Mathematical Preliminaries

  • Rohit M. Thanki
  • Ashish Kothari


This chapter covers various mathematical theories which can be used in data compression techniques. The information and properties of various image transforms are discussed in this chapter. The description of compressive sensing (CS) theory, image compression standard for medical images, and performance evaluation parameters for compression method are also covered in this chapter.


Fourier transform Compression ratio Compressive sensing DCT JPEG SVD Wavelet transform 


  1. 1.
    Fourier, J. (1822). Analytical theory of heat, by M. Fourier. At Firmin Didot, father, and son.Google Scholar
  2. 2.
    Shih, F. Y. (2017). Digital watermarking and steganography: Fundamentals and techniques. Boca Raton, FL: CRC Press.CrossRefGoogle Scholar
  3. 3.
    Brigham, E. O., & Brigham, E. O. (1988). The fast Fourier transform and its applications (Vol. 448). Englewood Cliffs, NJ: Prentice Hall.zbMATHGoogle Scholar
  4. 4.
    Jain, A. (1999). Fundamentals of digital image processing (pp. 151–153). New Jersey: Prentice Hall Inc.Google Scholar
  5. 5.
    Lopez, R., & Boulgouris, N. (2010). Compressive sensing and combinatorial algorithms for image compression. A project report. London: King’s College.Google Scholar
  6. 6.
    Gonzalez, R. C., & Woods, R. E. (2002). Digital image processing. Upper Saddle River, NJ: Pearson-Prentice-Hall.Google Scholar
  7. 7.
    Mertins, A., & Mertins, D. A. (1999). Signal analysis: Wavelets, filter banks, time-frequency transforms and applications. John Wiley & Sons, Inc. USA.Google Scholar
  8. 8.
    Thanki, R. M., Dwivedi, V. J., & Borisagar, K. R. (2018). Multibiometric watermarking with compressive sensing theory: Techniques and applications. Springer, Germany.CrossRefGoogle Scholar
  9. 9.
    Candès, E. J. (2006, August). Compressive sampling. In Proceedings of the international congress of mathematicians (Vol. 3, pp. 1433–1452). Madrid, Spain.Google Scholar
  10. 10.
    Donoho, D. L. (2006). Compressed sensing. IEEE Transactions on Information Theory, 52(4), 1289–1306.MathSciNetCrossRefGoogle Scholar
  11. 11.
    Baraniuk, R. G. (2007). Compressive sensing [lecture notes]. IEEE Signal Processing Magazine, 24(4), 118–121.CrossRefGoogle Scholar
  12. 12.
    Tropp, J. A., & Gilbert, A. C. (2007). Signal recovery from random measurements via orthogonal matching pursuit. IEEE Transactions on information theory, 53(12), 4655–4666. Dai, W., & Milenkovic, O. (2009). Subspace pursuit for compressive sensing signal reconstruction. IEEE Transactions on Information Theory, 55(5), 2230–2249.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Needell, D. (2009). Topics in compressed sensing Ph. D (Doctoral dissertation, Dissertation).Google Scholar
  14. 14.
    Duarte, M. F., & Eldar, Y. C. (2011). Structured compressed sensing: From theory to applications. IEEE Transactions on Signal Processing, 59(9), 4053–4085.MathSciNetCrossRefGoogle Scholar
  15. 15.
    Uthayakumar, J., Vengattaraman, T., & Dhavachelvan, P. (2018). A survey on data compression techniques: From the perspective of data quality, coding schemes, data type, and applications. Journal of King Saud University-Computer and Information Sciences.Google Scholar
  16. 16.
    Chandler, D. M., & Hemami, S. S. (2007). VSNR: A wavelet-based visual signal-to-noise ratio for natural images. IEEE Transactions on Image Processing, 16(9), 2284–2298.MathSciNetCrossRefGoogle Scholar
  17. 17.
    Elshaikh, M., Fadzil, M. F. M., Kamel, N., & Isa, C. M. N. C. (2012, March). Weighted signal-to-noise ratio average routing metric for dynamic sequence distance vector routing protocol in mobile ad-hoc networks. In Signal Processing and its Applications (CSPA), 2012 IEEE 8th International Colloquium on (pp. 329–334). IEEE.Google Scholar
  18. 18.
    Damera-Venkata, N., Kite, T. D., Geisler, W. S., Evans, B. L., & Bovik, A. C. (2000). Image quality assessment based on a degradation model. IEEE Transactions on Image Processing, 9(4), 636–650.CrossRefGoogle Scholar
  19. 19.
    Wang, Z., & Bovik, A. C. (2002). A universal image quality index. IEEE Signal Processing Letters, 9(3), 81–84.CrossRefGoogle Scholar
  20. 20.
    Wang, Z., Simoncelli, E. P., & Bovik, A. C. (2003, November). Multiscale structural similarity for image quality assessment. In the Thirty-Seventh Asilomar Conference on Signals, Systems & Computers, 2003 (Vol. 2, pp. 1398–1402). IEEE.Google Scholar
  21. 21.
    Wang, Z., Bovik, A. C., Sheikh, H. R., & Simoncelli, E. P. (2004). Image quality assessment: From error visibility to structural similarity. IEEE Transactions on Image Processing, 13(4), 600–612.CrossRefGoogle Scholar
  22. 22.
    Zhang, L., Zhang, L., Mou, X., & Zhang, D. (2011). FSIM: A feature similarity index for image quality assessment. IEEE Transactions on Image Processing, 20(8), 2378–2386.MathSciNetCrossRefGoogle Scholar
  23. 23.
    Zhang, L., & Li, H. (2012, September). SR-SIM: A fast and high performance IQA index based on spectral residual. In Image Processing (ICIP), 2012 19th IEEE International Conference on (pp. 1473–1476). IEEE.Google Scholar
  24. 24.
    Reisenhofer, R., Bosse, S., Kutyniok, G., & Wiegand, T. (2018). A Haar wavelet-based perceptual similarity index for image quality assessment. Signal Processing: Image Communication, 61, 33–43.Google Scholar
  25. 25.
    Berger, T. (1971). Rate distortion theory: A mathematical basis for data compression. London: Prentice Hall.zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Rohit M. Thanki
    • 1
  • Ashish Kothari
    • 2
  1. 1.Faculty of Technology and EngineeringC. U. Shah UniversityWadhwan CityIndia
  2. 2.Atmiya UniversityRajkotIndia

Personalised recommendations