Skip to main content
SpringerLink
Log in

Sampling in Image Representation and Compression

  • Chapter
  • First Online:
New Perspectives on Approximation and Sampling Theory

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

In recent years, interest has grown in the study of sparse solutions of underdetermined systems of linear equations because of their many and potential applications [11]. In particular, these types of solutions can be used to describe images in a compact form, provided one is willing to accept an imperfect representation.

Dedicated to Paul l. Butzer on the Occasion of his 85th Birthday

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Au-Yeung, E., Benedetto, J.J.: Balayage and short time Fourier transform frames. Proceedings of SampTA (2013)

    Google Scholar 

  2. Austin, D.: What is…JPEG? Notices of the AMS 55(2), 226–229 (2008)

    MathSciNet  Google Scholar 

  3. Benedetto, J.J.: Harmonic Analysis and Applications. CRC Press, Boca Raton (1997)

    Google Scholar 

  4. Benedetto, J.J., Frazier, M.W.: Wavelets: Mathematics and Applications. CRC Press, Boca Raton (1994)

    MATH  Google Scholar 

  5. Benedetto, J.J., Nava-Tudela, A.: Frame estimates for OMP. Preprint (2014)

    Google Scholar 

  6. Benedetto, J.J., Powell, A.M., Yilmaz, O.: Sigma-delta (\(\Sigma \Delta \)) quantization and finite frames. IEEE Trans. Inform. Theor. 52(5), 1990–2005 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Beurling A.: The Collected Works of Arne Beurling. Vol. 2. Harmonic Analysis. Birkhäuser, Boston (1989)

    MATH  Google Scholar 

  8. Beurling, A., Malliavin, P.: On Fourier transforms of measures with compact support. Acta Math. 107, 291–309 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  9. Beurling, A., Malliavin, P.: On the closure of characters and the zeros of entire functions. Acta Math. 118, 79–93 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  10. Briggs, W.L., Henson, V.E.: The DFT, an Owner’s Manual for the Discrete Fourier Transform. SIAM, Philadelphia (1995)

    Book  MATH  Google Scholar 

  11. Bruckstein, A.M., Donoho, D.L., Elad, M.: From sparse solutions of systems of equations to sparse modeling of signals and images. SIAM Rev. 51(1), 34–81 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  12. Candy, J.C., Temes, G.C. (eds.): Oversampling Delta-Sigma Data Converters. IEEE Press, New York (1992)

    Google Scholar 

  13. Casazza, P.G., Kovačević, J.: Uniform tight frames with erasures. Adv. Comput. Math. 18(2–4), 387–430 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Christensen, O.: An Introduction to Frames and Riesz Bases. Springer-Birkhäuser, New York (2003)

    Book  MATH  Google Scholar 

  15. Christopoulos, C., Skodras, A., Ebrahimi, T.: The JPEG 2000 still image coding system: an overview. IEEE Trans. Cons. Electron. 46(4), 1103–1127 (2000)

    Article  Google Scholar 

  16. Donoho, D., Johnstone, I., Rousseeuw, P., Stahel, W.: Discussion: projection pursuit. Ann. Stat. 13(2), 496–500 (1985)

    Article  Google Scholar 

  17. Duffin, R.J., Schaeffer, A.C.: A class of nonharmonic Fourier series. Trans. Amer. Math. Soc. 72, 341–366 (1952)

    Article  MATH  MathSciNet  Google Scholar 

  18. Gleick, J.: The Information: A History, a Theory, a Flood. Pantheon Books, New York (2011)

    Google Scholar 

  19. Gray, R.M., Neuhoff, D.L.: Quantization. IEEE Trans. Inform. Theor. 44(6), 2325–2383 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  20. Gröchenig, K.: Foundations of time-frequency analysis. In: Applied and Numerical Harmonic Analysis. Birkhäuser Boston Inc., Boston (2001)

    Book  MATH  Google Scholar 

  21. Herman, M.A., Strohmer, T.: High-resolution radar via compressed sensing. IEEE Trans. Signal Process. 57(6), 2275–2284 (2009)

    Article  MathSciNet  Google Scholar 

  22. Huber, P.J.: Projection pursuit. Ann. Stat. 13(2), 435–475 (1985)

    Article  MATH  Google Scholar 

  23. Huffman, W.C., Vera Pless: Fundamentals of Error-Correcting Codes. Cambridge University Press, New York (2010)

    Google Scholar 

  24. Jaffard, S.: A density criterion for frames of complex exponentials. Michigan Math. J. 38, 339–348 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  25. Landau, H.J.: Necessary density conditions for sampling and interpolation of certain entire functions. Acta Math. 117, 37–52 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  26. Mallat, S.G.: A Wavelet Tour of Signal Processing. Academic, San Diego (1998)

    MATH  Google Scholar 

  27. Mallat, S.G., Zhang, Z.: Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Proc. 41(12), 3397–3415 (1993)

    Article  MATH  Google Scholar 

  28. Natarajan, B.K.: Sparse approximate solutions to linear systems. SIAM J. Comput. 24(2), 227–234 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  29. Pati, Y., Rezaiifar, R., Krishnaprasad, P.: Orthogonal matching pursuit: recursive function approximation with application to wavelet decomposition. In: 27th Asilomar Conference on Signals, Systems and Computers, 1993, pp. 40–44 (1993)

    Google Scholar 

  30. Pfander, G.E.: Gabor frames in finite dimensions. In: Casazza, P.G., Kutyniok, G. (eds.) Finite Frames: Theory and Applications, pp. 193–239. Birkhäuser, New York (2013)

    Chapter  Google Scholar 

  31. Pless, V.S., Huffman, W.C. (eds.): Handbook of Coding Theory, vol. 1. Elsevier Science B.V., Amsterdam (1998)

    Google Scholar 

  32. Rao, K.R., Yip, P.: Discrete Cosine Transform: Algorithms, Advantages, Applications Academic Press Professional, Incorporation, San Diego (1990)

    MATH  Google Scholar 

  33. Resnikoff, H.L., Wells, Jr. R.O.: Wavelet Analysis. The Scalable Structure of Information. Springer, New York (1998) (Corrected 2nd printing)

    Google Scholar 

  34. Said, A., Pearlman, W.A.: A new, fast, and efficient image codec based on set partitioning in hierarchical trees. IEEE Trans. Circ. Syst. Video Tech. 6(3), 243–250 1996

    Article  Google Scholar 

  35. Seip, K.: On the connection between exponential bases and certain related sequences in L2(−π, π). J. Funct. Anal. 130, 131–160 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  36. Shannon, C.E.: A mathematical theory of communication. The Bell Syst. Tech. J. 27, 379–423, 623–656 (1948)

    Article  MathSciNet  Google Scholar 

  37. Shapiro, J.M.: Embedded image coding using zerotrees of wavelet coefficients. IEEE Trans. Signal Proc. 41(12), 3445–3462 (1993)

    Article  MATH  Google Scholar 

  38. Taubman, D.S., Marcellin, M.W.: JPEG 2000: Image Compression Fundamentals, Standards and Practice, 2nd edn. Kluwer Academic Publishers, Norwell (2002)

    Google Scholar 

  39. Tian, J., Wells, Jr. R.O.: A lossy image codec based on index coding. In: IEEE Data Compression Conference, DCC ’96, 456, 1996

    Google Scholar 

  40. Viterbi School of Engineering, University of Southern California: The USC-SIPI image database. http://sipi.usc.edu/database/ (2012)

  41. Wallace, G.K.: The JPEG still picture compression standard. Comm. ACM 34(4), 30–44 (1991)

    Article  Google Scholar 

  42. Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P: Image quality assessment: from error measurement to structural similarity. IEEE Trans. Image Proc. 13(1), 1–14 (2004)

    Google Scholar 

  43. Watson, A.B. (ed.): Digital Images and Human Vision. MIT Press, Cambridge (1993)

    Google Scholar 

  44. Wickerhauser, M.V.: Adapted Wavelet Analysis from Theory to Software. A K Peters Ltd., Massachusetts (1996)

    MATH  Google Scholar 

Download references

Acknowledgements

The first named author gratefully acknowledges the support of MURI-ARO Grant W911NF-09-1-0383, NGA Grant 1582-08-1-0009, and DTRA Grant HDTRA1-13-1-0015. The second named author gratefully acknowledges the support of the Institute for Physical Science and Technology at the University of Maryland, College Park. We are both appreciative of expert observations by Professor Radu V. Balan, Department of Mathematics and Center for Scientific Computation and Mathematical Modeling (CSCAMM), Professor Ramani Duraiswami, Department of Computer Science and University of Maryland Institute of Advanced Computer Studies (UMIACS), and Professor Wojciech Czaja, Department of Mathematics, all at the University of Maryland, College Park.

Author information

Authors and Affiliations

  1. Norbert Wiener Center, Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    John J. Benedetto

  2. Institute for Physical Science and Technology, University of Maryland, College Park, MD, 20742, USA

    Alfredo Nava-Tudela

Authors
  1. John J. Benedetto
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alfredo Nava-Tudela
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to John J. Benedetto .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, DePaul University, Chicago, Illinois, USA

    Ahmed I. Zayed

  2. Department of Mathematics, University of Erlangen-Nuremberg, Erlangen, Germany

    Gerhard Schmeisser

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Benedetto, J.J., Nava-Tudela, A. (2014). Sampling in Image Representation and Compression. In: Zayed, A., Schmeisser, G. (eds) New Perspectives on Approximation and Sampling Theory. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-08801-3_7

Download citation

Publish with us

Policies and ethics

Search

Navigation