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December  2019, 13(6): 1327-1348. doi: 10.3934/ipi.2019058

## A reproducing kernel Hilbert space framework for inverse scattering problems within the Born approximation

 Department of Mathematics and Statistics, Villanova University, Villanova, PA 19085, USA

Corresponding author: k.muller@villanova.edu

Received  January 2019 Revised  June 2019 Published  October 2019

In this work we develop a new reproducing kernel Hilbert space (RKHS) framework for inverse scattering problems using the Born approximation. We assume we have backscattered data of a field that is dependent on an unknown scattering potential. Our goal is to reconstruct or image this scattering potential. Assuming the scattering potential lies in a RKHS, we find that the imaging equation can be rewritten as the inner product of the desired unknown function with the adjoint of the forward operator applied to the kernel of the imaging operator. We therefore may choose the kernel of the imaging operator such that the adjoint applied to this kernel is precisely the reproducing kernel of the Hilbert space the reflectivity function lies in. In this way we are able to obtain an alternative definition of an ideal image. We will demonstrate this theory using synthetic aperture radar imaging as an example, though there are other applicable imaging modalities i.e. inverse diffraction and diffraction tomography [1,6]. We choose SAR as it was the motivating application for this work. We will compare the RKHS ideal imaging technique to the standard microlocal analytic ideal image from backprojection theory. Note this method requires a variation of the standard SAR data model with the assumption of a full two dimensional data collection surface as opposed to a one dimensional flight path, however we are able to perform imaging with a single frequency and avoid the approximations made in the backprojection imaging operator derivation.

Citation: Kaitlyn (Voccola) Muller. A reproducing kernel Hilbert space framework for inverse scattering problems within the Born approximation. Inverse Problems and Imaging, 2019, 13 (6) : 1327-1348. doi: 10.3934/ipi.2019058
##### References:
 [1] M. Bertero and C. De Mol, Stability problems in inverse diffraction, IEEE Transactions on Antennas and Propagation, 29 (1981), 368-372.  doi: 10.1109/TAP.1981.1142558. [2] M. Bertero, C. De Mol and E. R. Pike, Linear inverse problems with discrete data Ⅰ: General formulation and singular system analysis, Inverse Problems, 1 (1985), 301-330.  doi: 10.1088/0266-5611/1/4/004. [3] G. Beylkin, Imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized Radon transform, J. Math. Phys., 26 (1985), 99-108.  doi: 10.1063/1.526755. [4] M. Boyarsky, T. Sleasman, L. Pulido-Mancera, A. V. Diebold, M. F. Imani and D. R. Smth, Single-frequency 3D synthetic aperture imaging with dynamic metasurface antennas, Applied Optics, 57 (2018), 4123-4134.  doi: 10.1364/AO.57.004123. [5] M. Cheney and B. Borden, Fundamentals of Radar Imaging, CBMS-NSF Regional Conference Series in Applied Mathematics, 79. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2009. doi: 10.1137/1.9780898719291. [6] A. J. Devaney, A computer simulation study of diffraction tomography, IEEE Transactions on Biomedical Engineering, 30 (1983). [7] J. J. Duistermaat, Fourier Integral Operators, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2011. doi: 10.1007/978-0-8176-8108-1. [8] G. B. Folland, Introduction to Partial Differential Equations, Princeton University Press, Princeton, NJ, 1995. [9] M. Gilman, E. Smith and S. Tsynkov, A linearized inverse scattering problem for the polarized waves and anisotropic targets, Inverse Problems, 28 (2012), 085009, 38 pp. doi: 10.1088/0266-5611/28/8/085009. [10] A. Grigis and J. Sjöstrand, Microlocal Analysis for Differential Operators: An Introduction, London Mathematical Society Lecture Note Series, 196. Cambridge University Press, Cambridge, 1994. doi: 10.1017/CBO9780511721441. [11] K. Muller, The relationship between backprojection and best linear unbiased estimation in synthetic-aperture radar imaging, Inverse Problems and Imaging, 10 (2016), 549-561.  doi: 10.3934/ipi.2016011. [12] F. Natterer, A Sobolev space analysis of picture reconstruction, SIAM Journal on Applied Mathematics, 39 (1980), 402-411.  doi: 10.1137/0139034. [13] C. J. Nolan and M. Cheney, Microlocal analysis of synthetic aperture radar imaging, The Journal of Fourier Analysis and Applications, 10 (2004), 133-148.  doi: 10.1007/s00041-004-8008-0. [14] F. O'Sullivan, A statistical perspective on ill-posed inverse problems, Statistical Science, 1 (1986), 502-527.  doi: 10.1214/ss/1177013525. [15] T. Sleasman, M. Boyarsky, M. F. Imani, T. Fromenteze, J. N. Gollub and D. R. Smith, Single-frequency microwave imaging with dynamic metasurface apertures, Optical Physics, 34 (2017), 1713-1726.  doi: 10.1364/JOSAB.34.001713. [16] M. E. Taylor, Pseudodiferential Operators, Princeton University Press, 34. Princeton University Press, Princeton, N.J., 1981. [17] F. Tréves, Introduction to Pseudodifferential and Fourier Integral Operators, The University Series in Mathematics, Plenum Press, New York, 1980. [18] K. Voccola, Statistical and Analytical Techniques in Synthetic Aperture Radar Imaging, Thesis (Ph.D.)-Rensselaer Polytechnic Institute, 2011. [19] K. Voccola, B. Yazici, M. Cheney and M. Ferrara, On the relationship between the generalized likelihood ratio test and backprojection method for synthetic-aperture radar imaging, SPIE Defense, Security, and Sensing, International Society for Optics and Photonics, (2009), 73350I-73350I. [20] G. Wahba, Spline Models for Observational Data, CBMS-NSF Regional Conference Series in Applied Mathematics, 59. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990. doi: 10.1137/1.9781611970128. [21] X. G. Xia and M. Z. Nashed, The Backus-Gilbert method for signals in reproducing kernel Hilbert spaces and wavelet subspaces, Inverse Problems, 10 (1994), 785-804.  doi: 10.1088/0266-5611/10/3/018.

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##### References:
 [1] M. Bertero and C. De Mol, Stability problems in inverse diffraction, IEEE Transactions on Antennas and Propagation, 29 (1981), 368-372.  doi: 10.1109/TAP.1981.1142558. [2] M. Bertero, C. De Mol and E. R. Pike, Linear inverse problems with discrete data Ⅰ: General formulation and singular system analysis, Inverse Problems, 1 (1985), 301-330.  doi: 10.1088/0266-5611/1/4/004. [3] G. Beylkin, Imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized Radon transform, J. Math. Phys., 26 (1985), 99-108.  doi: 10.1063/1.526755. [4] M. Boyarsky, T. Sleasman, L. Pulido-Mancera, A. V. Diebold, M. F. Imani and D. R. Smth, Single-frequency 3D synthetic aperture imaging with dynamic metasurface antennas, Applied Optics, 57 (2018), 4123-4134.  doi: 10.1364/AO.57.004123. [5] M. Cheney and B. Borden, Fundamentals of Radar Imaging, CBMS-NSF Regional Conference Series in Applied Mathematics, 79. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2009. doi: 10.1137/1.9780898719291. [6] A. J. Devaney, A computer simulation study of diffraction tomography, IEEE Transactions on Biomedical Engineering, 30 (1983). [7] J. J. Duistermaat, Fourier Integral Operators, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2011. doi: 10.1007/978-0-8176-8108-1. [8] G. B. Folland, Introduction to Partial Differential Equations, Princeton University Press, Princeton, NJ, 1995. [9] M. Gilman, E. Smith and S. Tsynkov, A linearized inverse scattering problem for the polarized waves and anisotropic targets, Inverse Problems, 28 (2012), 085009, 38 pp. doi: 10.1088/0266-5611/28/8/085009. [10] A. Grigis and J. Sjöstrand, Microlocal Analysis for Differential Operators: An Introduction, London Mathematical Society Lecture Note Series, 196. Cambridge University Press, Cambridge, 1994. doi: 10.1017/CBO9780511721441. [11] K. Muller, The relationship between backprojection and best linear unbiased estimation in synthetic-aperture radar imaging, Inverse Problems and Imaging, 10 (2016), 549-561.  doi: 10.3934/ipi.2016011. [12] F. Natterer, A Sobolev space analysis of picture reconstruction, SIAM Journal on Applied Mathematics, 39 (1980), 402-411.  doi: 10.1137/0139034. [13] C. J. Nolan and M. Cheney, Microlocal analysis of synthetic aperture radar imaging, The Journal of Fourier Analysis and Applications, 10 (2004), 133-148.  doi: 10.1007/s00041-004-8008-0. [14] F. O'Sullivan, A statistical perspective on ill-posed inverse problems, Statistical Science, 1 (1986), 502-527.  doi: 10.1214/ss/1177013525. [15] T. Sleasman, M. Boyarsky, M. F. Imani, T. Fromenteze, J. N. Gollub and D. R. Smith, Single-frequency microwave imaging with dynamic metasurface apertures, Optical Physics, 34 (2017), 1713-1726.  doi: 10.1364/JOSAB.34.001713. [16] M. E. Taylor, Pseudodiferential Operators, Princeton University Press, 34. Princeton University Press, Princeton, N.J., 1981. [17] F. Tréves, Introduction to Pseudodifferential and Fourier Integral Operators, The University Series in Mathematics, Plenum Press, New York, 1980. [18] K. Voccola, Statistical and Analytical Techniques in Synthetic Aperture Radar Imaging, Thesis (Ph.D.)-Rensselaer Polytechnic Institute, 2011. [19] K. Voccola, B. Yazici, M. Cheney and M. Ferrara, On the relationship between the generalized likelihood ratio test and backprojection method for synthetic-aperture radar imaging, SPIE Defense, Security, and Sensing, International Society for Optics and Photonics, (2009), 73350I-73350I. [20] G. Wahba, Spline Models for Observational Data, CBMS-NSF Regional Conference Series in Applied Mathematics, 59. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990. doi: 10.1137/1.9781611970128. [21] X. G. Xia and M. Z. Nashed, The Backus-Gilbert method for signals in reproducing kernel Hilbert spaces and wavelet subspaces, Inverse Problems, 10 (1994), 785-804.  doi: 10.1088/0266-5611/10/3/018.
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