224 IEEE TRANSACTIONS ON COMPUTATIONAL IMAGING, VOL. 7, 2021 Electromagnetic Field Imaging in Arbitrary Scattering Environments Karteekeya Sastry , Chandan Bhat , Raffaele Solimene , Senior Member, IEEE, and Uday K. Khankhoje , Senior Member, IEEE Abstract—In this article, we propose a method to reconstruct the total electromagnetic field in an arbitrary two-dimensional scattering environment without any prior knowledge of the incident field or the permittivities of the scatterers. However, we assume that the region between the scatterers is homogeneous and that the approximate geometry describing the environment is known. Our approach uses field measurements and a compressive sensing inspired algorithm to estimate the incident field and the tangential electric and magnetic fields on the scatterers’ surfaces. These esti- mates are then used to predict the field everywhere using Huygens’ principle. Further, we identify the best measurement locations in the environment, which reduces the estimation error to approxi- mately half of the error obtained when using random locations. We show that in an indoor scenario with up to four scattering objects, the total electric field is recovered with less than 10% error when the number of measurements is just 0.3 times the number of unknowns in which the problem is formulated. The formulated problem is solved using ‘Total field - Compressive sensing based subspace optimization method’ – an algorithm that leverages the sparsity of the tangential fields in known transformation domains to obtain an optimal solution. Index Terms—Compressive sensing, electromagnetic fields, sensor placement, subspace optimization. I. INTRODUCTION K NOWLEDGE of the electromagnetic (EM) fields is useful in many applications like network planning, WiFi access point planning in indoor scenarios, and indoor localization [1]– [4]. Existing methods in the literature that address this problem are based on ray tracing [5], [6]. It is well known that the ray-tracing model fails to account for the effects of diffraction Manuscript received August 22, 2020; revised November 29, 2020; accepted January 27, 2021. Date of publication February 1, 2021; date of current version February 19, 2021. This work was supported by the Microwave Inverse Scat- tering for Breast Cancer Detection project through the Science and Engineering Research Board, Department of Science and Technology, Government of India, under Grant ECR/2018/001953. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Ilaria Catapano. (Karteekeya Sastry and Chandan Bhat contributed equally to this work.) (Cor- responding authors: Chandan Bhat; Uday K. Khankhoje.) Karteekeya Sastry is with the Department of Electrical Engineering, California Institute of Technology, Pasadena, CA 91125 USA (e-mail: karteekdhara98@gmail.com). Chandan Bhat and Uday K. Khankhoje are with the Department of Electrical Engineering, Indian Institute of Technology Madras, Chennai 600036, India (e-mail: chandanbhat21@gmail.com; uday@ee.iitm.ac.in). Raffaele Solimene is with the Universita della Campania Luigi Vanvitelli, Italy, and is also an Adjunct Faculty with the Department of Electrical Engi- neering, Indian Institute of Technology Madras, Chennai 600036, India (e-mail: raffaele.solimene@unicampania.it). Digital Object Identifier 10.1109/TCI.2021.3055982 Fig. 1. Problem depiction with a source (J ) radiating in the presence of objects, enclosed within a wall. Field measurements are made at strategic locations as indicated by filled dark circles. The objective is to predict the field at any point outside the object(s) and within the walls. To achieve this, tangential electromagnetic fields are estimated on object and wall boundaries. around the corners of the objects, waveguiding in the corridors, and multiple reflections [7, Fig. 3]. Another popular approach is to assume that the EM fields are sparse in the spatial Fourier domain, and to leverage this information to reconstruct the fields [8]. This assumption works well when the domain of interest is far from the scatterers and the sources. Moreover, in most of the literature on EM inverse problems, information on the scattered fields is needed to solve the problem, which implies that the incident field should also be known [9], [10]. In this paper, we present a method to reconstruct the total EM field from total field measurements at optimal sampling locations. We do not assume any prior knowledge of the incident field or the permittivities of the scatterers. However, we assume that the region between the scatterers is homogeneous and that the approximate geometry of the scatterers and the source is known. Our method is based on surface integral formulations and is therefore a significant improvement over existing ray tracing methods in the literature. In recent work [11], we presented a method for reconstructing the scattered EM fields by making measurements at random locations, but assumed knowledge of the incident field. In the current work, we generalize this method to recover the total field without any prior knowledge of the incident field. We also identify optimal sampling locations in the environment. The problem setup is graphically represented in Fig. 1. 2333-9403 © 2021 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See https://www.ieee.org/publications/rights/index.html for more information. Authorized licensed use limited to: CALIFORNIA INSTITUTE OF TECHNOLOGY. 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