Abstract
A 3D scalar electromagnetic imaging of dielectric objects buried under a rough surface is presented. The problem has been treated as a 3D scalar problem for computational simplicity as a first step to the 3D vector problem. The complexity of the background in which the object is buried is simplified by obtaining Green’s function of its background, which consists of two homogeneous half-spaces, and a rough interface between them, by using Buried Object Approach (BOA). Green’s function of the two-part space with planar interface is obtained to be used in the process. Reconstruction of the location, shape, and constitutive parameters of the objects is achieved by Contrast Source Inversion (CSI) method with conjugate gradient. The scattered field data that is used in the inverse problem is obtained via both Method of Moments (MoM) and Comsol Multiphysics pressure acoustics model.
1. Introduction
Imaging of objects embedded in a layer or beneath a rough surface has been a popular subject of remote sensing due to its wide range of application areas such as nondestructive testing, mine detection, medical imaging, and geophysical or archaeological exploration. In most of the cases, the interface separating two layers is rough, and the object of interest is embedded in the inaccessible layer beneath the rough surface. In order to image the object, the area in which the object is presumed to be located is illuminated by a number of antennas, and the field scattered by the object and the roughness is measured by receiving antennas above the surface. The measured data is then used to obtain the unknown properties of the object, such as shape, location, dielectric permittivity, and conductivity.
There have been a wide range of approaches introduced to investigate and reconstruct the object and the roughness. Interactions between the object and the rough surface are calculated in [1, 2]. The forward-backward (FB) method and propagation inside layer expansion (PILE) are used to calculate the scattering from an object above and below rough surface, respectively. In order to reconstruct the rough surface and the object buried beneath it simultaneously, an iterative method presented in [3] decreases the cost function by optimizing the boundary control parameters using least-squares and the Semianalytic Mode Matching (SAMM) forward model.
In [4], a two-step method is used to detect and characterize a target buried beneath a rough surface. First, the target is detected by analyzing the frequency-averaged Wigner-Ville function in order to filter out rough surface scattering, and then it is characterized by applying an iterative solution derived from the Newton-Kantorovich algorithm to the Wigner-Ville function. In [5], surface and buried mines are detected by using mid-wavelength infrared (MWIR) images and image segmentation based on a wavelet thresholding algorithm. A hybrid numerical method based on the parallel genetic algorithm (GA) and the finite difference time domain (FDTD) is used to obtain the location and dimensions of 2D inhomogeneous objects buried in a lossy ground in [6].
Surface impedance is obtained from the impedance boundary condition, by using the electric field and its normal derivative on the surface. By observing the surface impedance, the location of the dielectric objects buried beneath the surface is obtained in [7, 8]. Several methods also use standard, modified, or distorted Born iterative methods along with methods such as conjugate gradient, fast Fourier transform, and singular value decomposition (SVD) for imaging objects buried beneath a rough surface [9–12].
In this study, objects buried beneath a rough surface in a 3D geometry are reconstructed by using CSI [13]. To this aim, Green’s function of the background is obtained by using BOA [14] in order to simplify the complexity of the geometry. A similar approach is presented in [15] for 2D case.
Formulation of the problem is given in Section 2. In Section 3, Green’s function of the background is obtained. The inverse problem is explained in Section 4, and numerical results of simulations and conclusion are given in Sections 5 and 6, respectively.
Time dependence has been used and suppressed throughout this problem.
2. Formulation of the Problem
In this paper, the problem of the imaging of an unknown object buried under a rough surface has been treated as a 3D scalar problem for computational simplicity as a first step to the 3D vector problem. The configuration of the problem can be seen in Figure 1. In this configuration, the interface between the two half-spaces is a locally rough interface . The half-spaces and are composed of simple materials with constitutive parameters and , respectively. In the lower half-space, an arbitrary shaped object with parameters is located. Both half-spaces are assumed to be filled with nonmagnetic materials, whose magnetic permeabilities are equal to free-space permeability .
The region in which the object is assumed to be located is illuminated by microwave point sources at points , , on surface , and total electric field is measured on surface for each illumination. The incident scalar electric field function for each illumination, which is time harmonic with angular frequency , is , where represents the distance between the source and observation points. The inverse scattering problem here is reconstructing the object’s location, shape, and constitutive parameters by using the total electric field , , measured on the surface .
Let be the background field, the total field for the th illumination in the absence of the object. It can be computed by using BOA [14], which treats the roughness as a series of objects embedded in upper and lower half-spaces that can be seen in Figure 2. Therefore, if the field scattered by the object is represented by , the inverse problem can be expressed as the reconstruction of in the following system of integral equations:which are the object and data equations [13], respectively. In these equations, is Green’s function of the background system which consists of the two half-spaces and the rough interface that separates them. The function is the object or contrast function which is defined by where and are the wavenumbers defined byIt can be seen in (2) that the contrast function is zero outside the object, and its real and imaginary parts are related to and , respectively.
3. Green’s Function of the Background Medium
BOA is used to obtain , which is Green’s function of the background, for solving the inverse scattering problem given by (1). This approach considers the roughness as a series of scatterers whose parameters differ according to their position. The parameters of the sections that lie in the half-space and the sections that lie in the region are and , respectively.
, Green’s function of the background, can be expressed as the sum of the contribution of its two components: where is Green’s function of the two-part space with a planar interface and is the contribution of the roughness. can also be considered as the field scattered from the objects that form the roughness due to a point source with unit strength which is located at , and it satisfies the integral equationwhere the integral operator is defined by is the wavenumber of the two-part space with a planar interface, defined byEquation (5) can be solved for by applying the forward solution method given in [16]. , which is needed to solve this equation, can be given aswhereThe square root functions in (10) are defined on complex plane under the conditions
4. Inverse Problem
If we rewrite (1) in a symbolic form, we obtain the object equation and the data equationwhere the integral operators and are defined byBecause the product of the contrast function and the fields are seen in the integral equation system of the problem, the quantitywhich is known as a contrast source, can be replaced in (12) and (13) to obtainUsing CSI method, the inverse problem becomes the minimization of the cost functional defined aswhere and denote the norms on and , respectively. During the iteration process, the contrast sources and the object function are alternately constructed by using conjugate gradient method such that they minimize the whole cost functional until it reaches a predetermined value. The explicit expressions and the iteration steps for the CSI method can be found in [13].
5. Numerical Results
Throughout the simulations, the region is considered to be free-space, and dry soil with relative dielectric permittivity and conductivity . 25 equally spaced point sources are located on a square surface, which is defined by , , and , and the scattered field is measured on equally spaced positions on a surface which is defined by , , and , for each illumination, where is the wavelength of the soil. The reconstruction domain is divided by cubical cells of size on each side. The scattered field data are obtained synthetically by using Comsol Multiphysics pressure acoustics with attenuation model, or MoM, and 5% random noise; is added, where is the noise level and is a uniformly distributed random variable between 0 and 1, and therefore the corresponding signal to noise ratio is . All simulations were carried out at a frequency of 300 MHz.
According to Comsol Multiphysics pressure acoustics with attenuation model, the total acoustic pressure function created by a point source with strength satisfies the differential equationwhere is the wavenumber, and , , and are the density, speed of sound, and attenuation, respectively.
In this model, it can be seen from (18) that corresponds to scalar electric field of the problem at hand. If is matched to the relative magnetic permeability, and and are selected so that the real and imaginary parts of the wavenumbers match, this pressure acoustics model can be used to simulate scalar electromagnetic model.
MoM has been used to obtain Green’s function of the background in (4) and (5). A cell size of is selected.
For the first simulation, a simple roughness, which is located in the domain defined by , , and , is selected. The profile of the roughness can be seen in Figure 3(a). A spherical object with relative dielectric permittivity and conductivity and a radius of is centered at ). The scattered field was measured at 400 points. The reconstruction was carried out in a domain defined by , , and . The iteration is stopped at 500. The original and reconstructed relative dielectric permittivity are shown in Figure 3(b), and the original and reconstructed conductivity are shown in Figure 3(c). It has been observed that the shape, location, and the relative dielectric permittivity of the object are obtained quite well. However, the conductivity of the object is underestimated.
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A second simulation has been performed with the synthetic data obtained by solving the forward problem using MoM for the same configuration. The original and reconstructed values of relative dielectric permittivity and conductivity of the object can be seen in Figures 4(a) and 4(b), respectively. This simulation shows that, using MoM, instead of Comsol Multiphysics, although the object appears to be slightly above its actual position, its shape and location are still clear, and the reconstructed value of the relative permittivity improves; however, the reconstructed conductivity still does not improve significantly.
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A larger and more complex roughness, which is located in the domain defined by , , and , and that can be seen in Figure 5(a), is selected for the third simulation. The same object used in the previous examples is centered at the same location in order to investigate the effect of the roughness. The scattered field was measured at 625 points. After 500 iterations, the original and reconstructed relative dielectric permittivity and conductivity can be seen in Figures 5(b) and 5(c), respectively.
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It is observed that although the location and the value of the relative permittivity of the object are still well determined, the shape is not as clear, indicating that the method is sensitive to the complexity of the structure, and the higher the complexity is, the lower the resolution is with the given configuration of the reconstruction.
For the fourth simulation, two spherical objects with a radius of are assumed to be buried under the rough surface that can be seen in Figure 5(a). The center of the object with relative dielectric permittivity and conductivity is located at , and the center of the object with relative dielectric permittivity and conductivity is located at . The scattered field was measured at 900 points. After 500 iterations, the original and reconstructed relative dielectric permittivity and conductivity can be seen in Figures 6(a) and 6(b), respectively.
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It can be seen in Figure 6(a) that the value of the relative dielectric permittivity is determined, and although the value of the conductivity in Figure 6(b) is underestimated, the shape and location of the objects are quite clear. This simulation suggests that when the complexity of the geometry rises, the clarity and the precision of the reconstructions decrease.
As it can be seen in some of the simulations, the position of the object may appear slightly different from its actual position although its shape and constructive parameters remain in an acceptable range. In such cases, measuring the error based on pixel-by-pixel comparison may be misleading. Therefore, in order to prevent this, instead of a measure of the errors, the exact and the reconstructed profiles are given for visual interpretation.
6. Conclusion
A 3D scalar electromagnetic imaging of dielectric objects buried under a rough surface is presented. BOA is used to obtain Green’s function of the background, which includes the contribution of the roughness in order to be able to separate the contribution of the object from its background. Green’s function of the two-part space with a planar interface is obtained to be used in the process. CSI method with conjugate gradient is used to reconstruct the location, shape, and constitutive parameters of the object.
The numerical simulations with data obtained by using both MoM and Comsol Multiphysics pressure acoustics model show that the method shows satisfactory results although, due to the computational weight of the 3D structure, larger cell size, relative to the complexity of the problem, has been used.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.