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Bit-depth quantization and reconstruction error in digital images


Digital images can be considered as sparse or approximately sparse in the two-dimensional discrete cosine transform domain. According to the compressive sensing theory, these images can be recovered from a reduced set of pixels. Such reconstructions are influenced by the noise and the non-reconstructed coefficients for approximately sparse images. In the literature, this kind of reconstruction error is described by the error bound relations. In some hardware implementations, the pixels may be sensed with a low number of bits, causing quantization error. In this paper, we derive exact formula for the expected error energy in the reconstructed images, caused by the pixels quantization. It is validated trough numerical examples.

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  1. Donoho, D.: Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006)

    MathSciNet  Article  Google Scholar 

  2. Candès, E.J.: The restricted isometry property and its implications for compressed sensing. C.R. Math. 346(9), 589–592 (2008)

    MathSciNet  Article  Google Scholar 

  3. Baraniuk, R.: Compressive sensing. IEEE Signal Process. Mag. 24(4), 118–121 (2007)

    Article  Google Scholar 

  4. Candès, E., Romberg, J., Tao, T.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52(2), 489–509 (2006)

    MathSciNet  Article  Google Scholar 

  5. Wohlberg, B.: Noise sensitivity of sparse signal representations: reconstruction error bounds for the inverse problem. IEEE Trans. Signal Process. 51(12), 3053–3060 (2003)

    MathSciNet  Article  Google Scholar 

  6. Stanković, L., Stanković, S., Amin, M.: Missing samples analysis in signals for applications to L-estimation and compressive sensing. Sig. Proc. 94, 401–408 (2014)

    Article  Google Scholar 

  7. Stanković, L., Stanković, I., Daković, M.: Nonsparsity influence on the ISAR recovery from reduced data. IEEE Trans. Aerosp. Electron. Syst. 52(6), 3065–3070 (2016)

    Article  Google Scholar 

  8. Stanković, S., Orović, I., Stanković, L.: An automated signal reconstruction method based on analysis of compressive sensed signals in noisy environment. Sig. Proc. 104, 43–50 (2014)

    Article  Google Scholar 

  9. Elad, M.: Sparse and Redudant Representations: From Theory to Applications in Signal and Image Processing. Springer, Berlin (2010)

    Book  Google Scholar 

  10. Davenport, M., Duarte, M., Eldar, Y., Kutyniok, G.: Introduction to compressed sensing. In: Compressed Sensing: Theory and Applications, chap. 1, pp. 1–64. Cambridge University Press, Cambridge (2012)

  11. Needell, D., Tropp, J.A.: CoSaMP: iterative signal recovery from incomplete and inaccurate samples. Appl. Comput. Harmon. Anal. 20(3), 301–321 (2009)

    MathSciNet  Article  Google Scholar 

  12. Sejdić, E., Rothfuss, M.A., Gimbel, M.L., Mickle, M.H.: Comparative analysis of compressive sensing approaches for recovery of missing samples in an implantable wireless doppler device. IET Signal Proc. 8(3), 230–238 (2014)

    Article  Google Scholar 

  13. Sejdić, E., Cam, A., Chaparro, L.F., Steele, C.M., Chau, T.: Compressive sampling of swallowing accelerometry signals using TF dictionaries based on modulated discrete prolate spheroidal sequences. EURASIP J. Adv. Signal Process. (2012).

    Article  Google Scholar 

  14. Sejdic, E.: Time–frequency compressive sensing. In: Boashash, B. (ed.) Time–Frequency Signal Analysis and Processing, pp. 424–429. Academic Press, New York (2015)

    Google Scholar 

  15. Jokanovic, B., Amin, M.G., Zhang, Y.D., Ahmad, F.: Multi-window time–frequency signature reconstruction from undersampled continuous-wave radar measurements for fall detection. IET Radar Sonar Navig. 9(2), 173–183 (2015)

    Article  Google Scholar 

  16. Zhang, Z., Xu, Y., Yang, J., Li, X., Zhang, D.: A survey of sparse representation: algorithms and applications. IEEE Access 3, 490–530 (2015)

    Article  Google Scholar 

  17. Volaric, I., Sucic, V.: On the noise impact in the L1 based reconstruction of the sparse time–frequency distributions. In: International Conference on Broadband Communications for Next Generation Networks and Multimedia Applications (CoBCom), pp. 1–6, Graz (2016)

  18. Stanković, L., Orović, I., Stanković, S., Amin, M.: Robust time frequency analysis based on the L-estimation and compressive sensing. IEEE Signal Process. Lett. 20(5), 499–502 (2013)

    Article  Google Scholar 

  19. Stanković, L., Daković, M., Vujović, S.: Reconstruction of sparse signals in impulsive disturbance environments. Circuits Syst. Signal Process. 2016, 1–28 (2016).

    Article  MATH  Google Scholar 

  20. Li, X., Bi, G.: Image reconstruction based on the improved compressive sensing algorithm. In: IEEE International Conference on Digital Signal Processing (DSP), pp. 357–360, Singapore (2015)

  21. Musić, J., Marasović, T., Papić, V., Orović, I., Stanković, S.: Performance of compressive sensing image reconstruction for search and rescue. IEEE Geosci. Remote Sens. Lett. 13(11), 1739–1743 (2016)

    Article  Google Scholar 

  22. Stanković, I., Orović, I., Daković, M., Stanković, S.: Denoising of sparse images in impulsive disturbance environment. Multimed. Tools Appl. 77(5), 5885–5905 (2018)

    Article  Google Scholar 

  23. Stanković, L.: Digital Signal Processing with Selected Topics, CreateSpace Independent Publishing Platform, An Company, November 4, 2015

  24. Stanković, L., Brajović, M.: Influence of missing samples on signals sparse in the DCT domain with application to audio signals. IEEE Trans. Audio Speech Lang. Proc. 26(7), 1216–1231 (2018)

    Google Scholar 

  25. Jacques, L., Laska, J.N., Boufounos, P.T., Baraniuk, R.G.: Robust 1-bit compressive sensing via binary stable embeddings for sparse vectors. IEEE Trans. Inf. Theory 59(4), 2082–2102 (2011)

    MathSciNet  Article  Google Scholar 

  26. Dai, W., Milenkovic, O.: Information theoretical and algorithmic approaches to quantized compressive sensing. IEEE Trans. Commun. 59(7), 1857–1866 (2011)

    Article  Google Scholar 

  27. Wang, Y., Feng, S., Zhang, P.: Information estimations and acquisition costs for quantized compressive sensing. In: International Conference on Digital Signal Processing (DSP) 2015. Singapore, Singapore (2015)

  28. Shi, H.M., Case, M., Gu, X., Tu, S., Needell, D.: Methods for quantized compressed sensing. In: Information Theory and Applications Workshop (ITA) 2016, CA, USA (2016)

  29. More, J., Toraldo, G.: On the solution of large quadratic programming problems with bound constraints. SIAM J. Optim. 1, 93–113 (1991)

    MathSciNet  Article  Google Scholar 

  30. Davis, G., Mallat, S., Avellaneda, M.: Greedy adaptive approximation. J. Constr. Approx. 12, 57–98 (1997)

    Article  Google Scholar 

  31. Tibshirani, R.J.: Regression shrinkage and selection via the lasso: a retrospective. J. R. Stat. Soc. B 73, 273–282 (2011)

    MathSciNet  Article  Google Scholar 

  32. Brajović, M., Stanković, I., Daković, M., Ioana, C., Stanković, L.: Error in the reconstruction of nonsparse images. Math. Prob. Eng. (2018).

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Miloš Brajović.

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Stanković, I., Brajović, M., Daković, M. et al. Bit-depth quantization and reconstruction error in digital images. SIViP 14, 1545–1553 (2020).

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  • Compressed sensing
  • Image processing
  • Bit depth
  • Quantization
  • Reconstruction