May 21, 2014 - Block-based Comprssed Sensing (BCS). (L. Gan ... consecutive Projection Onto Convex Sets (POCS) (E. Candès & J. Romberg, 2005) and.
Block Compressed Sensing Images using Curvelet Transform Nasser Eslahi, Ali Aghagolzadeh, Seyed Mehdi Hosseini Andargoli Faculty of Electrical and Computer Engineering Babol University of Technology
May 2014
21/05/2014
The 22nd Iranian Conference on Electrical Engineering (ICEE 2014)
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Outline Basics of Compressed Sensing (CS)
Block Compressed Sensing (BCS) Curvelet Transform BCS using Accelerated Iterative Curvelet Thresholding (ICT) (Proposed) Numerical Experiments Conclusion 21/05/2014
N. Eslahi, A. Aghagolzadeh, S.M.H. Andargoli
BCS Images Using Curvelet Transform
ICEE2014
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Basics of Comprssed Sensing Compressed Sensing (CS)
Also called Compressive Sensing or Compressive Sampling
(E. Candès and T. Tao, 2006) (E. Candès, J. Romberg & T. Tao, 2006) (D. L. Donoho, 2006)
𝑓𝑚×1 = Φ𝑚×𝑛 𝑢𝑛×1
Underdetermined System of Measurement LatrixEquations (USLE) linear
Measurements
or
or
(𝜗 = Ψ𝑢)
Sparsifying transform domain basis (t = n) or frame (t > n) or even adaptive learned sparsifying basis
𝑢
21/05/2014
Sensing Matrix
Observations
Ψ ∈ ℝ𝑡×𝑛
(𝑢 = Ψ −1 𝜗)
N. Eslahi, A. Aghagolzadeh, S.M.H. Andargoli
(𝑚 ≪ 𝑛 )
(1)
or
𝜗 = argmin Sampling 𝜗 𝓁0 Matrix 𝑠. 𝑡. 𝑓 = ΦΨ−1 𝜗
(2)
𝜗
NP-hard 𝜗 = argmin 𝜗 𝜗
𝓁1
𝑠. 𝑡.
𝑓 = ΦΨ −1 𝜗
BCS Images Using Curvelet Transform
ICEE2014
(3)
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Basics of Comprssed Sensing In practice:
𝑓𝑚×1 = Φ𝑚×𝑛 𝑢𝑛×1 + n𝑚×1
(4)
basis pursuit denoising (BPDN) min 𝑓 − ΦΨ −1 𝜗 𝜗
2 𝓁2
+𝜆 𝜗
𝓁1
(5)
Iterative Shrinkage/Thresholding (IST) Algorithms (I. Daubechies et al., 2004)
(J. Bioucas-Dias & M. Figueiredo, 2007)
(A. Beck & M. Tebulle, 2009)
Bregman Iterative Algorithms (W. Yin, S. Osher, D. Goldfarm & J. Darbon, 2008) (T. Goldstein & S. Osher, 2009)
Projected Gradient Methods (M. A. T. Figueiredo, R. D. Nowak & S. J. Wright, 2007)
and etc. 21/05/2014
N. Eslahi, A. Aghagolzadeh, S.M.H. Andargoli
BCS Images Using Curvelet Transform
ICEE2014
4
Block-based Comprssed Sensing (BCS) Block Compressed Sensing (BCS) (L. Gan, 2007) Φ𝐵 Φ= ⋮ 0 𝐼𝑟
⋯ 0 ⋱ ⋮ ⋯ Φ𝐵
Φ𝐵 ∈ ℝ𝑚𝐵 ×𝑛 , 𝑚𝐵 =
𝑓𝑖 = Φ𝐵 𝑢𝑖 𝑚𝐵2 𝑛
, 𝑛 = 𝐼𝑟 𝐼𝑐
Block size 𝐵 × 𝐵 𝐼𝑐
Solve (5) using consecutive Projection Onto Convex Sets (POCS) (E. Candès & J. Romberg, 2005) and hard thresholding via a 2D Wiener filter. 21/05/2014
N. Eslahi, A. Aghagolzadeh, S.M.H. Andargoli
BCS Images Using Curvelet Transform
ICEE2014
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Block-based Comprssed Sensing (BCS) BCS using Directional Transform (S. Mun & J. E. Fowler, 2009)
Using Dual-tree Discrete Wavelet Transform (DDWT) [1] and [2] Contourlet Transform (CT) as the sparsifying transform matrix. [3] Adapting Bivariate Shrinkage to directional decomposition Solve (5) using Projected Landweber [4]iterations via a 2D Wiener filter. [1]
(N. G. Kingsbury, 2001)
[2]
(M. N. Do & M. Vetterli, 2005)
[3]
(L. Sendur & I. W. Selesnick, 2002)
[4]
(H. J. Trussell & M. R. Civanlar, 1985)
BCS-Smooth Projected Landweber
BCS-SPL 21/05/2014
N. Eslahi, A. Aghagolzadeh, S.M.H. Andargoli
BCS Images Using Curvelet Transform
ICEE2014
6
Curvelet Transform 1) a scale 2−𝑗 , 𝑗𝜖ℕ0 ; 2) an equispaced sequence of rotation angle 𝜃𝑗,𝑙 such that 0 ≤ 𝜃𝑗,𝑙 ≤ 2𝜋;
3) a position
(𝑗,𝑙) 𝑥𝑘 =𝑅𝜃−1 𝑗,𝑙
𝑘1 2−𝑗 , 𝑘2
𝑇
−𝑗2 2
𝑗,𝑙
Ψ𝑗,𝑙,𝑘 𝑥 ∶= Ψ𝑗 𝑅𝜃𝑗,𝑙 𝑥 − 𝑥𝑘
−𝑗2 =2𝜋𝑙. 2
(E. Candès & D. Donoho, 2004)
Discrete Curvelet Transform (DCuT)
2
−1
; 𝑘= 𝑘1 , 𝑘2 ϵℤ2 ,
𝑥 = 𝑥1 , 𝑥2 ϵℝ2
,
𝑢= Curvelet Transform
, 𝑙 = 0,1, … , 2
𝑗
𝑢, Ψ𝑗,𝑙,𝑘 Ψ𝑗,𝑙,𝑘
(6)
(7)
𝑗,𝑙,𝑘
𝜗𝑗,𝑙,𝑘 = 𝑢, Ψ𝑗,𝑙,𝑘
(8)
(E. Candès et al., 2006) 21/05/2014
N. Eslahi, A. Aghagolzadeh, S.M.H. Andargoli
BCS Images Using Curvelet Transform
ICEE2014
7
Iterative Curvelet Tresholding (ICT) 𝑓𝑚×1 = Φ𝑚×𝑛 𝑢𝑛×1 𝑢(𝑘+1) = 𝒮𝜏,𝒯 𝑢
Analysis formulation Synthesis 𝜗
𝑘
= Ψ𝑢
𝑘
Where:
𝑘
𝜗 (𝑘+1) = 𝒮𝜏,𝒯 𝜗
+ 𝛼 (𝑘) Φ𝑇 (𝑓 − Φ𝑢 𝑘
𝑘
)
+ 𝛼 (𝑘) ΨΦ𝑇 (𝑓 − ΦΨ −1 𝜗
𝒮𝜏,𝒯 𝑓 =
𝒯𝑇,𝜏 𝑓, Ψ𝑗,𝑙,𝑘
Ψ𝑗,𝑙,𝑘
(9)
𝑘
)
(11)
(10)
𝑗,𝑙,𝑘
𝒯𝑠,𝜏
𝑥 − 𝜏, 𝑥 = 0, 𝑥 + 𝜏,
𝑥≥𝜏 𝑥