[1] Zhang Xu, Wu Jiasong, Yang Guanyu, et al. L1-norm minimization for quaternion signals [J]. Journal of Southeast University (English Edition), 2013, 29 (1): 33-37. [doi:10.3969/j.issn.1003-7985.2013.01.007]

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L₁-norm minimization for quaternion signals()

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Journal of Southeast University (English Edition)[ISSN:1003-7985/CN:32-1325/N]

Volumn:

29

Issue:

2013 1

Page:

33-37

Research Field:

Computer Science and Engineering

Publishing date:

2013-03-20

Info

Title:

L₁-norm minimization for quaternion signals

Author(s):

Zhang Xu¹;  Wu Jiasong¹;  3;  Yang Guanyu¹;  3;  Lotfi Senahdji²;  3;  Shu Huazhong¹;  3

¹Laboratory of Image Science and Technology, Southeast University, Nanjing 210096, China
²LTSI, INSERM U 1099, Université de Rennes 1, Rennes 35000, France
³Centre de Recherche en Information Biomédicale Sino-français, Nanjing 210096, China

Keywords:

quaternion;  signal recovery;  compressed sensing

PACS:

TP391

DOI:

10.3969/j.issn.1003-7985.2013.01.007

Abstract:

An algorithm for recovering the quaternion signals in both noiseless and noise contaminated scenarios by solving an L₁-norm minimization problem is presented. The L₁-norm minimization problem over the quaternion number field is solved by converting it to an equivalent second-order cone programming problem over the real number field, which can be readily solved by convex optimization solvers like SeDuMi. Numerical experiments are provided to illustrate the effectiveness of the proposed algorithm. In a noiseless scenario, the experimental results show that under some practically acceptable conditions, exact signal recovery can be achieved. With additive noise contamination in measurements, the experimental results show that the proposed algorithm is robust to noise. The proposed algorithm can be applied in compressed-sensing-based signal recovery in the quaternion domain.

References:

[1] Candès E, Romberg J, Tao T. Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information [J]. IEEE Transactions on Information Theory, 2006, 52(2): 489-509.
[2] Donoho D. Compressed sensing [J]. IEEE Transactions on Information Theory, 2006, 52(4): 1289-1306.
[3] Candès E, Romberg J, Tao T. Stable signal recovery from incomplete and inaccurate measurements [J]. Communications on Pure and Applied Mathematics, 2006, 59(8):1207-1223.
[4] Winter S, Kellermann W, Sawada H, et al. MAP-based underdetermined blind source separation of convolutive mixtures by hierarchical clustering and L₁-norm minimization [J]. EURASIP Journal on Advances in Signal Processing, 2007, 2007(1): 81.
[5] Yu S, Khwaja A S, Ma J. Compressed sensing of complex-valued data[J]. Journal on Signal Processing, 2012, 92(2): 357-362.
[6] Hamilton W R. On quaternions [C]//Proceedings of Royal Irish Academy. Dublin, Ireland, 1844: 1-16.
[7] Ell T, Sangwine S. Hypercomplex Fourier transforms of color images [J]. IEEE Transactions on Image Processing, 2007, 16(1): 22-35.
[8] Vía J, Ramírez D, Vielva L. Properness and widely linear processing of quaternion random vectors [J]. IEEE Transactions on Information Theory, 2010, 56(7): 3502-3515.
[9] Jiang T, Wei M. Equality constrained least squares problem over quaternion field [J]. Applied Mathematics Letters, 2003, 16(6): 883-888.
[10] Jiang T, Zhao J, Wei M. A new technique of quaternion equality constrained least squares problem [J]. Journal of Computational and Applied Mathematics, 2008, 216(2): 509-513.
[11] Vía J, Palomar D P, Vielva L, et al. Quaternion ICA from second-order statistics [J]. IEEE Transactions on Signal Processing, 2011, 59(4): 1586-1600.
[12] Javidi S, Took C C, Mandic D P. A fast independent component analysis algorithm for quaternion signals [J]. IEEE Transactions on Neural Networks, 2011, 22(12): 1967-1978.
[13] Le Bihan N, Buchholz S. Quaternionic independent component analysis using hypercomplex nonlinearities [C]//7th International Conference on Mathematics of Signal Processing. Cirencester, UK, 2006.
[14] Le Bihan N, Sangwine S J. Quaternion principal component analysis of color images [C]//International Conference on Image Processing. Barcelona, Spain, 2003: 808-812.
[15] Sturm J F. Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones [J]. Optimization Methods and Software, 1999, 11(1): 625-653.
[16] Candès E, Recht B. Exact matrix completion via convex optimization [J]. Foundations of Computational Mathematics, 2009, 9(6): 717-772.
[17] Candès E, Li X, Ma Y, et al. Robust principal component analysis? [J]. Journal of ACM, 2011, 58(3): 1-37.

Memo

Memo:

Biographies: Zhang Xu(1984—), male, graduate; Shu Huazhong(corresponding author), male, doctor, professor, shu.list@seu.edu.cn.
Foundation items: The National Basic Research Program of China(973 program)(No. 2011CB707904), the National Natural Science Foundation of China(No. 61073138, 61271312, 61201344, 81101104, 60911130370), the Research Fund for the Doctoral Program of Higher Education of Ministry of Education of China(No. 20110092110023, 20120092120036), the Natural Science Foundation of Jiangsu Province(No.BK2012329, BK2012743).
Citation: Zhang Xu, Wu Jiasong, Yang Guanyu, et al. L₁-norm minimization for quaternion signals[J].Journal of Southeast University(English Edition), 2013, 29(1):33-37.[doi:10.3969/j.issn.1003-7985.2013.01.007]

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