volume | PIER Journals

Abstract

Quantitative active microwave imaging for breast cancer screening and therapy monitoring applications requires adequate reconstruction algorithms, in particular with regard to the nonlinearity and ill-posedness of the inverse problem. We employ a fully vectorial three-dimensional nonlinear inversion algorithm for reconstructing complex permittivity profiles from multi-view single-frequency scattered field data, which is based on a Gauss-Newton optimization of a regularized cost function. We tested it before with various types of regularizing functions for piecewise-constant objects from Institut Fresnel and with a quadratic smoothing function for a realistic numerical breast phantom. In the present paper we adopt a cost function that includes a Huber function in its regularization term, relying on a Markov Random Field approach. The Huber function favors spatial smoothing within homogeneous regions while preserving discontinuities between contrasted tissues. We illustrate the technique with 3D reconstructions from synthetic data at 2 GHz for realistic numerical breast phantoms from the University of Wisconsin-Madison UWCEM online repository: we compare Huber regularization with a multiplicative smoothing regularization and show reconstructions for various positions of a tumor, for multiple tumors and for different tumor sizes, from a sparse and from a denser data configuration.

1. Joines, W. T., Y. Zhang, C. Li, and R. L. Jirtle, "The measured electrical properties of normal and malignant human tissues from 50 to 900 MHz," Med. Phys., Vol. 21, No. 4, 547-550, 1994.
doi:10.1118/1.597312 Google Scholar

2. Lazebnik, M. D., et al., "A large-scale study of the ultrawideband microwave dielectric properties of normal, benign and malignant breast tissues obtained from cancer surgeries," Phys. Med. Biol., Vol. 52, No. 20, 6093-6115, 2007.
doi:10.1088/0031-9155/52/20/002 Google Scholar

3. Grzegorczyk, T. M., et al., "Fast 3D tomographic microwave imaging for breast cancer detection," IEEE Trans. Med. Imaging, Vol. 31, No. 8, 1584-1592, 2012.
doi:10.1109/TMI.2012.2197218 Google Scholar

4. Larsen, L. E. and J. H. Jacobi, "Microwave scattering parameter imagery of an isolated canine kidney," Med. Phys., Vol. 6, No. 5, 394-403, 1979.
doi:10.1118/1.594595 Google Scholar

5. Pichot, C., L. Jofre, G. Peronnet, and J.-C. Bolomey, "Active microwave imaging of inhomogeneous bodies," IEEE Trans. Antennas Propag., Vol. 33, No. 4, 416-425, 1985.
doi:10.1109/TAP.1985.1143603 Google Scholar

6. Jofre, L., et al., "Medical imaging with a microwave tomographic scanner," IEEE Trans. Biomed. Eng., Vol. 37, No. 3, 303-312, 1990.
doi:10.1109/10.52331 Google Scholar

7. Li, X., E. J. Bond, B. D. Van Veen, and S. C. Hagness, "An overview of ultra-wideband microwave imaging via space-time beamforming for early-stage breast-cancer detection," IEEE Antennas Propag. Mag., Vol. 47, No. 1, 19-34, 2005.
doi:10.1109/MAP.2005.1436217 Google Scholar

8. Klemm, M., et al., "Radar-based breast cancer detection using a hemispherical antenna array --- Experimental results," IEEE Trans. Antennas Propag., Vol. 57, No. 6, 1692-1704, 2009.
doi:10.1109/TAP.2009.2019856 Google Scholar

9. Porter, E., A. Santorelli, and M. Popovic, "Time-domain microwave radar applied to breast imaging: Measurement reliability in a clinical setting," Progress In Electromagnetics Research, Vol. 149, 119-132, 2014.
doi:10.2528/PIER14080503 Google Scholar

10. De Zaeytijd, J., C. Conmeaux, and A. Franchois, "Three-dimensional linear sampling applied to microwave breast imaging," Proc. XXIXth URSI General Assembly, Chicago, Aug. 7-16 2008. Google Scholar

11. Chew, W. C. and Y. M. Wang, "Reconstruction of two-dimensional permittivity distribution using the distorted born iterative method," IEEE Trans. Med. Imaging, Vol. 9, No. 2, 218-225, 1990.
doi:10.1109/42.56334 Google Scholar

12. Kleinman, R. E. and P. M. van den Berg, "A modified gradient method for two-dimensional problems in tomography," J. Comput. Appl. Math., Vol. 42, 17-35, 1992.
doi:10.1016/0377-0427(92)90160-Y Google Scholar

13. Joachimowicz, N., C. Pichot, and J.-P. Hugonin, "Inverse scattering: An iterative numerical method for electromagnetic imaging," IEEE Trans. Antennas Propag., Vol. 39, No. 12, 1742-1752, 1991.
doi:10.1109/8.121595 Google Scholar

14. Meaney, P. M., K. D. Paulsen, and T. P. Ryan, "Two-dimensional hybrid element image reconstruction for TM illumination," IEEE Trans. Antennas Propag., Vol. 43, No. 3, 239-247, 1995.
doi:10.1109/8.371992 Google Scholar

15. Franchois, A. and C. Pichot, "Microwave imaging-complex permittivity reconstruction with a Levenberg-Marquardt method," IEEE Trans. Antennas Propag., Vol. 45, 203-215, 1997.
doi:10.1109/8.560338 Google Scholar

16. Abubakar, A., P. M. van den Berg, and J. J. Mallorqui, "Imaging of biomedical data using a multiplicative regularized contrast source inversion Method," IEEE Trans. Microw. Theory Tech., Vol. 50, No. 7, 1761-1771, 2002.
doi:10.1109/TMTT.2002.800427 Google Scholar

17. Zhang, Z. Q. and Q. H. Liu, "Three-dimensional nonlinear image reconstruction for microwave biomedical imaging," IEEE Trans. Biomed. Eng., Vol. 51, No. 3, 544-548, 2004.
doi:10.1109/TBME.2003.821052 Google Scholar

18. Bulyshev, A. E., et al., "Three-dimensional vector microwave tomography: Theory and computational experiments," Inverse Probl., Vol. 20, No. 4, 1239-1259, 2004.
doi:10.1088/0266-5611/20/4/013 Google Scholar

19. De Zaeytijd, J., A. Franchois, C. Eyraud, and J.-M. Geffrin, "Full-wave three-dimensional microwave imaging with a regularized Gauss-Newton method --- Theory and experiment," IEEE Trans. Antennas Propag., Vol. 55, No. 11, 3279-3292, 2007.
doi:10.1109/TAP.2007.908824 Google Scholar

20. Takenaka, T., H. Zhou, and T. Tanaka, "Inverse scattering for a three-dimensional object in the time domain," J. Opt. Soc. Am. A, Vol. 20, No. 10, 1867-1874, 2003.
doi:10.1364/JOSAA.20.001867 Google Scholar

21. Rekanos, I. T., "Time-domain inverse scattering using Lagrange multipliers: An iterative FDTDbased optimization technique," Journal of Electromagnetic Waves and Applications, Vol. 17, No. 2, 271-289, 2003. Google Scholar

22. Fhager, A., P. Hashemzadeh, and M. Persson, "Reconstruction quality and spectral content of an electromagnetic time-domain inversion algorithm," IEEE Trans. Biomed. Eng., Vol. 53, No. 8, 1594-1604, 2006. Google Scholar

23. Meaney, P. M., K. D. Paulsen, A. Hartov, and R. K. Crane, "Microwave imaging for tissue assessment: Initial evaluation in multitarget tissue-equivalent phantoms," IEEE Trans. Biomed. Eng., Vol. 43, No. 9, 878-890, 1996. Google Scholar

24. Franchois, A., A. Joisel, C. Pichot, and J.-C. Bolomey, "Quantitative microwave imaging with a 2.45-GHz planar microwave camera," IEEE Trans. Med. Imaging, Vol. 17, No. 4, 550-561, 1998. Google Scholar

25. Joachimowicz, N., J. J. Mallorqui, J.-C. Bolomey, and A. Broquetas, "Convergence and stability assessment of Newton-Kantorovich reconstruction algorithms for microwave tomography," IEEE Trans. Med. Imaging, Vol. 17, No. 4, 562-570, 1998. Google Scholar

26. Mojabi, P. and J. LoVetri, "Microwave biomedical imaging using the multiplicative regularized Gauss-Newton inversion," IEEE Ant. Wireless Prop. Lett., Vol. 8, 645-648, 2009. Google Scholar

27. Fhager, A., M. Gustafsson, and S. Nordebo, "Image reconstruction in microwave tomography using a dielectric Debye model," IEEE Trans. Biomed. Eng., Vol. 59, No. 1, 156-166, 2012. Google Scholar

28. Semenov, S. Y., et al., "Three-dimensional microwave tomography: Initial experimental imaging of animals," IEEE Trans. Biomed. Eng., Vol. 49, No. 1, 55-63, 2002. Google Scholar

29. Yu, C., et al., "Active microwave imaging II: 3-D system prototype and image reconstruction from experimental data," IEEE Trans. Microw. Theory Tech., Vol. 56, No. 4, 991-1000, 2008. Google Scholar

30. Abubakar, A., T. M. Habashy, G. Pan, and M.-K. Li, "Application of the multiplicative regularized Gauss-Newton algorithm for three-dimensional microwave imaging," IEEE Trans. Antennas Propag., Vol. 60, No. 5, 2431-2441, 2012. Google Scholar

31. Catapano, I., L. Di Donato, L. Crocco, and O. M. Bucci, A. F. Morabito, T. Isernia, R. Massa, "On quantitative microwave tomography of female breast," Progress In Electromagnetics Research, Vol. 97, 75-93, 2009. Google Scholar

32. Meaney, P. M., et al., "A clinical prototype for active microwave imaging of the breast," IEEE Trans. Microw. Theory Tech., Vol. 48, No. 11, 1841-1853, 2000. Google Scholar

33. Rubaek, T., 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 Trans. Antennas Propag., Vol. 55, No. 8, 2320-2331, 2007. Google Scholar

34. Henriksson, T., et al., "Quantitative microwave imaging for breast cancer detection using a planar 2.45 GHz system," IEEE Trans. Instrum. Meas., Vol. 59, No. 10, 2691-2699, 2010. Google Scholar

35. Bulyshev, A. E., et al., "Computational modeling of three-dimensional microwave tomography of breast cancer," IEEE Trans. Biomed. Eng., Vol. 48, No. 9, 1053-1056, 2001. Google Scholar

36. Zhou, H., T. Takenaka, J. E. Johnson, and T. Tanaka, "A breast imaging model using microwaves and a time domain three dimensional reconstruction method," Progress In Electromagnetics Research, Vol. 93, 57-70, 2009. Google Scholar

37. Winters, D. W., 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 Trans. Med. Imaging, Vol. 28, No. 7, 969-981, 2009. Google Scholar

38. Winters, D. W., et al., "A sparsity regularization approach to the electromagnetic inverse scattering problem," IEEE Trans. Antennas Propag., Vol. 58, No. 1, 145-154, 2010. Google Scholar

39. Shea, J. D., P. Kosmas, B. D. Van Veen, and S. C. Hagness, "Contrast-enhanced microwave imaging of breast tumors: A computational study using 3D realistic numerical phantoms," Inverse Probl., Vol. 26, No. 7, Article ID 074009 2010. Google Scholar

40. Shea, J. D., 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," Med. Phys., Vol. 37, No. 8, 4210-4226, 2010. Google Scholar

41. De Zaeytijd, J., On the 3D electromagnetic quantitative inverse scattering problem: Algorithms and regularization, Ph.D. thesis, Ghent University, 2009.

42. Bai, F., A. Franchois, J. De Zaeytijd, and A. Pizurica, "Three-dimensional quantitative microwave imaging of realistic numerical breast phantoms using Huber regularization," Proc. 35th Annual Int. Conf. IEEE Eng. Med. Biol. Society (EMBS), 5135-5138, Osaka, Japan, Jul. 3–7 2013. Google Scholar

43. De Zaeytijd, J. and A. Franchois, "A subspace preconditioned LSQR Gauss-Newton method with a constrained linesearch path applied to 3D biomedical microwave imaging," Int. J. Antennas Propag., Article ID 924067 2015. Google Scholar

44. Azghani, M., P. Kosmas, and F. Marvasti, "Fast microwave medical imaging based on iterative smoothed adaptive thresholding," IEEE AWP Letters, Vol. 14, 438-441, 2015. Google Scholar

45. Golnabi, A. H., et al., "Comparison of no-prior and soft-prior regularization in biomedical microwave imaging," J. Med. Phys., Vol. 36, No. 3, 159-170, 2011. Google Scholar

46. Tijhuis, A. G., K. Belkebir, A. C. S. Litman, and B. P. De Hon, "Theoretical and computational aspects of 2-D inverse profiling," IEEE Trans. Geosci. Remote Sens., Vol. 39, 1316-1330, 2001. Google Scholar

47. Habashy, T. M. and A. Abubakar, "A general framework for constraint minimization for the inversion of electromagnetic measurements," Progress In Electromagnetics Research, Vol. 46, 265-312, 2004. Google Scholar

48. De Zaeytijd, J., A. Franchois, and J.-M. Geffrin, "A new value picking regularization strategy — Application to the 3-D electromagnetic inverse scattering problem," IEEE Trans. Antennas Propag., Vol. 57, No. 4, 1133-1149, 2009. Google Scholar

49. Van den Bulcke, S., A. Franchois, and D. De Zutter, "Piecewise smoothed value picking regularization applied to 2-D TM and TE inverse scattering," IEEE Trans. Antennas Propag., Vol. 61, No. 6, 3261-3269, 2013. Google Scholar

50. Bai, F., et al., "Weakly convex discontinuity adaptive regularization for microwave imaging," IEEE Trans. Antennas Propag., Vol. 61, No. 12, 6242-6246, 2013. Google Scholar

51. Bai, F., et al., "Weakly convex discontinuity adaptive regularization for 3D quantitative microwave tomography," Inverse Probl., Vol. 30, No. 8, Article ID 085005 2014. Google Scholar

52. Lobel, P., L. Blanc-Feraud, Ch. Pichot, and M. Barlaud, "A new regularization scheme for inverse scattering," Inverse Probl., Vol. 13, No. 2, 403-410, 1997. Google Scholar

53. Caorsi, S., G. L. Gragnani, S. Medicina, M. Pastorino, and G. Zunino, "Microwave imaging based on a Markov random field model," IEEE Trans. Antennas Propag., Vol. 42, No. 3, 293-303, 1994. Google Scholar

54. Baussard, A., D. Premel, and O. Venard, "A Bayesian approach for solving inverse scattering from microwave laboratory-controlled data," Inverse Probl., Vol. 17, 1659-1669, 2001. Google Scholar

55. Ferraiuolo, G. and V. Pascazio, "The effect of modified markov random fields on the local minima occurrence in microwave imaging," IEEE Trans. Geosci. Remote Sens., Vol. 41, 1043-1055, 2003. Google Scholar

56. De Zaeytijd, J. and A. Franchois, "Three-dimensional quantitative microwave imaging from measured data with multiplicative smoothing and value picking regularization," Inverse Probl., Vol. 25, No. 2, 024004, 2009. Google Scholar

57. Li, S. Z., Markov Random Field Modeling in Image Analysis, Springer Publishing Company, Incorporated, 1995.

58. Huber, P. J., "Robust estimation of a location parameter," Ann. Math. Statist., Vol. 35, 73-101, 1964. Google Scholar

59. Schultz, R. R. and R. L. Stevenson, "A Bayesian approach to image expansion for improved definition," IEEE Trans. Image Process., Vol. 3, No. 3, 233-242, 1994. Google Scholar

60. Geffrin, J.-M., P. Sabouroux, and C. Eyraud, "Free space experimental scattering database continuation: Experimental set-up and measurement precision," Inverse Probl., Vol. 21, No. 6, S117, 2005. Google Scholar

61. Geffrin, J.-M. and P. Sabouroux, "Continuing with the Fresnel database: Experimental setup and improvements in 3D scattering measurements," Inverse Probl., Vol. 25, No. 2, 024001, 2009. Google Scholar

62. Bai, F., Spatial priors for tomographic reconstructions from limited data, Ph.D. thesis , Ghent University 2014.

63. Zastrow, E., et al., Database of 3D grid-based numerical breast phantoms for use in computational electromagnetics simulations, https://uwcem.ece.wisc.edu/MRIdatabase/InstructionManual.pdf.

64. De Zaeytijd, J., I. Bogaert, and A. Franchois, "An efficient hybrid MLFMA-FFT solver for the volume integral equation in case of sparse 3D inhomogeneous dielectric scatterers," J. Comput. Phys., Vol. 227, No. 14, 7052-7068, 2008. Google Scholar

65. Chaumet, P. C. and K. Belkebir, "Three-dimensional reconstruction from real data using a conjugate gradient-coupled dipole method," Inverse Probl., Vol. 25, No. 2, 024003, 2009. Google Scholar

66. Da Cunha, R. D. and T. Hopkins, "The parallel iterative methods (PIM) package for the solution of systems of linear equations on parallel computers," Appl. Num. Math., Vol. 19, 33-50, 1995. Google Scholar

67. Fletcher, R., Practical Methods of Optimization, 2nd Ed., John Wiley, 1990.

68. Zastrow, E., et al., "Development of anatomically realistic numerical breast phantoms with accurate dielectric properties for modeling microwave interactions with the human breast," IEEE Trans. Biomed. Eng., Vol. 55, No. 12, 2792-2800, 2008. Google Scholar

69. Bucci, O. M. and T. Isernia, "Electromagnetic inverse scattering: retrievable information and measurement strategies," Radio Sci., Vol. 32, 2123-2137, 1997. Google Scholar

Dr. Funing Bai

Dr. Ann Franchois

Dr. Aleksandra Pizurica