Table of Contents Author Guidelines Submit a Manuscript
International Journal of Antennas and Propagation
Volume 2015 (2015), Article ID 924067, 21 pages
http://dx.doi.org/10.1155/2015/924067
Research Article

A Subspace Preconditioned LSQR Gauss-Newton Method with a Constrained Line Search Path Applied to 3D Biomedical Microwave Imaging

Department of Information Technology (INTEC), Ghent University, Sint-Pietersnieuwstraat 41, 9000 Gent, Belgium

Received 19 December 2014; Accepted 24 March 2015

Academic Editor: Amelie Litman

Copyright © 2015 Jürgen De Zaeytijd and Ann Franchois. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. N. Joachimowicz, C. Pichot, and J.-P. Hugonin, “Inverse scattering: an iterative numerical method for electromagnetic imaging,” IEEE Transactions on Antennas and Propagation, vol. 39, no. 12, pp. 1742–1752, 1991. View at Google Scholar
  2. H. Harada, M. Tanaka, and T. Takenaka, “Image reconstruction of a three-dimensional dielectric object using a gradient-based optimization,” Microwave and Optical Technology Letters, vol. 29, no. 5, pp. 332–336, 2001. View at Publisher · View at Google Scholar · View at Scopus
  3. A. Abubakar, P. M. van den Berg, and J. J. Mallorqui, “Imaging of biomedical data using a multiplicative regularized contrast source inversion method,” IEEE Transactions on Microwave Theory and Techniques, vol. 50, no. 7, pp. 1761–1771, 2002. View at Publisher · View at Google Scholar · View at Scopus
  4. Z. Q. Zhang and Q. H. Liu, “Three-dimensional nonlinear image reconstruction for microwave biomedical imaging,” IEEE Transactions on Biomedical Engineering, vol. 51, no. 3, pp. 544–548, 2004. View at Publisher · View at Google Scholar · View at Scopus
  5. A. E. Bulyshev, A. E. Souvorov, S. Y. Semenov, V. G. Posukh, and Y. E. Sizov, “Three-dimensional vector microwave tomography: theory and computational experiments,” Inverse Problems, vol. 20, no. 4, pp. 1239–1259, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. G. Franceschini, D. Franceschini, and A. Massa, “Full-vectorial three-dimensional microwave imaging through the iterative multiscaling strategy—a preliminary assessment,” IEEE Geoscience and Remote Sensing Letters, vol. 2, no. 4, pp. 428–432, 2005. View at Publisher · View at Google Scholar · View at Scopus
  7. I. Catapano, L. Crocco, M. D'Urso, and T. Isernia, “A novel effective model for solving 3-D nonlinear inverse scattering problems in lossy scenarios,” IEEE Geoscience and Remote Sensing Letters, vol. 3, no. 3, pp. 302–306, 2006. View at Publisher · View at Google Scholar · View at Scopus
  8. J. De Zaeytijd, A. Franchois, C. Eyraud, and J.-M. Geffrin, “Full-wave three-dimensional microwave imaging with a regularized Gauss-Newton method-theory and experiment,” IEEE Transactions on Antennas and Propagation, vol. 55, no. 11, pp. 3279–3292, 2007. View at Publisher · View at Google Scholar · View at Scopus
  9. P. C. Chaumet and K. Belkebir, “Three-dimensional reconstruction from real data using a conjugate gradient-coupled dipole method,” Inverse Problems, vol. 25, no. 2, Article ID 024003, 17 pages, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. C. Eyraud, A. Litman, A. Hérique, and W. Kofman, “Microwave imaging from experimental data within a Bayesian framework with realistic random noise,” Inverse Problems, vol. 25, no. 2, Article ID 024005, 16 pages, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. A. Abubakar, T. M. Habashy, G. Pan, and M.-K. Li, “Application of the multiplicative regularized Gauss-Newton algorithm for three-dimensional microwave imaging,” IEEE Transactions on Antennas and Propagation, vol. 60, no. 5, pp. 2431–2441, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. J. C. Bolomey, L. Jofre, and G. Peronnet, “On the possible use of microwave-active imaging for remote thermal sensing,” IEEE Transactions on Microwave Theory and Techniques, vol. 31, no. 9, pp. 777–781, 1983. View at Publisher · View at Google Scholar · View at Scopus
  13. S. Gabriel, R. W. Lau, and C. Gabriel, “The dielectric properties of biological tissues. II. Measurements in the frequency range 10 Hz to 20 GHz,” Physics in Medicine and Biology, vol. 41, no. 11, pp. 2251–2269, 1996. View at Publisher · View at Google Scholar · View at Scopus
  14. M. Lazebnik, L. McCartney, D. Popovic et al., “A large-scale study of the ultrawideband microwave dielectric properties of normal breast tissue obtained from reduction surgeries,” Physics in Medicine and Biology, vol. 52, no. 10, pp. 2637–2656, 2007. View at Publisher · View at Google Scholar · View at Scopus
  15. M. Lazebnik, D. Popovic, L. McCartney et al., “A large-scale study of the ultrawideband microwave dielectric properties of normal, benign and malignant breast tissues obtained from cancer surgeries,” Physics in Medicine and Biology, vol. 52, no. 20, pp. 6093–6115, 2007. View at Publisher · View at Google Scholar · View at Scopus
  16. C. Yu, M. Yuan, J. Stang et al., “Active microwave imaging II: 3-D system prototype and image reconstruction from experimental data,” IEEE Transactions on Microwave Theory and Techniques, vol. 56, no. 4, pp. 991–1000, 2008. View at Publisher · View at Google Scholar · View at Scopus
  17. D. W. Winters, J. D. Shea, P. Kosmas, B. D. Van Veen, and S. C. Hagness, “Three-dimensional microwave breast imaging: dispersive dielectric properties estimation using patient-specific basis functions,” IEEE Transactions on Medical Imaging, vol. 28, no. 7, pp. 969–981, 2009. View at Publisher · View at Google Scholar · View at Scopus
  18. J. D. Shea, P. Kosmas, B. D. Van Veen, and S. Hagness, “Contrast-enhanced microwave imaging of breast tumors: a computational study using 3D realistic numerical phantoms,” Inverse Problems, vol. 26, no. 7, Article ID 074009, 22 pages, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. T. M. Grzegorczyk, P. M. Meaney, P. A. Kaufman, R. M. Diflorio-Alexander, and K. D. Paulsen, “Fast 3-D tomographic microwave imaging for breast cancer detection,” IEEE Transactions on Medical Imaging, vol. 31, no. 8, pp. 1584–1592, 2012. View at Publisher · View at Google Scholar · View at Scopus
  20. F. Bai, A. Franchois, J. de Zaeytijd, and A. Pizurica, “Three-dimensional quantitative microwave imaging of realistic numerical breast phantoms using Huber regularization,” in Proceedings of the 35th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBS '13), pp. 5135–5138, Osaka, Japan, July 2013. View at Publisher · View at Google Scholar · View at Scopus
  21. R. E. Kleinman and P. M. van den Berg, “A modified gradient method for two-dimensional problems in tomography,” Journal of Computational and Applied Mathematics, vol. 42, no. 1, pp. 17–35, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  22. I. Catapano, L. Di Donato, L. Crocco et al., “On quantitative microwave tomography of female breast,” Progress in Electromagnetics Research, vol. 97, pp. 75–93, 2009. View at Publisher · View at Google Scholar · View at Scopus
  23. J. De Zaeytijd, A. Franchois, and J.-M. Geffrin, “A new value picking regularization strategy—application to the 3-D electromagnetic inverse scattering problem,” IEEE Transactions on Antennas and Propagation, vol. 57, no. 4, pp. 1133–1149, 2009. View at Publisher · View at Google Scholar · View at Scopus
  24. J. De Zaeytijd and A. Franchois, “3D quantitative microwave imaging from measured data with multiplicative smoothing and value picking regularization,” Inverse Problems, vol. 25, no. 2, Article ID 024004, 26 pages, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. F. Bai, A. Pižurica, B. Truyen, W. Philips, and A. Franchois, “Weakly convex discontinuity adaptive regularization for 3D quantitative microwave tomography,” Inverse Problems, vol. 30, no. 8, Article ID 085005, 28 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  26. Z. Q. Peng and A. G. Tijhuis, “Transient scattering by a lossy dielectric cylinder: marching-on-in-frequency approach,” Journal of Electromagnetic Waves and Applications, vol. 7, no. 5, pp. 739–763, 1993. View at Publisher · View at Google Scholar
  27. A. G. Tijhuis, M. C. van Beurden, and A. P. M. Zwamborn, “Iterative solution of field problems with a varying physical parameters,” in Ultra-Wideband, Short-Pulse Electromagnetics 6, pp. 223–230, Springer US, 2003. View at Google Scholar
  28. A. G. Tijhuis, K. Belkebir, A. C. S. Litman, and B. P. de Hon, “Theoretical and computational aspects of 2-D inverse profiling,” IEEE Transactions on Geoscience and Remote Sensing, vol. 39, no. 6, pp. 1316–1330, 2001. View at Publisher · View at Google Scholar · View at Scopus
  29. J. de Zaeytijd, I. Bogaert, and A. Franchois, “An efficient hybrid MLFMA-FFT solver for the volume integral equation in case of sparse 3D inhomogeneous dielectric scatterers,” Journal of Computational Physics, vol. 227, no. 14, pp. 7052–7068, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  30. P. Zwamborn and P. M. van den Berg, “The three dimensional weak form of the conjugate gradient FFT method for solving scattering problems,” IEEE Transactions on Microwave Theory and Techniques, vol. 40, no. 9, pp. 1757–1766, 1992. View at Publisher · View at Google Scholar · View at Scopus
  31. Z. Q. Zhang, Q. H. Liu, C. Xiao, E. Ward, G. Ybarra, and W. T. Joines, “Microwave breast imaging: 3-D forward scattering simulation,” IEEE Transactions on Biomedical Engineering, vol. 50, no. 10, pp. 1180–1189, 2003. View at Publisher · View at Google Scholar · View at Scopus
  32. W. C. Chew and Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Transactions on Medical Imaging, vol. 9, no. 2, pp. 218–225, 1990. View at Publisher · View at Google Scholar · View at Scopus
  33. P. M. Meaney, K. D. Paulsen, and T. P. Ryan, “Two-dimensional hybrid element image reconstruction for TM illumination,” IEEE Transactions on Antennas and Propagation, vol. 43, no. 3, pp. 239–247, 1995. View at Publisher · View at Google Scholar · View at Scopus
  34. A. Franchois and C. Pichot, “Microwave imaging-complex permittivity reconstruction with a levenberg-marquardt method,” IEEE Transactions on Antennas and Propagation, vol. 45, no. 2, pp. 203–215, 1997. View at Publisher · View at Google Scholar · View at Scopus
  35. R. F. Remis and P. M. van den Berg, “On the equivalence of the Newton-Kantorovich and distorted Born methods,” Inverse Problems, vol. 16, no. 1, pp. L1–L4, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  36. T. Henriksson, N. Joachimowicz, C. Conessa, and J.-C. Bolomey, “Quantitative microwave imaging for breast cancer detection using a planar 2.45–GHz system,” IEEE Transactions on Instrumentation and Measurement, vol. 59, no. 10, pp. 2691–2699, 2010. View at Publisher · View at Google Scholar · View at Scopus
  37. T. Rubæk, P. M. Meaney, P. Meincke, and K. D. Paulsen, “Nonlinear microwave imaging for breast-cancer screening using Gauss-Newton’s method and the CGLS inversion algorithm,” IEEE Transactions on Antennas and Propagation, vol. 55, no. 8, pp. 2320–2331, 2007. View at Publisher · View at Google Scholar · View at Scopus
  38. P. Mojabi and J. Lovetri, “Enhancement of the Krylov subspace regularization for microwave biomedical imaging,” IEEE Transactions on Medical Imaging, vol. 28, no. 12, pp. 2015–2019, 2009. View at Publisher · View at Google Scholar · View at Scopus
  39. A. Franchois and A. G. Tijhuis, “A quasi-Newton reconstruction algorithm for a complex microwave imaging scanner environment,” Radio Science, vol. 38, no. 2, pp. 8011–8013, 2003. View at Google Scholar · View at Scopus
  40. T. M. Habashy and A. Abubakar, “A general framework for constraint minimization for the inversion of electromagnetic measurements,” Progress in Electromagnetics Research, vol. 46, pp. 265–312, 2004. View at Publisher · View at Google Scholar · View at Scopus
  41. P. Mojabi and J. LoVetri, “Microwave biomedical imaging using the multiplicative regularized gauss-Newton inversion,” IEEE Antennas and Wireless Propagation Letters, vol. 8, pp. 645–648, 2009. View at Publisher · View at Google Scholar · View at Scopus
  42. A. H. Golnabi, P. M. Meaney, S. D. Geimer, and K. D. Paulsen, “Comparison of no-prior and soft-prior regularization in biomedical microwave imaging,” Journal of Medical Physics, vol. 36, no. 3, pp. 159–170, 2011. View at Publisher · View at Google Scholar · View at Scopus
  43. P. Mojabi and J. LoVetri, “Overview and classification of some regularization techniques for the Gauss-Newton inversion method applied to inverse scattering problems,” IEEE Transactions on Antennas and Propagation, vol. 57, no. 9, pp. 2658–2665, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  44. M. Jacobsen, P. C. Hansen, and M. A. Saunders, “Subspace preconditioned LSQR for discrete ill-posed problems,” BIT Numerical Mathematics, vol. 43, no. 5, pp. 975–989, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  45. J. D. Shea, P. Kosmas, S. C. Hagness, and B. D. Van Veen, “Three-dimensional microwave imaging of realistic numerical breast phantoms via a multiple-frequency inverse scattering technique,” Medical Physics, vol. 37, no. 8, pp. 4210–4226, 2010. View at Publisher · View at Google Scholar · View at Scopus
  46. R. Fletcher, Practical Methods of Optimization, John Wiley, New York, NY, USA, 2nd edition, 1990. View at MathSciNet
  47. A. van den Bos, “Complex gradient and Hessian,” IEE Proceedings: Vision, Image and Signal Processing, vol. 141, no. 6, pp. 380–382, 1994. View at Publisher · View at Google Scholar · View at Scopus
  48. R. D. da Cunha and T. Hopkins, “The Parallel Iterative Methods (PIM) package for the solution of systems of linear equations on parallel computers,” Applied Numerical Mathematics, vol. 19, no. 1-2, pp. 33–50, 1995. View at Publisher · View at Google Scholar · View at Scopus
  49. J. de Zaeytijd, On the 3D electromagnetic quantitative inverse scattering problem: algorithms and regularization [Ph.D. thesis], Ghent University, Ghent, Belgium, 2009.
  50. T. K. Sarkar, E. Arvas, and S. M. Rao, “Application of FFT and the conjugate gradient method for the solution of electromagnetic radiation from electrically large and small conducting bodies,” IEEE Transactions on Antennas and Propagation, vol. 34, no. 5, pp. 635–640, 1986. View at Google Scholar · View at Scopus
  51. H. Gan and W. C. Chew, “A discrete BCG-FFT algorithm for solving 3D inhomogeneous scatterer problems,” Journal of Electromagnetic Waves and Applications, vol. 9, no. 10, pp. 1339–1357, 1995. View at Google Scholar · View at Scopus
  52. M. Hanke and C. R. Vogel, “Two-level preconditioners for regularized inverse problems. I. Theory,” Numerische Mathematik, vol. 83, no. 3, pp. 385–402, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  53. C. C. Paige and M. A. Saunders, “LSQR: An algorithm for sparse linear equations and sparse least squares,” ACM Transactions on Mathematical Software, vol. 8, no. 1, pp. 43–71, 1982. View at Publisher · View at Google Scholar · View at MathSciNet
  54. G. Strang, “The discrete cosine transform,” SIAM Review, vol. 41, no. 1, pp. 135–147, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  55. M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proceedings of the IEEE, vol. 93, no. 2, pp. 216–231, 2005. View at Publisher · View at Google Scholar · View at Scopus
  56. O. M. Bucci and T. Isernia, “Electromagnetic inverse scattering: retrievable information and measurement strategies,” Radio Science, vol. 32, no. 6, pp. 2123–2137, 1997. View at Publisher · View at Google Scholar · View at Scopus
  57. A. E. Bulyshev, A. E. Souvorov, S. Y. Semenov et al., “Three-dimensional microwave tomography. Theory and computer experiments in scalar approximation,” Inverse Problems, vol. 16, no. 3, pp. 863–875, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  58. D. W. Winters, E. J. Bond, B. D. van Veen, and S. C. Hagness, “Estimation of the frequency-dependent average dielectric properties of breast tissue using a time-domain inverse scattering technique,” IEEE Transactions on Antennas and Propagation, vol. 54, no. 11, pp. 3517–3528, 2006. View at Publisher · View at Google Scholar · View at Scopus
  59. M. Klemm, I. J. Craddock, J. A. Leendertz, A. Preece, and R. Benjamin, “Radar-based breast cancer detection using a hemispherical antenna array—experimental results,” IEEE Transactions on Antennas and Propagation, vol. 57, no. 6, pp. 1692–1704, 2009. View at Publisher · View at Google Scholar · View at Scopus
  60. D. Colton, H. Haddar, and M. Piana, “The linear sampling method in inverse electromagnetic scattering theory,” Inverse Problems, vol. 19, pp. S105–S137, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  61. J. de Zaeytijd, C. Lanza Conmeaux, and A. Franchois, “Three-dimensional linear sampling applied to microwave breast imaging,” in Proceedings of the 29th General Assembly of URSI, CD-ROM, Chicago, Ill, USA, August 2008.
  62. E. Zastrow, S. Davis, M. Lazebnik, F. Kelcz, B. van Veen, and S. Hagness, “Database of 3D grid-based numerical breast phantoms for use in computational electromagnetics simulations,” http://uwcem.ece.wisc.edu/home.htm.
  63. E. Zastrow, S. K. Davis, M. Lazebnik, F. Kelcz, B. D. van Veen, and S. C. Hagness, “Development of anatomically realistic numerical breast phantoms with accurate dielectric properties for modeling microwave interactions with the human breast,” IEEE Transactions on Biomedical Engineering, vol. 55, no. 12, pp. 2792–2800, 2008. View at Publisher · View at Google Scholar