informИнформатикаInformatics1816-03012617-6963UIIP NASBinform-213Research ArticleОБРАБОТКА СИГНАЛОВ, ИЗОБРАЖЕНИЙ, РЕЧИ, ТЕКСТА И РАСПОЗНАВАНИЕ ОБРАЗОВSIGNAL, IMAGE, SPEECH, TEXT PROCESSING AND PATTERN RECOGNITIONСИНГУЛЯРНОЕ РАЗЛОЖЕНИЕ МАТРИЦ В АНАЛИЗЕ ЦИФРОВЫХ ИЗОБРАЖЕНИЙSINGULAR VALUE DECOMPOSITION IN DIGITAL IMAGE ANALYSISСтаровойтовВ. В.StarovoitovV. V.valerys@newman.bas-net.byОбъединенный институт проблем информатики НАН БеларусиRussian Federation20171506201702(54)7083Copyright © Старовойтов В.В., 20172017Старовойтов В.В.Starovoitov V.V.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://inf.grid.by/jour/article/view/213

Описываются новые свойства сингулярных чисел, вычисляемых для матриц цифровых изо-бражений. Показано, что перестановка строк или столбцов матрицы и ее поворот на 90° не меняют множества сингулярных чисел, однако изменение значения одного элемента или перестановка местами двух элементов матрицы могут привести к изменению всего множества сингулярных чисел. Приводятся примеры повышения резкости и контраста изображений путем модификации множества сингулярных чисел.

The paper describes new properties of the singular matrix decomposition. It is shown that permutation of rows or columns of the matrix or matrix rotation by 90 degrees does not change the set of its singular numbers. However, variation the value of at least one matrix element or permutation of any two matrix elements leads to a modification of the whole set of the singular numbers. Examples of image sharpening and contrast enhancement by modification of the singular numbers are given.

ReferencesGolub, G. Calculating the singular values and pseudo-inverse of a matrix / G. Golub, W. Kahan // Journal of the Society for Industrial and Applied Mathematics. Series B: Numerical Analysis.– 1965. – Vol 2, no. 2. – P. 205–224.Golub, G. Calculating the singular values and pseudo-inverse of a matrix / G. Golub, W. Kahan // Journal of the Society for Industrial and Applied Mathematics. Series B: Numerical Analysis.– 1965. – Vol 2, no. 2. – P. 205–224.Demmel, J. Accurate singular values of bidiagonal matrices / J. Demmel, W. Kahan // SIAM Journal on Scientific and Statistical Computing. – 1990. – Vol. 11, no. 5. – P. 873–912.Demmel, J. Accurate singular values of bidiagonal matrices / J. Demmel, W. Kahan // SIAM Journal on Scientific and Statistical Computing. – 1990. – Vol. 11, no. 5. – P. 873–912.Gerbrands, J.J. On the relationships between SVD, KLT and PCA / J.J. Gerbrands // Pattern Recognition. – 1981. – Vol. 14, no. 1–6. – P. 375–381.Gerbrands, J.J. On the relationships between SVD, KLT and PCA / J.J. Gerbrands // Pattern Recognition. – 1981. – Vol. 14, no. 1–6. – P. 375–381.Moonen, M. SVD and signal processing: algorithms, applications and architectures /M. Moonen, B. De Moor. – Elsevier Science, 1995. – 485 р.Moonen, M. SVD and signal processing: algorithms, applications and architectures /M. Moonen, B. De Moor. – Elsevier Science, 1995. – 485 р.Хорн, Р. Матричный анализ / Р. Хорн, Ч. Джонсон ; пер. с англ. – М. : Мир, 1989. – 655 с.Хорн, Р. Матричный анализ / Р. Хорн, Ч. Джонсон ; пер. с англ. – М. : Мир, 1989. – 655 с.Jha, S.K. Denoising by singular value decomposition and its application to electronic nose data processing / S.K. Jha, R.D.S. Yadava // IEEE Sensors Journal. – 2011. – Vol. 11, no. 1. – P. 35–44.Jha, S.K. Denoising by singular value decomposition and its application to electronic nose data processing / S.K. Jha, R.D.S. Yadava // IEEE Sensors Journal. – 2011. – Vol. 11, no. 1. – P. 35–44.Liu, R. An SVD-based watermarking scheme for protecting rightful ownership / R. Liu, T. Tan // IEEE transactions on multimedia. – 2002. – Vol. 4, no. 1. – P. 121–128.Liu, R. An SVD-based watermarking scheme for protecting rightful ownership / R. Liu, T. Tan // IEEE transactions on multimedia. – 2002. – Vol. 4, no. 1. – P. 121–128.Andrews, H. Singular value decomposition (SVD) image coding / H. Andrews, C. Patterson // IEEE transactions on Communications. – 1976. – Vol. 24, no. 4. – P. 425–432.Andrews, H. Singular value decomposition (SVD) image coding / H. Andrews, C. Patterson // IEEE transactions on Communications. – 1976. – Vol. 24, no. 4. – P. 425–432.Narwaria, M. SVD-based quality metric for image and video using machine learning / M. Narwaria, W. Lin // IEEE Trans. on Systems, Man, and Cybernetics. Part B. – 2012. – Vol. 42, no. 2. – P. 347–364.Narwaria, M. SVD-based quality metric for image and video using machine learning / M. Narwaria, W. Lin // IEEE Trans. on Systems, Man, and Cybernetics. Part B. – 2012. – Vol. 42, no. 2. – P. 347–364.Targhi, A.T. Clustering of singular value decomposition of image data with applications to texture classification / A.T. Targhi, A. Shademan // Proc. of Intern. Conf. on Visual Communications and Image Processing. – Lugano, Switzerland, 2003. – Vol. 5150. – P. 972–979.Targhi, A.T. Clustering of singular value decomposition of image data with applications to texture classification / A.T. Targhi, A. Shademan // Proc. of Intern. Conf. on Visual Communications and Image Processing. – Lugano, Switzerland, 2003. – Vol. 5150. – P. 972–979.Zhang, D. A new face recognition method based on svd perturbation for single example image per person / D. Zhang, S. Chen, Z.-H. Zhou // Applied Mathematics and Computation. – 2005. – Vol. 163, no. 2. – P. 95–907.Zhang, D. A new face recognition method based on svd perturbation for single example image per person / D. Zhang, S. Chen, Z.-H. Zhou // Applied Mathematics and Computation. – 2005. – Vol. 163, no. 2. – P. 95–907.No-reference image blur index based on singular value curve / Q. Sang [et al.] // Journal of Visual Communication and Image Representation. – 2014. – Vol. 25, no. 7. – P. 1625–1630.No-reference image blur index based on singular value curve / Q. Sang [et al.] // Journal of Visual Communication and Image Representation. – 2014. – Vol. 25, no. 7. – P. 1625–1630.Singular value decomposition based sample diversity and adaptive weighted fusion for face recognition / G. Zhang [et al.] // Digital Signal Processing. – 2017. – Vol. 62, no. 3. – P. 150–156.Singular value decomposition based sample diversity and adaptive weighted fusion for face recognition / G. Zhang [et al.] // Digital Signal Processing. – 2017. – Vol. 62, no. 3. – P. 150–156.Карчевский, Е.М. Лекции по линейной алгебре и аналитической геометрии /Е.М. Карчевский, М.М. Карчевский. – Казань : Казанский гос. ун-т, 2014. – 352 с.Карчевский, Е.М. Лекции по линейной алгебре и аналитической геометрии /Е.М. Карчевский, М.М. Карчевский. – Казань : Казанский гос. ун-т, 2014. – 352 с.Белоусов, И.В. Матрицы и определители : учеб. пособие по линейной алгебре. – 2-е изд. – Кишинев : Изд-во Ин-та прикладной физики АН Республики Молдова, 2006. – 101 с.Белоусов, И.В. Матрицы и определители : учеб. пособие по линейной алгебре. – 2-е изд. – Кишинев : Изд-во Ин-та прикладной физики АН Республики Молдова, 2006. – 101 с.

The authors declare that there are no conflicts of interest present.