### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Statement of the Problem
- 3. Inversion Scheme
- 4. Numerical Results
- 5. Conclusion
- Appendix A:: Updating Directions
- References

[1] This paper deals with an iterative approach to solve the electromagnetic inverse scattering problem from single view transient data. The measurement configuration consists in illuminating a two-dimensional scattering system with an electromagnetic transient source and in measuring the time domain response all around the investigated region. The aim is then to determine the electromagnetic properties of the target from the measurements. The problem is formulated in the frequency domain for a large number of frequencies, such that the incident pulse is accurately sampled, rather than in the time domain. The parameters of interest, namely, the relative permittivity and the conductivity profiles, are built up iteratively by minimizing a cost functional involving the discrepancy between the measured scattered fields and those that would be obtained via a forward model. Numerical examples are presented to prove the efficiency of the suggested method and its robustness against noise. The influence of the central frequency and of the bandwidth of the incident field on the achievable performances is also illustrated.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Statement of the Problem
- 3. Inversion Scheme
- 4. Numerical Results
- 5. Conclusion
- Appendix A:: Updating Directions
- References

[2] Inverse scattering problems play an important role in many applications, which require a non invasive characterization of unknown objects. These applications bring different areas together, like geophysical probing, medical imaging or nondestructive evaluation. The aim of inverse problems is to retrieve the features (i.e., position, geometry and constitutive materials) of unknown targets from the knowledge of their scattered fields.

[3] The most popular strategy for solving this ill-posed and nonlinear problem is to reconstruct the parameters of interest iteratively, by minimizing a cost functional involving the discrepancy between the data and the scattered fields that would be obtained via a forward model from the best available estimation of the parameter. In the frequency domain, a theoretical analysis of the dynamic range as well as of the spatial resolution of linearized schemes [*Tijhuis et al.*, 2001a] shows that the convergence is better at low frequencies, with however a “poor” resolution, while at higher frequencies the resolution is improved but the convergence is not guaranteed.

[4] To circumvent the foreseen difficulties, a multiple frequency approach, known as frequency-hopping approach [*Chew and Lin*, 1995] has been proposed. In this approach, the iterative inversion process starts at low frequencies in order to get a trend that is used as an initial guess for the inversion at higher frequencies, thus making possible to retrieve the object under test with high resolution. Successful results have been obtained from synthetic data [*Tijhuis et al.*, 2001a] as well as experimental data [*Tijhuis et al.*, 2001b].

[5] Another way to overcome these difficulties is to use transient data. In this case, targets are illuminated by a short time duration pulse, that is to say a wideband incident field, so that both low frequencies, required to ensure convergence of the algorithm, and high frequencies, which improve the spatial resolution, are considered at the same time in the inversion scheme.

[6] The inversion of transient fields may also be tackled in two ways depending on the formulation of the scattering problem, either in the frequency domain [*Moghaddam and Chew*, 1992, 1993] or directly in the time domain [*Tijhuis*, 1981; *Weedon and Chew*, 1993]. In the present work, the first representation has been preferred because it has some advantages like the restriction of the computational domain to the scattering domain, boundaries conditions inherently satisfied through the appropriate Green function and possibility to take easily into account the dispersion of the background media as well as the target under test, etc. The main bottleneck solving the transient scattering problem in the frequency domain is that repeated field computations require an excessive amount of computation time. However, the computational time may be reduced drastically by using a special extrapolation procedure as suggested by *Peng and Tijhuis* [1993] and *Tijhuis et al.* [2001a]. Therefore Maxwell's equations as well as data are expressed in the frequency domain thanks to a temporal Fourier transform. This allows us to investigate two different strategies. The first one is the frequency hopping in which inversions are performed sequentially starting from low frequencies and using final results as initial guesses for inversion at higher frequencies. In the second approach, inversion of transient data, entire spectra of data are inverted once. This paper is restricted to two-dimensional targets embedded in a homogeneous background.

### 2. Statement of the Problem

- Top of page
- Abstract
- 1. Introduction
- 2. Statement of the Problem
- 3. Inversion Scheme
- 4. Numerical Results
- 5. Conclusion
- Appendix A:: Updating Directions
- References

[8] A right-handed Cartesian coordinate frame (*O*, **u**_{x}, **u**_{y}, **u**_{z}) is defined, with *z* axis parallel to the direction of invariance. The position of a point *M* is given by:

The investigated region is illuminated by a line source parallel to the *z* axis and fed by a current ℐ(*t*), thus generating a transient incident field ℰ^{inc}(**r**, *t*). A short Gaussian pulse is used, leading to a bandwidth to central frequency ratio Δ*f*/*f*_{0}≃ 0.5, as shown in Figure 2. The scattered field is measured by *N*_{r} receivers located on a line Γ surrounding the target. The study is restricted to the transverse magnetic (TM) polarization case. Therefore only the *z* component of the involved electrical fields is considered.

[9] By taking into account the Parseval theorem, the inverse scattering problem can be formulated in the frequency domain rather then in the time domain, provided that the transient fields are replaced by the corresponding time-harmonic ones and the involved frequency range is sampled according to Shannon's theorem. However, this may lead to consider a large and redundant number *L* of time harmonics problems. As discussed by *Bucci et al.* [2000], the use of multifrequency data (instead of monochromatic ones) offers the possibility to enlarge the amount of available independent information and then to improve the local stability and robustness against false solutions of the inverse approach. On the other hand, recording the data at two different frequencies does not means to double up the amount of independent information, while it certainly leads to increase the processing times required by the inversion procedure. Then, a key issue is to understand how many frequencies have to be considered in a given frequency range in order to improve the achievable performances, while keeping acceptable the computational effort. Some useful advises on this point are given by *Bucci et al.* [2000] and *Catapano et al.* [2006]. Let us define the time Fourier transform as:

*E*(**r**, *ω*) is discretized with respect to frequency into *L* components and *l*th frequency component of *E*(**r**, *ω*) is from now on denoted as *E*_{l}(**r**).

[10] The electromagnetic scattering problem can then be formulated, for each frequency *f*_{l}, with *l* = 1, ⋯, *L*, as two coupled contrast-source equations. The observation equation (equation (3)) expresses the scattered field *E*_{l}^{d} at **r** on Γ and the state equation (equation (4)) expresses the total field in Ω.

In these two equations, *χ*_{l} represents the contrast of complex relative permittivity at *f*_{l}, *χ*_{l}(**r**) = _{r,l}(**r**) − _{rb,l}, and *k*_{0,l} is the wave number of vacuum. Since *χ*_{l} changes with the frequency, as far as the inverse scattering problem is concerned, some hypothesis on the constitutive relationship of the involved media is required in order to take advantage of the frequency diversity, without neglecting the dispersive effect of background and scatterers. In several applications in the microwave domain a reliable guess is to consider an an Ohmic dispersion model with conductivity *σ* and neglecting the dispersion of the real part of the permittivity [*Lambert et al.*, 1998]. Then, the complex permittivity at frequency *f*_{l} is given by the following relationship:

This leads to assume as actual unknowns of the inverse scattering problem the relative permittivity ɛ_{r} and the conductivity *σ*, which do not depend on the frequency. On the other hand, equivalent frequency independent unknowns can be introduced also in the case of a more sophisticated model then the Ohmic one, supposed that the nature of the involved media is known.

[11] Kernels *G*_{l,Ω} and *G*_{l,Γ} in equations (4) and (3) involve the free space Green function. The convolution structure of the integral equation in equation (4) is exploited numerically by using a fast Fourier transform (FFT) algorithm. For sake of simplicity, symbolic notations are introduced. Equations (3) and (4) are rewritten as

### 3. Inversion Scheme

- Top of page
- Abstract
- 1. Introduction
- 2. Statement of the Problem
- 3. Inversion Scheme
- 4. Numerical Results
- 5. Conclusion
- Appendix A:: Updating Directions
- References

[12] The aim of the inverse problem is to determine both the contrast of permittivity κ = ɛ_{r}(**r**) − ɛ_{rb} and the contrast of conductivity *σ*(**r**) − *σ*_{b} in Ω, such that the associated scattered field matches the measured one ℰ^{mes} on Γ. Many iterative methods have been developed to solve such inverse scattering problems. Starting from an initial guess, parameters of interest are adjusted gradually by minimizing a cost functional representing the discrepancy between the data and the scattered fields computed from the best available estimation of the parameters. The minimization is carried out by means of a hybrid method [*Belkebir and Tijhuis*, 2001] combining advantages of linearized methods [*Chew and Wang*, 1990], in which the field in the test domain is considered as fixed at each iteration step, and of a nonlinearized method [*Kleinman and van den Berg*, 1992], in which the field is considered as an auxiliary unknown determined together with the parameter of interest during the minimization procedure. This hybrid method is described by *Belkebir and Tijhuis* [2001] for the multistatic (multiple transmitters and multiple receivers) and time harmonic configuration. We extend herein this method to the case of transient fields.

[13] The basic idea underlying the inversion algorithm is to build up three sequences *E*_{l,n}, *ξ*_{n} and *η*_{n} related to the total field in Ω, the real part and the imaginary part of the relative permittivity distribution. In addition, a priori information is incorporated. This information consists in stating that the real part of the relative permittivity ɛ_{r} is greater than unity (positivity of the electrical susceptibility), and the conductivity *σ* is positive. This is done through the transformations ɛ_{r}(**r**) = 1 + *ξ*^{2}(**r**), and *σ*(**r**) = *η*^{2}(**r**). The three sequences mentioned above are built up through the following recursive relations

where *v*_{l,n}, *w*_{l,n} are updating directions related to the field, while *d*_{n;ξ} and *d*_{n;η} are updating directions related to the electrical susceptibility and the conductivity, respectively. These updating directions are specified in Appendix A. The complex coefficients *α*_{l,n;v} and *α*_{l,n;w} and the real coefficients *β*_{n,ξ} and *β*_{n,η} are scalars obtained by minimizing at each iteration step a cost function ℱ_{n} which writes as

The subscripts Ω and Γ, included in the norm ∥.∥ and later in the inner product 〈., .〉, indicate the domain of integration. The normalization coefficients *W*_{Ω} and *W*_{Γ} are given by

Functions *h*_{l}^{(1)} and *h*_{l}^{(2)} are residual errors computed from the observation (equation (6)) and from the state equations (equation (7)), respectively.

[14] Once all the updating directions *v*_{l,n}, *w*_{l,n}, *d*_{n,ξ} and *d*_{n,η} are determined, ℱ_{n} is a non linear function of 2 × *L* complex variables (*α*_{l,n;v} and *α*_{l,n;w}) and two real variables (*β*_{n;ξ} and *β*_{n;η}). The minimization of ℱ_{n} is achieved thanks to the standard Polak-Ribière conjugate gradient method [*Press et al.*, 1986]. Note that minimizing the cost function equation (11) is equivalent, via Parseval's theorem, to minimizing the same cost function where the time harmonic quantities are replaced by the associated transient ones and where the sum is over time instead of frequency.