# American Institute of Mathematical Sciences

June  2020, 28(2): 1107-1121. doi: 10.3934/era.2020061

## Generating geometric body shapes with electromagnetic source scattering techniques

 1 Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong SAR, China 2 School of Mathematics and Statistics, Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, Shandong, China 3 School of Economics, Changchun University of Finance and Economics, Changchun 130122, Jilin, China 4 Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong SAR, China 5 School of Astronautics, Harbin Institute of Technology, Harbin 150001, Heilongjiang, China

* Corresponding author: Xianchao Wang

Received  May 2020 Revised  May 2020 Published  June 2020

In this paper, we are concerned with the three-dimensional (3D) geometric body shape generation with several well-selected characteristic values. Since 3D human shapes can be viewed as the support of the electromagnetic sources, we formulate a scheme to regenerate 3D human shapes by inverse scattering theory. With the help of vector spherical harmonics expansion of the magnetic far field pattern, we build on a smart one-to-one correspondence between the geometric body space and the multi-dimensional vector space that consists of all coefficients of the spherical vector wave function expansion of the magnetic far field pattern. Therefore, these coefficients can serve as the shape generator. For a collection of geometric body shapes, we obtain the inputs (characteristic values of the body shapes) and the outputs (the coefficients of the spherical vector wave function expansion of the corresponding magnetic far field patterns). Then, for any unknown body shape with the given characteristic set, we use the multivariate Lagrange interpolation to get the shape generator of this new shape. Finally, we get the reconstruction of this unknown shape by using the multiple-frequency Fourier method. Numerical examples of both whole body shapes and human head shapes verify the effectiveness of the proposed method.

Citation: Youzi He, Bin Li, Tingting Sheng, Xianchao Wang. Generating geometric body shapes with electromagnetic source scattering techniques. Electronic Research Archive, 2020, 28 (2) : 1107-1121. doi: 10.3934/era.2020061
##### References:
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##### References:
 [1] D. Anguelov, P. Srinivasan, D. Koller, S. Thrun, J. Rodgers and J. Davis, SCAPE: Shape completion and animation of people, ACM Transactions on Graphics (TOG), 24 (2005), 408-416.  doi: 10.1145/1186822.1073207.  Google Scholar [2] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, Vol. 93. Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03537-5.  Google Scholar [3] J. Li, H. Liu, Z. Shang and H. Sun, Two single-shot methods for locating multiple electromagnetic scatterers, SIAM J. Appl. Math., 73 (2013), 1721-1746.  doi: 10.1137/130907690.  Google Scholar [4] J. Li, H. Liu, W.-Y. Tsui and X. Wang, An inverse scattering approach for geometric body generation: A machine learning perspective, Mathematics in Engineering, 1 (2019), 800-823.  doi: 10.3934/mine.2019.4.800.  Google Scholar [5] J. Li, H. Liu and J. Zou, Locating multiple multiscale acoustic scatterers, Multiscale Model. Simul., 12 (2014), 927-952.  doi: 10.1137/13093409X.  Google Scholar [6] I. V. Lindell, TE/TM decomposition of electromagnetic sources, IEEE Trans. Antennas and Propagation, 36 (1988), 1382-1388.  doi: 10.1109/8.8624.  Google Scholar [7] H. Liu, A global uniqueness for formally determined inverse electromagnetic obstacle scattering, Inverse Problems, 24 (2008), 035018, 13 pp. doi: 10.1088/0266-5611/24/3/035018.  Google Scholar [8] H. Liu, L. Rondi and J. Xiao, Mosco convergence for $H(\rm curl)$ spaces, higher integrability for Maxwell's equations, and stability in direct and inverse EM scattering problems, J. Eur. Math. Soc. (JEMS), 21 (2019), 2945-2993.  doi: 10.4171/JEMS/895.  Google Scholar [9] H. Liu, M. Yamamoto and J. Zou, Reflection principle for the Maxwell's equations and its application to inverse electromagnetic scattering problem, Inverse Problems, 23 (2007), 2357-2366.  doi: 10.1088/0266-5611/23/6/005.  Google Scholar [10] H. Liu and J. Zou, Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers, Inverse Problems, 22 (2006), 515-524.  doi: 10.1088/0266-5611/22/2/008.  Google Scholar [11] T. Sauer and Y. Xu, On multivariate Lagrange interpolation, Math. Comp., 64 (1995), 1147-1170.  doi: 10.1090/S0025-5718-1995-1297477-5.  Google Scholar [12] H. Seo, F. Cordier and N. Magnenat-Thalmann, Synthesizing animatable body models with parameterized shape modifications, Proceedings of the 2003 ACM SIGGRAPH/Eurographics Symposium on Computer Animation. Eurographics Association, (2003), 120–125. Google Scholar [13] M. Takada and T. Esaki, Method and apparatus for measuring human body or the like, U.S. Patent, (1983), 406–544. Google Scholar [14] G. Wang, F. Ma, Y. Guo and J. Li, Solving the multi-frequency electromagnetic inverse source problem by the Fourier method, J. Differential Equations, 265 (2018), 417-443.  doi: 10.1016/j.jde.2018.02.036.  Google Scholar [15] X. Wang, M. Song, Y. Guo, H. Li and H. Liu, Fourier method for identifying electromagnetic sources with multi-frequency far-field data, J. Comput. Appl. Math., 358 (2019), 279-292.  doi: 10.1016/j.cam.2019.03.013.  Google Scholar [16] D. Zhang and Y. Guo, Fourier method for solving the multi-frequency inverse source problem for the Helmholtz equation, Inverse Problems, 31 (2015), 035007, 30 pp. doi: 10.1088/0266-5611/31/3/035007.  Google Scholar [17] D. Zhang, Y. Guo, J. Li and H. Liu, Retrieval of acoustic sources from multi-frequency phaseless data, Inverse Problems, 34 (2018), 094001, 21 pp. doi: 10.1088/1361-6420/aaccda.  Google Scholar [18] D. Zhang, Y. Guo, J. Li and H. Liu, Locating multiple multipolar acoustic sources using the direct sampling method, Commun. Comput. Phys., 25 (2019), 1328-1356.  doi: 10.4208/cicp.oa-2018-0020.  Google Scholar [19] D. Zhang, F. Sun, Y. Guo and H. Liu, Unique determinations in inverse scattering problems with phaseless near-field measurements, Inverse Problems and Imaging, 14 (2020), 569-582.   Google Scholar [20] Make Human Community: Open Source tool for making 3D characters, Available from: http://www.makehumancommunity.org. Google Scholar
Isosurface plots of some random training body data with different characteristic values
Isosurface plots of the exact and reconstucted body shapes. (a) The exact child body, (b) the reonstucted child body with the given characteristic values
Isosurface plots of some random training head data with different characteristic values
Plots of the exact and reconstruted head shapes with the given characteristic values. The left column: the exact head shapes, the right column: the reconstruted head shapes
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