Image reconstruction for a partially immersed perfectly conducting cylinder using the steady state genetic algorithm

Authors


Abstract

[1] This paper presents a computational approach to the imaging of a partially immersed perfectly conducting cylinder by the steady state genetic algorithm. A conducting cylinder of unknown shape scatters the incident transverse magnetic (TM) wave in free space while the scattered field is recorded in free space. Based on the boundary condition and the measured scattered field, a set of nonlinear integral equations is derived and the imaging problem is reformulated into an optimization problem. An improved steady state genetic algorithm is employed to search for the global extreme solution. Numerical results demonstrate that, even when the initial guess is far away from the exact one, good reconstruction can be obtained. Numerical results also show that the reconstructed image in the free space area is better than that in the medium area.

1. Introduction

[2] The image problem of conducting objects has been a subject of considerable importance in noninvasive measurement, medical imaging, and biological application. In the past two decades, many techniques have been developed to determine the shape and locations of two-dimensional (2-D) cylindrical conducting objects. However, inverse problems of this type are difficult to solve because they are ill-posed and nonlinear. As a result, many inverse problems are reformulated as optimization problems. General speaking, two main kinds of approaches have been developed. The first is based on gradient searching schemes such as the Newton-Kantorovitch method [Roger, 1981; Tobocman, 1989; Chiu and Kiang, 1991], the Levenberg-Marguart algorithm [Colton and Monk, 1986; Kirsch et al., 1988; Hettlich, 1994], and the successive-overrelaxation method [Kleiman and van den Berg, 1994]. These methods are highly dependent on the initial guess and tend to get trapped in a local extreme. In contrast, the second approach is based on the evolutionary searching schemes [Chiu and Liu, 1996; Chiu and Chen, 2000a, 2000b]. They tend to converge to the global extreme of the problem and are less dependent on the initial estimate [Goldberg, 1989]. Recently, researchers have applied GA together with electromagnetic solver to attack the inverse scattering problem mainly in two ways. One is surface reconstruction approach, the other is volume reconstruction approach. Chiu and Liu [1996] first applied the GA for the inversion of a perfectly conducing cylinder with the geometry described by a Fourier series (surface reconstruction approach), while Takenaka et al. [1997], Meng et al. [1998], and Zhou and Ling [2002] used the concept of local shape function to describe the conducting objects (volume reconstruction approach). Alternatively, Zhou et al. [2003], Qing [2001], and Barkeshli et al. [2001] used B-splines to describe the geometry of a metallic cylinder (surface reconstruction approach). Most of the conducing objects are placed in a homogeneous space, while a buried imperfect conductor is reconstructed using GA by Chiu and Chen [2000a, 2000b]. The scattering of a partially immersed object is discussed recently [Ling and Ufimtsev, 2001]. In this case, the scattering object is not immersed in a single medium, but instead is located right at the interface of two mediums, the theoretical and numerical analysis of the scattering problem become much more difficult. One has to deal with not only the usual target surface boundary condition, but also the media interface boundary condition. To our knowledge, there is still no investigation on the electromagnetic imaging of partially immersed objects. In this paper, the electromagnetic imaging of a partially immersed perfectly conducting cylinder is first reported using GA. GA exhibits several interesting features: (1) it is robust enough to reach the global extreme; (2) it is easy to incorporate a lot of a priori information; (3) the use of parallel computers to accelerate the searching procedure is possible. Although GA's are well suited in searching for the global extreme, they suffer from slow convergence as compared to a local search algorithm when they are implemented on serial computers. A natural improvement of speeding up the typical GA is to hybridize the simple GA with a local search algorithm [Zhou et al., 2003]. However, the hybridization usually yields a population that is highly clustered around the local extremes. Such clustering then leads to the re-exploration of the same extreme regions, which is quite wasteful. In this paper, an improved SSGA using nonuniform probability density function (pdf) is proposed to enhance the convergence and increase the converging rate of finding the global extreme of the inverse scattering problems. It is found the improved steady state genetic algorithm can reduce the calculation time of the image problem compared with the typical steady state genetic algorithm. In section 2, the relevant theory and formulation are presented. In section 3, the details of the improved SSGA are given. Numerical results for reconstructing objects of different shapes are shown in section 4. Finally, some conclusions are drawn in section 5.

2. Theoretical Formulation

[3] Let us consider a perfectly conducting cylinder which is partially immersed in a lossy homogeneous half-space, as shown in Figure 1. Media in regions 1 and 2 are characterized by permittivities and conductivities (ɛ1, σ1) and (ɛ2, σ2) respectively. A perfectly conducting cylinder is illuminated by a transverse magnetic (TM) plane wave. The cylinder is of an infinite extent in the z direction, and its cross-section is described in polar coordinates in the x, y plane by the equation ρ = F(θ), i.e., the object is of a star-like shape. We assume that time dependence of the field is harmonic with the factor exp(jωt). Let Einc denote the incident field from region 1 with incident angle ϕ1. Owing to the interface between regions 1 and 2, the incident plane wave generates two waves that would exist in the absence of the conducting object. Thus, the unperturbed field is given by

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Figure 1.

(a) Geometry of the problem in (x, y) plane for the case a > 0. (b) Geometry of the problem in (x, y) plane for the case a < 0.

[4] Since the cylinder is partially immersed, the equivalent current exists both in the upper half space and the lower half space. As a result, the details of Green's function are given first as follows:

[5] 1. When the equivalent current exists in the upper half space, the Green's function for the line source in the region 1 can be expressed as

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where

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[6] 2. When the equivalent current exists in the lower half space, the Green's function for the line source in the region 2 is

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where

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[7] For programming purposes, the scattered field can be expressed according to the following two cases:[case 1] if a > 0(θ1 > θ2), as shown in Figure 1a

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[case 2] if a < 0(θ1 < θ2), as shown in Figure 1b

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with

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[8] Here Js(θ) is the induced surface current density which is proportional to the normal derivative of electric field on the conductor surface. Note that G1 and G2 denote the Green's function for the line source in the region 1 and region 2, respectively. H0(2) is the Hankel function of the second kind of order zero. The boundary condition on the surface of the scatterer states that the total tangential electric field must be zero and this yields an integral equation for J(θ):[case 1] if a > 0(θ1 > θ2), as shown in Figure 1a

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[case 2] if a < 0(θ1 < θ2), as shown in Figure 1b

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[9] For the direct scattering problem, the scattered field ES is calculated by assuming that the shape is known. This can be achieved by first solving J in (6) and (7) and calculating ES in (4) and (5) for the cases 1 and 2, respectively. For the inverse problem, we assume the approximate center of the scatterer, which in fact can be any point inside the scatterer, is known. Then, the shape function F(θ) can be expanded as:

equation image

where Bn and Cn are real numbers to be determined, and N + 1 is the number of unknowns for the shape function. Note that the discretisation number of J(θ) for the inverse problem must be different from that for the direct problem. In our simulation, the discretisation number for the direct problem is twice that for the inverse problem. Since it is crucial that the synthetic data generated through a direct solver are not like those obtained by the inverse solver. The steady state genetic algorithm is used to maximize the following fitness function:

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where Mt is the total number of the measurements. Esexp(equation image) and Escal(equation image) are the measured scattered field and the calculated scattered field, respectively. The factor α∣F′(θ)∣2 can be interpreted as the smoothness requirement for the boundary F(θ). The optimal value of α is mostly dependent on the dimensions of the geometry. One can always choose a large enough value to ensure the convergence, although overestimation will result in a very smooth reconstruction.

[10] The parameters Bn and Cn are coded using Gray code [Lee and Foster, 1992], and the processes of reproduction, mutation and crossover are employed to optimize Bn and Cn. Here, we use an improved steady state genetic algorithm for our image problem. The details of the improved efficient SSGA are addressed in next section.

3. Efficient Steady State Genetic Algorithm

[11] Since there is no general GA theory that can guarantee the global convergence, some guidelines have been listed for the crossover rate and mutation rate and for the choice of various selection schemes in the past years [Weile and Michielssen, 1997; Johnson and Rahmat-Samii, 1997]. However, when addressing real valued optimization problems, the binary representation traditionally used in GA has some drawbacks, especially, for multidimensional, high precision numerical optimization problems [Michalewicz, 1994]. One way to resolve this is to use real parameters directly to represent the solutions as in the other areas of evolutionary computations, evolutionary programming (EP) and evolutionary strategies (ES), where nonuniform pdfs are usually used. As an example, Gaussian distribution is used for Gaussian mutation. To the best of our knowledge, most (if not all) GA's use the uniform pdf to generate the random numbers needed during the course of offspring generation. In this paper, we propose an improved efficient SSGA version, called NU-SSGA, for which the bit string representation is kept and nonuniform beta distributions are introduced to help control the generation of offspring. Thus, in NU-SSGA, the control over the granularity of representation is still possible, while the crossover and mutation operators are modified by incorporated the beta distribution such that the convergence speed are improved for high precision numerical optimization problems.

[12] In general, a typical GA optimizer must be able to perform seven basic tasks [Johnson and Rahmat-Samii, 1997]: (1) encode the solution parameters as genes, (2) create a string of the genes to form a chromosome, (3) initialize a starting population, (4) evaluate and assign fitness values to individuals in the population, (5) perform reproduction through some selection scheme, (6) perform recombination of genes to produce offspring, and (7) perform mutation of genes to produce offspring.

[13] In an NU-SSGA optimizer, task 2 is omitted, i.e., the strings of the parameters are not accumulated to form a chromosome as a typical GA optimizer does. For each individual, different parameters remain separated. In this case, the crossover operator for task 6 needs to be modified. For demonstration, a single-point crossover operator is shown in Figure 2. Note that the new offspring is created by swapping the genetic materials of the corresponding parameters instead of the chromosomes; the latter is used in a typical GA [Johnson and Rahmat-Samii, 1997]. Task 6 may be regarded as a modified N-point crossover version of the typical GA with one crossover point per parameter, where N is the number of parameters.

Figure 2.

A single point crossover operator for NU-SSGA uses one crossover point per parameter (marked by arrows).

[14] The key distinction between an NU-SSGA and a typical GA is on the location of random crossover points. In a typical GA, the crossover points are randomly determined through an uniform probability density function [Johnson and Rahmat-Samii, 1997], while, for an NU-SSGA, the crossover points are randomly determined through other pdf's in a nonuniform way. The beta distributions are used in this paper, of which the pdf's with different parameter pair p and q are plotted in Figure 3 [Rohatgi and Ehsanes Saleh, 2001]. Without loss of generality, we assume aL and bL in Figure 2 to be the most significant bits (MSB's), while a1 and b1 to be the less significant bits (LSB's). Then, by changing the parameter pair p and q adaptively, we are able to move the crossover points between the MSB regions and LSB regions for different individuals. For those individuals that the crossover points are around the MSB regions, NU-SSGA is in the phase of searching through the solution space with the parameters as largely spaced as possible, by which the diversity of the genetic distribution is maintained. On the other hand, when the crossover points are around the LSB regions, NU-SSGA is in the phase of speeding convergence, which is analog to the mechanism of a local search algorithm. Similarly, the same distribution must be applied to the mutation operator for task 7 in accordance with the crossover operator. To adaptively change the parameter pair p and q in a meaningful way, an integer M indexing only these parameter pairs (p, q) in Figure 3 is introduced to associate with each individual. When any two individuals are picked to perform the crossover operation, the two associated integers M's themselves would perform the crossover first such that two new indexing integers are created, while only one of them is kept. Then the usual crossover operation is executed according to the parameter pair p and q indexed by the new integer M. Thus, the crossover points are actually moved between MSB regions and LSB regions adaptively such that all the binary bits of each parameter can converge. Finally, it should be mentioned that uniform pdf is still used for all the other tasks such as the choice of parents for crossover and creation of initial population, etc.

Figure 3.

Beta distributions with various parameters p and s: (0.5, 2), (2, 4), (3, 3), (4, 2) and (2, 0.5).

4. Numerical Results

[15] At first, the proposed NU-SSGA is applied to find the maximum of a sixty-dimensional function f(xi) given as follows [Johnson and Rahmat-Samii, 1997; Chen, 1998]:

equation image

where the variables xi, i = 1 ∼ 60, are within the range 0 ≤ xi ≤ 5. The results with 2 different coding lengths are shown in Figure 4 to demonstrate the superior characteristics of NU-SSGA. In this paper, a temporary population is created during the course of the NU-SSGA searching procedure. The temporary population consists of the parent population and the new individuals generated by the crossover and mutation operators. The offspring individuals are then reproduced using rank selection scheme until the original population size is reached again. The population size 100 is used, and the rank = 20 is set. The crossover rate is set low at 0.1 for the NU-SSGA, while the probability of mutation is set at 0.05 for each individual. The cost function is defined as the distance away from the optimum value. The results are averaged over 10 independent trials and are compared to those produced by a typical uniform SSGA using uniform pdf, for which the same parameters are used. It is obvious that the NU-SSGA optimizer outperformed the typical uniform SSGA using uniform pdf, for either case of coding length 8 or 16. Note that there are 480 bits for a chromosome in total to converge for the case with coding length 8, while, for the case of coding length 16 there are 960 bits in total to converge. It is observed that when the coding length is raised to obtain high precision, the convergence rate (the slope) is not affected for the NU-SSGA. On the other hand, the typical GA suffers quite a lot in this situation, especially in the early searching stage. As a conclusion, NU-SSGA is well suited to optimize those problems with a large number of parameters. The efficient NU-SSGA is then applied to enhance the convergence and increase the converging rate of finding the global extreme of the inverse scattering problems. For a typical uniform SSGA, low crossover rate can result in faster convergence [Johnson and Rahmat-Samii, 1997]. In our experience, the crossover rate between 0.02 and 0.1 is suited for the cases of image reconstruction.

Figure 4.

The cost function versus function evaluation number for the NU-SSGA and a typical GA with coding lengths 8 and 16, respectively.

[16] Let us consider a perfectly conducting cylinder which is partially immersed in a lossless half-space (σ1 = σ2 = 0) and the parameter a is set to zero. The permittivity in region 1 and region 2 is characterized by ɛ1 = ɛ0 and ɛ2 = 2.56ɛ0, respectively. The frequency of the incident wave are chosen to be 1 GHz, with incident angles ϕ1 equal to 45° and 315°, respectively. For each incident wave 8 measurements are made at the points equally separated on a semi-circle with the radius of 3 m in region 1. Therefore, there are totally 16 measurements in each simulation. The number of unknowns is set to be 9 (i.e., N + 1 = 9), to save the computation time. The population size of 100 is chosen and rank selection scheme is used with the top 30 individuals being reproduced according to the rank. The searching range for the unknown coefficients is chosen from 0 to 0.2. The extreme values of the coefficient of the shape function can be determined by the prior knowledge of the objects. The crossover rate is set to 0.1 such that only 10 iterations are performed per generation. The mutation probability is set to 0.05 and the value of α in (9) is chosen to be 0.001. In the below examples the size of the scatter is about one wavelength, so the frequency is in the resonance range.

[17] In the first example, the shape function is chosen to be F(θ) = (0.05 − 0.01 cos 3θ + 0.01 sin 3θ) m. The reconstructed shape function of the best population member (chromosome) is plotted in Figure 5a with the shape error shown in Figure 5b. Here, the shape function discrepancy DR is defined as

equation image

where N′ is set to 360. The quantity DR provides the measure of how well Fcal(θ) approximates F(θ). From Figure 5, it is observed that there is some notable error for the buried part of the object in the region 2. The physical reason is that incident waves are from region 1 and the scattered fields are collected all in region 1. In other words, the scattered fields carry less information for the buried part of the object than the unburied part in free space.

Figure 5.

(a) Shape function for example 1. The solid curve represents the exact shape, and the others are the calculated shape function in the iterative process. (b) Shape function error in each generation.

[18] To investigate the sensitivity of the imaging algorithm against random noise, two independent Gaussian noises with zero mean are added to the real and imaginary parts of the simulated scattered fields. Normalized standard deviations of 10−4, 10−3, 10−2 and 10−1, respectively, are used in the simulations. The normalized standard deviation is defined as the ratio of the standard deviation of the Gaussian noise divided by the R.M.S. value of the scattered fields. Thus, the signal-to-noise ratio (SNR) is inversely proportional to the normalized standard deviation. The numerical result for example 1 is plotted in Figure 6. It is noted that the effect of noise is negligible for normalized standard deviations below 10−2.

Figure 6.

Shape function error as a function of noise levels for example 1.

[19] In the second example, a peanut shape function F(θ) = (0.1 + 0.02 cos 2θ + 0.02 sin θ − 0.025 sin 2θ) m is tested. The purpose of this example is to demonstrate that the proposed method is able to reconstruct a scatterer whose shape function has two concavities. The reconstructed shape function of the best population member (chromosome) is plotted in Figure 7a with the DR shown in Figure 7b.

Figure 7.

(a) Shape function for example 2. The solid curve represents the exact shape, and the others are the calculated shape function in the iterative process. (b) Shape function error in each generation.

[20] In the third example, the shape function is selected to be F(θ) = (0.05 − 0.01 cos 4θ + 0.01 sin 4θ) m. This example shows that the proposed scheme can reconstruct a more complex scatterer whose shape function has four concavities. The best population member (chromosome) is plotted in Figure 8a with the DR shown in Figure 8b. Again, the shape error for the buried part is larger than that for the unburied part.

Figure 8.

(a) Shape function for example 3. The solid curve represents the exact shape, and the others are the calculated shape function in the iterative process. (b) Shape function error in each generation.

[21] It should be noted that the calculation of the Green's function is quite computational expensive. The steady state genetic algorithm has not only the characteristic of faster convergence [Chen, 1998; Vavak and Fogarty, 1996], but also the lower rate of crossover. As a result, it is a suitable scheme to effectively save the calculation time for the inverse problem as compared with the generational GA.

5. Conclusions

[22] We have presented a study of applying the steady state genetic algorithm to reconstruct the shape of a partially immersed conducting cylinder through knowledge of scattered fields. Based on the boundary condition and measured scattered fields, we have derived a set of nonlinear integral equations and reformulated the imaging problem into an optimization problem. By using the steady state genetic algorithm, the shape of the object can be reconstructed from the scattered fields in free space. Numerical results illustrate that the shape function error decreases as the generation increases, and reconstruction of the unburied part of the cylinder in free space is better than the buried part of the cylinder in the medium. Good images can be reconstructed as long as the normalized noise level is less than 10−2.

Acknowledgments

[23] This work was supported by the National Science Council, Republic of China, under grant NSC-91-2213-E032-022.

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