Adaptive Corrected Procedure for TVL1 Image Deblurring under Impulsive Noise

Adaptive Corrected Procedure for TVL1 Image Deblurring under Impulsive Noise Minru Bai(x T) College of Mathematics and Econometrics Hunan University Joint work with Xiongjun Zhang, Qianqian Shao June 30, 2018 Minru Bai(x T) (HNU) Adaptive Corrected Procedure 1 / 29

Outline 1 Introduction 2 Adaptive corrected procedure 3 Algorithm 4 Numerical experiments 5 Conclusions Minru Bai(x T) (HNU) Adaptive Corrected Procedure 2 / 29

Outline 1 Introduction 2 Adaptive corrected procedure 3 Algorithm 4 Numerical experiments 5 Conclusions Minru Bai(x T) (HNU) Adaptive Corrected Procedure 3 / 29

Image denoising and deblurring Let x R n2 be an original image concatenated into an n 2 -vector, K R n2 n 2 be a blurring operator, and f R n2 be an observation of x satisfying the relationship f = N imp (K x), where N imp represents the degradation by impulse noise. Noisy and blurry observation Minru Bai(x T) (HNU) Adaptive Corrected Procedure 4 / 29

Two types of impulsive noise Let y R n2 denote an original image. The dynamic range of y is in [d min, d max ], i.e. d min y i d max for all i. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 5 / 29

Two types of impulsive noise Let y R n2 denote an original image. The dynamic range of y is in [d min, d max ], i.e. d min y i d max for all i. Salt-and-pepper noise f i = d min, with probability r 2, d max, with probability r 2, y i, with probability 1 r, where f i, y i are the i-th pixel values of f and y, respectively, 0 r 1. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 5 / 29

Two types of impulsive noise Let y R n2 denote an original image. The dynamic range of y is in [d min, d max ], i.e. d min y i d max for all i. Salt-and-pepper noise f i = d min, with probability r 2, d max, with probability r 2, y i, with probability 1 r, where f i, y i are the i-th pixel values of f and y, respectively, 0 r 1. Random-valued noise { di, with probability r, f i = y i, with probability 1 r, where d i are the identically and uniformly distributed random numbers in [d min, d max ], 0 r 1. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 5 / 29

Maximum a posteriori (MAP) estimator How to recover the original image from the noisy and blurred image? 1 Chen F, Shen L, Xu Y, Zeng X 2014 The Moreau envelope approach for the L1/TV image denoising. Inverse Problem and Imaging, 8 53-77. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 6 / 29

Maximum a posteriori (MAP) estimator How to recover the original image from the noisy and blurred image? Based on MAP estimator for a unknown image, Chen et al. 1 obtained the denoising reconstruction model in discrete form as TVL0 model: min x n 2 i=1 D i x + µ Kx f 0, 1 Chen F, Shen L, Xu Y, Zeng X 2014 The Moreau envelope approach for the L1/TV image denoising. Inverse Problem and Imaging, 8 53-77. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 6 / 29

Maximum a posteriori (MAP) estimator How to recover the original image from the noisy and blurred image? Based on MAP estimator for a unknown image, Chen et al. 1 obtained the denoising reconstruction model in discrete form as TVL0 model: min x n 2 i=1 D i x + µ Kx f 0, where for each i, D i x R 2 denotes a certain local first-order finite difference of x at pixel i in both horizontal and vertical directions; 0 denotes the number of non-zero elements in a vector; K is a blurry operator; µ is a regularization parameter; f R n2 be an observation of original image x. 1 Chen F, Shen L, Xu Y, Zeng X 2014 The Moreau envelope approach for the L1/TV image denoising. Inverse Problem and Imaging, 8 53-77. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 6 / 29

Convex relaxation Unfortunately, the function 0 is not convex, and it is a NP-hard problem. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 7 / 29

Convex relaxation Unfortunately, the function 0 is not convex, and it is a NP-hard problem. It is well known that l 1 norm is a nice convex approximation of l 0 norm. So replacing 0 by 1 yields the following TVL1 model min x n 2 i=1 D i x + µ Kx f 1. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 7 / 29

Outline 1 Introduction 2 Adaptive corrected procedure 3 Algorithm 4 Numerical experiments 5 Conclusions Minru Bai(x T) (HNU) Adaptive Corrected Procedure 8 / 29

Motivation Nikolova 2 pointed out from the view of MAP that the solutions of the TVL1 model substantially deviate from both the data-acquisition model and the prior model. 2 Nikolova M 2007 Model distortions in bayesian MAP reconstruction Inverse Problems and Imaging 1 399-422. 3 Tibshirani R 1996 Regression Shrinkage and Selection via the Lasso Journal of the Royal Statistical Society, Ser. B (Methodological) 267-88. 4 Hui Zou 2006 The Adaptive Lasso and Its Oracle Properties Journal of the American Statistical Association 101(476): 1418-1429. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 9 / 29

Motivation Nikolova 2 pointed out from the view of MAP that the solutions of the TVL1 model substantially deviate from both the data-acquisition model and the prior model. The l 1 -norm penalty has long been known to yield biased estimators for simultaneous estimation 3. 2 Nikolova M 2007 Model distortions in bayesian MAP reconstruction Inverse Problems and Imaging 1 399-422. 3 Tibshirani R 1996 Regression Shrinkage and Selection via the Lasso Journal of the Royal Statistical Society, Ser. B (Methodological) 267-88. 4 Hui Zou 2006 The Adaptive Lasso and Its Oracle Properties Journal of the American Statistical Association 101(476): 1418-1429. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 9 / 29

Motivation Nikolova 2 pointed out from the view of MAP that the solutions of the TVL1 model substantially deviate from both the data-acquisition model and the prior model. The l 1 -norm penalty has long been known to yield biased estimators for simultaneous estimation 3. Adaptive lasso was proposed by Zou 4, where adaptive weights are used for penalizing different coefficients in l 1 -norm penalty. 2 Nikolova M 2007 Model distortions in bayesian MAP reconstruction Inverse Problems and Imaging 1 399-422. 3 Tibshirani R 1996 Regression Shrinkage and Selection via the Lasso Journal of the Royal Statistical Society, Ser. B (Methodological) 267-88. 4 Hui Zou 2006 The Adaptive Lasso and Its Oracle Properties Journal of the American Statistical Association 101(476): 1418-1429. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 9 / 29

Motivation Nikolova 2 pointed out from the view of MAP that the solutions of the TVL1 model substantially deviate from both the data-acquisition model and the prior model. The l 1 -norm penalty has long been known to yield biased estimators for simultaneous estimation 3. Adaptive lasso was proposed by Zou 4, where adaptive weights are used for penalizing different coefficients in l 1 -norm penalty. The key to overcome the limit of TVL1 is how to improve the sparsity of the l 1 term. 2 Nikolova M 2007 Model distortions in bayesian MAP reconstruction Inverse Problems and Imaging 1 399-422. 3 Tibshirani R 1996 Regression Shrinkage and Selection via the Lasso Journal of the Royal Statistical Society, Ser. B (Methodological) 267-88. 4 Hui Zou 2006 The Adaptive Lasso and Its Oracle Properties Journal of the American Statistical Association 101(476): 1418-1429. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 9 / 29

Motivation 15000 15000 10000 10000 Frequency Frequency 5000 5000 0 0.1 0.05 0 0.05 0.1 0.15 x x (a) 0 0.1 0.05 0 0.05 0.1 0.15 x x (b) Figure: Histograms of x x for House image corrupted by Average blur and salt-and-pepper noise or random-valued noise, where x is the original image and x is the recovered image by TVL1. (a) Salt-and-pepper with noise level 30%. (b) Random-valued noise with noise level 40%. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 10 / 29

Motivation 15000 15000 10000 10000 Frequency Frequency 5000 5000 0 0.1 0.05 0 0.05 0.1 0.15 x x (a) 0 0.1 0.05 0 0.05 0.1 0.15 x x (b) Figure: Histograms of x x for House image corrupted by Average blur and salt-and-pepper noise or random-valued noise, where x is the original image and x is the recovered image by TVL1. (a) Salt-and-pepper with noise level 30%. (b) Random-valued noise with noise level 40%. TVL1 model can effectively remove abnormal value noise signals. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 10 / 29

Motivation 15000 15000 10000 10000 Frequency Frequency 5000 5000 0 0.1 0.05 0 0.05 0.1 0.15 x x (a) 0 0.1 0.05 0 0.05 0.1 0.15 x x (b) Figure: Histograms of x x for House image corrupted by Average blur and salt-and-pepper noise or random-valued noise, where x is the original image and x is the recovered image by TVL1. (a) Salt-and-pepper with noise level 30%. (b) Random-valued noise with noise level 40%. TVL1 model can effectively remove abnormal value noise signals. These observations imply that small biased estimates may contain some information of the sparsity of Kx f to a certain extent. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 10 / 29

Corrected model Variable substitution min x,z n 2 i=1 D i x + µ z 1 s.t. z = Kx f. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 11 / 29

Corrected model Variable substitution Corrected TVL1(CTVL1): min x,z min x,z n 2 i=1 n 2 i=1 D i x + µ z 1 s.t. z = Kx f. s.t. z = Kx f, D i x + µ( z 1 F( z), z ) Minru Bai(x T) (HNU) Adaptive Corrected Procedure 11 / 29

Corrected model Variable substitution Corrected TVL1(CTVL1): min x,z min x,z n 2 i=1 n 2 i=1 D i x + µ z 1 s.t. z = Kx f. s.t. z = Kx f, D i x + µ( z 1 F( z), z ) where µ > 0 is the regularization parameters which depends on noise level and blur operator of corrupted image. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 11 / 29

Corrected model F : R n2 R n2 5 is an operator defined as F i (z) = { φ( z i z ), z R n2 \{0}, 0, z = 0. The scalar function φ : R R is defined as φ(t) := sgn(t)(1 + ε τ t τ ) t τ + ε τ, t R, for some τ > 0 and ε > 0. z is a reasonable initial estimator. In particular, when F 0, CTVL1 reduces to TVL1. 5 Miao W, Pan S and Sun D 2015 A rank-corrected procedure for matrix completion with fixed basis coefficients Math. Program., Ser. A 1-50. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 12 / 29

Analysis of correction term Notice that if z 0, n 2 z 1 F( z), z = ( z i φ(t i )z i ), where t i = z i / z and 0 t i 1 for i = 1,..., n 2. i=1 Minru Bai(x T) (HNU) Adaptive Corrected Procedure 13 / 29

Analysis of correction term Notice that if z 0, n 2 z 1 F( z), z = ( z i φ(t i )z i ), where t i = z i / z and 0 t i 1 for i = 1,..., n 2. If t i is near zero, then the fidelity (sparsity) becomes important and z i φ(t i )z i z i as t i 0. i=1 Minru Bai(x T) (HNU) Adaptive Corrected Procedure 13 / 29

Analysis of correction term Notice that if z 0, n 2 z 1 F( z), z = ( z i φ(t i )z i ), where t i = z i / z and 0 t i 1 for i = 1,..., n 2. If t i is near zero, then the fidelity (sparsity) becomes important and z i φ(t i )z i z i as t i 0. If t i is more near 1, then the TV term (smoothness) becomes more important and z i φ(t i )z i 0 as t i 1. i=1 Minru Bai(x T) (HNU) Adaptive Corrected Procedure 13 / 29

Adaptive corrected procedure Initialization: Input f, K. Step 1: Compute x by solving TVL1. Step 2: Let z = K x f. Step 3: Compute x by solving CTVL1. Step 4: If necessary, x = x and go to Step 2. Else return x and the procedure stop. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 14 / 29

Outline 1 Introduction 2 Adaptive corrected procedure 3 Algorithm 4 Numerical experiments 5 Conclusions Minru Bai(x T) (HNU) Adaptive Corrected Procedure 15 / 29

Proximal alternating direction method of multipliers Introduce an auxillary variable to enforce the constraint on x, corrected TVL1 model can be written as: min y,z,x n 2 i=1 y i 2 + µ( z 1 F( z), z ) s.t. z = Kx f, y i = D i x, i = 1,..., n 2. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 16 / 29

Proximal alternating direction method of multipliers Introduce an auxillary variable to enforce the constraint on x, corrected TVL1 model can be written as: min y,z,x n 2 i=1 y i 2 + µ( z 1 F( z), z ) s.t. z = Kx f, y i = D i x, i = 1,..., n 2. The augmented Lagrangian function L(x, y, z, λ 1, λ 2 ) = n 2 i=1 y i 2 λ T 1 (y Dx) + β 1 n 2 2 i=1 y i D i x 2 2 +µ( z 1 F( z), z ) λ T 2 [z (Kx f )] + β 2 2 z (Kx f ) 2 2, where β 1, β 2 > 0 are penalty parameters, and λ = (λ T 1, λt 2 )T R 2n2 is the Lagrangian multiplier. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 16 / 29

Proximal alternating direction method of multipliers 6 Iterative procedure: y k+1 = arg min y { L(x k, y, z k, λ k 1, λk 2 )}, { z k+1 = arg min L(x k, y k, z, λ k z 1, λk 2 )}, { x k+1 = arg min L(x, y k+1, z k+1, λ k x 1, λk 2 ) + 1 2 x } xk 2 S, λ k+1 1 = λ k 1 γβ 1(y k+1 Dx k+1 ), λ k+1 2 = λ k 2 γβ 2[z k+1 (Kx k+1 f )]. 6 Fazel M, Pong T K, Sun D and Tseng P 2013 Hankel matrix rank minimization with applications in system identification and realization SIAM J. Matrix Anal. Appl. 34 946-977. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 17 / 29

The details of PADMM Compute y: y k+1 i = max { D i x k + λk 1 β 1 2 1 β 1, 0 } D i x k + (λ k 1 ) i/β 1 D i x k + (λ k 1 ) i/β 1 2, i = 1, 2,..., n 2. Compute z: z k+1 = sgn(kx k f +[λ k 2 +µf( z)]/β 2) max{ Kx k f +[λ k 2 +µf( z)]/β 2 µ/β 2, 0}. Compute x: (β 1 D T D+β 2 K T K +S)x = D T (β 1 y k+1 λ k 1 )+KT (β 2 z k+1 λ k 2 )+β 2K T f +Sx k. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 18 / 29

Convergence of PADMM Theorem Assume that S + β 1 D T D + β 2 K T K is positive definite. Let {y k, z k, x k, λ k 1, λk 2 } be generated from Algorithm 1. If γ (0, (1 + 5)/2), then the sequence {y k, z k, x k } converges to an optimal solution of CTVL1 and {λ k 1, λk 2 } converges to an optimal solution to the dual problem of CTVL1. 7 Yang J, Zhang Y and Yin W 2009 An efficient TVL1 algorithm for deblurring multichannnel images corrupted by implusive nosie SIAM J. Sci. Comput. 31 2842-65. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 19 / 29

Convergence of PADMM Theorem Assume that S + β 1 D T D + β 2 K T K is positive definite. Let {y k, z k, x k, λ k 1, λk 2 } be generated from Algorithm 1. If γ (0, (1 + 5)/2), then the sequence {y k, z k, x k } converges to an optimal solution of CTVL1 and {λ k 1, λk 2 } converges to an optimal solution to the dual problem of CTVL1. Theorem removes the condition N(D) N(K) = {0} of convergence results in J. Yang et al. [Theorem 3.4] 7. 7 Yang J, Zhang Y and Yin W 2009 An efficient TVL1 algorithm for deblurring multichannnel images corrupted by implusive nosie SIAM J. Sci. Comput. 31 2842-65. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 19 / 29

Convergence of PADMM Theorem Assume that S + β 1 D T D + β 2 K T K is positive definite. Let {y k, z k, x k, λ k 1, λk 2 } be generated from Algorithm 1. If γ (0, (1 + 5)/2), then the sequence {y k, z k, x k } converges to an optimal solution of CTVL1 and {λ k 1, λk 2 } converges to an optimal solution to the dual problem of CTVL1. Theorem removes the condition N(D) N(K) = {0} of convergence results in J. Yang et al. [Theorem 3.4] 7. The assumption condition of S + β 1 D T D + β 2 K T K is very easy to be satisfied. If β 1 D T D + β 2 K T K are positive definite, then we can choose S = 0. If β 1 D T D + β 2 K T K are positive semidefinite, then we can choose positive semidefinite matrix S such that where α σ max (β 1 D T D + β 2 K T K). S + β 1 D T D + β 2 K T K = αi, 7 Yang J, Zhang Y and Yin W 2009 An efficient TVL1 algorithm for deblurring multichannnel images corrupted by implusive nosie SIAM J. Sci. Comput. 31 2842-65. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 19 / 29

Outline 1 Introduction 2 Adaptive corrected procedure 3 Algorithm 4 Numerical experiments 5 Conclusions Minru Bai(x T) (HNU) Adaptive Corrected Procedure 20 / 29

Sparsity comparison Define the sparse rate of Kx f as follows: s := {i : (Kx f ) i < 10 4, i = 1,, n 2 } n 2 where denotes the number of elements in the set. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 21 / 29

Sparsity comparison Define the sparse rate of Kx f as follows: s := {i : (Kx f ) i < 10 4, i = 1,, n 2 } n 2 where denotes the number of elements in the set. Table: Sparse rate s(%) of Kx f for various methods on Lena and House images corrupted by Average blur and salt-and-pepper noise (sp) or random value noise (rv). sp rv noise Lena House level Original TVL1 CTVL1 Original TVL1 CTVL1 30% 69.98% 25.71% 60.37% 69.78% 28.29% 62.25% 50% 49.98% 6.09% 46.35% 49.89% 9.77% 34.65% 70% 29.93% 1.68% 12.12% 29.88% 2.22% 12.14% 25% 74.99% 30.35% 66.71% 74.85% 46.16% 68.06% 40% 59.97% 11.70% 46.35% 59.83% 9.01% 45.85% Minru Bai(x T) (HNU) Adaptive Corrected Procedure 21 / 29

Deblurring image with random-valued noise Corruption: 40% SNR: 14.38 SNR: 21.40 SNR: 21.43 SNR: 21.47 Corruption: 40% SNR: 17.59 SNR: 22.58 SNR: 22.65 SNR: 22.72 Figure: Recovered images (with SNR(dB)) of TVL1 and CTVL1 on Cameraman and Lena images corrupted by Average blur and random-valued noise with noise level 40%. First column: Corrupted images. Second column: The restored image by TVL1. From third to fifth columns: The restored image by the first correction step, the second correction step, the third correction step. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 22 / 29

Deblurring image with random-valued noise Corruption: 70% SNR: 8.15 SNR: 9.93 SNR: 11.64 SNR: 13.10 SNR: 14.21 SNR: 15.12 SNR: 15.59 SNR: 16.07 SNR: 16.32 SNR: 16.53 SNR: 16.58 Figure: Recovered images of TVL1 and CTVL1 on Lena image corrupted by Average blur and random-valued noise with noise level 70%. From left to right and top to bottom: Corrupted image. The restored image by TVL1, the first correction step until to the tenth correction step. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 23 / 29

Deblurring image with random-valued noise Table: SNR(dB) values for deblurring and denoising results of different methods for the test images corrupted by Average blur or Gaussian blur and random-valued noise. Boat House Man Goldhill noise Average Gaussian level TVL1 Two-phase 8 CTVL1 TVL1 Two-phase CTVL1 25% 20.92 22.65 25.46 20.86 22.54 25.20 40% 16.59 20.53 22.72 16.74 20.42 22.74 70% 6.24 10.66 15.91 6.27 10.54 15.84 25% 22.57 24.53 27.39 22.71 24.00 27.33 40% 18.52 23.04 25.18 18.22 22.41 24.99 70% 8.07 12.05 17.84 8.20 12.25 18.42 25% 18.29 21.24 23.69 18.17 21.12 23.45 40% 15.64 18.72 21.26 15.60 18.68 21.39 70% 6.53 11.19 14.77 6.61 10.76 14.75 25% 19.19 21.95 23.52 19.11 21.99 23.36 40% 16.52 19.70 21.40 16.54 19.69 21.53 70% 6.91 11.55 15.63 6.94 11.66 16.86 8 Cai J, Chan R H and Nikolova M 2008 Two-phase approach for deblurring images corrupted by impulse plus Gaussian noise Inverse Problems and Imaging 2 187-204. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 24 / 29

Deblurring image with salt-and-pepper noise Table: SNR(dB) values for deblurring and denoising results of different methods for the test images corrupted by Average blur or Gaussian blur and salt-and-pepper noise. Boat House Man Goldhill noise Average Gaussian level TVL1 Two-phase CTVL1 TVL1 Two-phase CTVL1 30% 19.83 22.63 25.96 19.85 22.50 25.61 50% 15.39 20.36 22.33 15.41 20.32 22.13 70% 11.88 17.56 18.02 12.16 17.55 18.24 90% 6.40 12.18 11.56 6.21 13.36 12.36 30% 21.16 24.44 27.06 20.92 24.07 26.52 50% 18.02 22.23 24.21 17.97 21.00 24.15 70% 13.37 18.67 20.97 13.73 18.72 20.94 90% 5.12 14.04 15.97 5.18 14.21 13.04 30% 18.01 21.33 23.42 17.99 21.24 23.39 50% 14.82 18.77 20.46 14.83 18.85 20.42 70% 11.09 16.25 17.25 11.68 16.32 17.27 90% 6.09 13.04 13.10 6.29 13.12 12.30 30% 18.29 22.41 23.86 18.85 22.42 23.80 50% 15.78 20.15 21.01 15.82 20.21 21.00 70% 12.52 17.72 18.25 12.63 17.77 18.33 90% 7.17 13.79 12.69 6.81 13.73 12.75 Minru Bai(x T) (HNU) Adaptive Corrected Procedure 25 / 29

Outline 1 Introduction 2 Adaptive corrected procedure 3 Algorithm 4 Numerical experiments 5 Conclusions Minru Bai(x T) (HNU) Adaptive Corrected Procedure 26 / 29

Conclusions Propose a corrected model for TVL1. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 27 / 29

Conclusions Propose a corrected model for TVL1. Present the Proximal ADMM to solve the corrected model. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 27 / 29

Conclusions Propose a corrected model for TVL1. Present the Proximal ADMM to solve the corrected model. The accuracy of our proposed method is verified by numerical examples. Minru Bai(x T) (HNU) Adaptive Corrected Procedure 27 / 29

References Minru Bai, Xiongjun Zhang, Qianqian Shao, Adaptive correction procedure for TVL1 image deblurring under impulse noise, Inverse Problems, 2016, 32(2016):085004(23pp). Minru Bai, Xiongjun Zhang, Guyan Ni, Chunfeng Cui, An adaptive correction approach for tensor completion, SIAM J. Imaging Sciences, 2016, 9(3): 1298-1323 Minru Bai(x T) (HNU) Adaptive Corrected Procedure 28 / 29

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