On the Theory and Application of Singular Value Decomposition in Image Processing and Data Analysis
DOI:
https://doi.org/10.51408/1963-0144Keywords:
Singular Value Decomposition, Adaptive rank selection, Cumulative energy, Elbow detection, Image denoising, Image compression, Principal component analysisAbstract
Singular Value Decomposition (SVD) is a fundamental factorization method central to image processing, dimensionality reduction, and numerical linear algebra. Its practical strength lies in truncating small singular values, thereby reducing storage while preserving the essential structure. The key challenge is choosing the truncation rank, which determines the trade-off between accuracy, efficiency, and stability.
We introduce an adaptive rank selection method that combines cumulative-energy thresholds with automated elbow detection, yielding a principled alternative to ad hoc rules. The framework is validated on image compression, denoising, and principal component analysis (PCA), with benchmarks against eigenvalue decomposition (EVD) and QR factorization. The results show that the adaptive rule improves fidelity and reproducibility, especially for noisy or ill-conditioned data.
All code, figures, and tables are released as open source, ensuring that the experiments can be reproduced in a clean environment. This unifies theory, automation, and reproducibility, reaffirming that SVD is both mathematically optimal and practically versatile for contemporary data analysis.
References
G.H. Golub and C.F. Van Loan, Matrix Computations, Johns Hopkins University Press, 4th edition edition, 2013.
P.C. Hansen, The truncated svd as a method for regularization”, BIT Numerical Mathematics, vol. 27, no. 4, pp. 534–553, 1987. doi: 10.1007/BF02579243.
M.E. Wall, A. Rechtsteiner, and L.M. Rocha, “Singular value decomposition and principal component analysis”, A Practical Approach to Microarray Data Analysis, pp. 91–109, 2003.
C. Eckart and G. Young, “The approximation of one matrix by another of lower rank”, Psychometrika, vol. 1, no. 3, pp.211–218, 1936. doi: 10.1007/BF02288367.
H. C. Andrews and C. L. Patterson, “Singular value decomposition (svd) image coding”, IEEE Transactions on Communications, vol. 24, no. 4, pp. 425–432, 1976. doi: 10.1109/TCOM.1976.1093311.
H.R. Swathi and G. Sulochana, “Image compression using svd”, IOP Conference Series: Materials Science and Engineering, vol. 263, 042082, 2017. doi: 10.1088/1757-899X/263/4/042082.
P. C. Hansen, Discrete Inverse Problems: Insight and Algorithms, SIAM, 2010. [8] Q. Guo, C. Zhang, Y. Zhang and H. Liu, “An efficient svd-based method for image denoising”, IEEE Transactions on Circuits and Systems for Video Technology, vol. 26, no. 5, pp. 868–880, 2016, doi: 10.1109/TCSVT.2015.2416631.
S. Gu, L. Zhang, W. Zuo and X. Feng, “Weighted nuclear norm minimization with application to image denoising”, Proc. IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 2862–2869, 2014, doi: 10.1109/CVPR.2014.366.
E.J. Cands, X. Li, Y. Ma and J. Wright, “Robust principal component analysis?”, Journal of the ACM, vol. 58, no. 3, pp. 1–37, 2011. doi: 10.1145/1970392.1970395.
I.T. Jolliffe and J. Cadima, Principal Component Analysis, Springer, 2nd edition edition, 2016.
N. Halko, P.G. Martinsson and J.A. Tropp, “Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions”, SIAM Review, vol. 53, no. 2, pp. 217–288, 2011. doi: 10.1137/090771806.
Z. Lin, “A review on low-rank models in data analysis”, Big Data and Information Analytics, vol. 1, no. 2-3, pp. 139–161, 2016. doi: 10.3934/bdia.2016001.
P.C. Hansen, Rank-deficient and Discrete Ill-posed Problems: Numerical Aspects of Linear Inversion, SIAM, 1998.
M. Gavish and D.L. Donoho, “The optimal hard threshold for singular values is 4/√3”, Biometrika, vol. 101, no.3, pp. 587–602, 2014. doi: 10.1093/biomet/asu056.
V. Satopaa, J. Albrecht, D. Irwin and B. Raghavan, “Finding a “kneedle” in a haystack: Detecting knee points in system behavior”, 31st International Conference on Distributed Computing Systems Workshops (ICDCSW), pp. 166–171. IEEE, 2011. doi: 10.1109/ICDCSW.2011.20.
L.N. Trefethen and D. Bau, Numerical Linear Algebra, SIAM, 1997.
R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, 2nd edition edition, 2012.
L. Mirsky, “Symmetric gauge functions and unitarily invariant norms”, Quarterly Journal of Mathematics, vol. 11, no. 1, pp. 50–59, 1960. doi: 10.1093/qmath/11.1.50.
V. Klema and A. Laub, “The singular value decomposition: Its computation and some applications”, IEEE Transactions on Automatic Control, vol. 25, no. 2, pp. 164–176, 1980. doi: 10.1109/TAC.1980.1102314.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2026 Gayane G. Ghazaryan & Artashes L. Ghazaryan
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
