Abstract
Fractal image compression is one of the efficient structure-based methods in applications where images are compressed only once but decoded several times due to its resolution-independent feature and fast reconstruction time. However, it has high computational complexity restricting practical use most of the time. Although several methods have been developed to speed up the compression process, these do not satisfy the compression time or the decoded image quality requirements. The affine transforms of image blocks used in fractal coding require a huge number of multiplications and additions and are very expensive in computation that may also slow down the compression process. This paper presents a novel fractal image compression using a fast affine transform and hierarchical classification scheme. The applied affine transform computation algorithm of image blocks uses relationships among neighboring pixels of transformed image block that significantly reduces the number of multiplication and addition operations. Then, this strategy with hierarchical classification and class-wise domain sorting is applied in fractal coding with quad-tree and horizontal vertical partitioning schemes to reduce compression time. Experimental results show that the quad-tree-based fractal coding with the proposed scheme can significantly speed up the compression process keeping image quality and compression ratio almost unchanged.
Similar content being viewed by others
References
Wallace, GK.: The JPEG still picture compression standard. IEEE Trans. Consum. Electron. 38(1), xviii–xxxiv (1992)
Barnsley, M.F.: Fractals Everywhere, 2nd edn. Academic Press, New York (1993)
Jacquin, A.E.: Image coding based on a fractal theory of iterated contractive image transformations. IEEE Trans. Image Process. 1(1), 18–30 (1992)
Barnsley, M.F., Jacquin, A.E.: Application of recurrent iterated function systems to images. Proc SPIE 1001, 122–131 (1988)
Fisher, Y.: Fractal Image Compression: Theory and Application. Springer, New York (1995)
Wang, S.S., Tsai, S.L.: Automatic image authentication and recovery using fractal code embedding and image inpainting. Pattern Recognit. 41(2), 701–712 (2008)
Lin, T.K., Yeh, S.L.: Encrypting image by assembling the fractal image addition method and the binary encoding method. Opt. Commun. 285(9), 2335–2342 (2012)
Tang, X., Qu, C.: Facial image recognition based on fractal image encoding. Bell Labs Tech. J. 15(1), 209–214 (2010)
Ghazel, M., Freeman, G.H., Vrscay, E.R.: Fractal image denoising. IEEE Trans. Image Process. 12(12), 1560–1578 (2003)
Papathomas, T.V., Julesz, B.: Animation with fractals from variations on the mandelbrot set. Vis. Comput. 3, 23–26 (1987). https://doi.org/10.1007/BF02153648
Davern, P., Scott, M.: Fractal based image steganography. In: Anderson, R. (ed.) Information Hiding, pp. 279–294. Springer, Berlin (1996)
Liao, X., Wen, Q., Song, T., Zhang, J.: Quantum steganography with high efficiency with noisy depolarizing channels. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E96.A(10), 2039–2044 (2013). https://doi.org/10.1587/transfun.E96.A.2039
Liao, X., Wen, Q., Zhang, J.: Improving the adaptive steganographic methods based on modulus function. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. e96.A(12), 2731–2734 (2013). https://doi.org/10.1587/transfun.E96.A.2731
Liao, X., Shu, C.: Reversible data hiding in encrypted images based on absolute mean difference of multiple neighboring pixels. J. Vis. Commun. Image Represent. 28, 21–27 (2015). https://doi.org/10.1016/j.jvcir.2014.12.007
Truong, T.K., Kung, C.M., Jeng, J.H., Hsieh, M.L.: Fast fractal image compression using spatial correlation. Chaos Solitons Fractals 22(5), 1071–1076 (2004)
He, C., Xu, X., Li, G.: Improvement of fast algorithm based on correlation coefficients for fractal image encoding. Comput. Simul. 12(4), 60–63 (2005)
Wang, X., Wang, Y., Yun, J.: An improved fast fractal image compression using spatial texture correlation. Chin. Phys. B 20(10), 104202-1-104202–11 (2011)
Wang, J., Zheng, N.: A novel fractal image compression scheme with block classification and sorting based on Pearson’s correlation coefficient. IEEE Trans. Image Process. 22(9), 3690–3702 (2013)
Wang, J., Cheg, P.: Fast sparse fractal image compression. PLoS ONE 12(9), e0184408 (2017)
Zhou, Y., Zhang, C., Zhang, Z.: An efficient fractal image coding algorithm using unified feature and DCT. Chaos Solitons Fractals 39(4), 1823–1830 (2009)
Schwartz, W.R., Pedrini, H.: Improved fractal image compression based on robust feature descriptors. Int. J. Image Graph 11(4), 571–587 (2011)
Lai, A.C., Lam, K., Siu, W.: A fast fractal image coding based on kick-out and zero contrast conditions. IEEE Trans. Image Process. 12(11), 1398–1403 (2003)
Chen, H.N., Chung, K.L., Hung, J.E.: Novel fractal image encoding algorithm using normalized one-norm and kick-out condition. Image Vis. Comput. 28(3), 518–525 (2010)
Jeng, J., Tseng, C., Hsieh, J.: Study on huber fractal image compression. IEEE Trans. Image Process. 18(5), 995–1003 (2009)
Lin, Y.: Robust estimation of parameter for fractal inverse problem. Comput. Math. Appl. 60, 2099–2108 (2010)
Lu, J., Ye, Z., Zou, Y.: Huber fractal image coding based on a fitting plane. IEEE Trans. Image Process. 22(1), 134–145 (2013)
Distasi, R., Nappi, M., Riccio, D.: A range/domain approximation error-based approach for fractal image compression. IEEE Trans. Image Process. 15(1), 89–97 (2006)
Xing, C., Ren, Y., Li, X.: A hierarchical classification matching scheme for fractal image compression. In: 2008 Congress on Image and Signal Processing, IEEE, pp. 283–286 (2008)
Kovacs, T.: A fast classification based method for fractal image encoding. Image Vis. Comput. 26(8), 1129–1136 (2008)
Wang, X., Zhang, D.: Discrete wavelet transform-based simple range classification strategies for fractal image coding. Nonlinear Dyn. 75(3), 439–448 (2014)
Bhattacharya, N., Roy, S., Nandi, U., Banerjee, S.: Fractal image compression using hierarchical classification of sub-images. In: 10th International Conference on Computer Vision Theory and Applications (VISAPP 2015), SCITEPRESS, pp. 46–53 (2015)
Nandi, U., Mandal, J.K.: Efficiency of adaptive fractal image compression with archetype classification and its modifications. Int. J. Comput. Appl. 38(2–3), 156–163 (2016)
Nandi, U., Mandal, J.K.: Fractal image compression with adaptive quardtree partitioning and archetype classification. In: IEEE International Conference on Research in Computational Intelligence and Communication Networks (ICRCICN 2015), Kolkata, pp. 56–60 (2015). https://doi.org/10.1109/ICRCICN.2015.7434209
Nandi, U.: An adaptive fractal-based image coding with hierarchical classification strategy and its modifications. Innov. Syst. Softw. Eng. 15(1), 35–42 (2019)
Nandi, U., Mandal, J.K.: A novel hierarchical classification scheme for adaptive quardtree partitioning based fractal image coding. In: Mandal, J.K., Sinha, D. (eds) Social Transformation—Digital Way. CSI 2018. Communications in Computer and Information Science, Springer Singapore, Kolkata, vol. 836, pp. 603–615 (2018)
Wang, X.Y., Wang, Y.X., Yun, J.J.: An improved no-search fractal image coding method based on a fitting plane. Image Vis. Comput. 28(8), 1303–1308 (2010)
Gupta, R., Mehrotra, D., Tyagi, R.K.: Adaptive searchless fractal image compression in DCT domain. Imaging Sci. J. 64(7), 374–380 (2016)
Zhao, Y., Yuan, B.: A new affine transformation: its theory and application to image coding. IEEE Trans. Circuits Syst. Video Technol. 8(3), 269–274 (1998)
Liu, S., Fu, W., Liqiang, H., et al.: Distribution of primary additional errors in fractal encoding method. Multimed. Tools Appl. 76, 5787–5802 (2017)
Liu, S., Zhang, Z., Qi, L., et al.: A fractal image encoding method based on statistical loss used in agricultural image compression. Multimed. Tools Appl. 75, 15525–15536 (2016)
Liu, S., Pan, Z., Cheng, X.: A novel fast fractal image compression method based on distance clustering in high dimensional sphere surface. Fractals 25(4), 1740004-1-1740004–11 (2017)
Roy, S., Kumar, S., Chanda, B., et al.: Fractal image compression using upper bound on scaling parameter. Chaos Solitons Fractals 106, 16–22 (2017)
Menassel, R., Nini, B., Mekhaznia, T.: An improved fractal image compression using wolf pack algorithm. J. Exp. Theor. Artif. Intell. 30(3), 429–439 (2018). https://doi.org/10.1080/0952813X.2017.1409281
Nandi, U.: Fractal image compression with adaptive quadtree partitioning and non-linear affine map. Multimed. Tools Appl. 79, 26345–26368 (2020)
Al Sideiri, A., Alzeidi, N., Al Hammoshi, M.: Cuda implementation of fractal image compression. J. Real-Time Image Proc. 17, 1375–1387 (2020). https://doi.org/10.1007/s11554-019-00894-7
Menassel, R., Gaba, I., Titi, K.: Introducing bat inspired algorithm to improve fractal image compression. Int. J. Comput. Appl. 42(7), 697–704 (2020). https://doi.org/10.1080/1206212X.2019.1638631
Nandi, U., Laya, B., Ghorai, A., Singh, M.M.: Three-level hierarchical classification scheme: its application to fractal image compression technique. In: Satapathy, S.C., Zhang, Y.D., Bhateja, V., Majhi, R. (eds.) Intelligent Data Engineering and Analytics, pp. 123–132. Springer, Singapore (2021)
Nandi, U., Ghorai, A., Laya, B., Singh, M.M.: A fast partitioning strategy: its application to fractal image coding. In: Sherpa, K.S., Bhoi, A.K., Kalam, A., Mishra, M.K. (eds.) Advances in Smart Grid and Renewable Energy, pp. 237–247. Springer, Singapore (2021)
Svynchuk, O., Barabash, O., Nikodem, J., Kochan, R., Laptiev, O.: Image compression using fractal functions. Fractal Fract. (2021). https://doi.org/10.3390/fractalfract5020031
Lee, S., Lee, G., Gang, E.S., Kim, W.: Fast affine transform for real-time machine vision applications. In: Intelligent Computing, Springer, Berlin, pp. 1180–1190 (2006)
Weber, G.: USC-SIPI Image Database: Version 4. University Southern California, Los Angeles, CA, USA, Department of Electrical Engineering-System, Technical Report (1993)
Wang, Q., Bi, S.: Prediction of the PSNR quality of decoded images in fractal image coding. Math. Probl. Eng. 2016, 1–13 (2016)
Hore, A., Ziou, D.: Image quality metrics: PSNR versus SSIM. In: International Conference on Pattern Recognition Proceedings, IEEE, pp. 2366–2369 (2010)
Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P.: Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 600–612 (2004)
Acknowledgements
We’d like to thank to the Dept. of Computer Science, Vidyasagar University, Paschim Medinipur, for providing infrastructure.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Nandi, U. Fractal image compression using a fast affine transform and hierarchical classification scheme. Vis Comput 38, 3867–3880 (2022). https://doi.org/10.1007/s00371-021-02226-y
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00371-021-02226-y