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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

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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

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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