volume | PIER Journals

Abstract

In this paper, the Compressive Sampling Matching Pursuit Algorithm (CoSaMP) is applied to microwave reconstruction of a 2-dimensional non-sparse object. First, an adaptive discretization method, DistMesh method, is applied to discretize the image domain based on the region of interest. The dual-mesh method is able to provide denser and smaller discretized cells in more important areas of the object and larger cells in other areas, thereby providing more details in the interest domain and keeping the computational burden at a reasonable level. Another benefit of using the dual-mesh method is that it automatically generates size functions and adapts to the curvature and the feature size of the geometry. In addition, the size of each cell changes gradually. Next, the inverse scattering problem is solved in frame of Distorted Born Iterative Method (DBIM). During each iteration of DBIM, the near field scattering problem is modeled as a set of linear equations. Furthermore, a compressive sensing (CS) method called the Compressive Sampling Matching Pursuit Algorithm is applied to solve the nonlinear inverse problem. During this process, two innovative steps are applied. First, for the reconstruction of the non-sparse object, the signal input to our algorithm is processed via a wavelet transformation to obtain sparsity. Second, as the dual-mesh method discretizes more important cells in smaller sizes, these cells have high potential to be filtered by the threshold of CoSaMP. As a result, a regularization matrix is introduced to reduce the effect of size. Finally, we present numerical experiment results based on our dual-mesh method combined with the regularized CoSaMP algorithm.

1. Chandra, R., H. Zhou, I. Balasingham, and R. M. Narayanan, "On the opportunities and challenges in microwave medical sensing and imaging," IEEE Transactions on Biomedical Engineering, Vol. 62, No. 7, 1667-1682, 2015.
doi:10.1109/TBME.2015.2432137 Google Scholar

2. Zhou, H., R. Narayanan, I. Balasingham, and R. Chandra, "Radar for disease detection and monitoring," Radar for Indoor Monitoring: Monitoring: Detection, Localization and Assessment, edited by M. G. Amin, 301–335, CRC Press, Boca Raton, FL, 2017. Google Scholar

3. Chandra, R., H. Zhou, I. Balasingham, and R. M. Narayanan, "Medical microwave imaging and analysis," Medical Image Analysis and Informatics: Computer-Aided Diagnosis and Therapy, P. M. de Azevedo-Marques, A. Mencattini, M. Salmeri, and R. M. Rangayyan (eds.), 451–466, CRC Press, Boca Raton, FL, 2018. Google Scholar

4. Colton, D. L. and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer- Verlag, Berlin, 1992.
doi:10.1007/978-3-662-02835-3

5. Franchois, A. and C. Pichot, "Microwave imaging-complex permittivity reconstruction with a Levenberg-Marquardt method," IEEE Transactions on Antennas and Propagation, Vol. 45, No. 2, 203-215, 1997.
doi:10.1109/8.560338 Google Scholar

6. Hohage, T., "Logarithmic convergence rates of the iteratively regularized Gauss-Newton method for an inverse potential and an inverse scattering problem," Inverse Problems, Vol. 13, No. 5, 1279-1299, Jan. 1997.
doi:10.1088/0266-5611/13/5/012 Google Scholar

7. Zhou, H., R. M. Narayanan, R. Chandra, and I. Balasingham, "Microwave imaging of circular phantom using the Levenberg-Marquardt method," Proc. SPIE Conf. Radar Sensor Technology XIX and Active and Passive Signatures VI, Baltimore, MD, Apr. 2015, doi: 10.1117/12.2176754. Google Scholar

8. Van Den Berg, P. M. and A. Abubakar, "Contrast source inversion method: State of art," Progress In Electromagnetics Research, Vol. 34, 189-218, 2001.
doi:10.2528/PIER01061103 Google Scholar

9. Li, L., H. Zheng, and F. Li, "Two-dimensional contrast source inversion method with phaseless data: TM case," IEEE Transactions on Geoscience and Remote Sensing, Vol. 47, No. 6, 1719-1736, 2009.
doi:10.1109/TGRS.2008.2006360 Google Scholar

10. Hajihashemi, M. R. and M. El-Shenawee, "Shape reconstruction using the level set method for microwave applications," IEEE Antennas and Wireless Propagation Letters, Vol. 7, 92-96, 2008.
doi:10.1109/LAWP.2008.920464 Google Scholar

11. Irishina, N., D. Alvarez, O. Dorn, and M. Moscoso, "Structural level set inversion for microwave breast screening," Inverse Problems, Vol. 26, No. 3, 035015, 2010.
doi:10.1088/0266-5611/26/3/035015 Google Scholar

12. Candes, E. and M. Wakin, "An introduction to compressive sampling," IEEE Signal Processing Magazine, Vol. 25, No. 2, 21-30, 2008.
doi:10.1109/MSP.2007.914731 Google Scholar

13. Pati, Y. C., R. Ramin, and P. S. Krishnaprasad, "Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition," Proc. 27th Asilomar Conference on Signals, Systems and Computers, 40-44, Pacific Grove, CA, Nov. 1993. Google Scholar

14. Davis, G., S. Mallat, and M. Avellaneda, "Adaptive greedy approximations," Constructive Approximation, Vol. 13, No. 1, 57-98, 1997.
doi:10.1007/BF02678430 Google Scholar

15. Azghani, M., P. Kosmas, and F. Marvasti, "Microwave medical imaging based on sparsity and an iterative method with adaptive thresholding," IEEE Transactions on Medical Imaging, Vol. 34, No. 2, 357-365, 2014.
doi:10.1109/TMI.2014.2352113 Google Scholar

16. Zhou, H. and R. M. Narayanan, "Microwave imaging of nonsparse object using dual-mesh method and iterative method with adaptive thresholding," IEEE Transactions on Antennas and Propagation, Vol. 67, No. 1, 504-512, 2019.
doi:10.1109/TAP.2018.2876164 Google Scholar

17. Blumensath, T. and M. E. Davies, "Iterative thresholding for sparse approximations," Journal of Fourier Analysis and Applications, Vol. 14, No. 5–6, 629-654, 2008.
doi:10.1007/s00041-008-9035-z Google Scholar

18. Blumensath, T. and M. E. Davies, "Iterative hard thresholding for compressed sensing," Applied and Computational Harmonic Analysis, Vol. 27, No. 3, 265-274, 2009.
doi:10.1016/j.acha.2009.04.002 Google Scholar

19. Needell, D. and J. A. Tropp, "CoSaMP: Iterative signal recovery from incomplete and inaccurate samples," Applied and Computational Harmonic Analysis, Vol. 26, No. 3, 301-321, 2009.
doi:10.1016/j.acha.2008.07.002 Google Scholar

20. Persson, P.-O., "Mesh size functions for implicit geometries and PDE-based gradient limiting," Engineering with Computers, Vol. 22, No. 2, 95-109, 2006.
doi:10.1007/s00366-006-0014-1 Google Scholar

21. Zhou, H., R. M. Narayanan, and I. Balasingham, "Microwave reconstruction method using a circular antenna array cooperating with an internal transmitter," Proc. SPIE Conf. Radar Sensor Technology XX, Baltimore, MD, Apr. 2016, doi: 10.1117/12.2228287. Google Scholar

22. Paulsen, K., P. Meaney, M. Moskowitz, and J. Sullivan, "A dual-mesh scheme for finite element based reconstruction algorithms," IEEE Transactions on Medical Imaging, Vol. 14, No. 3, 504-514, 1995.
doi:10.1109/42.414616 Google Scholar

23. Brassarote, G. O. N., E. M. Souza, and J. F. G. Monico, "Non-decimated wavelet transform for a shift-invariant analysis," Tendencias em Matematica Aplicada e Computacional, Vol. 19, No. 1, 93-110, 2018.
doi:10.5540/tema.2018.019.01.93 Google Scholar

24. Loboda, N. S., A. V. Glushkov, V. N. Khokhlov, and L. Lovett, "Using non-decimated wavelet decomposition to analyse time variations of North Atlantic oscillation, eddy kinetic energy, and Ukrainian precipitation," Journal of Hydrology, Vol. 322, No. 1–4, 14-24, 2006.
doi:10.1016/j.jhydrol.2005.02.029 Google Scholar

25. Balanis, C. A., Advanced Engineering Electromagnetics, J. Wiley & Sons, Hoboken, NJ, 2012.

26. Chandra, R., A. J. Johansson, M. Gustafsson, and F. Tufvesson, "A microwave imaging-based technique to localize an in-body RF source for biomedical applications," IEEE Transactions on Biomedical Engineering, Vol. 62, No. 5, 1231-1241, 2015.
doi:10.1109/TBME.2014.2367117 Google Scholar

27. Chew, W. and Y. Wang, "Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method," IEEE Transactions on Medical Imaging, Vol. 9, No. 2, 218-225, 1990.
doi:10.1109/42.56334 Google Scholar

28. Davenport, M. A., D. Needell, and M. B. Wakin, "Signal space CoSaMP for sparse recovery with redundant dictionaries," IEEE Transactions on Information Theory, Vol. 59, No. 10, 6820-6829, 2013.
doi:10.1109/TIT.2013.2273491 Google Scholar

29. Sandhu, A. I. and H. Bagci, "A modified CoSaMP algorithm for electromagnetic imaging of two dimensional domains," Proc. 2017 International Applied Computational Electromagnetics Society Symposium-Italy (ACES), Florence, Italy, Mar. 2017, doi: 10.23919/ROPACES.2017.7916412. Google Scholar

30. Geffrin, J.-M., P. Sabouroux, and C. Eyraud, "Free space experimental scattering database continuation: Experimental set-up and measurement precision," Inverse Problems, Vol. 21, No. 6, S117-S130, 2005.
doi:10.1088/0266-5611/21/6/S09 Google Scholar

Dr. Huiyuan Zhou

Prof. Ram M. Narayanan