 Gif-sur-Yvette, FRANCE. djafari@lss.supelec.fr http://djafari.free.fr http://www.lss.supelec.fr
A. Mohammad-Djafari, Advanced Signal and Image Processing
Huazhong & Wuhan Universities, September 2012,
3/59
Representation of signals and images
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Signal: f (t ), f (x ), f (ν )
|g (ω )|2 is called the spectrum of the signal f (t ) For real valued signals f (t ), |g (ω )| is symetric f (t ) exp {−j ω0 t } sin(ω0 t ) cos(ω0 t ) exp −t 2 1 (t − m)2 /σ 2 exp − 2 exp {−t /τ } , t > 0 1 if |t | < T /2 g (ω ) δ(ω − ω0 ) ? ? ? ? ? ?
g (s) =
D
f (r ) h(r , s) dr
f (r ) −→ h(r , s) −→ g (s)
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1–D : g (t ) =
D
f (t ′ ) h(t , t ′ ) dt ′ f (x ′ ) h(x , x ′ ) dx ′
D
g (x ) =
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2–D : g (x , y ) =
D
f (x ′ , y ′ ) h(x , y ; x ′ , y ′ ) dx ′ dy ′ f (x , y ) h(x , y ; r φ) dx dy
Content
1. Introduction: Signals and Images, Linear transformations (Convolution, Fourier, Laplace, Hilbert, Radon, ..., Discrete convolution, Z transform, DFT, FFT, ...) 2. Modeling: parametric and non-parametric, MA, AR and ARMA models 3. Parameter Estimation: Deterministic (LS, WLS) and Probabilistic methods (ML and Bayesian) 4. Bayesian estimation 5. Kalman Filtering and smoothing 6. Case study: Signal deconvolution 7. Case study: Image restoration 8. Case study: Image reconstruction and Computed Tomography
A. Mohammad-Djafari, Advanced Signal and Image Processing
Huazhong & Wuhan Universities, September 2012,
2/59
1. Introduction
1. Representation of signals and images 2. Linear Transformations 3. Convolution 4. Fourier Transform (FT) 5. Laplace Transform (LT) 6. Hilbert, Melin, Abel, ... 7. Radon Transform (RT) 8. Link between Different Linear Transforms 9. Discrete signals and transformations 10. Discrete convolution, Z Transform, DFT, FFT
f (x , y ) exp {−j (ωx x + ωy y )} dx dy
h(x , y , ωx , ωy ) = h1 (ωx x ) h2 (ωy y ) exp {−j (ωx x + ωy y )} = exp {−j (ωx x )} exp {−j (ωy y )}
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nD Fourier Transform g (ω ) = f (x) exp −j ω ′ x) dx
A. Mohammad-Djafari, Advanced Signal and Image Processing
Linear Transformations: Separable systems
g (s) =
D
f (r ) h(r , s) dr hj (rj , sj )
j
h(r , s) = Examples: ◮ 2D Fourier Transform g (ωx , ωy ) =
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A. Mohammad-Djafari, Advanced Signal and Image Processing
Huazhong & Wuhan Universities, September 2012,
Fourier Transform
[Joseph Fourier, French Mathematicien (1768-1830)] ◮ 1D Fourier: F1 g (ω ) = f (t ) exp {−j ω t } dt
2 = ( 21 π)
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nD Fourier: Fn g (ω ) = f (x)
f (x) exp −j ω ′ x
dx dω
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n = ( 21 π)
g (ω ) exp +j ω ′ x
A. Mohammad-Djafari, Advanced Signal and Image Processing
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f (t ) Variation of temperature in a given position as a function of time t f (x ) Variation of temperature as a function of the position x on a line f (ν ) Variation of temperature as a function of the frequency ν f (x , y ) Distribution of temperature as a function of the position (x , y ) f (x , t ) Variation of temperature as a function of x and t ... f (x , y , z ) Distribution of temperature as a function of the position (x , y , z ) f (x , y , z , t ) Variation of temperature as a function of (x , y , z ) and t ...
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Examples:
A. Mohammad-Djafari, Advanced Signal and Image Processing
2D Fourier Transform: F2
g (ωx , ωy ) = f (x , y )
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f (x , y ) exp {−j (ωx x + ωy y } dx dy g (ωx , ωy ) exp {+j (ωx x + ωy y } dωx dωy
2 = ( 21 π)
|g (ωx , ωy )|2 is called the spectrum of the image f (x , y ) ◮ For real valued image f (x , y ), |g (ωx , ωy )| is symetric with respect of the two axis ωx and ωy . Examples: f (x , y ) exp {−j (ωx 0 x + ωy 0 y )} exp −(x 2 + y 2 ) 1 2 + (y − m )2 /σ 2 ] exp − 2 [(x − mx )2 /σx y y exp {−(|x | + |y |)} 1 if |x | < Tx /2 & |y | < Ty /2 1 if (x 2 + y 2 ) < a
Huazhong & Wuhan Universities, September 2012, 4/59
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Image: f (x , y ), f (x , t ), f (ν, t ), f (ν1 , ν2 )
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3D, 3D+t, 3D+ν , ... signals: f (x , y , z ), f (x , y , t ), f (x , y , z , t )