Abstract
An iterative approach, based on the linear sampling method (LSM) and the contrast source inversion (CSI) method, is proposed to improve the recovered images of multiple targets and targets with layered or continuous profile, including shape and distribution of electric properties. The difficulties in dealing with large targets or high contrast are partly overcome with this approach. Typical targets studied in the literatures are chosen for simulations and comparison.
1. Introduction
Electromagnetic inverse techniques have been widely explored in geophysical survey, target detection, nondestructive testing, medical imaging, and so forth to retrieve the electric properties of possible targets. The linear sampling method (LSM) has been proposed to estimate the target shape [1, 2], implemented with the techniques of singular value decomposition (SVD) and Tikhonov regularization [3]. A level set process (LSP) has also been developed to highlight the target shape more accurately, especially when it has a high conductivity [4, 5].
When the target shape is properly acquired, the contrast source inversion (CSI) method can be applied to retrieve the permittivity and conductivity in both the target and the background medium. An IE-CSI (integral equation CSI) method was developed to recover targets immersed in homogeneous or layered background media; and an FD-CSI (finite difference CSI) method was claimed to be suitable for recovering inhomogeneous targets embedded in an inhomogeneous background medium [6–8]. A typical CSI algorithm, facilitated with the SVD technique, usually takes many iterations to converge [1].
The LSM and the CSI method have also been applied to recover multiple targets. For example, nine square cylinders, of width , , and mS/m, were placed in free space and probed at 1 GHz [9]. The corners of each cylinder look blurred, and the recovered electric properties appear uneven. It was observed that the permittivity of dielectric targets could be underestimated [10].
In [11], four different inverse methods are compared on the Fresnel dataset [12]. When applying the CSI method to two separated cubes, the dielectric constant in the gap between the cubes is overestimated, especially at lower frequencies. In the case of two jointed spheres, the dielectric constant near the joint is strongly overestimated.
Similarly, in other cases of multiple lossy dielectric cylinders, the electric properties in between cylinders appear to be overestimated, and those near the center of cylinders are underestimated [13, 14]. When the radius of a cylinder is increased beyond half a wavelength, the dielectric constant near the center of the cylinder is severely underestimated [14].
In [15], a layered target was considered, which is composed of a cylinder of radius , , and mS/m, enclosed within a shell of inner radius , outer radius , , and mS/m. In [16, 17], the same geometry was studied, with larger permittivity of both the cylinder and the shell. In all these cases, the permittivity of the shell appears to be underestimated. In [18], two lossy dielectric cylinders were placed within a shell. It was found that the recovered shell image appears to shift inwards, with its dielectric constant being underestimated. On the other hand, the dielectric constant of the two cylinders is overestimated, which may be attributed to the blockage by the shell.
In [19], a coated cube of inner width and outer width , with and 1.3, respectively, was considered. The recovered permittivity near the center of the inner cube appears to be underestimated. Similar types of geometry reported in the literatures indicate that the permittivity within a target tends to be underestimated, especially when its electric size is large. With a larger permittivity difference between the target and the background, or between adjacent targets, the recovered permittivity profile becomes less accurate.
In this work, we propose an iterative approach, based on the LSM and the CSI method, to improve the recovered shape and the profile of electric properties of multiple targets, layered targets, and targets with a continuous profile. This paper is organized as follows. The iterative approach is described in Section 2 and simulations results of multiple targets, layered targets, and targets with a continuous profile are presented and discussed in Sections 3, 4, and 5, respectively. Finally, some conclusions are drawn in Section 6.
2. Description of Iterative Approach
The inverse process is executed in two stages. In the first stage, apply the linear sampling method (LSM) to estimate the geometrical shape of the target embedded in the detection domain, , as shown in Figure 1(a). The probes are deployed on the perimeter , which is outside the detection domain and encloses all the targets. Then, apply the contrast source inversion (CSI) method to estimate the electric parameters in the target domains, and .
(a)
(b)
In the second stage, as shown in Figure 1(b), select part of the target domain, , for example, in the first stage and treat it as part of the background. Then, the detection domain is reduced from to ; the background field and the scattered field are updated accordingly. Next, apply the LSM to the new detection domain to estimate the geometrical shape of the remaining targets; then apply the CSI method to estimate the electric parameters in the remaining target domain.
The LSM and the CSI algorithms are the same as those in [1], which will be briefly described in Sections 2.1 and 2.2. The iterative strategy proposed to improve the accuracy of the target shape and to reduce the rippling artifacts in the inverse results will be presented in Section 2.3.
2.1. LSM in Stage 1
The scattered field is used to estimate the target shape with the LSM [1]. Given an excitation source at , the scattered field can be expressed aswhere is the scattered field at , is the total field at , is the contrast function of the medium, and is the two-dimensional Green’s function, which satisfies the wave equationIt has the explicit form , where is the zeroth-order Hankel’s function of the second kind and is the wavenumber of the background medium. The contrast function is defined aswhere and are the complex permittivity of the target and the background medium, respectively. The complex permittivity can be expressed as , where is the real dielectric constant and is related to the conductivity as .
Define an adjoint field , which satisfies the adjoint equationwhere all possible sources are located in , which is practically outside of . To solve (4) for , consider excitation probes placed at ’s, with [1]. Hence, (4) can be discretized intowhere is a weighting factor associated with the th excitation probe.
The detection domain is divided into cells, with the center of the th cell being at . For a specific , (5) can be discretized into a matrix form aswhere , , and , with , . Then, apply the singular value decomposition (SVD) and the Tikhonov regularization techniques to solve (6) for .
An LSM indicator for cell centered at is calculated asIf is smaller than a threshold value, the cell centered is categorized into part of the target.
2.2. CSI in Stage 1
The right-hand side of (4) can be viewed as an adjoint scattered field, , which is the scattered field operated by . Similarly, define an adjoint incident field and an adjoint total field asThe last equality holds because . By substituting (1) into (4) and using (9), we haveTo solve for in the detection domain , (10) is transformed to a matrix formwhere , ,…,,…,, andThe explicit form of iswith , , . Similar to the discretization of (4) to derive (5), can be calculated by discretizing (9) as
Note that is the number of cells in the target domain, which is smaller than . The SVD and the Tikhonov regularization techniques can then be applied to solve (11) for .
2.3. LSM and CSI in Stage 2
The definition of scattered field is extended towhere is total field and is the background field, which is the total field in a given background medium. The background field will reduce to the incident field if the background medium is free space.
A portion of the target area can be selected and merged into the background medium. For example, the shape of , as shown in Figure 1, and the electric parameters are estimated in stage 1 and merged as part of the background. The background field is then numerically calculated and stored as .
Then, the scattered field is updated asBased on , an adjoint field is defined, which satisfies the adjoint equationThe LSM as used in stage 1 is applied to solve (17) for , which is then used to estimate the target shape.
Similar to the CSI method in stage 1, the right-hand side of (17) can be viewed as an adjoint scattered field, . The corresponding adjoint incident field and adjoint total field can be defined as
Next, substitute (16) into (17) to havefrom which in domain can be solved by applying the SVD and Tikhonov regularization techniques as used in stage 1, where is calculated as
3. Multiple Targets
Figure 2 shows three types of targets that have been commonly tested in the literatures. The efficacy of the proposed strategy will be studied by simulations on these types in the following three sections, respectively.
(a)
(b)
(c)
Figure 2(a) shows five cylindrical targets placed in free space. The radius of the detection domain is m and probes are used. At the operating frequency of 300 MHz, the separation between two adjacent probes is about . The cell size is , and .
The recovered distributions of permittivity and conductivity after stage 1 are shown in Figures 3(a) and 4(a), respectively. In stage 2, the domain slightly larger than the center cylinder is selected to be part of the background. The results after stage 2 are shown in Figures 3(b) and 4(b), respectively; and the distributions at are shown in Figure 5. The recovered images after stage 1 and stage 2 look similar. The spacing between the center cylinder and the other four seems to be large enough to allow sufficient probing signals to reach all the five cylinders.
(a)
(b)
(a)
(b)
(a)
(b)
Next, the separation between adjacent cylinders is reduced from m to m. The recovered permittivity and conductivity distributions after stage 1 are shown in Figures 6(a) and 7(a), respectively. The permittivity in the center cylinder is obviously underestimated.
(a)
(b)
(a)
(b)
In stage 2, an annular domain, enclosing the four outer cylinders but excluding the center one, is selected to be part of the background. The results after stage 2 are shown in Figures 6(b) and 7(b), respectively; and the distributions at are shown in Figure 8.
(a)
(b)
In this case, the separation between the center cylinder and the other four seems to be too small. Some of the probing signals are blocked by the outer four cylinders from reaching the center one. Hence, the permittivity of the center cylinder is underestimated, and that over the gap between the center cylinder and the other four is overestimated.
In summary, when m, the results shown in Figures 4 and 5 indicate that both the conventional method and the proposed approach give similar results. When m, the results shown in Figures 7 and 8 indicate that the conventional method underestimates the permittivity of the center cylinder because the outer four cylinders block some of the probing waves. After merging the four blocking cylinders to the background, the center cylinder is better observed with the probing waves.
To quantify the improvement of accuracy by using the proposed strategy, we define the error indices on target shape, permittivity, and conductivity aswhere is the shape-error index and and are the error indices of permittivity and conductivity, respectively, within the target; is the number of cells in the target, is the number of cells in the target which are misrecognized as part of the background, and the superscripts and indicate the estimated value and the actual value, respectively.
By comparing the results after stages 1 and 2 as shown in Figures 6 and 7, it is observed that is reduced from to , is reduced from to , and is reduced from to .
4. Layered Targets
Figure 2(b) shows a cylinder target enclosed by a cylindrical shell. The cylinder has a radius of m, and the shell has an internal radius of m and an external radius of m. The radius of the detection domain is chosen to be m, and probes are used. At the operating frequency of 300 MHz, the spacing between two adjacent probes is about . The cell size is .
The solid curves in Figure 9 are the results after stage 1, by applying the conventional LSM and CSI method. Three different permittivities of target are simulated, and the position of the shell appears to shift inwards in all three cases.
(a)
(b)
(c)
When the iterative approach is applied, the electrical parameters of and around the center cylinder estimated in stage 1 are treated as part of the background medium in stage 2. The recovered distributions at after stage 2, with and 2.5, become closer to the original, as compared to the conventional method. For the case with , the permittivity in the shell is underestimated, but the shell position is also closer to the original than that predicted with the conventional method. By comparing the results after stages 1 and 2, it is found that is reduced from to , and is reduced from to .
As the shell is placed too close to the cylinder, the recovered position of the former may be shifted when using the conventional method. Next, we compare the effects of shell-cylinder separation by simulating cases with m and m, with the distributions at shown in Figures 10 and 11, respectively. In both cases, the position shift of shell becomes less severe than the previous case with m. The iterative approach not only improves the shell position as in the previous case but also obtains a better estimation of permittivity in the shell.
(a)
(b)
(a)
(b)
The thickness of the shell may affect the estimation of shell position and permittivity. Hence, a thicker shell with m is simulated. Figure 12 shows the recovered distributions at , where position shift is barely observable.
(a)
(b)
In summary, by applying the conventional LSM and CSI method, the shell position will be shifted if the cylinder and the shell are put too close or if the shell is too thin. Using the LSM indicator cannot completely separate the shell from the cylinder in some cases. When the shell is thin, more probing waves can reach the internal cylinder and the permittivity of the latter can be well recovered. However, the position of the shell is shifted and its permittivity is underestimated. On the other hand, if the shell is thick, less probing waves can reach the internal cylinder, leading to underestimation of permittivity of the cylinder. The iterative approach seems capable of overcoming this problem by merging the internal cylinder into the background to improve the image of the external shell.
5. Targets with Continuous Profile
Figure 2(c) shows a cylinder with a continuous permittivity profile. The radius of the cylinder is m, the radius of the detection domain is m, and probes are used. At the operating frequency of 300 MHz, the spacing between two adjacent probes is about . The cell size is . Figures 13 and 14 show the recovered distributions of relative dielectric constant and conductivity at . The relative dielectric constant has a linear profile, with the maximum of 2.4 and 3.0, respectively. Each profile is approximated as a piecewise-constant function of 4, 8, and 15 stairs, respectively. The recovered results of the case with appear closer to the original profile than those with . As shown in Figure 14, the permittivity in the internal portion of the 4-stair approximation is underestimated.
(a)
(b)
(a)
(b)
Figure 15 shows the recovered distributions with , , , and 0.5 m, respectively. A linear permittivity profile is assumed, with . With or 0.5 m, the permittivity in the internal portion is seriously underestimated.
(a)
(b)
Similar recovered images were observed in [20], where two concentric square cylinders were immersed in a background medium with and mS/m at the operating frequency of 400 MHz. The external square cylinder has the width of 0.5 m, , and mS/m. The internal square cylinder has the width of 0.25 m, , and mS/m. The results using the contrast source-extended Born (CS-EB) approach are consistent with the original, but the results using the contrast source inversion approach of [8] are inconsistent. The permittivity in the internal portion is seriously underestimated, similar to those shown in Figures 14 and 15.
The incident waves are partially reflected at the interfaces between adjacent stairs, and smaller discontinuity of over the interfaces leads to less reflection. Hence, more incident waves can reach the internal portion of the target domain and get more information to estimate the internal permittivity.
Figure 16 shows the recovered distribution of relative dielectric constant at , using the iterative approach. A linear profile with is assumed, and m. By merging the first few external layers to the background medium in stage 2, the internal portion is recovered more accurately. The recovered image with merged as the background medium matches the most with the original one. However, the recovered distribution at the interface between the background medium and the target becomes less accurate because the contrast function jumps from zero to a finite number at the interface.
In order to avoid the discontinuity of , the permittivity at the interface estimated in stage 1 is used as the background permittivity inside the interface in stage 2. As shown in Figure 17, the discontinuity problem is reduced, if not completely removed. When is selected as the background, is reduced from to and is reduced from to by comparing the results after stages 1 and 2.
6. Conclusion
An iterative approach, based on LSM and CSI method, is proposed to improve the accuracy of recovered images for multiple targets, layered targets, and targets with a continuous permittivity profile. For multiple targets, when a target is partially blocked by other targets, its permittivity tends to be underestimated, and that of the gap between targets tends to be overestimated. For layered targets, the external layers tend to be shifted inwards, especially when the gap between layers is small or the external layer is thin. For a cylinder with continuous permittivity profile, when the radius or the spatial change rate of permittivity is large, the permittivity in the internal portion tends to be underestimated. All these symptoms can be relieved with the proposed iterative approach, which are validated by simulations.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was sponsored by the Ministry of Science and Technology, Taiwan, under Contract NSC 101-2221-E-002-129 and the Ministry of Education, Taiwan, under Aim for Top University Project 103R3401-1.