Block Compressed Sensing Images using Curvelet ...

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)

1

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

2

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)


(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


ICEE2014

(3)

3

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



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



ICEE2014

5

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



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



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