An image compression method based on Ramanujan Sums and measures of central dispersion
DOI:
https://doi.org/10.23755/rm.v41i0.683Keywords:
Ramanujan Sum cq(n), lossy image compression, mean, standard deviationAbstract
This paper introduces a simple lossy image compression method based on Ramanujan Sums cq(n) and the statistical measures of numerical data such as mean and standard deviation. The Ramanujan Sum cq(n) has been used in digital signal processing for a variety of applications nowadays. Some of them include the recently developed image kernels for edge detection, extraction of periodicity from signals, etc. The presented compression algorithm is an extension of the edge detection algorithm using an integer image kernel based on Ramanujan Sums. We propose a block-based compression algorithm that detects edges in the images using this image kernel and then compresses the image by storing kernel operation values, the mean and standard deviation for each block instead of pixel values. The proposed method has the advantage of low computational complexity and shows its ability in fast reconstruction and high compression that can be achieved for different block sizes.References
References
Tom M Apostol. Analytic number theory. 1976.
Eckford Cohen. A class of arithmetical functions. Proceedings of the National Academy of Sciences of the United States of America, 41(11):939, 1955.
Eckford Cohen. Representations of even functions (mod r), arithmetical identities. Duke Mathematical Journal, 25(3):401–421, 1958.
Murray Eden, Michael Unser, and Riccardo Leonardi. Polynomial representation of pictures. Signal Processing, 10(4):385–393, 1986.
Robert Gallager. Variations on a theme by huffman. IEEE Transactions on Information Theory, 24(6):668–674, 1978.
Rafael C Gonzalez and Richard E Woods. Digital image processing. Nueva Jersey, 2008.
Vivek K Goyal. Theoretical foundations of transform coding. IEEE Signal Processing Magazine, 18(5):9–21, 2001.
Anil K Jain. Fundamentals of digital image processing. Prentice-Hall, Inc., 1989.
Madhuri A Joshi. Digital image processing: An algorithmic approach. PHI Learning Pvt. Ltd., 2018.
P Krishnaprasad and Ajeesh Ramanujan. Ramanujan sums based image kernels for computer vision. In 2016 International Conference on Electrical, Electronics, and Optimization Techniques (ICEEOT), pages 835–839. IEEE, 2016.
Christoph H Lampert. Kernel methods in computer vision. Now Publishers Inc,2009.
Francesca Odone, Annalisa Barla, and Alessandro Verri. Building kernels from binary strings for image matching. IEEE Transactions on Image Processing, 14(2):169-180,2005.
William B Pennebaker and Joan L Mitchell. JPEG: Still image data compression
standard. Springer Science & Business Media, 1992.
Srinivasa Ramanujan. On certain trigonometrical sums and their applications in the theory of numbers. Trans. Cambridge Philos. Soc, 22(13):259–276, 1918.
I Sadeh. Polynomial approximation of images. Computers & Mathematics with Applications, 32(5):99–115, 1996.
S Sajikumar and AK Anilkumar. Image compression using chebyshev polynomial surface fit. Int. J. Pure Appl. Math. Sci, 10:15–27, 2017.
David Salomon. A concise introduction to data compression. Springer Science & Business Media, 2007.
Khalid Sayood. Introduction to data compression. Newnes, 2012.
David Taubman and Michael Marcellin. JPEG2000 Image Compression Fundamentals, Standards and Practice: Image Compression Fundamentals, Standards and Practice, volume 642. Springer Science & Business Media, 2012.
PP Vaidyanathan. Ramanujan sums in the context of signal processingpart i: Fundamentals. IEEE transactions on signal processing, 62(16):4145–4157, 2014.
Ian H Witten, Radford M Neal, and John G Cleary. Arithmetic coding for data compression. Communications of the ACM, 30(6):520–540, 1987.
Haichao Zhang, Jianchao Yang, Yanning Zhang, and Thomas S Huang. Non-local kernel regression for image and video restoration. In European Conference onComputer Vision, pages 566–579. Springer, 2010.
Downloads
Published
Issue
Section
License
This journal provides immediate open access to its content on the principle that making research freely available to the public supports a greater global exchange of knowledge. The journal is committed to real and immediate open access for academic work. All of the journal articles are free to access immediately from the date of publication. There are no charge for readers to download articles and reviews for their own scholarly use. Benefits of open access for authors, include:
- Free access for all users worldwide
- Increased visibility and readership
- Rapid publication
- No spatial constraints
- Authors can share their research papers on their personal web-pages (i.e. ResearchGate, etc.)
Ratio Mathematica publishes open access articles under the terms of the Creative Commons Attribution (CC BY) License. The Creative Commons Attribution License (CC-BY) allows users to copy, distribute and transmit an article, adapt the article and make commercial use of the article. The CC BY license permits commercial and non-commercial re-use of an open access article, as long as the author is properly attributed. Copyright on any research article published by Ratio Mathematica is retained by the author(s). Authors grant Ratio Mathematica a license to publish the article and identify itself as the original publisher. Authors also grant any third party the right to use the article freely as long as its original authors, citation details and publisher are identified.
For further information visit: https://creativecommons.org/licenses/by/3.0/. The following logo will appear in any paper that will be published from the volume number 31/2016 onwards.
