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A new Sylvester-type quaternion matrix equation model for color image data transmission

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Abstract

In this paper, we investigate an encryption method for color image data transmission based on Sylvester-type quaternion matrix equations. We first study the solvability conditions and general solution to a system of Sylvester-type quaternion matrix equations. We derive some practical necessary and sufficient conditions for the existence of a solution to this system by using a simultaneous decomposition of quaternion matrices. We present the expression of the general solution to the system when the solvability conditions are satisfied. Based on the form of this system, we consider the applications of this system in color image data transmission. Moreover, we provide some algorithms and examples to illustrate the main results.

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References

  • Castelan EB, Silva VG (2005) On the solution of a Sylvester equation appearing in descriptor systems control theory. Syst Control Lett 54(2):109–117

    Article  MathSciNet  Google Scholar 

  • Chen Y, Xiao X, Zhou Y (2019) Low-rank quaternion approximation for color image processing. IEEE Trans Image Process 29:1426–1439

    Article  MathSciNet  Google Scholar 

  • Chu MT, Funderlic RE, Golub GH (1997) On a variational formulation of the generalized singular value decomposition. SIAM J Matrix Anal Appl 18(4):1082–1092

    Article  MathSciNet  Google Scholar 

  • Darouach M (2006) Solution to Sylvester equation associated to linear descriptor systems. IEEE Control Syst Lett 55(10):835–838

    Article  MathSciNet  Google Scholar 

  • De Leo S, Scolarici G (2000) Right eigenvalue equation in quaternionic quantum mechanics. J Phys A-Math Gen 33(15):2971

    Article  MathSciNet  Google Scholar 

  • De Moor B, Van Dooren P (1992) Generalizations of the singular value and QR decompositions. SIAM J Matrix Anal Appl 13(4):993–1014

    Article  MathSciNet  Google Scholar 

  • De Moor B, Golub GH (1991) The restricted singular value decomposition: properties and applications. SIAM J Matrix Anal Appl 12(3):401–425

    Article  MathSciNet  Google Scholar 

  • De Moor B, Zha H (1991) A tree of generalization of the ordinary singular value decomposition. Linear Algebra Appl 147:469–500

    Article  MathSciNet  Google Scholar 

  • De Schutter B, De Moor B (2002) The QR Decomposition and the singular value decomposition in the symmetrized max-plus algebra revisited. SIAM Rev 44(3):417–454

    Article  MathSciNet  Google Scholar 

  • De Teran F, Dopico FM, Guillery N, Montealegre D, Reyes N (2013) The solution of the equation \(AX +X^{*}B = 0\). Linear Algebra Appl 438(7):2817–2860

    MathSciNet  Google Scholar 

  • Dehghan M, Hajarian M (2011) Analysis of an iterative algorithm to solve the generalized coupled Sylvester matrix equations. Appl Math Model 35(7):3285–3300

    Article  MathSciNet  Google Scholar 

  • Futorny V, Klymchuk T, Sergeichuk VV (2016) Roth’s solvability criteria for the matrix equations \(AX-{\widehat{X}}B=C\) and \(X-A{\widehat{X}}B=C\) over the skew field of quaternions with an involutive automorphism \(q\rightarrow {\hat{q}}\). Linear Algebra Appl 510:246–258

    Article  MathSciNet  Google Scholar 

  • He ZH (2019a) Pure PSVD approach to Sylvester-type quaternion matrix equations. Electron J Linear Algebra 35:265–284

    Article  MathSciNet  Google Scholar 

  • He ZH (2019b) A system of coupled quaternion matrix equations with seven unknowns and its applications. Adv Appl Clifford Algebras 29:38

    Article  MathSciNet  Google Scholar 

  • He ZH (2021) Some new results on a system of Sylvester-type quaternion matrix equations. Linear Multilinear Algebra 69(16):3069–3091

    Article  MathSciNet  Google Scholar 

  • He ZH, Wang M (2021) A quaternion matrix equation with two different restrictions. Adv Appl Clifford Algebras 31:25

    Article  MathSciNet  Google Scholar 

  • He ZH, Agudelo OM, Wang QW, De Moor B (2016) Two-sided coupled generalized Sylvester matrix equations solving using a simultaneous decomposition for fifteen matrices. Linear Algebra Appl 496:549–593

    Article  MathSciNet  Google Scholar 

  • He ZH, Liu J, Tam TY (2017) The general \(\phi \)-Hermitian solution to mixed pairs of quaternion matrix Sylvester equations. Electron J Linear Algebra 32:475–499

    Article  MathSciNet  Google Scholar 

  • He ZH, Wang QW, Zhang Y (2018) A system of quaternary coupled Sylvester-type real quaternion matrix equations. Automatica 87:25–31

    Article  MathSciNet  Google Scholar 

  • He ZH, Wang QW, Zhang Y (2019) A simultaneous decomposition for seven matrices with applications. J Comput Appl Math 349:93–113

    Article  MathSciNet  Google Scholar 

  • He ZH, Qin WL, Wang XX (2021) Some applications of a decomposition for five quaternion matrices in control system and color image processing. Comput Appl Math 40(6):205

    Article  MathSciNet  Google Scholar 

  • He ZH, Navasca C, Wang XX (2022) Decomposition for a quaternion tensor triplet with applications. Adv Appl Clifford Algebras 32:9

    Article  MathSciNet  Google Scholar 

  • He ZH, Wang XX, Zhao YF (2023) Eigenvalues of quaternion tensors with applications to color video processing. J Sci Comput 94(1):1

    Article  MathSciNet  Google Scholar 

  • Jia Z, Ng MK (2021) Structure preserving quaternion generalized minimal residual method. SIAM J Matrix Anal Appl 42(2):616–634

    Article  MathSciNet  Google Scholar 

  • Jia Z, Ng MK, Song GJ (2019a) Robust quaternion matrix completion with applications to image inpainting. Numer Linear Algebra Appl 26(4):e2245

    Article  MathSciNet  Google Scholar 

  • Jia Z, Ng MK, Song GJ (2019b) Lanczos method for large-scale quaternion singular value decomposition. Numer Algorithms 82:699–717

    Article  MathSciNet  Google Scholar 

  • Kyrchei I (2014) Determinantal representations of the Drazin inverse over the quaternion skew field with applications to some matrix equations. Appl Math Comput 238:193–207

    MathSciNet  Google Scholar 

  • Kyrchei I (2018) Cramer’s rules for Sylvester quaternion matrix equation and its special cases. Adv Appl Clifford Algebras 28:90

    Article  MathSciNet  Google Scholar 

  • Le Bihan N, Sangwine SJ (2003) Quaternion principal component analysis of color images. ICIP 1:I–809

  • Li S, Li Y (2014) Nonlinearly activated neural network for solving time-varying complex Sylvester equation. IEEE Trans Cybern 44(8):1397–1407

    Article  Google Scholar 

  • Li S, Chen S, Liu B (2013) Accelerating a recurrent neural network to finite-time convergence for solving time- varying Sylvester equation by using a sign-bi-power activation function. Neural Process Lett 37:189–205

    Article  Google Scholar 

  • Ng MK, Weiss P, Yuan X (2010) Solving constrained total-variation image restoration and reconstruction problems via alternating direction methods. SIAM J Sci Comput 32(5):2710–2736

    Article  MathSciNet  Google Scholar 

  • Ng MK, Yuan X, Zhang W (2013) Coupled Variational Image Decomposition and Restoration Model for Blurred Cartoon-Plus-Texture Images With Missing Pixels. IEEE Trans Image Process 22(6):2233–2246

    Article  MathSciNet  Google Scholar 

  • Rodman L (2014) Topics in quaternion linear algebra. Princeton University Press, Princeton

    Book  Google Scholar 

  • Song C, Chen G (2011) On solutions of matrix equations \(XF-AX=C\) and \(XF-A{\widetilde{X}}=C\) over quaternion field. J Appl Math Comput 1(37):57–88

    Article  Google Scholar 

  • Took CC, Mandic DP, Zhang FZ (2011) On the unitary diagonalization of a special class of quaternion matrices. Appl Math Lett 24(11):1806–1809

    Article  MathSciNet  Google Scholar 

  • Wang QW, He ZH (2013) Solvability conditions and general solution for the mixed Sylvester equations. Automatica 49(9):2713–2719

    Article  MathSciNet  Google Scholar 

  • Wang QW, He ZH, Zhang Y (2019) Constrained two-sided coupled Sylvester-type quaternion matrix equations. Automatica 101:207–213

    Article  MathSciNet  Google Scholar 

  • Wei Q, Dobigeon N, Tourneret JY (2015) Fast fusion of multi-band images based on solving a Sylvester equation. IEEE Trans Image Process 24(11):4109–4121

    Article  MathSciNet  Google Scholar 

  • Wimmer HK (1994) Consistency of a pair of generalized Sylvester equations. IEEE Trans Autom Control 39(5):1014–1016

    Article  MathSciNet  Google Scholar 

  • Wu AG, Duan GR, Zhou B (2008) Solution to generalized Sylvester matrix equations. IEEE Trans Autom Control 53(3):811–815

    Article  MathSciNet  Google Scholar 

  • Yu SW, He ZH, Qi TC, Wang XX (2021) The equivalence canonical form of five quaternion matrices with applications to imaging and Sylvester-type equations. J Comput Appl Math 393:113494

    Article  MathSciNet  Google Scholar 

  • Zhang Y, Wang B (2008) Optical image encryption based on interference. Opt Lett 33(21):2443–2445

    Article  Google Scholar 

  • Zhou B, Duan GR (2006) A new solution to the generalized Sylvester matrix equation \(AV-EVF=BW\). Syst Control Lett 55(3):193–198

    Article  Google Scholar 

Download references

Acknowledgements

This research was supported by the National Natural Science Foundation of China (Grant nos. 12271338 and 12371023) and Shanghai Oriental Talent Program (Youth Program).

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Authors and Affiliations

  1. Department of Mathematics and Newtouch Center for Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China

    Zhuo-Heng He, Wei-Lu Qin & Jie Tian

  2. Department of Mathematics and Statistics, University of Nevada, Reno, NV, 89557, USA

    Xiang-Xiang Wang

  3. Department of Mathematics, University of Manitoba, Winnipeg, MB, R3T 2N2, Canada

    Yang Zhang

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  1. Zhuo-Heng He
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  2. Wei-Lu Qin
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  3. Jie Tian
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  4. Xiang-Xiang Wang
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  5. Yang Zhang
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Correspondence to Zhuo-Heng He.

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He, ZH., Qin, WL., Tian, J. et al. A new Sylvester-type quaternion matrix equation model for color image data transmission. Comp. Appl. Math. 43, 227 (2024). https://doi.org/10.1007/s40314-024-02732-4

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  • DOI: https://doi.org/10.1007/s40314-024-02732-4

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