An Efficient of Overlapping Grid Method with Scattering Technique in Time Domain for Numerical Modeling

An Overlapping Grid Method (OGM) with Biquadratic Spline Interpolation in scattering technique was developed to solve the direct and inverse scattering issues. A two-dimensional (2D) numerical image model was used to analyze the accuracy of the proposed method in a direct scattering process. It was discovered that when the sub-grid, s x ∆ increased, the absolute error for the electric field amplitude will also increase. The results also discovered that as the grid size ratio increased, the absolute error of the amplitude Z E will also increase. The findings show that smaller grid spacing and a finer grid size can produce more accurate results. The Overlapping Grid Method (OGM) with Biquadratic Spline Interpolation was expanded by incorporating with Forward-Backward Time Stepping (FBTS) technique to solve inverse scattering issues. Homogenous embedded objects with a square and circular shape are used to validate the efficiency of the proposed method. The findings showed that the proposed numerical method could detect and reconstruct embedded objects in different shapes. The efficiency of the proposed method was examined by Mean Square Error (MSE) and normalizing the functional error. The findings revealed that the MSE of dielectric profiles for the proposed method were lower than the FDTD method in FBTS. The relative permittivity and conductivity profile differed by 27.06% and 20%, respectively. Hence, it was proven that the proposed method successfully solved a known drawback to the FDTD method and produced more accurate and efficient results.

Active microwave imaging is a wave-based noninvasive imaging method involving two principles; tomographic and confocal radar.Microwave tomography is divided into two categories: qualitative and quantitative imaging.The qualitative microwave imaging method generates a qualitative profile, such as a reflectivity feature or a qualitative picture representing a hidden item.The quantitative imaging approach is used to obtain the electrical and magnetic properties distribution to obtain the geometrical parameters of an imaged object.The spatial distribution of the complex permittivity is calculated using the transmitted (incident) and received (scattered) fields (Nikolova 2011).
The FDTD algorithm is a simple and effective way of addressing Electromagnetic (EM) interaction problems (Baek, Kim, & Jung 2018;Rahman & Rather 2020).Moreover, it can analyse a wide range of frequencies without using additional computer resources.Therefore, the time-domain inversion method is suitable for improving the detection and reconstruction of embedded objects (Narayan 2017;Okada 2014;Schneide 2016).However, this algorithm is limited to intrinsic orthogonal grids due to it is based on a Cartesian coordinate system.Hence, it is challenging to build the meshes for modelling curved borders and microscopic structures (E.Jiménez-Mejía, & Herrera-Murcia, J. 2015; Nilavalan 2002).Several approaches for improving the efficiency of the FDTD method have been published in the literature, including non-uniform (E.Jiménez-Mejía & Herrera-Murcia 2015), sub-gridding (Cabello et al. 2017), and sub-cell algorithm (Navarro et al. 2021).Nevertheless, those approaches still have some disadvantages, such as requiring a long calculation time and additional memory, as well as the Courant-Friedrichs-Lewy (CFL) stability requirement restricting time step or cell size (De Moura & Kubrusly 2013).
Therefore, an Overlapping Grid Method (OGM) with Biquadratic Spline Interpolation in Forward-Backward Time Stepping (FBTS) Technique was proposed and developed in this paper.In order to calculate the dispersed fields for an embedded object, two-dimensional (2D) numerical simulations for electromagnetic (EM) field analysis in various ratios for the sub-grid were carried out.Then, the proposed method incorporated the FBTS inverse scattering technique for detecting and reconstructing embedded objects with different shapes.Finally, the efficiency of the proposed method was examined by Mean Square Error (MSE) and by normalizing the functional error.An Overlapping Grid method is a grid embedding technique that provides a theoretically direct domain decomposition mechanism (Yang, Shao, Li, & Chen 2019).This method divides the computing domain into two or more subdomains in the measured EM region.The advantages of an Overlapping Grid method are it can reduce complex geometric problems and produce the quality and spatial resolution of imaging.An Overlapping grid comprises two grids, as seen in Figure 1(a).The Finite-difference time-domain (FDTD) lattice, also known as Layer 1 (main-grid), is utilized to cover the whole computational domain.Layer 2 (sub-grid) is an overlapping field used to model an undefined numerical entity.The Biquadratic Spline interpolation approach will be used to determine the overlapping region between the main-grid and the sub-grid.The area where the main-grid and subgrid overlap can be determined by using Biquadratic Spline Interpolation.Then, unknown values at the border points can be retrieved in the overlapping region.Biquadratic Spline interpolation (Boor, 1978;Schumaker, 2015;Späth, 1995) will provide more accurate results due to the fact that it is constructing new data points within the range of a discrete set of known data points (Han 2013;Singh 2016).As a result, it can generate more accurate interpolation for determining curved borders and microscopic features.

DIRECT AND INVERSE SCATTERING TECHNIQUE
There are two types of electromagnetic scattering problems: direct and inverse scattering.The distribution of dispersed fields can be calculated using the direct scattering technique.In contrast, the inverse scattering issue is used to deduce an object's attributes (e.g., structure, location, and dielectric properties) based on scattered field measurement findings.
Direct scattering formulates the problem in the time or frequency domain using microwave signals.As seen in Figure 2, direct scattering is used to calculate an object's dispersed area using two antennas.In free space, the target is implanted with transmitter (T x ) and reception (R x ) antennas inside the Region of Interest (ROI).Each antenna sequentially broadcasts microwave signals, while the remaining antennas act as receivers, absorbing scattered fields.Colton and Kress (Colton 1998) defined the inverse scattering problem as an inverse problem that was highly difficult to solve.However, this scattering problem can be solved by using Forward-Backward Time Stepping (FBTS) technique (Takenaka 2015), as illustrated in Figure 3.The FBTS technique can reconstruct an image based on its electrical characteristics for solving the nonlinear inverse problem.

FIGURE 3. FBTS Inverse scattering technique
Microwave tomography collects information about an object by sending microwave pulses and dispersed pulses for data analysis, also known as a computationally intensive image reconstruction algorithm.The overlapping grid method with the FBTS scattering technique is described in detail in (Wee 2020).

Forward Scattering
i. Forward time-stepping setup ii.Each antenna serves as a transmitter that send pulses one at a time, while the other 15 antennas take on the role of receivers that collect the dispersed electromagnetic fields until a set of transmitter or receiver data for 16 antenna combinations was produced.iii.Interpolate the EM field by using OGM with Biquadratic spline interpolation iv.Measure and store the reconstruction data in Forward scattering

Backward Scattering
i. Backward time-stepping setup ii.Each antenna serves as a transmitter that send pulses one at a time, while the other 15 antennas take on the role of receivers that collect the dispersed electromagnetic fields until a set of transmitter or receiver data for 16 antenna combinations was produced.iii.Interpolate the EM field by using OGM with Biquadratic spline interpolation iv.Measure and store the reconstruction data in Backward scattering The OGM with Biquadratic Spline Interpolation in FBTS Inverse Scattering algorithm is shown in Figure 4.All parameters needed for this simulation were specified, and values were assigned.The measuring procedure for the dispersed field was then established.The dielectric characteristics of embedded objects and the Region of Interest (ROI) were included in the actual profiles.In the simulations, an excitation signal was a sinusoidal modulated Gaussian pulse.Each antenna functioned as a transmitter, successively transmitting a pulse, while the remaining 15 antennas served as receivers, collecting dispersed electromagnetic fields.
FBTS is a technique for determining important quantitative information about embedded objects.The forward step begins with applying a Gaussian pulse signal to the predicted profile of an embedded object.Then, the dispersed fields for forwarding time-stepping were calculated by using the OGM approach.In the OGM lattice, the EM field was interpolated using Biquadratic Spline Interpolation for the overlapped region.The forward timestepping reconstructions data gathered at the receiving location was then compared to the measurement data.
The adjoint fields with backward time-stepping were calculated by using the OGM with Biquadratic Spline Interpolation where there was backward scattering.Data from the receiving point's backward time-stepping reconstructions were gathered and compared to measurement data.Finally, the original profile was irradiated with the difference between the observed and estimated scattered fields as a source.
The cost functional was utilized to reduce the number of FBTS reconstruction iterations.It was calculated by comparing the estimated and actual profiles.Finally, MATLAB was used to illustrate the reconstructed relative permittivity and conductivity images in the two-dimensional.Mean Square Error (MSE) was used to compare the reconstructed image with the original image to evaluate the accuracy of the proposed method.

RESULTS AND DISCUSSION
This paper focuses on the development of the OGM with spline interpolation and demonstrates its efficiency.We have previously published the study related to the noise for the proposed inverse scattering technique in (Elizabeth et al., 2015;Munawwarah Ibrahim et al., 2016).In this research, a study on the performance of OGM with spline interpolation in direct and inverse scattering was evaluated in Case A, Case B and Case C. CASE A: DIRECT SCATTERING A 2D numerical model was used to analyse the effectiveness of the OGM with the spline interpolation technique in a direct scattering process.The spline interpolation technique will transfer the data between the main-grid and the sub-grid.Figure 5 shows the sub-grid was configured to converge on top of the main-grid, with L s and W s denoting the length and width for sub-grid, respectively.L m and W m are the length and width of the main-grid, respectively.The values of L s , W s , L m and W m were consistent with (i, j) and (m, n).In this study, the sub-grid was set to 50 mm x 50 mm grids, while the main-grid was set to 190 mm x 190 mm grids.In addition, the cell size was set to 1 mm x 1 mm for the maingrid and the sub-grid.A range of ratios of the grid size, R between the main-grid and sub-grid, was utilized to investigate the precision of the proposed numerical method.As shown in Figure 6, the ratios of the grid size among the main grid and the subgrid were . In this analysis, the number of grids on the main grid,   7. The computational time versus the ratio of grid sizes for the sub-grid Figure 7 illustrates the computing time versus grid size ratio for sub-grid.The computational time decreased when the ratio of grid size increased.The results showed that the smallest grid size ratio required the highest computational time.
FIGURE 8.The number of steps versus the amplitude of E z field Figure 8 shows the amplitude of E z field with a different ratio of grid size versus the number of steps.The solid line colour (red, yellow, green, cyan, blue, purple, and dark) represented the ratio of the grid, R = 0.1 until 0.7. Figure 9 shows the maximum amplitude of E z field versus the ratio of grid size.It was found that the maximum amplitude of E z field increased as the grid size ratio increased.The absolute error, Absolute Error = |E R (t) -E R1.0 (t)| was used to examine the error analysis for a received signal at the observation point, where E R1.0 (t) is an electric field in an overlapping lattice for R=0.1 as the benchmark, and E R (t) is an electric field in an overlapping lattice for R values ranging from 0.1 to 0.7. Figure 10 depicts the absolute inaccuracy of the amplitude of the E z field versus the grid size ratio.The findings revealed that as the grid size ratio increased, the absolute error of the amplitude E z also increased.Consequently, it may be stated that a more refined grid size can yield more accurate findings.3. The dispersed fields for an embedded object were examined with error analysis to examine the stability and effectiveness of the OGM approach with spline interpolation.The comparison of transmitted and received signals for embedded objects in free space using the FDTD method, OGM with the Bilinear Interpolation, and OGM with the Spline Interpolation is shown in Figures 11 and 12  where, E 0 (t) is the electric field in FDTD lattice, E z (t) is the electric field in OGM with bilinear interpolation or OGM with spline interpolation, and N is the total number of time steps.Table 4 presents the error analysis for received signals in free space.The maximum amplitude peak of electric field, E 0 (t) for FDTD method was 1.01627 × 10 -4 and is used as a reference in this analysis.This is due to the difference between OGM method with bilinear interpolation and OGM method with spline interpolation can be compared and analyzed.The OGM method with spline interpolation showed lower relative error than the OGM method with bilinear interpolation of difference.Besides, the MSE for the OGM method with spline interpolation was also lower than the OGM method bilinear interpolation based on Table 4. Hence, it proves that the OGM method with spline interpolation could to measure the scattered fields around an embedded object accurately as compared with OGM method with bilinear interpolation.The electrical profiles for the numerical model of multiple objects are summarised in Table 5.The dielectric characteristics of ROI represent fatty tissues, and the square object is assumed to represent fibroglandular tissues in the breast.The main-backdrop lattices are supposed to be free space.Figure 14 depicts the real dielectric profiles of multiple simple-shaped objects.Figure 15 demonstrates the reconstructed dielectric profiles of the FDTD method in FBTS and it was used as a reference in this analysis.The FDTD approach in FBTS is limited to intrinsic orthogonal grids because it cannot describe curved borders and minor characteristics of several basic objects.Therefore, an Overlapping Grid Method with scattering technique was proposed in this case study.Figure 16 shows the reconstructed dielectric profiles of the proposed algorithm.The proposed algorithm effectively reconstructed the dielectric profiles of the circular and square objects.The findings present that the proposed algorithm can generate more accurate and efficient interpolation in the reconstruction images to detect curved borders and minor features.Figure 17 and Figure 18 show a 1D cross-section of the original and reconstructed dielectric profiles.Crosssectional views of the actual and reconstructed dielectric profiles for a circular embedded object are shown in Fig. 15, while for a square embedded object are shown in Fig. 16.These images show the stark difference in dielectric characteristics between the buried items and ROI.The findings demonstrated that the proposed numerical method could distinguish between square and circular objects in ROI while estimating dielectric values accurately.
The proposed method's efficacy was assessed by using the Mean Square Error (MSE), as shown in Table 6.According to this investigation, the MSE of reconstructed dielectric profiles utilizing the proposed method produced much lower values than the FDTD method in FBTS.The relative permittivity profile differed by 27.06%, whereas the conductivity profile differed by 20%. Figure 19 depicts the normalized functional error vs iterations count.A gradient optimization with 100 iterations was used to reconstruct the numerous simple-shaped objects.As observed, the normalized functional error was reduced as the number of iterations increased.It was also discovered that in FBTS, the OGM with Biquadratic Spline Interpolation showed lower values of normalised functional error than the FDTD approach.At 100 th iterations, the difference in normalised functional error between these two techniques was 1.41 × 10 -7 .Compared to the FDTD approach in FBTS, the results showed that the proposed method could accurately reconstruct numerous simple-shaped objects.Consequently, the proposed method successfully addressed a recognized problem in the FDTD method.It was challenging to simulate the curving borders and tiny features due to inherent orthogonal grid restrictions.

CONCLUSION
An Overlapping Grid Method with Biquadratic Spline Interpolation in direct scattering method was proposed as a new numerical method for measuring the scattered fields around an embedded object.The findings show that smaller grid spacing or cell sizes can produce more accurate results.The results also indicated that when the grid size ratio increases, the absolute error of the amplitude Ez will also be increased.Consequently, it may be stated that a more refined grid size can yield more accurate findings.The FBTS inverse approach was combined with the Overlapping Grid Method with Biquadratic Spline Interpolation to solve inverse scattering issues.The findings demonstrated that the proposed technique was excellent at recognizing buried items and rebuilding them into various forms.The location, shape, and interior content of buried artefacts were determined quantitatively.As opposed to the FDTD method in FBTS, the proposed approach generates sharper and better-reconstructed images.
FIGURE 1. Overlapping Grid Method Figure 1(b) depicts a four (4) point Biquadratic Spline interpolation derived from Figure 1(a).P, Q, R, and S are the four known value points from the main-grid, whereas Z' is an unknown value point from the sub-grid.The unknown value of Z'(x z , y z ) is interpolated by using the A(x a , y a ) and B(x b , y b ) at the y-axis as in Equation (1).

FIGURE
FIGURE 2. Direct scattering Scattering Technique Jurnal Kejuruteraan 35(4) 2023: xxx-xxx https://doi.org/10.17576/jkukm-2023-35(4)-12OGM method with Spline Interpolation in FBTS Inverse Scattering Technique Start Parameters setting Measurement setup i. Create or load image for simulation ii.Transmitter and Receiver antennae setting Direct Scattering i. Interpolate the EM field by using Biquadratic Spline interpolation in OGM lattice ii.Measured the EM fields for an arbitrary shaped multiple objects in time-domain Plot the two dimensional reconstruction images (Relative Permittivity and Conductivity) Satisfactory End

FIGURE 5 .
FIGURE 5.The numerical model setup for square object

Figure 5
Figure5shows the numerical model of the Overlapping method with spline interpolation in free space.The observation point is in the middle of the sub-grid and maingrid.The spaces of main-grid,

FIGURE 6 .
FIGURE 6.The absolute error of the amplitude of E z field versus space increment of sub-grid set based on Table2.The number of grids in the sub-grid decreased when the ratio of grid size increased.The ratio of grid size, R = 1.0 and number of the grids 50 mm for sub-grid were used as a reference in this analysis.All these simulations were carried out in free space.

FIGURE 9 .
FIGURE 9.The ratio of grid size versus the maximum amplitude of E z field

FIGURE 10 .
FIGURE 10.The ratio of grid size versus the absolute error of the amplitude of E z field .

FIGURE 11 .
FIGURE 11.Transmitted signal for electric field in Case B Figure13illustrates the numerical model of microwave tomography for the square object in 2D.Layer 1 was equipped with 190 mm x 190 mm grids.Layer 2 was configured with 50 mm x 50 mm grids.The sub-grid was created as multiple objects and located on top of the main grid with cell size, Δx = 1 mm and Δy = 1 mm.The ROI region was immersed in free space as the backdrop media.A sinusoidal modulated Gaussian pulse with a core frequency of 2.0GHz and a bandwidth of 1.3 GHz was utilized in this work as an excitation signal.The ROI was irradiated by sixteen (16) point source antennas.Each point source antenna served as a transmitter, generating one Gaussian pulse at a time.The remaining fifteen (15) antennas were converted into receivers to collect dispersed fields in the

FIGURE 13 .
FIGURE 13.The numerical model of multiple embedded objects

Position
FIGURE 14.Real dielectric profiles

FIGURE 19 .
FIGURE 19.Normalized functional error vs iterations count Table 1 presents the dielectric profile setting in free space.

TABLE 2 .
The ratio of grid sizes between 0.1 until 0.7

TABLE 3 .
Dielectric profiles for Case B

TABLE 4 .
Error analysis for received signal in Case B

TABLE 5 .
Dielectric Profiles Setting

TABLE 6 .
MSE of dielectric profiles