INVERSE PROBLEMS IN ELECTROMAGNETICS: CASE STUDIES OF DEEP LEARNING-BASED SOURCE RECONSTRUCTION AND OBJECT COUNTING

Abstract

In this thesis, two case-specific Electromagnetic (EM) inverse problems are investigated. The first inverse problem is the prediction of the number of wires randomly distributed within a square-shaped region of interest (ROI). The predictions are made based on the Electric Field (E-field) values calculated at 40 points at the outer perimeter of the ROI uponilluminating each Randomized Wire Distribution (RWD) within the ROI with a linearly-polarized plane wave at a given frequency f = 10 MHz. The calculation of the E-field component parallel to the wire orientation is accomplished using the self-consistent approach described in Ref. 1. This semi-analytic approach made it possible to generate the E-field data for millions of RWDs at a relatively low computational cost. A Convolutional Neural Network (CNN) of sequential topology was utilized as a data-driven inverse solver. Under the assumption that the upper limit to the number of wires, Nmax , is known, the CNN is structured such that when the trained CNN is fed with the numerically-generated E-field values, it outputs the likelihood of the presence of N wires within the ROI, where 1 ⩽ N ⩽ Nmax . The accuracy of the predictions were examined upon adding data channels in the input vector, with each channel containing the E-field data acquired upon illuminating the RWD with a plane wave (PW) at a different angle of incidence. A novel technique is developed based on this analysis which enables the visualization of the “blind spots” of the predictor within the ROI. Moreover, in an effort to gain physical insights into the inner workings or the inverse solver, the CNN diffraction limit was deduced from the statistical analyses performed on CNN predictions made on 3 × 106 test samples and comparing the f = 10 MHz results with the f = 5 MHz case. The second inverse problem is the data-driven reconstruction of the coarse surface reflectivity patterns formed on a Printed Circuit Board (PCB). This reconstruction is merely based on the Radio Frequency (RF) snapshots taken in the far-zone while the PCB is under illumination with quasi-planar EM radiation within 9–11 GHz. In particular, the term “RF snapshot” is used here to refer to the set of S21 traces acquired at the 8 Receiver (Rx) antennas in the frequency domain. In essence, an RF snapshot is an undersampled∗ EM image with multiple frequency channels. On the PCB side, the reflectivity patterns are formed dynamically using a new and novel technique proposed in this thesis. The hardware developed in this work, which is referred to as “RF display,” makes it possible to generate 8-by-8-pixel reflectivity patterns at a relatively high speed by means of electronic switching. As a result, large datasets of distinct 8-by-8-pixel randomized patterns can be generated along with their corresponding RF snapshots. The resulting datasets are subsequently used to train various CNNs of U-net2 architechture. A reflectivity pattern that is outside the training dataset can then be reconstructed upon feeding its corresponding RF snapshot into the trained U-net. Some of the Data Acquisition (DAQ) cycles were run inside an anechoic chamber and some cycles were run in a cluttered room, which accumulatively led to the acquisition of ∼ 1.5 million samples, with each sample containing a randomly-generated reflectivity pattern and its corresponding experimentally-acquired RF snapshot. The U-nets that are separately trained on the resulting “in-chamber” and “cluttered-room” datasets led to highly accurate predictions when tested on independent test samples. This empirically proves the U-net-based pattern decoding strategy to be effective even when the RF snapshots are acquired in a cluttered room. Prior to the design and fabrication of the RF display, preliminary numerical investigations were carried out to explore the feasibility of this U-net-based decoding strategy. To do so, a semi-analytic model was developed to calculate the EM-field of a full patch antenna array (PAA) by superposing the EM-field due to individual elements. Owing to analytic expressions for the EM-field due to a single patch antenna,3 millions of randomized on-off configurations for an 8-by-8 PAAs are generated within a time scale of ∼ 103 sec—a numerical task that could not have been achieved through the use of full EM simulations. The resulting numerically-generated samples were used to train the U-nets, and the trained U-nets were capable of making high-accuracy predictions on independent numerically-generated test samples. It is worth noting that, (i) this superposition-based model can only be used for PAAs with a large enough nearest-neighbor distance, (ii) this model is developed for an electromagnetically active∗ (EM-active) array whereas the fabricated RF display is an EM-passive array, i.e., it does not radiate by itself, and (iii) the EM-field data produced by this model is valid only within the far-field zone of the array. The parameters of interest which are inferred from EM-field measurements in the first and second cases are the number of vertically-oriented wires inside the ROI and the overall on-off state of the RF pixel array (the pattern), respectively. In the former case, the inverse solver is built based on the prior knowledge of (i) the number of points at which the E-field is sampled, nRx , (ii) the number of swept frequency points, (iii) the number of distinct illumination angles, ninc , and (iv) the upper limit to the number of wires within the ROI, Nmax . Similarly, in the latter case, the construction of the inverse solver requires the prior knowledge of (i) the number of points at which the E-field is sampled, nRx , (ii) the numberof swept frequency points, nf , and (iii) the dimension of the array, d. The common aspect between these two scenarios is the inference of cause from the effect; hence the name inverse.