doi: 10.3934/jimo.2021211
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A new hybrid $ l_p $-$ l_2 $ model for sparse solutions with applications to image processing

1. 

Department of Mathematics, Shanghai University, Shanghai 200444, China

2. 

Edward P. Fitts Department of Industrial and, Systems Engineering, North Carolina State University, Raleigh, NC 27695-7906, USA

3. 

School of Management Science and, Engineering, Dongbei University of Finance and Economics, Dalian 116025, China

4. 

School of Mathematics, Physics and, Statistics, Shanghai University of Engineering Science, Shanghai 201620, China

* Corresponding author: Yanqin Bai (yqbai@shu.edu.cn)

Received  August 2021 Revised  October 2021 Early access December 2021

Fund Project: This work has been sponsored by the National Natural Science Foundations of China Grant 12171307, 11901382 and 71701035; The US Army Research Office Grant W911NF-15-1-0223; Chinese NNF Grant 71620107003 for Major International Joint Research Project

Finding sparse solutions to a linear system has many real-world applications. In this paper, we study a new hybrid of the $ l_p $ quasi-norm ($ 0 <p< 1 $) and $ l_2 $ norm to approximate the $ l_0 $ norm and propose a new model for sparse optimization. The optimality conditions of the proposed model are carefully analyzed for constructing a partial linear approximation fixed-point algorithm. A convergence proof of the algorithm is provided. Computational experiments on image recovery and deblurring problems clearly confirm the superiority of the proposed model over several state-of-the-art models in terms of the signal-to-noise ratio and computational time.

Citation: Xuerui Gao, Yanqin Bai, Shu-Cherng Fang, Jian Luo, Qian Li. A new hybrid $ l_p $-$ l_2 $ model for sparse solutions with applications to image processing. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021211
References:
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W. F. CaoJ. Sun and Z. B. Xu, Fast image deconvolution using closed-form thresholding formulas of $L_q(q=\frac{1}{2}, \frac{2}{3})$ regularization, Journal of Visual Communication and Image Representation, 24 (2013), 31-41.   Google Scholar

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X. J. Chen, Smoothing methods for nonsmooth, nonconvex minimization, Math. Program., 134 (2012), 71-99.  doi: 10.1007/s10107-012-0569-0.  Google Scholar

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X. J. ChenD. D. GeZ. Z. Wang and Y. Y. Ye, Complexity of unconstrained $L_2-L_p$ minimization, Math. Program., 143 (2014), 371-383.  doi: 10.1007/s10107-012-0613-0.  Google Scholar

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M. J. LaiY. Xu and W. T. Yin, Improved iteratively reweighted least squares for unconstrained smoothed $l_q$ minimization, SIAM J. Numer. Anal., 51 (2013), 927-957.  doi: 10.1137/110840364.  Google Scholar

[22]

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[23]

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[24]

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[25]

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[30]

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Z. B. XuX. Y. ChangF. M. Xu and H. Zhang, $L_{1/2}$ regularization: A thresholding representation theory and a fast solver, IEEE Transactions on Neural Networks and Learning Systems, 23 (2012), 1013-1027.   Google Scholar

[32]

B. C. ZhangW. Hong and Y. R. Wu, Sparse microwave imaging: Principles and applications, Sci. China Inf. Sci., 55 (2012), 1722-1754.  doi: 10.1007/s11432-012-4633-4.  Google Scholar

[33]

C. ZhangJ. J. Wang and N. H. Xiu, Robust and sparse portfolio model for index tracking, J. Ind. Manag. Optim., 15 (2019), 1001-1015.   Google Scholar

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H. Zou, The adaptive lasso and its oracle properties, J. Amer. Statist. Assoc., 101 (2006), 1418-1429.  doi: 10.1198/016214506000000735.  Google Scholar

show all references

References:
[1]

E. Amaldi and V. Kann, On the approximability of minimizing nonzero variables or unsatisfied relations in linear systems, Theoret. Comput. Sci., 209 (1998), 237-260.  doi: 10.1016/S0304-3975(97)00115-1.  Google Scholar

[2]

A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), 183-202.  doi: 10.1137/080716542.  Google Scholar

[3]

A. M. BrucksteinD. L. Donoho and M. Elad, From sparse solutions of systems of equations to sparse modeling of signals and images, SIAM Rev., 51 (2009), 34-81.  doi: 10.1137/060657704.  Google Scholar

[4]

W. F. CaoJ. Sun and Z. B. Xu, Fast image deconvolution using closed-form thresholding formulas of $L_q(q=\frac{1}{2}, \frac{2}{3})$ regularization, Journal of Visual Communication and Image Representation, 24 (2013), 31-41.   Google Scholar

[5]

E. J. Candès, The restricted isometry property and its implications for compressed sensing, C. R. Math. Acad. Sci. Paris, 346 (2008), 589-592.  doi: 10.1016/j.crma.2008.03.014.  Google Scholar

[6]

E. J. Candès and T. Tao, Decoding by linear programming, IEEE Trans. Inform. Theory, 51 (2005), 4203-4215.  doi: 10.1109/TIT.2005.858979.  Google Scholar

[7]

E. J. CandèsM. B. Wakin and S. P. Boyd, Enhancing sparsity by reweighted $l_1$ minimization, J. Fourier Anal. Appl., 14 (2008), 877-905.  doi: 10.1007/s00041-008-9045-x.  Google Scholar

[8]

X. J. Chen, Smoothing methods for nonsmooth, nonconvex minimization, Math. Program., 134 (2012), 71-99.  doi: 10.1007/s10107-012-0569-0.  Google Scholar

[9]

X. J. ChenD. D. GeZ. Z. Wang and Y. Y. Ye, Complexity of unconstrained $L_2-L_p$ minimization, Math. Program., 143 (2014), 371-383.  doi: 10.1007/s10107-012-0613-0.  Google Scholar

[10]

R. A. DeVoreB. Jawerth and B. J. Lucier, Image compression through wavelet transform coding, IEEE Trans. Inform. Theory, 38 (1992), 719-746.  doi: 10.1109/18.119733.  Google Scholar

[11]

D. L. Donoho, De-noising by soft-thresholding, IEEE Trans. Inform. Theory, 41 (1995), 613-627.  doi: 10.1109/18.382009.  Google Scholar

[12]

D. L. Donoho, Compressed sensing, IEEE Trans. Inform. Theory, 52 (2006), 1289-1306.  doi: 10.1109/TIT.2006.871582.  Google Scholar

[13]

M. DuarteM. DavenportD. TakharJ. LaskaT. SunK. Kelly and R. Baraniuk, Single-pixel imaging via compressive sampling, IEEE Signal Processing Magazine, 25 (2008), 83-91.  doi: 10.1109/MSP.2007.914730.  Google Scholar

[14]

E. Elhamifar and R. Vidal, Sparse subspace clustering: Algorithm, theory, and applications, IEEE Transactions on Pattern Analysis and Machine Intelligence, 35 (2013), 2765-2781.  doi: 10.1109/TPAMI.2013.57.  Google Scholar

[15]

J. Q. Fan and R. Z. Li, Variable selection via nonconcave penalized likelihood and its oracle properties, J. Amer. Statist. Assoc., 96 (2001), 1348-1360.  doi: 10.1198/016214501753382273.  Google Scholar

[16]

D. Foster and E. George, The risk inflation criterion for multiple regression, Ann. Statist., 22 (1994), 1947-1975.  doi: 10.1214/aos/1176325766.  Google Scholar

[17]

X. R. GaoY. Q. Bai and Q. Li, A sparse optimization problem with hybrid $L_2$-$L_p$ regularization for application of magnetic resonance brain images, J. Combinatorial Optimization, 42 (2019), 760-784.  doi: 10.1007/s10878-019-00479-x.  Google Scholar

[18]

S. JiangS.-C. Fang and Q. W. Jin, Sparse solutions by a quadratically constrained $l_q (0 < q < 1)$ minimization model, Informs J. Comput., 33 (2021), 511-530.  doi: 10.1287/ijoc.2020.1004.  Google Scholar

[19]

S. JiangS.-C. FangT. T. Nie and W. X. Xing, A gradient descent based algorithm for $l_p$ minimization, European J. Oper. Res., 283 (2020), 47-56.  doi: 10.1016/j.ejor.2019.11.051.  Google Scholar

[20]

M. J. Lai and J. Y. Wang, An unconstrained $l_q$ minimization with $0 < q \leq 1$ for sparse solution of underdetermined linear systems, SIAM J. Optim., 21 (2011), 82-101.  doi: 10.1137/090775397.  Google Scholar

[21]

M. J. LaiY. Xu and W. T. Yin, Improved iteratively reweighted least squares for unconstrained smoothed $l_q$ minimization, SIAM J. Numer. Anal., 51 (2013), 927-957.  doi: 10.1137/110840364.  Google Scholar

[22]

Q. LiY. BaiC. Yu and Y.-X. Yuan, A new piecewise quadratic approximation approach for $L_0$ norm minimization problem, Sci. China Math., 62 (2019), 185-204.  doi: 10.1007/s11425-017-9315-9.  Google Scholar

[23]

Y. F. LouP. H. YinQ. He and J. Xin, Computing sparse representation in a highly coherent dictionary based on difference of $L_1$ and $L_2$, J. Sci. Comput., 64 (2015), 178-196.  doi: 10.1007/s10915-014-9930-1.  Google Scholar

[24]

N. Meinshausen and B. Yu, Lasso-type recovery of sparse representations for high-dimensional data, Ann. Statist., 37 (2009), 246-270.  doi: 10.1214/07-AOS582.  Google Scholar

[25]

D. MerhejC. DiabM. Khalil and R. Prost, Embedding prior knowledge within compressed sensing by neural networks, IEEE Transactions on Neural Networks, 22 (2011), 1638-1649.  doi: 10.1109/TNN.2011.2164810.  Google Scholar

[26]

B. Natraajan, Sparse approximate solutions to linear systems, SIAM J. Comput., 24 (1995), 227-234.  doi: 10.1137/S0097539792240406.  Google Scholar

[27]

I. Selesnick, Sparse regularization via convex analysis, IEEE Trans. Signal Process., 65 (2017), 4481-4494.  doi: 10.1109/TSP.2017.2711501.  Google Scholar

[28]

H. TakedaS. Farsiu and P. Milanfar, Deblurring using regularized locally adaptive kernel regression, IEEE Trans. Image Process., 17 (2008), 550-563.  doi: 10.1109/TIP.2007.918028.  Google Scholar

[29]

R. Tibshirani, Regression shrinkage and selection via the lasso, J. Roy. Statist. Soc. Ser. B, 58 (1996), 267-288.  doi: 10.1111/j.2517-6161.1996.tb02080.x.  Google Scholar

[30]

Y. WangW. Q. Liu and G. L. Zhou, An efficient algorithm for non-convex sparse optimization, J. Ind. Manag. Optim., 15 (2019), 2009-2021.  doi: 10.3934/jimo.2018134.  Google Scholar

[31]

Z. B. XuX. Y. ChangF. M. Xu and H. Zhang, $L_{1/2}$ regularization: A thresholding representation theory and a fast solver, IEEE Transactions on Neural Networks and Learning Systems, 23 (2012), 1013-1027.   Google Scholar

[32]

B. C. ZhangW. Hong and Y. R. Wu, Sparse microwave imaging: Principles and applications, Sci. China Inf. Sci., 55 (2012), 1722-1754.  doi: 10.1007/s11432-012-4633-4.  Google Scholar

[33]

C. ZhangJ. J. Wang and N. H. Xiu, Robust and sparse portfolio model for index tracking, J. Ind. Manag. Optim., 15 (2019), 1001-1015.   Google Scholar

[34]

H. Zou, The adaptive lasso and its oracle properties, J. Amer. Statist. Assoc., 101 (2006), 1418-1429.  doi: 10.1198/016214506000000735.  Google Scholar

Figure 1.  Comparison for $ Q(x) $ with $ p = 1/3 $ by taking different $ \alpha $
Figure 2.  Comparison for $ Q^*({{\mathit{\boldsymbol{x}}}}) $ with different $ p $
Figure 3.  Comparison between $ Q^*(x) $ and $ Q^*_t(x) $ with $ p = 1/3 $
Figure 4.  Contours of $ F(x) $ with different $ \lambda $ in two dimension
Figure 5.  256 $ \times $ 256 Shepp-Logan head phantom and the images recovered by PLAFPA$ _p $
Figure 6.  Comparison results of recoverability
Figure 7.  The original images and the observed blurred noise images
Figure 8.  The values of SNR obtained by PLAFPA$ _p $ with different $ p $ at different numbers of iterations for image deblurring
Figure 9.  The values of SNR obtained by tested methods at different numbers of iterations for image deblurring
Figure 10.  The results of "Cameraman" and "Lena" image deblurring obtained by different tested methods
Figure 11.  The choice of $ \lambda $ for each tested method and the values of SNR obtained by tested methods at different numbers of iterations for the "Cameraman" image deblurring
Figure 12.  SNR value trend graph with different $ p $ for the method PLAFPA$ _p $
Figure 13.  $ \Phi(\eta) $ for $ \eta>0 $
Table 1.  Results of image recovery for the Shepp-Logan head phantom
Method $ l/n_s $=0.51 $ l/n_s $=0.39 $ l/n_s $=0.25
SNR time Iter SNR time Iter SNR time Iter
Soft 136.60 77.70 813 122.18 103.09 1000 89.72 94.73 1000
Half 173.20 10.20 94 169.09 15.87 139 161.68 33.04 306
PQA 160.46 18.36 183 155.84 27.23 285 52.80 42.77 438
H$ L_2 $-$ L_{p} $ $ p=1/5 $ 173.80 10.06 94 169.43 13.63 135 162.50 28.59 298
$ p=1/2 $ 173.53 11.10 94 169.25 13.95 137 161.76 29.70 301
$ p=4/5 $ 173.92 9.66 97 169.64 13.91 143 162.10 31.96 322
PLAFPA$ _{p} $ $ p=1/5 $ 175.08 6.19 76 170.14 8.63 113 163.09 21.10 256
$ p=1/2 $ 175.53 6.36 77 171.70 9.57 113 163.14 20.31 251
$ p=4/5 $ 175.35 6.30 80 171.58 9.86 121 163.99 21.98 279
Method $ l/n_s $=0.51 $ l/n_s $=0.39 $ l/n_s $=0.25
SNR time Iter SNR time Iter SNR time Iter
Soft 136.60 77.70 813 122.18 103.09 1000 89.72 94.73 1000
Half 173.20 10.20 94 169.09 15.87 139 161.68 33.04 306
PQA 160.46 18.36 183 155.84 27.23 285 52.80 42.77 438
H$ L_2 $-$ L_{p} $ $ p=1/5 $ 173.80 10.06 94 169.43 13.63 135 162.50 28.59 298
$ p=1/2 $ 173.53 11.10 94 169.25 13.95 137 161.76 29.70 301
$ p=4/5 $ 173.92 9.66 97 169.64 13.91 143 162.10 31.96 322
PLAFPA$ _{p} $ $ p=1/5 $ 175.08 6.19 76 170.14 8.63 113 163.09 21.10 256
$ p=1/2 $ 175.53 6.36 77 171.70 9.57 113 163.14 20.31 251
$ p=4/5 $ 175.35 6.30 80 171.58 9.86 121 163.99 21.98 279
Table 2.  Results of image deblurring by PLAFPA$ _{8/9} $ at 500 iterations
t 10 20 30 40 50 60 70 80 90 100
"Cameraman" SNR 12.10 12.80 13.05 13.15 13.18 13.19 13.17 13.15 13.12 13.09
time 10.47 12.53 14.09 15.46 15.92 17.11 17.68 18.31 19.26 20.62
"Lena" t 10 20 30 40 50 60 70 80 90 100
SNR 13.96 14.69 15.03 15.17 15.21 15.19 15.15 15.09 15.03 14.97
time 49.38 54.88 60.33 64.82 68.99 72.83 76.09 78.31 80.59 83.25
t 10 20 30 40 50 60 70 80 90 100
"Cameraman" SNR 12.10 12.80 13.05 13.15 13.18 13.19 13.17 13.15 13.12 13.09
time 10.47 12.53 14.09 15.46 15.92 17.11 17.68 18.31 19.26 20.62
"Lena" t 10 20 30 40 50 60 70 80 90 100
SNR 13.96 14.69 15.03 15.17 15.21 15.19 15.15 15.09 15.03 14.97
time 49.38 54.88 60.33 64.82 68.99 72.83 76.09 78.31 80.59 83.25
Table 3.  "Cameraman" image deblurring at different numbers of iterations
Iter 100 200 300 400 500 600 700 800 900 1000
Soft SNR 11.84 12.30 12.46 12.51 12.52 12.52 12.50 12.49 12.47 12.45
time 1.68 2.75 3.98 4.95 6.12 7.29 8.43 9.51 10.57 11.84
Half Iter 100 200 300 400 500 600 700 800 900 1000
SNR 11.37 11.55 11.55 11.51 11.46 11.41 11.36 11.31 11.27 11.23
time 3.67 5.24 7.68 10.30 13.50 15.88 18.47 20.42 23.21 25.10
PQA Iter 100 200 300 400 500 600 700 800 900 1000
SNR 11.98 12.55 12.74 12.77 12.72 12.64 12.54 12.42 12.30 12.18
time 1.92 2.67 3.91 5.13 6.20 7.46 9.76 9.96 11.27 13.58
H$ L_2 $-$ L_{p} $ with $ p=8/9 $ Iter 100 200 300 400 500 600 700 800 900 1000
SNR 11.99 12.60 12.81 12.88 12.87 12.82 12.76 12.68 12.59 12.50
time 6.25 11.89 16.77 21.83 26.35 29.46 34.17 38.54 42.34 46.18
PLAFPA$ _{p} $
with $ p=8/9 $
Iter 100 200 300 400 500 600 700 800 900 1000
SNR 11.87 12.56 12.88 13.05 13.15 13.20 13.22 13.28 13.22 13.22
time 5.30 8.71 11.65 14.87 15.92 19.57 22.27 24.01 28.26 29.52
Iter 100 200 300 400 500 600 700 800 900 1000
Soft SNR 11.84 12.30 12.46 12.51 12.52 12.52 12.50 12.49 12.47 12.45
time 1.68 2.75 3.98 4.95 6.12 7.29 8.43 9.51 10.57 11.84
Half Iter 100 200 300 400 500 600 700 800 900 1000
SNR 11.37 11.55 11.55 11.51 11.46 11.41 11.36 11.31 11.27 11.23
time 3.67 5.24 7.68 10.30 13.50 15.88 18.47 20.42 23.21 25.10
PQA Iter 100 200 300 400 500 600 700 800 900 1000
SNR 11.98 12.55 12.74 12.77 12.72 12.64 12.54 12.42 12.30 12.18
time 1.92 2.67 3.91 5.13 6.20 7.46 9.76 9.96 11.27 13.58
H$ L_2 $-$ L_{p} $ with $ p=8/9 $ Iter 100 200 300 400 500 600 700 800 900 1000
SNR 11.99 12.60 12.81 12.88 12.87 12.82 12.76 12.68 12.59 12.50
time 6.25 11.89 16.77 21.83 26.35 29.46 34.17 38.54 42.34 46.18
PLAFPA$ _{p} $
with $ p=8/9 $
Iter 100 200 300 400 500 600 700 800 900 1000
SNR 11.87 12.56 12.88 13.05 13.15 13.20 13.22 13.28 13.22 13.22
time 5.30 8.71 11.65 14.87 15.92 19.57 22.27 24.01 28.26 29.52
Table 4.  "Lena" image deblurring at different numbers of iteration
Iter 100 200 300 400 500 600 700 800 900 1000
Soft SNR 14.86 14.81 14.67 14.53 14.43 14.35 14.29 14.23 14.18 14.13
time 7.68 15.21 21.94 27.80 37.62 42.08 48.75 56.46 63.33 69.26
Half Iter 100 200 300 400 500 600 700 800 900 1000
SNR 14.02 13.85 13.70 13.55 13.43 13.34 13.26 13.19 13.13 13.07
time 12.50 24.50 35.50 47.05 59.63 71.80 83.47 94.40 107.80 118.27
PQA Iter 100 200 300 400 500 600 700 800 900 1000
SNR 15.13 15.15 14.82 14.41 14.00 13.62 13.27 12.94 12.65 12.36
time 7.63 14.91 22.60 29.37 33.68 43.74 52.27 57.88 65.31 73.68
H$ L_2 $-$ L_{p} $
with $ p=8/9 $
Iter 100 200 300 400 500 600 700 800 900 1000
SNR 15.18 15.28 15.02 14.69 14.35 14.02 13.72 13.44 13.18 12.94
time 27.11 47.28 66.64 85.65 95.97 123.68 143.52 159.69 176.60 192.09
PLAFPA$ _{p} $
with $ p=8/9 $
Iter 100 200 300 400 500 600 700 800 900 1000
SNR 15.13 15.45 15.43 15.33 15.21 15.09 14.96 14.84 14.73 14.62
time 24.76 38.04 50.43 62.75 68.99 85.64 95.82 105.12 116.96 120.00
Iter 100 200 300 400 500 600 700 800 900 1000
Soft SNR 14.86 14.81 14.67 14.53 14.43 14.35 14.29 14.23 14.18 14.13
time 7.68 15.21 21.94 27.80 37.62 42.08 48.75 56.46 63.33 69.26
Half Iter 100 200 300 400 500 600 700 800 900 1000
SNR 14.02 13.85 13.70 13.55 13.43 13.34 13.26 13.19 13.13 13.07
time 12.50 24.50 35.50 47.05 59.63 71.80 83.47 94.40 107.80 118.27
PQA Iter 100 200 300 400 500 600 700 800 900 1000
SNR 15.13 15.15 14.82 14.41 14.00 13.62 13.27 12.94 12.65 12.36
time 7.63 14.91 22.60 29.37 33.68 43.74 52.27 57.88 65.31 73.68
H$ L_2 $-$ L_{p} $
with $ p=8/9 $
Iter 100 200 300 400 500 600 700 800 900 1000
SNR 15.18 15.28 15.02 14.69 14.35 14.02 13.72 13.44 13.18 12.94
time 27.11 47.28 66.64 85.65 95.97 123.68 143.52 159.69 176.60 192.09
PLAFPA$ _{p} $
with $ p=8/9 $
Iter 100 200 300 400 500 600 700 800 900 1000
SNR 15.13 15.45 15.43 15.33 15.21 15.09 14.96 14.84 14.73 14.62
time 24.76 38.04 50.43 62.75 68.99 85.64 95.82 105.12 116.96 120.00
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