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Frame Fundamental Sensor Modeling and Stability of One-Sided Frame Perturbation

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Abstract

We demonstrate that for all linear devices and/or sensors, signal requisition and reconstruction is naturally a mathematical frame expansion and reconstruction issue, whereas the measurement is carried out via a sequence generated by the exact physical response function (PRF) of the device, termed sensory frame {h n }. The signal reconstruction, on the other hand, will be carried out using the dual frame \(\{\tilde{h}^{a}_{n}\}\) of an estimated sensory frame {h a n }. This consequently results in a one-sided perturbation to a frame expansion, which resides in each and every signal and image reconstruction problem. We show that the stability of such a one-sided frame perturbation exits. Examples of image reconstructions in de-blurring are demonstrated.

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References

  1. Aldroubi, A.: Portraits of frames. Proc. Am. Math. Soc. 123, 1661–1668 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  2. Aldroubi, A., Sun, Q., Tang, W.S.: p-frames and shift invariant subspaces of L p. J. Fourier Anal. Appl. 7(1), 1–21 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Balan, R.: Equivalence relations and distances between Hilbert frames. Proc. Am. Math. Soc. 127(8), 2353–2366 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Balan, R., Casazza, P., Heil, C., Landau, Z.: Deficits and excesses of frames. Adv. Comput. Math. 18, 93–116 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  5. Benedetto, J.J.: Frame decompositions, sampling, and uncertainty principle inequalities. In: Benedetto, J.J., Frazier, M.W. (eds.) Wavelets: Mathematics and Applications. CRC Press, Boca Raton (1994). Chap. 7

    Google Scholar 

  6. Benedetto, J.J., Walnut, D.F.: Gabor frames for L 2 and related spaces. In: Benedetto, J.J., Frazier, M.W. (eds.) Wavelets: Mathematics and Applications. CRC Press, Boca Raton (1994). Chap. 3

    Google Scholar 

  7. Casazza, P.G., Christensen, O.: Perturbation of operators and applications to frame theory. J. Fourier Anal. Appl. 3, 543–557 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  8. Casazza, P.G., Christensen, O.: Frames containing a Riesz basis and preservation of this property under perturbation. SIAM J. Math. Anal. 29(1), 266–278 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  9. Christensen, O.: Frame perturbations. Proc. Am. Math. Soc. 123, 1217–1220 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  10. Christensen, O., Heil, C.: Perturbations of Banach frames and atomic decompositions. Math. Nachr. 185, 33–47 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  11. Daubechies, I.: The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Inf. Theory 36(5), 961–1005 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  12. Daubechies, I.: Ten lectures on wavelets (1992)

  13. Daubechies, I., Grossmann, A., Meyer, Y.: Painless nonorthogonal expansions. J. Math. Phys. 27, 1271–1283 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  14. DeVito, C.: Functional Analysis and Linear Operator Theory. Addison-Wesley, Reading (1990)

    MATH  Google Scholar 

  15. Duffin, R., Schaeffer, A.: A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72, 341–366 (1952)

    Article  MATH  MathSciNet  Google Scholar 

  16. Favier, S.J., Zalik, R.A.: On the stability of frames and Riesz bases. Appl. Comput. Harmon. Anal. 2, 160–173 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  17. Feichtinger, H.G., Gröchenig, K.: Gabor wavelets and the Heisenberg group: Gabor expansions and short time Fourier transform from the group theoretical point of view. In: Chui, C.K. (ed.) Wavelets: A Tutorial in Theory and Applications, vol. 2, pp. 359–398. Academic Press, San Diego (1992)

    Google Scholar 

  18. Feichtinger, H.G., Kaiblinger, N.: Varying the time-frequency lattice of Gabor frames. Trans. Am. Math. Soc. 356, 2001–2023 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  19. Han, D.: Approximations for Gabor and wavelet frames. Trans. Am. Math. Soc. 355, 3329–3342 (2003)

    Article  MATH  Google Scholar 

  20. Han, D.: Tight frame approximation for multi-frames and super-frames. J. Approx. Theory 129(1), 78–93 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. Heil, C., Walnut, D.: Continuous and discrete wavelet transforms. SIAM Rev. 31, 628–666 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  22. Larson, D.: Frames and wavelets from an operator-theoretical point of view. Contemp. Math. 228, 201–218 (1998)

    MathSciNet  Google Scholar 

  23. Larson, D., Han, D.: Frames, bases and group representations. Mem. Am. Math. Soc. 147(697) (2000)

  24. Li, S.: On general frame decompositions. Numer. Funct. Anal. Optim. 16(9 & 10), 1181–1191 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  25. Li, S., Ogawa, H.: Pseudoframes for subspaces with applications. J. Fourier Anal. Appl. 10(4), 409–431 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  26. Li, S., Yao, Z., Yi, W.: Frame fundamental super-resolution image fusion. Preprint (2009)

  27. Naylor, A.W., Sell, G.R.: Linear Operator Theory in Engineering and Science. Springer, Berlin (1982)

    MATH  Google Scholar 

  28. Sun, Q.: Frames in spaces with finite rate of innovation. Adv. Comput. Math. 28, 301–329 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, San Francisco State University, San Francisco, CA, 94132, USA

    Shidong Li

  2. Department of Information Sciences, Graduate University of the Chinese Academy of Sciences, Beijing, 100049, People’s Republic of China

    Dunyan Yan

Authors
  1. Shidong Li
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  2. Dunyan Yan
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Correspondence to Shidong Li.

Additional information

S. Li is supported in part by NSF Grants of USA DMS-0406979 and DMS-0709384.

D. Yan is supported in part by the Natural Science Foundation of China under grants 10571014 and 10631080.

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Li, S., Yan, D. Frame Fundamental Sensor Modeling and Stability of One-Sided Frame Perturbation. Acta Appl Math 107, 91–103 (2009). https://doi.org/10.1007/s10440-008-9419-8

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  • DOI: https://doi.org/10.1007/s10440-008-9419-8

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