# Layers Image Compression and Reconstruction by Fuzzy Transforms

Chapter

## Abstract

Recently we proved that fuzzy transforms ($$F$$-transforms) are useful in coding/decoding images, showing that the resulting peak-signal-to-noise-ratio (PSNR) is better than the one obtained using fuzzy relation equations and comparable with that obtained using the JPEG method. Recently some authors have explored a new image compression/reconstruction technique: the range interval [0,1] is partitioned in a finite number of subintervals of equal width in such a way that each subinterval corresponds to a image-layer of pixels. Each image-layer is coded using the direct $$F$$-transform, and afterwards all the inverse $$F$$-transforms are put together to reconstruct the whole initial image. We modify slightly this process: the pixels of the original image are normalized [15] with respect to the length of the gray scale, and thus are seen as a fuzzy matrix $$R$$, which we divide into (possibly square) submatrices $$R_{B}$$, called blocks. Hence we divide [0,1] into subintervals by adopting the quantile method, so that each subinterval contains the same number of normalized pixels of every block $$R_{B}$$, then we apply the $$F$$-transforms to each block-layer. In terms of quality of the reconstructed image, our method is better than that one based on the standard $$F$$-transforms.

## References

1. 1.
Di Martino, F., Loia, V., Sessa, S.: A method for coding/decoding images by using fuzzy relation equations. In: Fuzzy Sets and Systems (IFSA 2003). Lecture Notes in Artificial Intelligence, vol. 2715, pp. 436–441. Springer, Berlin (2003)Google Scholar
2. 2.
Di Martino, F., Loia, V., Sessa, S.: A method in the compression/decompression of images using fuzzy equations and fuzzy similarities. In: Proceedings of the 10th IFSA World Congress, Istanbul, pp. 524–527 (2003)Google Scholar
3. 3.
Di Martino, F., Nobuhara, H., Sessa, S.: Eigen fuzzy sets and image information retrieval. In: Proceedings of the International Conference on Fuzzy Information Systems, Budapest, vol. 3, pp. 1385–1390 (2004)Google Scholar
4. 4.
Di Martino, F., Sessa, S.: Digital watermarking in coding/decoding processes with fuzzy relation equations. Soft Comput. 10, 238–243 (2006)
5. 5.
Di Martino, F., Sessa, S.: Compression and decompression of images with discrete fuzzy transforms. Inf. Sci. 177, 2349–2362 (2007)
6. 6.
Di Martino, F., Loia, V., Perfilieva, I., Sessa, S.: An image coding/decoding method based on direct and inverse fuzzy transforms. Int. J. Approx. Reason. 48(1), 110–131 (2008)
7. 7.
Di Martino, F., Loia, V., Sessa, S.: Direct and inverse fuzzy transforms for coding/ decoding color images in YUV space. J. Uncertain Syst. 2(1), 11–30 (2009)Google Scholar
8. 8.
Di Martino, F., Loia, V., Sessa, S.: Multidimensional fuzzy transforms for attribute dependencies. In: Proceedings of IFSA/EUSFLAT 2009, Lisbon, pp. 53–57 (2009)Google Scholar
9. 9.
Di Martino, F., Loia, V., Sessa, S.: A segmentation method for images compressed by fuzzy transforms. Fuzzy Sets Syst. 161, 56–74 (2010)
10. 10.
Di Martino, F., Loia, V., Sessa, S.: Fuzzy transform method and attribute dependency in data analysis. Inf. Sci. 180, 493–505 (2010)
11. 11.
Di Martino, F., Loia, V., Sessa, S.: Fuzzy transforms for compression and decompression of color videos. Inf. Sci. 180, 3914–3931 (2010)
12. 12.
Di Nola, A., Pedrycz, W., Sanchez, E., Sessa, S.: Fuzzy Relation Equations and Their Applications to Knowledge Engineering. Kluwer Academic Publishers, Dordrecht (1989)
13. 13.
Gronscurth, H.M.: Fuzzy data compression for energy optimization models. Energy 23(1), 1–9 (1998)
14. 14.
Hirota, K., Pedrycz, W.: Data compression with fuzzy relational equations. Fuzzy Sets Syst. 126, 325–335 (2002)
15. 15.
Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht (2000)
16. 16.
Loia, V., Pedrycz, W., Sessa, S.: Fuzzy relation calculus in the compression and decompression of fuzzy relations. Int. J. Image Graph. 2, 1–15 (2002)
17. 17.
Loia, V., Sessa, S.: Fuzzy relation equations for coding/decoding processes of images and videos. Inf. Sci. 171, 145–172 (2005)
18. 18.
Nobuhara, H., Pedrycz, W., Hirota, K.: Fast solving method of fuzzy relational equation and its application to lossy image compression. IEEE Trans. Fuzzy Syst. 8(3), 325–334 (2000)Google Scholar
19. 19.
Nobuhara, H., Pedrycz, W., Hirota, K.: Relational image compression: optimizations through the design of fuzzy coders and YUV color space. Soft Comput. 9(6), 471–479 (2005)
20. 20.
Nobuhara, H., Hirota, K., Di Martino, F., Pedrycz, W., Sessa, S.: Fuzzy relation equations for compression/decompression processes of colour images in the RGB and YUV colour spaces. Fuzzy Optim. Decis. Mak. 4(3), 235–246 (2005)
21. 21.
Nobuhara, H., Hirota, K., Pedrycz, W., Sessa, W.: A motion compression/ reconstruction method based on max t-norm composite fuzzy relational equations. Inf. Sci. 176, 2526–2552 (2006)
22. 22.
Novak, V., Perfilieva, I.: Fuzzy transform in the analysis of data. Int. J. Approx. Reason. 48(1), 36–46 (2008)
23. 23.
Perfilieva, I.: Fuzzy transforms: application to reef growth problem. In: Demicco, R.B., Klir, G.J. (eds.) Fuzzy Logic in Geology, pp. 275–300. Academic Press, Amsterdam (2003)Google Scholar
24. 24.
Perfilieva, I.: Fuzzy transforms. Fuzzy Sets Syst. 157, 993–1023 (2006)
25. 25.
Perfilieva, I., Chaldeeva, E.: Fuzzy transformation. In: Proceedings of the 9th IFSA, World Congress and 20th NAFIPS International Conference, pp. 1946–1948. Vancouver (2001)Google Scholar
26. 26.
Perfilieva, I., Novak, V., Pavliska, V., Dvork, A., Stepnicka, M.: Analysis and prediction of time series using fuzzy transform. In: Proceedings World Congress on Computational Intelligence/FUZZ-IEEE, pp. 3875–3879. Hong Kong (2008)Google Scholar
27. 27.
Perfilieva, I., Valek, R.: Fuzzy approach to data compression. In: Proceedings 8th Czech-Japan Seminar on Data Analysis and Decision Making under Uncertainty, pp. 91–100. Praga (2005)Google Scholar
28. 28.
Perfilieva, I., Pavliska, V., Vajgl, M., De Baets, B.: Advanced image compression on the basis of fuzzy transforms. In: Proceedings of IPMU, pp. 1167–1174. Malaga (2008)Google Scholar
29. 29.
Perfilieva, I.: Fuzzy transforms and their applications to image compression. In: Bloch, I., Petrosino, A.,Tettamanzi, A. (eds.) Fuzzy Logic and Applications. LNAI, vol. 3849, pp. 19–31. Springer, Heidelberg (2006)Google Scholar
30. 30.
Perfilieva, I., De Baets, B.: Fuzzy transforms of monotone functions with application to image compression. Inf. Sci. 180, 3304–3315 (2010)
31. 31.
Perfilieva, I.: Fuzzy transform in image compression and fusion. Acta Mathematica Universitatis Ostraviensis 15, 27–37 (2007)
32. 32.
Stepnicka, M., Valasek, R.: Fuzzy transforms and their application to wave equation. J. Electr. Eng. 55(12), 7–10 (2004)
33. 33.
Stepnicka, M., Pavliska, V., Novak, V., Perfilieva, I.: Time series analysis and prediction based on fuzzy rules and the fuzzy transform. In: Proceedings of IFSA/EUS- FLAT, pp. 1601–1605. Lisbon (2009)Google Scholar
34. 34.
The International Telegraph And Telephone Consultative Committee. Information Technology—Digital Compression And Coding of Continuous-Tone Still Images-Requirements and Guidelines, Recommendation T81 (1992)Google Scholar