# A reconstruction procedure for microwave nondestructive evaluation based on a numerically computed green's function

## Summary (2 min read)

### A Reconstruction Procedure for Microwave Nondestructive Evaluation based on a Numerically Computed Green's Function

- In the framework of NDE/NDT, this paper presents a new microwave diagnostic tool.
- Keywords: Imaging Systems, Imaging Processing, Material Characterization, Green’s Function, Genetic Algorithms I. INTRODUCTION.
- In very few cases, the objects under test are weak enough to allow the practical use of simplified [8][9] or closed-form solutions [10].
- Recently, the development of reconstruction procedures for microwave tomography has been addressed by resorting to the numerical discretization of the integral equations of the inverse scattering problem.
- Section III gives a description of the optimization procedure based on a customized genetic algorithm pointing out the keypoints of its application to the NDE/NDT framework.

### II. MATHEMATICAL FORMULATION

- These operators are defined in details in [11]-[13] for the case in which the kernel is the free space Green's function [14].
- A numerically computed Green's function for the unperturbed configuration (the configuration without the defect) is considered.
- Equation (3) can be solved off-line and once for all by means of the moment method.
- As an example of these computations, Figure 1 shows the amplitudes of the inhomogeneous Green’s function for a point source located at the center of the investigation domain and in correspondence with different host medium configurations.
- After discretization of the continuous model, ( ) ( ){ }rEr toti,γℑ (equation (2)) is minimized by means of a suitable GA-based procedure [15] able to efficiently exploit the features of the proposed approach.

### III. GA-BASED PROCEDURE

- Binary tournament selection [16] and real/binary double point crossover [15][17] are used for the selection and the crossover, respectively.
- The mutation is performed with probability mP on an individual and consists in perturbing one element of its genetic sequence.

### IV. NUMERICAL VALIDATION

- In order to assess the effectiveness of the proposed approach (in the following indicated by IGA) pointing out its efficiency by a computational point of view, some numerical simulations have been performed.
- Figure 3 shows the images of the reconstructed distributions at different iterations.
- Finally, in order to quantify the computational effectiveness of the proposed approach, the following parameter is evaluated: 100* ** ×−=∆ FGA IGAFGA conv K KK (7) and the obtained results are reported in Figure 5.
- As can be observed, it results that, on an average, an IGA iteration took approximately 52 of the time necessary for the FGA iterative step.

### V. CONCLUSIONS AND FUTURE DEVELOPMENTS

- An innovative approach for the crack detection in known host medium has been presented.
- In order to fully exploit the knowledge of the scenario under test, a new formulation based on the numerical computation of the Green’s function of the unperturbed configuration has been proposed.
- This fact, confirmed by several numerical simulations, clearly indicates a possibility for the quasi real-time implementation of the proposed technique in real-word monitoring of industrial processes.
- To this end, further improvements and generalizations are mandatory.
- Let us consider the extension to more general crack shapes (and consequently the need for more complete and complicated crack parameterizations), or the possibility of dealing with multiple defects in the same host medium, or (in some specific industrial applications) the increase of the resolution capabilities.

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### Cites background from "A reconstruction procedure for micr..."

...Microwave tomographic approaches have also been considered for several other applications in industrial fields as well as in medical fields (Li and Hagness 2001, Caorsi et al. 2004, Bozza et al. 2007, Kharkovsky and Zoughi 2007b)....

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##### References

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### "A reconstruction procedure for micr..." refers background in this paper

...In very few cases, the objects under test are weak enough to allow the practical use of simplified [8], [9] or closed-form solutions [10]....

[...]

855 citations

### "A reconstruction procedure for micr..." refers methods in this paper

...Binary tournament selection [16] and real/binary double-point crossover [15], [17] are used for the selection and the crossover, respectively....

[...]

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##### Frequently Asked Questions (9)

###### Q2. What future works have the authors mentioned in the paper "A reconstruction procedure for microwave nondestructive evaluation based on a numerically computed green's function" ?

This fact, confirmed by several numerical simulations, clearly indicates a possibility for the quasi real-time implementation of the proposed technique in real-word monitoring of industrial processes. Let us consider the extension to more general crack shapes ( and consequently the need for more complete and complicated crack parameterizations ), or the possibility of dealing with multiple defects in the same host medium, or ( in some specific industrial applications ) the increase of the resolution capabilities. To this end, further improvements and generalizations are mandatory.

###### Q3. What is the main advantage of the GA?

The exploitation of a-priori information is very important in practical applications allowing a reduction of the search space sampled by the optimization procedure and, consequently, an increase of the convergence rate of the iterative process.

###### Q4. Why is a GA able to deal with integer as well as real variables at the same?

Due to its flexibility, a GA is able to deal with integer as well as real variables at the same time and does not require neither differentiability nor continuityof the cost function to be minimized.

###### Q5. What is the main interest of the paper?

the development of reconstruction procedures for microwave tomography has been addressed by resorting to the numerical discretization of the integral equations of the inverse scattering problem.

###### Q6. What is the main novelty of the proposed approach?

In this framework, the main novelty of the proposed approach lies in the use of the Green's function for the unperturbed geometry, which can be numerically computed off-line and once for all.

###### Q7. What is the main drawback of the GA?

Although the continuous increasing in the computational power of new computers tends to alleviate this problem, a pixel representation of the cross section of an unknown complex object is still a difficult task.

###### Q8. How many times did the IGA iteration take?

As can be observed, it results that, on an average, an IGA iteration took approximately 52 of the timenecessary for the FGA iterative step.

###### Q9. What is the main feature of the proposed method?

In this Section, selected numerical results are presented in order to demonstrate the two main features of the approach:(a) the accuracy in the crack detection and estimation; (b) the reduction of the computational load with respect to the use of the method (namely the FGA) presented in [11].