Imaging by convolutional neural networks in frequency domain 

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Sparsity constrained method

The imaging performances are expected to be better if more prior information about the probed object is provided. Here, the missing rods can be treated as equivalent (fictitious) sources of unknown location inside the structure. The equivalence theory provides a link between the collected data and the expansion coefficients of equivalent sources, the non-zero elements of which indicate the index of a missing rod. With data from sources and receivers in use, the solution is achieved by sparsity-constrained method hereafter.
The possibility of finding missing rods by analysing its electromagnetic responses relies on the perturbation of the background field due to the missing rods. The evaluation of this perturbation is performed with the Lippman-Schwinger integral formulation: eE (r) − E(r) = XL l Z Dl G(r, r′)(k2 l − k2ϵr)eE (r′).

Results of sparsity-constrained method

The results shown in Fig. 3.3 correspond to three different cases when SNR = 30dB: missing one rod, missing two rods, missing three rods in the micro-structure, for two rod distributions: 36 rods, 64 rods. Super-localization appears well realized by using the sparsity information.
Some other tests for different values of d and c are also carried out. Fig. 3.4 shows different radii of rod, Fig. 3.5 shows the results of different distances between rods. It turns out that the proposed method still works.
Different ϵr values are also used to guarantee the robustness of the sparsity-constrained method. As shown in Fig. 3.6, when the ϵ of rod is 7.5, the sparsity-constrained method succeed in locating the missing rod for the 36 rods case with a higher value of M, while it does not succeed for the 64 rods case even with a higher value of M, which shows that the proposed method still has limitations.

Results of binary-specialized contrast source inversion

In the present section, the binary-specialized CSI is tested for different cases of interest, wherein the radius r and the distance d that characterize the micro-structure are kept constant, as λ/12 and λ/4 respectively, λ wavelength in air. The size of the region of interest (the ROI) is set at 1.75λ × 1.75λ, and is discretized into 60 × 60 cells in the direct problem, and into 50 × 50 cells in the inverse problem. 36 receivers and 36 sources are placed in a regular fashion on a circle of radius 7.2 λ. The single operating frequency is in practice fixed at 3 GHz (λ = 0.1 m). The collected fields are used to reconstruct the contrast map of the damaged micro-structure by the binary-specialized CSI method. The micro-structure is involving 4 × 4 or 6 × 6 rods. Two cases are considered: one with no prior information on the positions of the rods, the other with that prior position information, the signal to noise ratios being taken as 30dB. From Fig.3.7, without the prior position information, the binary-specialized CSI method cannot realize the reconstruction of the contrast map for both 16 and 36 rods, the distance between two rods d equals λ/4, which is smaller than λ/2, may lead to this failure. From Fig. 3.8 and Fig. 3.9, one concludes that the binary-specialized CSI can realize the reconstruction with the prior position information.

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CI for reconstruction of random contrast distribution

The contents described earlier in this chapter are about the binary case of the structure. While, the contrast of rods can be different, so, deleting the binary prior, the traditional CSI method is used to reconstruct the distribution of ROI whose contrast value is random Illustrative results of application of traditional CSI to the micro-structure are shown in Fig. 3.10. About the test, two choices have been made: (i) using the CSI method without any information on the positions of rods, (ii) using the CSI method with information on those. Without prior position information, one cannot really reconstruct the contrast map, while helped by prior positions, it works slightly better but still cannot achieve precise retrieval.

Table of contents :

List of figures
List of tables
1 Introduction 
1.1 Research background
1.2 Outline
2 Modelling of the forward problem 
2.1 Multiple scattering method
2.2 Method of moments
2.3 Validation of the modelling
3 Sparsity constrained inversion and contrast source inversion 
3.1 Sparsity constrained method
3.1.1 Results of sparsity-constrained method
3.2 Contrast source inversion
3.2.1 Results of binary-specialized contrast source inversion
3.3 CSI for reconstruction of random contrast distribution
4 Imaging by convolutional neural networks in frequency domain 
4.1 CNN architecture
4.2 Loss function
4.3 Training method
4.4 Training dataset
4.5 Implementation
4.6 The binary-specialized CNN: a reference example
4.6.1 Different configurations for the test
4.6.2 Single frequency vs. multiple frequencies of operation
4.6.3 Different data noise ratios
4.6.4 Different values of contrast
4.6.5 Additional results for different numbers of missing rods and different shapes using binary-specialized CNN
4.6.6 Extension to random contrast distribution
5 Imaging by recurrent neural networks in time domain 
5.1 Motivation of using RNN
5.2 LSTM structure
5.2.1 Dataset
5.2.2 Training process
5.2.3 Results
5.3 Comparison with imaging by convolutional neural networks
5.3.1 CNN architecture
5.3.2 CNN results
5.3.3 Comparison between CNN and RNN
5.4 Validation on laboratory-controlled Data
5.4.1 The configuration of experiments
5.4.2 Results on experimental data
6 Imaging by convolutional-recurrent neural networks 
6.1 Architecture of proposed CRNN
6.2 Training process
6.3 Results of CRNN
7 Direct imaging method: time reversal 
7.1 Time reversal for localization of source
7.2 Time Reversal for localization of missing rods
8 Conclusion 
8.1 Summary of the work as completed
8.2 Potential work
Appendix A Graf’s addition theorem 
Appendix B The Lippman-Schwinger formulation 
Appendix C Further complementary material on time reversal I
Appendix D Further complementary material on time reversal II 


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