The use of the linear sampling method for obtaining super-resolution effects in Born approximation

Journal of Computational and Applied Mathematics

Volumn 203, Issue 1, 2007, Pages 145-158

  Brignone, M ^a   Coyle, J ^b   Piana, M ^c  

^a UNIVERSITY OF GENOA   (Italy)

^b MONMOUTH UNIVERSITY   (United States)

^c UNIVERSITY OF VERONA   (Italy)

Author keywords

Born approximation; Inverse scattering problems; Iterative algorithms; Regularization

Indexed keywords

EXTRAPOLATION; LINEAR SYSTEMS; NONLINEAR EQUATIONS; PROBLEM SOLVING; SAMPLING;

INVERSION METHODS; LINEAR SAMPLING; LINEAR SAMPLING METHODS; SUPER-RESOLUTION EFFECTS;

APPROXIMATION THEORY;

EID: 33847384977     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2006.03.014     Document Type: Article

References (32)

1. Albanese R.A., Medina R.L., Penn J.W., and Mathematics. medicine and microwaves. Inverse Problems 10 (1994) 995-1007
2. Arridge S.R. Optical tomography in medical imaging. Inverse Problems 15 (1999) R41-R93
3. Bertero M., and Boccacci P. An Introduction to Inverse Problems in Imaging (1998), IOPP, Bristol
4. Bertero M., and De Mol C. Super-resolution by data inversion. In: Wolf E. (Ed). Progress in Optics vol. XXXVI (1996), Elsevier, Amsterdam
5. Bialy H. Iterative behandlung linearer funktionalgleichungen. Arch. Rational Mech. Anal. 4 (1959) 166-176
6. Brignone M., and Piana M. The use of constraints for solving inverse scattering problems: physical optics and the linear sampling method. Inverse Problems 21 (2005) 207-222
7. Bulyshev A.E., Souvorov A.E., Semenov S.Y., Posukh V.G., and Sizov Y.E. Three-dimensional vector microwave tomography: theory and computational experiments. Inverse Problems 20 (2004) 1239-1259
8. Cakoni F., and Colton D. On the mathematical basis of the linear sampling method. Georg. Math. J. 10 (2003) 911-925
9. Cakoni F., and Colton D. Qualitative Methods in Inverse Scattering Theory (2005), Springer, Berlin
10. Chen B., and Stamnes J.J. Validity of diffraction tomography based on the first Born and the first Rytov approximations. Appl. Optics 37 (1998) 2996-3006
11. Colton D., Haddar H., and Piana M. The linear sampling method in inverse electromagnetic scattering theory. Inverse Problems 17 (2003) S105-S138
12. Colton D., and Kirsch A. A simple method for solving inverse scattering problems in the resonance region. Inverse Problems 12 (1996) 383-393
13. Colton D., and Kress R. Inverse Acoustic and Electromagnetic Scattering Theory (1998), Springer, Berlin
14. Colton D., and Monk P. Target identification of coated objects. IEEE Trans. Antennas Propagation 54 (2006) 1232-1242
15. Colton D., Piana M., and Potthast R. A simple method using Morozov's discrepancy principle for solving inverse scattering problems. Inverse Problems 12 (1997) 1477-1493
16. Coyle J., and Monk P. Scattering of time-harmonic electromagnetic waves by anisotropic inhomogeneous scatterers or impenetrable obstacles. SIAM J. Numer. Anal. 37 (2000) 1590-1617
17. Devaney A.J. Diffraction tomography. In: Boerner W.M., et al. (Ed). Inverse Methods in Electromagnetic Imaging (1985), Reidel, Dordrecht
18. Engl H.W., Hanke M., and Neubauer A. Regularization of Inverse Problems (1996), Kluwer, Dordrecht
19. Eskin G. The inverse scattering problem in two dimensions at fixed energy. Comm. Partial Differential Equations 26 (2001) 1055-1090
20. Frieden B.R. Evaluation, design and extrapolation methods for optical signals, based on the use of prolate functions. In: Wolf E. (Ed). Progress in Optics vol. IX (1971), Elsevier, Amsterdam
21. Gerchberg R.W. Super-resolution through error energy reduction. Optica Acta 21 (1974) 709-721
22. Habashy T.M., Groom R.W., and Spies B.R. Beyond the Born and Rytov approximations: a nonlinear approach to electromagnetic scattering. J. Geophys. Res. 98 (1993) 1759-1775
23. Hadamard J. Lectures on Cauchy's Problem in Partial Differential Equations (1923), Yale University Press, New Haven
24. Keller J.B. Accuracy and validity of the Born and Rytov approximation. J. Opt. Soc. Amer. 59 (1969) 1003-1004
25. Lagendijk R., Biemond J., and Boekee D. Regularized iterative image restoration with ringing reduction. IEEE Trans. Acoust. Speech Signal Process. 36 (1988) 1874-1888
26. Miller K. Least-squares methods for ill-posed problems with a prescribed bound. SIAM J. Math. Anal. 1 (1970) 52-74
27. Natterer F. An error bound for the Born approximation. Inverse Problems 20 (2004) 447-452
28. Natterer F., and Wubbeling F. Mathematical Methods in Image Reconstruction (2001), SIAM Publications, Philadelphia
29. Paley R.E.A.C., and Wiener N. Fourier Transforms in the Complex Domain (1934), American Mathematical Society, Providence
30. Piana M., and Bertero M. Projected Landweber algorithm and preconditioning. Inverse Problems 13 (1997) 441-463
31. Slepian D. Prolate spheroidal wave functions, Fourier analysis, and uncertainty-IV. Bell System Tech. J. 43 (1964) 3009
32. Wolter H. On basic analogies and principal differences between optical and electronic information. In: Wolf E. (Ed). Progress in Optics vol. I (1961), Elsevier, Amsterdam