Deep Learning Assisted Linear Sampling Method for the Reconstruction of Perfect Electric Conductors

. In this study, a linear approach, linear sampling method (LSM) is used to reconstruct the shape of perfectly electric conductors (PEC) with the help of deep learning as a post-processing technique. In microwave imaging, the LSM is a simple and reliable linear inversion technique for determining the morphological features of unknown objects under investigation. However, the output of this method depends on the frequency of operation, the choice of regularization parameter,and it is unable to produce satisfactory results for objects with complex shapes. To overcome this drawback, a deep learning approach is used in this work, which can produce a better output in terms of accuracy, resolution. Here, the rough estimate of the PEC scatterer obtained using LSM is used to train the U-Net based convolutional neural network, which maps this output with the corresponding ground truth profiles. The proposed hybrid model is validated using several examples of synthetic and experimental data.


Introduction
Microwave imaging (MI) is a non-contact and noninvasive methodology to estimate concealed or implanted objects in a structure using electromagnetic waves in the microwave range.MI is widely used in medical imaging, nondestructive testing, geophysical explorations etc. [1], [2].One of the challenges in microwave imaging is the reconstruction of perfectly electric conductors (PEC).The detection of PEC is essential in various applications, like concealed weapon detection [1][2][3][4], localization of metallic contaminants in food products [5] and landmine detection [6], [7] etc. PEC reconstruction techniques are broadly classified as volume based and surface-based methods.A volumebased method, such as the local shape function, assigns a Boolean variable to each pixel to indicate whether it is PEC or not [8][9][10].Genetic algorithms (GAs) are used to solve the PEC inverse scattering problem (ISP) in [11] and [12].The main drawback of GA is that the computational cost increases exponentially with sample size [13].In [14][15][16], a different volume-based approach for mixed targets that uses integral scattering equations in that each pixel has been given a complex permittivity value, and metallic objects have been modelled as dielectrics with a substantial but finite imaginary part.PEC scatterers are reconstructed by surface-based methods using the Newton-Kantorovich method [17], the Differential Evolution optimizer [18], [19], and the GAs [20][21][22][23][24].The Level Set (LS) approach has been utilised successfully in [25] to produce the PEC contour without any prior knowledge of the scattering scenario.The subspace optimization methodbased inversion process has been used to categorise each segment as PEC or non-PEC [26], [27].
LSM is one of the qualitative imaging procedures used for the shape reconstruction of dielectric as well as metallic targets.This method is rapid, reliable, and computationally efficient [28].It has been applied for various applications like detection of coated objects, buried objects, objects behind walls, and breast cancer cells etc., [29][30][31][32][33][34].LSM able to estimate applied by considering the targets to be either dielectric or perfect electric conducting (PEC) objects.The main drawback of this method is that the quality of output depends on frequency of operation and choice of regularization parameter [35], [36].To overcome these limitations, a modified LSM indicator based on a multi-frequency approach is introduced.However, the improved results are inadequate.Hence, in this work, we propose another methodology based on deep-learning to tackle the above issues.
To the best of author's knowledge, there are very few literatures on the applicability of deep learning for the reconstruction of PEC objects [37].An effective CNN-DL architecture is implemented through the modelling of the rough surface variation in terms of convenient spline type base functions.In [38], a deep learning-assisted linear sampling method (DLSM) is proposed for the reconstruction of multilayered dielectric objects with cylindrical and rectangular cross-sections.The relative permittivity is randomly chosen from the interval [1], [5].The background of the detection domain is assumed to be free space.So far, this is the only work reported on LSM with deep-learning.The novelty of the paper lies in the proposed hybrid model, which combines conventional LSM and deep learning.Furthermore, the limitations of LSM, such as its output depending on the frequency of operation and the choice of regularization parameter, are also addressed using deep learning.Various examples of synthetic and experimental data sets [39]

Problem Formulation
The paper considers a scenario involving two dimensional transverse magnetic (TM) characteristics and circular measurement configuration [40] as shown in Fig. 1.The unknown object is embedded within the investigation domain Ω.This object resides within a medium characterized by a homogeneous background.TM waves created by the  tx number of transmitting antennas denoted as  x , illuminate the unknown scatterers.The scattered field is measured by the  rx number of receiving antennas denoted as  x .
The integral equations for the relationship between scattered fields and the induced currents are as follows [41], [42]: The incident field, the scattered field, and the total field are denoted by  inc ,  scat , and , respectively.,  ′ ∈ (, ) are spatial variables that indicate the Cartesian coordinates of the receivers and emitter, respectively. b denotes the background's permittivity,  b denotes the magnetic permeability, and  denotes the working frequency. c is the impressed conduction current density. c is related to the boundary conditions of PEC.For PEC objects, the scattered fields depend on the object's shape and wave's frequency.
The free space Green's function is denoted by  ,  ′ and defined as where  (2)  0 (.) is the Hankel function of second kind and zeroth order and  b is the wave number.In PEC, the induced currents are present only at the boundary .
The solution to ( 1) and ( 2) can be found by applying the method of moments to transform them into discrete matrix equations by discretizing the integral equations and then calculating unknown  c and  scat .The related mathematical details can be found in [41], [42].

Linear Sampling Method
The LSM provides an estimate of the targets support by solving an auxiliary linear inverse problem rather than the nonlinear one formulated through (1) and (2).With respect to the scenario described above, the auxiliary problem is cast as [43] where  is the point source,  s ∈ Ω denotes a point of an arbitrary grid that samples the region under test Ω ,  is the unknown to be determined and F :  2 (Γ) →  2 (Γ) is the far-field operator [8].
Due to compactness of F , equation ( 4) corresponds to a linear ill-posed inverse problem [43].Hence, a stable solution of (4) in the generic sampling point requires regularization.Usually, this is done by considering the Tikhonov regularization, and singular value decomposition (SVD), the final form of solution is where {  ,   ,   } denotes SVD of F and  is the Tikhonov parameter / regularization parameter.
The estimated support is achieved by evaluating the energy (i.e., the  2 norm) of , ∀  s ∈ Ω , as this assumes its lowest values in the points of the investigated region belonging to the target, while it diverges in points external to it [43].Therefore, the support is simply determined by plotting the LSM indicator over Ω and associating the sampling points where the indicator is low to the unknown objects.
From (5), one achieves the explicit expression of the LSM indicator whose evaluation is computationally straightforward as it requires a single evaluation of the SVD of F .Moreover, the Tikhonov parameter can be determined only once for all sampling points [44][45][46][47].The regularization parameter employed in this proposed technique is the same for all sampling points and is set in accordance with the physics-based criteria described in [31], which eliminates the need to understand the level of noise present in the observed data.Regularization parameter is set as  = 0.0001 1 , where  1 is the leading singular value.

Deep Learning Assisted Linear Sampling Method for Inverse Problem of PEC
In recent years, deep learning has become one of the most powerful methods in the field of regression and classification problems.These methods are much faster and produce better image quality compared to the conventional iterative methods based on optimization [48].With a powerful nonlinear matching ability and a one-step testing procedure, deep learning methods are becoming the most popular to use in microwave imaging techniques to obtain an inverse solution in real-time.
In this regard, U-Net architecture having an encoder depth of 3 is used in the proposed model.The Proposed scheme for embedding deep learning in LSM is as shown in Fig. 2. The detail of the U-Net architecture with an encoder depth of 3 is as shown in Fig. 2(a).Since the U-Net architecture was originally designed for segmentation, it is modified such that the Softmax layer is removed and the segmentation layer is replaced with a regression layer.It consists of 3 × 3 convolutional, rectified linear units (ReLU) and 2 × 2 a maxpooling layers (contraction path), followed by successive up sampling and convolutional layers (expansion path).Additionally, in the expansive path, there are three concatenations with the corresponding feature maps from the contracting path.The purpose of this path is to combine precise localization with contextual information from the contracting path.More details about U-Net can be found in [49].

Training and Testing of the Network
The schematic for training and testing U-Net architecture is as shown in Fig. 2(b) and Fig. 2(c), respectively.The initial contrast function derived by solving LSM is used to train the U-Net with true contrast as the network's target (label).The training data contains 11600 reference PEC objects of various shapes like circular cylinders, rectangle, L shape, T shape, E shape, F shape, H shape and U shape PEC objects.The DOI is discretized into 40 × 40 small square cells.The background of the detection domain is assumed to be free space (contrast function is set to zero).The learning rate determines the step size during optimization.If the learning rate is excessively high, it can cause rapid convergence or divergence, while a very low learning rate may lead to slow convergence or getting stuck in local minima [56].Empirically, we have set the learning rate at 0.0001.The batch size determines the number of samples used in each iteration of training.Larger batch sizes can expedite the training process, but if they are too large, they may result in out-of-memory errors.Based on the recommendation in [57] and considering the dataset size of 11600, we have chosen a batch size of 40 (batch size × no. of iterations = total no. of training data).The number of epochs represents how many times the model goes through the entire training dataset.Too few epochs may result in underfitting, while too many may lead Priori information about the PEC scatterer like approximate position, size etc., is not required for this approach.In the reconstructed result, negative values are made zero as post processing (background of DOI is free space).The various results of testing are discussed in the subsequent section.

Results and Discussion
The imaging setup consists of a square investigation domain of side 20 cm × 20 cm.PEC objects are reconstructed using the noisy scattered field.MATLAB's Deep Learning toolbox is used to build the U-Net architecture.The proposed scheme is tested for various test images.Here 24 transceivers are used, which are located uniformly on a circular measurement domain of 0.6 m radius.The operating frequency is 3 GHz.The scattered field of synthetic data have been corrupted with additive Gaussian noise at the level of a signal-to-noise ratio (SNR) of 30 dB.The proposed scheme is tested for the various letters shaped PEC objects and the reconstruction results for some representative examples are shown in Figs.3-8.The results reveal that the suggested method outperforms the LSM algorithm in reconstructing high-quality images.Example 4: Consists of E-shape objects with four arms.
Example 5: Consists of U and rectangular shape objects of Fresnel experimental data.

Example 1: Consists of Circular and Rectangular Cylinders
A PEC circular cylinder of radius 3.5 cm centred at (3 cm,-2 cm) as shown in Fig. 3(a) is considered.The reconstruction result of LSM is shown in Fig. 3(b).This is given as input to the trained CNN and the corresponding output is as shown in Fig. 3(c).As the object is placed in a free space, the contrast function for background is zero and cannot be negative.So, as a post processing, any value less than 0.3 is set to 0 and the corresponding result is shown in Fig. 3(d).For rest of the examples, the post processed output is shown as there is no much difference in the object shape and location.
The rectangular strip of width 7.5 cm and height of 5.0 cm is positioned as shown in Fig. 4(a).The reconstruction result of LSM and CNN is as shown in Fig. 4(b) and 4(c), respectively.The LSM reconstruction looks like circular objects as sharp corners are rounded.Whereas, proposed method conserves the sharp corners and reconstruct the correct shape and size.

Example 2: Consists of L and T Shape Objects with Two Arms
In this study, a L-shape object is taken having both arm's length and breadth of 9.5 cm and 2.5 cm, respectively, positioned as shown in Fig. 5(a).The reconstruction result of LSM and CNN is as shown in Fig. 5(b) and 5(c), respectively.A T-shape object having both arm's length and breadth of 7.0 cm and 2.0 cm, respectively, positioned as shown in Fig. 5(d).The reconstruction result of LSM and CNN is as shown in Fig. 5(e) and 5(f), respectively.In LSM, the internal sharp corner is severely rounded and cannot reconstruct the correct shape and size of the object.However, the CNN result recovers the internal sharp covers and is able to identify the shape and size more accurately.

Example 3 : Consists of U, H and F Shape Objects with Three Arms
A U-shape object having both arm's length and breadth of 10.0 cm and 6.0 cm ,respectively, positioned as shown in Fig. 6(a).The reconstruction result of LSM and CNN is as shown in Fig. 6(b) and 6(c), respectively.A H-shape object having both arm's length and breadth of 8.5 cm and 7.5 cm, respectively, positioned as shown in Fig. 6(d).The reconstruction result of LSM and CNN is as shown in Fig. 6(e) and 6(f), respectively.A F-shape object having both arm's length and breadth of 10.0 cm and 5.0 cm, respectively, positioned as shown in Fig. 6(g).The reconstruction result of LSM and CNN is as shown in Fig. 6(h) and 6(i), respectively.In letter U, F and H, the reconstruction of internal edges is relatively thin due to multiple scattering effects.However, the shape can be approximated.

Example 4 : Consists of E Shape Objects with Four Arms
A E-shape object having both arm's length and breadth of 8.0 cm and 14.0 cm, respectively, positioned as shown in Fig. 7(a).The reconstruction result of LSM and CNN is as shown in Fig. 7(b) and 7(c), respectively.The reconstruction of internal edges in an E-shaped object is comparatively thin due to multiple scattering effects since its size and arms are larger than in Example 3.However, it's possible to fairly estimate the shape.
A U-shape object having both arm's length and breadth of 5.0 cm and 8.0 cm, respectively, positioned as shown in Fig. 8(a).The reconstruction result of LSM and CNN is as shown in Fig. 8(b) and 8(a), respectively.A rectangular object having both arm's length and breadth of 2.45 cm and 1.27 cm, respectively, positioned as shown in Fig. 8(d).The reconstruction result of LSM and CNN is as shown in Fig. 8(e) and 8(f), respectively.In summary, it has been shown that the proposed method can successfully reconstruct the shape of the more complex PEC objects.The proposed method can be more effectively recover some geometrical aspects that can't be faithfully restored using the traditional LSM method.

Effect of Regularization Parameter
The main drawback of the LSM is that the quality of output depends on the choice of regularization parameter [35], [36].To overcome this limitation, a multiple regularization parameter approach is used, where the CNN is trained using LSM reconstructed results obtained at various regularization parameters like  = 0.1 1 , 0.01 1 , 0.001 1 and 0.0001 1 , where  1 is the leading singular value.The trained CNN is tested for various PEC profiles like L-shape, T-shape and U-shape.The reconstruction result by LSM and CNN at various regularization parameters is as shown in Figs.9-11.From the result, it can be concluded that, firstly, although the training is done at regularization parameters  = 0.1 1 , 0.01 1 , 0.001 1 and 0.0001 1 , the reconstruction results at  = 0.1547 1 , 0.0786 1 and 0.0561 1 are also satisfactory, indicating the robustness and generalization of the proposed method.Secondly, the LSM fails to reconstruct at certain regularization parameters, but the deep learningassisted LSM can recover satisfactorily at various untrained regularization parameters.

Effect of Frequency
The reconstruction quality of the LSM deteriorates at higher frequencies.To overcome this limitation, a multiple frequency approach is used, where the CNN is trained with LSM reconstructions obtained at multiple frequencies like 1 GHz, 3 GHz, 5 GHz and 7 GHz.The trained CNN is tested for PEC profiles like L-shape and U-shape.The reconstruction result by LSM and CNN at various frequencies is as shown in Figs. 12 and 13.The two observations can be made from the result.Firstly, although the training is done 1 GHz, 3 GHz, 5 GHz and 7 GHz, the reconstruction results at 4 GHz and 8 GHz is also satisfactory indicating the robustness and generalization of the proposed method.Secondly, the LSM fails to reconstruct at higher frequencies but deep learning assisted LSM can recover satisfactorily also at higher frequencies.

Conclusion
In this paper, a deep learning based LSM approach is introduced to reconstruct perfectly electric conductors.In this work, U-Net CNN architecture is used where an initial guess by the LSM is fed to CNN.The network is trained by generating the reference profiles of circular cylinders, rectangle, L shape, T shape, E shape; F shape, H shape and U shape PEC objects.Thereafter, this trained model is tested on various examples including synthetic and experimental data sets.It is also found that the proposed scheme outperforms conventional LSM in terms of frequency of operation and choice of regularization parameter.A more advanced architecture can yield even better results, which will be explored in our subsequent studies.

Fig. 2 .
Fig. 2. (a) Architecture details of U-Net for the proposed model.(b) Training the CNN using LSM reconstruction as input.(c) Reconstruction of PEC objet using LSM and U-Net-CNN.

Example 1 :
Consists of circular and rectangular cylinders.

Example 2 :
Consists of L and T shape objects with two arms.Example 3: Consists of U, H and F shape objects with three arms.

Fig. 13 .
Fig. 13.Reconstruction of U-shape PEC object as shown in Example 3, Fig. 6(a), by LSM and CNN at 8 GHz and 6 GHz.
are used to validate the proposed model's performance.The results are excellent in terms of convergence rate and reconstruction quality.