Compressive Imaging by Generalized Total Variation Minimization Jie - - PowerPoint PPT Presentation

Compressive Imaging by Generalized Total Variation Minimization

Jie Yan and Wu-Sheng Lu

Department of Electrical and Computer Engineering University of Victoria, Victoria, BC, Canada

Aug 10, 2012

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OUTLINE

1

Introduction

2

Generalized TV and Weighted TV

3

A Power-Iterative Reweighting Strategy

4

WTV-Regularized Minimization: The Splitting Technique

5

Performance on Image Reconstruction

6

Conclusions

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Introduction

Signal recovery from compressive sensing (CS) measurements conventionally achieved by minimizing the ℓ1 norm. Several authors recently reported nonconvex ℓp regularization improves recovery performance, using a p less than 1. Iteratively reweighted ℓ1 minimization (IRL1) proves successful in tackling the ℓp problem without having to perform nonconvex

• ptimization.

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Introduction

A conventional total variation (TV) minimization for image recovery min

U

TV(U) s.t. R ◦ (FU) − B2

F < σ2

(1) where ◦ is the Hadamard product operator, F denotes the 2-D Fourier transform operator, R represents a random sampling matrix whose entries are either 1 or 0, and B stores the compressive sampled measurements. Inspired by the relationship between ℓp and ℓ1 norms, we generalize the concept of TV to a pth-power type TV with 0 ≤ p ≤ 1. Inspired by the IRL1, we propose an iteratively reweighted TV minimization algorithm to approach the solution that minimizes the pth-power TV.

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OUTLINE

1

Introduction

2

Generalized TV and Weighted TV

3

A Power-Iterative Reweighting Strategy

4

WTV-Regularized Minimization: The Splitting Technique

5

Performance on Image Reconstruction

6

Conclusions

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Anisotropic TV

Anisotropic total variation (TV) of a digital image U ∈ Rn×n is defined as TV(U) =

n−1

• i=1

n−1

• j=1

(|Ui,j − Ui+1,j| + |Ui,j − Ui,j+1|) +

n−1

• i=1

|Ui,n − Ui+1,n| +

n−1

• j=1

|Un,j − Un,j+1| (2) Under the periodic boundary condition, TV can be expressed in the form TV(U) = DU1 + UDT1 (3) with D ∈ Rn×n as a circulant matrix with the first row [1 − 1 0 · · · 0]. The notation X1 denotes the sum of magnitudes

• f all the entries in X, i.e., |xi,j|.

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Generalized pth power TV

We extend the concept of TV by defining a generalized pth power TV (GTV) as TVp(U) = DUp + UDTp The newly introduced notation Xp resembles an ℓp norm as it expresses the sum of pth power magnitudes of all the entries in X, i.e., |xi,j|p. We are inspired to investigate the GTV, i.e., TVp(U), with 0 ≤ p ≤ 1, for CS image recovery.

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Weighted TV

However attractive TVp appears in promoting a sparser TV,

• ptimization of TVp-related problem is nonconvex, nonsmooth,

and so far no algorithms have been discovered in handling such a problem to the authors’ knowledge. Instead, we introduce a weighted TV (WTV) as TVw(U) = Wx ◦ (DU)1 + Wy ◦ (UDT)1 TVw(U) becomes the conventional TV(U) when all entries in Wx and Wy are ones.

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OUTLINE

1

Introduction

2

Generalized TV and Weighted TV

3

A Power-Iterative Reweighting Strategy

4

WTV-Regularized Minimization: The Splitting Technique

5

Performance on Image Reconstruction

6

Conclusions

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A Power-Iterative Reweighting Strategy

We adopt a power-iterative strategy to approach the solution for a TV0-regularized problem.

1

Set p = 1, l = 1, Wx = Wy = 1.

2

Solve the WTV-regularized problem for U(l) min

U

TVw(U) s.t. R ◦ (FU) − B2

F < σ2

(4)

3

Terminate if p = 0; otherwise, set p = p − 0.1 and update the weights Wx and Wy as Wx = |DU(l) + ǫ|.p−1, Wy = |U(l)DT + ǫ|.p−1 (5) Then set l = l + 1 and repeat from step 2.

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A Power-Iterative Reweighting Strategy

Note that by Eq. (5), TVw(U) essentially becomes TVp(U) for U in a neighborhood of iterate U(l). Consequently, nonconvex minimization of TVp(U) can practically be achieved by a series of convex minimization of TVw(U) using the above strategy.

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OUTLINE

1

Introduction

2

Generalized TV and Weighted TV

3

A Power-Iterative Reweighting Strategy

4

WTV-Regularized Minimization: The Splitting Technique

5

Performance on Image Reconstruction

6

Conclusions

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WTV-Regularized Minimization

The analysis has led to the WTV-regularized problem min

U

TVw(U) s.t. R ◦ (FU) − B2

F < σ2

(6) We propose to solve the problem using a Split Bregman approach, but with important changes. Using Bregman iteration, we reduce the problem to U(k+1) = argmin

U

TVw(U) + µ 2 R ◦ (FU) − B(k)2

F

(7a) B(k+1) = B(k) + B − R ◦ (FU(k+1)) (7b)

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WTV-Regularized Minimization

A splitting strategy applied to (7a) leads to the formulation min

U Wx ◦ Dx1 + Wy ◦ Dy1 + µ

2 R ◦ (FU) − B(k)2

F

(8a) s.t. Dx = DV, Dy = VDT, U = V (8b) Here we apply the Split Bregman to split Dx = DV and Dy = VDT, but also introduce an additional split as U = V. Such a split allows us to decompose the most computationally expensive step of the algorithm into two much simpler steps.

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WTV-Regularized Minimization

Enforcing the constraints in (8b), we mimize the following function with respect to {U, V, Dx, Dy} min Wx ◦ Dx1 + Wy ◦ Dy1 + µ 2 R ◦ (FU) − B(k)2

F

+ λ 2 Dx − DV − E(h)

x 2 F + λ

2 Dy − VDT − E(h)

y 2 F

+ ν 2U − V − G(h)2

F

E(h)

x , E(h) y

and G(h) are updated through Bregman iterations.

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WTV-Regularized Minimization

At the hth iteration we solve four subproblems U(h+1) = argmin

U

µ 2 R ◦ (FU) − B(k)2

F

+ν 2U − V(h) − G(h)2

F

(9) V(h+1) = argmin

V

ν 2V − U(h+1) + G(h)2

F

+ λ 2 DV + E(h)

x

− D(h)

x 2 F + λ

2 VDT + E(h)

y

− D(h)

y 2 F

(10) D(h+1)

x

= argmin

Dx

Wx ◦ Dx1 + λ 2 Dx − DV(h+1) − E(h)

x 2 F

(11a) D(h+1)

y

= argmin

Dy

Wy ◦ Dy1 + λ 2 Dy − V(h+1)DT − E(h)

y 2 F

(11b)

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WTV-Regularized Minimization

The problems in (11a) and (11b) can be solved simply by soft shrinkage operations as D(h+1)

x

= TWx/λ(DV(h+1) + E(h)

x )

(12a) D(h+1)

y

= TWy/λ(V(h+1)DT + E(h)

y )

(12b) Solving problems (9) and (10) are however far from trivial

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WTV-Regularized Minimization

We write first-order optimality condition of (9) as µFTR ◦ FU + νU = µFTR ◦ Bk + ν(V(h) + G(h)) (13) Multiply both sides of (13) by F on the left, we obtain solution of (13) as U(h+1) = FT µR ◦ Bk + νF(V(h) + G(h))

• /(µR + ν)
• (14)

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WTV-Regularized Minimization

To solve (10), we write its first-order optimality condition as νV + λDTDV + λVDTD = C(h) (15) where C(h) = ν(U(h+1) − G(h)) + λDT(D(h)

x

− E(h)

x ) + λ(D(h) y

− E(h)

y )D

(16) Circulant matrix D can be diagonalized by the 2-D Fourier transform F as D = FTΛF.

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WTV-Regularized Minimization

Multiply both sides of (15) by F on the left and FT on the right, we have ν ˜ V + λ(T˜ V + ˜ VT) = FC(h)FT (17) where ˜ V = FVFT and T = Λ∗Λ. T is a diagonal matrix, we can further express (17) as (ν + λTr + λTc) ◦ ˜ V = FC(h)FT (18) where Tr has each element in its ith row as Ti,i while Tc has each element in its ith column as Ti,i. Therefore, we obtain solution of (10) as V(h+1) = FT (FC(h)FT) ◦ /(ν + λTr + λTc)

• F

(19)

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OUTLINE

1

Introduction

2

Generalized TV and Weighted TV

3

A Power-Iterative Reweighting Strategy

4

WTV-Regularized Minimization: The Splitting Technique

5

Performance on Image Reconstruction

6

Conclusions

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MRI of the Shepp-Logan Phantom

A normalized Shepp-Logan phantom of size 256 × 256, was measured at 2521 locations in the 2D Fourier plane (k-space). The sampling pattern was a star-shaped pattern consisting of only 10 radial lines. Recover the image based on the 2521 star-shaped 2D Fourier samples.

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MRI of the Shepp-Logan Phantom

(a) Star-shaped sampling pattern (b) Minimum energy reconstruction (c) Minimum TV reconstruction (d) Minimum GTV reconstruction with p = 0

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Recovered Shepp-Logan Phantom

Minimum GTV reconstruction with p = 0 The signal to noise (SNR) ratio: 16.3 dB. Computation time on a PC laptop with a 2.67 GHz Intel quad-core processor: 770.7 seconds. Minimum TV reconstruction SNR: 8.8 dB. Computation time: 756.8 seconds.

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OUTLINE

1

Introduction

2

Generalized TV and Weighted TV

3

A Power-Iterative Reweighting Strategy

4

WTV-Regularized Minimization: The Splitting Technique

5

Performance on Image Reconstruction

6

Conclusions

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Conclusions

Proposal of a reweighted TV minimization algorithm to obtain the solution approaching a pth power TV minimizer. A weighted TV-regularized problem has been solved in the Split Bregman framework, but with important additional splitting technique. A power-iterative strategy is utilized by gradually reducing the power p from 1 to 0. For compressive imaging, the proposed algorithm outperforms the conventional TV minimization method in terms of the reconstructed image quality.

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