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Keywords:

  • Nonlinear inversion;
  • inverse profiling;
  • regularization;
  • total variation

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Inversion Problem
  5. 3. Inversion Algorithm
  6. 4. Numerical Test Examples
  7. 5. Conclusions
  8. Acknowledgments
  9. References

[1] In this paper we discuss a new type of regularization technique for the nonlinear inverse scattering problem, namely the multiplicative technique. The main advantage is that we do not have to determine the regularization parameter before the inversion process is started. We consider different norms of the total variation as regularization factor. Specifically, we investigate a weighted L2-norm, and by using an appropriate updating scheme we show that this multiplicative regularization factor does not increase the nonlinearity of the inversion problem. Numerical examples using synthetic and experimental data demonstrate the robustness of the presented method.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Inversion Problem
  5. 3. Inversion Algorithm
  6. 4. Numerical Test Examples
  7. 5. Conclusions
  8. Acknowledgments
  9. References

[2] A central problem in target identification, non-destructive testing, medical imaging and numerous other areas of application concerns the determination of the shape, location and constitutive parameters, such as complex index of refraction or local sound speed, of a object or local inhomogeneity from measurements of the scattered field, when the object is illuminated successively by a number of known incident electromagnetic or acoustic waves. This problem is nonlinear and ill-posed, but during the years useful reconstruction algorithms have been developed. Comprehensive overviews of several results are given by Habashy et al. [1994], Chew [1999], Lesselier and Duchêne [1996], Colton et al. [2000] and Sabatier [2000] and the results of testing various inversion algorithms against experimental data can be found in the special section in the Inverse Problems journal organized by Belkebir and Saillard [2001]. Most of these algorithms make use of the domain integral equation for the field inside the scattering object as well as the related integral representation for the field outside the object.

[3] Our starting point is the contrast source inversion method [van den Berg and Kleinman, 1997], where the unknown contrast sources and the unknown contrast are updated by an iterative minimization of an appropriate cost functional. Inspired by the successful implementation of the minimization of total variation and other edge-preserving algorithms in image restoration and inverse scattering, previously van den Berg et al. [1999] have investigated the use of these image-enhancement techniques as an extra regularization. The drawback of adding a regularization term to the cost functional is the presence of an artificial weighting parameter in the cost functional, which can only be determined through considerable numerical experimentation and a priori information about the desired profile. Therefore, we take the regularization as a multiplicative constraint and show that the weighting parameter is now completely prescribed by the error norm of the data equation and the object equation. This multiplicative regularization technique is originally introduced by van den Berg et al. [1999], using the L1- and L2-norms total variation [Rudin et al., 1992] as constraints. Subsequently, van den Berg and Abubakar [2001] have shown that a weighted L2-norm total variation factor improves the reconstruction results significantly. The advantage is that the updating parameters in the contrast source inversion method can be determined explicitly, without the usual line minimization. A possible drawback of the introduction of a multiplicative regularization factor is the danger of increasing the nonlinearity of the inversion problem, viz., the loss of convexity of the cost criterion for updating the contrast.

[4] In this paper we have modified slightly the updating scheme of the contrast. This modification allows us to derive a convexity criterion of the multiplicative cost functional using the weighted L2-norm of the total variation as regularization factor. In our numerical experiments we observe that the present implementation of the contrast update leads always to a convex problem. Moreover the modification of the contrast updating scheme shows a further improvement of the reconstructions. In the numerical experiments using both synthetic and experimental data, we illustrate the reconstruction results with three different regularization factors, the L1-, L2-, and weighted L2-norm regularization factor, respectively. The numerical results clearly indicate that the use of the latter one yields a very robust inversion scheme.

2. Inversion Problem

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Inversion Problem
  5. 3. Inversion Algorithm
  6. 4. Numerical Test Examples
  7. 5. Conclusions
  8. Acknowledgments
  9. References

[5] We consider an object, B, of arbitrary bounded cross section. Let D denote the interior of a bounded domain with piecewise smooth discontinuity interfaces. Spatial points are denoted as x. We assume that the unknown scatterer, B, is contained in the domain D. The material contrast of the object with respect to its embedding medium is denoted as χ(x). Clearly χ(x) = 0 at those points in D exterior to the actual scattering object B. We assume that the fields vary sinusoidally in time with frequency ω. The corresponding wavelength is denoted by λ. We also assume that we know the Green function, g(x, x′), as the fundamental solution in the embedding medium. We further assume that the object is irradiated successively by a number of known incident fields ujinc(x), j = 1,…, originating from difference source positions. For each incident field, the total field will be denoted by uj(x).

[6] In the inverse scattering problem, the scattered field values, fj = ujsct, will be measured on some surface S outside D. The scattered field is modeled through a domain integral representation. For points exterior to D it is written symbolically as the data equation,

  • equation image

while for points in the interior of D it is written symbolically as the object equation,

  • equation image

where the operator GS is an operator mapping from L2(D) into L2(S) and GD is an operator mapping L2(D) into itself. These source-type integral operators are given by

  • equation image

when x ∈ {S, D}. Here the contrast sources

  • equation image

are introduced.

[7] The inverse scattering problem consists of determining χ(x) from a knowledge of the incident fields, ujinc(x), on D and the scattered fields, fj(x), on S. This inverse problem is non-linear.

3. Inversion Algorithm

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Inversion Problem
  5. 3. Inversion Algorithm
  6. 4. Numerical Test Examples
  7. 5. Conclusions
  8. Acknowledgments
  9. References

[8] We start with the so-called Contrast Source Inversion (CSI) method introduced by van den Berg and Kleinman [1997]. In this method, one chooses to reconstruct the material contrast χ and the contrast sources wj. Using the contrast sources defined in (4), the data equation becomes

  • equation image

while the object equation becomes

  • equation image

Substituting (6) into (4), we obtain an object equation for the contrast sources rather than for the fields, viz.,

  • equation image

[9] Then, in the CSI method, the sequences of the contrast sources wj,n and the contrast χn for n = 1, 2, ⋯, are iteratively found by minimizing a cost functional,

  • equation image

where

  • equation image

and

  • equation image

where ∥ · ∥S,D2 = 〈 · , · 〉S,D denotes the squared norm on S or D.

[10] This CSI method starts with back propagation as the initial estimates for the contrast sources and the contrast [van den Berg and Kleinman, 1997]. In each iteration, we first update the contrast sources wj = wj,n using a conjugate gradient step

  • equation image

where the functions vj,n is the Polak-Ribière conjugate gradient directions and the constant αn is found as minimizer of

  • equation image

Note that the functional is a quadratic function of α, and only one minimizer is arrived at. Subsequently, we compute the field

  • equation image

after which an updated approximation of the contrast χn is found as

  • equation image

and it is found explicitly as

  • equation image

provided ∑juj,n2 ≠ 0 on D.

3.1. Extra Regularization

[11] Although the inclusion of the object equation in the second term of (8) can be considered as a physical regularization of the ill-posed data equation in the first term of (8), the inversion results may be improved by taking into account a priori information about the contrast profile. The standard way to include this a priori information is to modify the functional by introducing an extra penalty function, viz.,

  • equation image

As known in the literature the addition of the regularization term FnR to the cost functional has a very positive effect on the quality of the reconstruction. The drawback is the presence of the positive weighting parameter γ2 in the cost functional, which, with the present knowledge, can only be determined through considerable numerical experimentation and a priori information of the desired reconstruction [van den Berg and Kleinman, 1995]. Further, numerical experiments have shown that the results improve when we let the parameter γ2 decrease with increasing number of iterations. In fact, a good choice seems to take this parameter proportional to the value of the cost functional Fn−1 of the previous iteration. This numerical experimentation has led us to the idea of multiplicative regularization technique, see van den Berg et al. [1999], viz.

  • equation image

Minimization of this functional with respect to changes in the contrast will change the minimizer χn given in (15) to χnR. Our aim is not to change the updating procedure of the contrast sources wj,n. At the beginning of each iteration we have to replace the quantity χn−1 in (10) and (12) by χn−1R, but the remainder of the contrast source updating procedure is not changed, when we keep the regularization factor to be equal to one during this part of the iteration. Then, only the updating of the contrast (for given wj = wj,n) has to be modified. Instead of taking the previous iterate of the contrast as done in our previous papers [van den Berg et al., 1999] and [van den Berg and Abubakar, 2001], we now take the analytic value of (15) as starting value. From this point we make an additional minimization step,

  • equation image

where χn is now given by (15) and dn is the conjugate gradient direction

  • equation image

We remark that we prefer now a line minimization around the minimum of the cost functional FD,n (physical cost criterion). In view of (15) we take gnR as

  • equation image

being a preconditioned gradient of the regularization factor FnR with respect to changes in the contrast around the point χ = χn. In view of the previous minimization step, the gradient of FD,n with respect to changes in the contrast around the point χ = χn vanishes. Hence, the gradient with respect to the contrast, in contrary to the previous approaches of the CSI method, contains only a contribution of the regularization additionally imposed. This simplifies the algorithm. In general, the real parameter βn is found from a line minimization as minimizer of

  • equation image

The structure of this minimization procedure is such that it will minimize the regularization factor with a large weighting parameter in the beginning of the optimization process, because the value of FS + FD,n is still large, and that it will gradually minimize more and more the error in the data and object equations when the regularization factor FnR remains a nearly constant value close to one. If noise is present in the data, the data error term FS will remain at a large value during the optimization and therefore, the weight of the regularization factor will be more significant. Hence, the noise will, at all times, be suppressed in the reconstruction process and we automatically fulfill the need of a larger regularization when the data contains noise as suggested by Chan and Wong [1998] and Rudin et al. [1992]. After we have obtained a new estimate χnR for the contrast, we update the contrast sources starting with χn−1 = χn−1R of the previous iteration.

3.2. Regularization Factors

[12] Using the modified gradient method, van den Berg and Kleinman [1995] have performed some experiments with the additive L1-norm of the total variation (TV). Later, van den Berg et al. [1999] have introduced this norm as a multiplicative regularization factor. In our present analysis the L1-norm that has a value equal to one at χ = χn is given by

  • equation image

where denotes the spatial differentiation with respect to x. Although the constant parameter δn2 is introduced for restoring differentiability of the TV regularizator, it also controls the influence of the regularization. We therefore have chosen to increase the regularization as a function of the number of iterations by decreasing this parameter δn2. Since the object error term will decrease as a function of the number of iterations, we choose

  • equation image

where equation image denotes the reciprocal mesh size of the discretized domain D, and FD,n is the normalized norm of the error in the object equation, before the extra regularization. Its choice is inspired by the idea that in the first few iterations, we do not need the minimization of the total variation and as the iterations proceed we want to increase the effect of the total variation. A disadvantage of the present regularization factor, is that we cannot show whether the minimization problem of (21) remains convex when we use the extra TV regularization factor in (22).

[13] We therefore consider the L2-norm of the total variation. Then our regularization factor is taken as,

  • equation image

For this extra regularization we are able to show that under very weak conditions for the choice of the parameter δn2 the minimization problem remains convex (the proof is very similar to the one given below). In contrary to the L1-norm in the TV-factor that favors piecewise constant contrasts, the L2-norm in the TV-factor favors a smooth profile. Hence, edge preserving effects of the regularization will be lost.

[14] Inspired by the edge-preserving algorithms in image restoration [Charbonnier et al., 1996] and in inverse scattering [Lobel et al., 1997], [Dourthe et al., 2000], [Zhdanov and Hursan, 2000], we now consider the TV-factor as a weighted norm on L2(D), in which the weighting favors flat parts and non-flat parts of the contrast profile almost equally. Instead of (24) we choose the regularization factor as in van den Berg and Abubakar [2001],

  • equation image

where V = ∫D dv(x) denotes the area of the test domain D. The latter regularization factor can be written in terms of norms as

  • equation image

where

  • equation image

The minimization of the multiplicative cost functional can now be performed analytically. The cost functional is a fourth-degree polynomial in β, viz.,

  • equation image

with

  • equation image
  • equation image
  • equation image
  • equation image
  • equation image

In the next subsection, we investigate under which conditions this minimization problem is convex with one minimum. Then, differentiation with respect to β yields a cubic equation with one real root and two complex conjugate roots. The real root is the desired minimizer βn. Note that the present analysis is valid for both real- and complex-valued of the contrast χ.

3.3. Convexity for Real β

[15] The cost functional ℱn is a convex function of real β and has one minimum for real β as its second derivative with respect to β is positive. The second derivative is obtained as

  • equation image

For β = 0 this function is positive, and remains positive when the right-hand side has no real zero's. This is the case if

  • equation image

Since both A and B are functions of wj,n, we rather want a criterion in which only functions of contrast quantities occur. Further the term AZ2/B is always non-negative, hence, a sufficient condition that the second derivative is a positive function of β is

  • equation image

or

  • equation image

However, from Cauchy-Schwarz inequality we know that

  • equation image

This means that

  • equation image

is a sufficient condition for the cost functional to be a convex function with one minimum. If the choice for the parameter δn2 of (23) is less than the right-hand side of (39), we replace the value of δn2 by the right-hand side of (39) in which we take bn = V−1/2(∣χn2 + δn−12)−1/2. We have refrained from using this value for δn2 in the whole iteration procedure, because from our numerical observations it appears this value has a large variation in the beginning of optimization procedure.

4. Numerical Test Examples

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Inversion Problem
  5. 3. Inversion Algorithm
  6. 4. Numerical Test Examples
  7. 5. Conclusions
  8. Acknowledgments
  9. References

[16] For our numerical examples, the test domain D consists of a square. The homogeneous embedding is chosen to be lossless and λ denotes the wavelength. The discrete form of the algorithm is obtained by dividing the test domain into subsquares, assuming the contrast and fields to be piecewise constant. The incident fields are chosen to be excited by line sources parallel to the axis of the scatterer. The sources and receivers are taken to be equally spaced on a measurement circle S. The measured data are simulated by solving the direct scattering problem with the help of a conjugate gradient method [van den Berg, 1981]. After generation of synthetic data, 10% random white noise is added using the following procedure:

  • equation image

where ran1 and ran2 are two random numbers varying from −1 up to 1, and ζ = 0.1 is the amount of noise.

4.1. Concentric Squares

[17] We first consider a scattering object that consists of concentric square cylinders, an inner cylinder of dimension λ by λ, with complex contrast χ = 0.6 + 0.2i, surrounded by an outer cylinder, 2λ by 2λ, with contrast χ = 0.3 + 0.4i. The test domain D is a square of dimension 3λ by 3λ and is discretized into 29 × 29 subsquares. The circle S, where both 29 sources and 29 receivers are located, has a radius of 3λ. The discretized real and imaginary parts of the exact contrast profile are shown in Figure 1a. Using the CSI method, the reconstructions after 512 iterations are shown in Figure 1b. We observe that the reconstructed profile without using the regularization factor is not acceptable due to the large amount of noise present in the data.

image

Figure 1. Original profile of concentric squares (a) and the reconstructed profiles from CSI (b), CSI with L1-norm TV (c), CSI with L2-norm TV (d) and CSI with weighted L2-norm TV (e).

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[18] The reconstruction results using L1- and L2-norms as regularization factors are given in Figures 1c and 1d. We clearly observe that the results using the CSI method with L1-norm produces a ‘blocky’ profile and while the CSI method with L2-norm produces a smooth profile. The results using the weighted L2-norm regularization factor is given in Figure 1e. Obviously, the results are significantly improved. Note that by using a variant of the TV factor, the so-called Cartesian TV [van den Berg et al., 1999], we can obtain nearly perfectly reconstructed concentric squares, but we have then used a priori information that the object is oriented along the Cartesian directions.

[19] Finally, in Figure 2, we show that in this case our choice of the quantity δn2 always satisfies the convexity criterion of (39).

image

Figure 2. Test of the convexity criterion for the concentric squares example: the value of δn2 of (23) actually used (dashed line) and the right-hand side of criterion (39) (solid line) as function of the number of iterations n.

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4.2. Sinus Profile

[20] As the next example we examined a smooth, non-“blocky”, contrast to see whether the weighted L2-norm TV factor, which tends to flatten oscillations, would adversely affect the good reconstruction using the CSI method without any regularization factor. The actual profile was

  • equation image

see Figure 3a. The reconstructions after 128 iterations using the CSI method are shown in Figure 3b and while the one using the CSI method with weighted L2-norm TV factor are shown in Figure 3c. From these results, it is clear that the weighted L2-norm TV factor, rather than affecting the reconstruction adversely, actually improves the quality both damping out unwanted oscillations in the imaginary part and more closely approximating the contrast at the boundary of the test domain.

image

Figure 3. Original profile of sinus profile (a) and the reconstructed profiles from CSI without (b) and with weighted L2-norm TV (c).

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4.3. “Austria” Profile

[21] In our third example, a number of single objects are contained in a test domain with side-length 2 m. These objects consist of two disks and one ring. The disks of radius 0.2 m are centered at (0.3, 0.6) m and (−0.3, 0.6) m. The ring has an exterior radius of 0.6 m and an inner radius of 0.3 m, and is centered at (0, 0.2) m. The electromagnetic case is considered, where the objects have a relative permittivity of 2 (χ = 1). This ö profile is referred to as the ‘Österreich’ profile by Belkebir and Tijhuis [1996]. They used a distorted Born method together with a 'marching-on-in frequency' technique from 100, 200, 300 to 400 MHz. For each frequency, the data were treated separately. The initial guess corresponds to the result of the last iteration of the previously treated frequency. A similar frequency-hopping technique has been applied by Litman et al. [1998], but using a controlled evolution of a level set for binary objects. They have used 64 sources and 65 receivers on a circle of radius 3 m centered at (0,0), while the test domain was discretized into 30 × 30 cells.

[22] We have observed that the present CSI method yields good reconstruction results for this case, using only a single frequency. Here, we suffice with presenting the results using only the highest frequency data, i.e., 400 MHz. To obtain more detail at higher frequencies, we discretize the test domain into 63 × 63 cells, but we take only 48 source/receiver stations. Hence, it seems that we have under-sampled our problem, but the reconstructions show that it does not reduce the quality of reconstruction. We should keep in mind that next to the contrast unknowns, the fields inside the reconstruction domain D, which amount to 63 × 63 × 48, are also unknowns, but we have 63 × 63 × 48 known incident field quantities as an extra information. After generation of synthetic data, 10% random white noise is added. The real and imaginary parts of the original ö profile is presented in Figure 4a. The reconstructed results using the non-regularized CSI method is given in Figure 4b and while the results using the L1-norm, L2-norm and weighted L2-norm regularization factors are given in Figures 4c, 4d, and 4e. Again, we observe the advantage of using the weighted L2-norm factor, while in Figure 5 we observe that in this case our choice of the quantity δn2 always satisfies the convexity criterion of (39).

image

Figure 4. Original profile of Austria profile (a) and the reconstructed profiles from CSI (b), CSI with L1-norm TV (c), CSI with L2-norm TV (d) and CSI with weighted L2-norm TV (e).

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image

Figure 5. Test of the convexity criterion for the “Austria” profile example: the value of δn2 of (23) actually used (dashed line) and the right-hand side of criterion (39) (solid line) as function of the number of iterations n.

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[23] Further we carried out also some tests in order to investigate the noise supressions ability of the multiplicative regularization technique. To that end we present in Figure 6 inversion results when the synthetic data are corrupted by increasing the amount of random additive white noise (using the procedure given in (40)). We observe even when the data are corrupted by 40% noise, an acceptable reconstruction image still can be obtained.

image

Figure 6. Original profile of Austria profile (a) and the reconstructed profiles from CSI method with weighted L2-norm TV factor when the data are corrupted by 10% (b), 20% (c) and 40% (d) random additive white noise.

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4.4. Institute Fresnel Experimental Data

[24] In order to validate our present inversion procedure, we now consider inversion of experimental dataset as measured by the Institute Fresnel, Marseille, France [Belkebir et al., 2000]. The experimental setup consists of a horn transmitting antenna which generates an electromagnetic wavefield at various measurement frequencies and a receiving antenna which is of the same type. The objects are all very large in the direction perpendicular to the plane in which the antennas are located. Therefore a two-dimensional TM electromagnetic inversion model is allowed. In the plane of illumination, we choose a square two-dimensional test domain D containing the objects. The transmitting antenna illuminates the objects from 36 different locations distributed equidistantly around the object. The receiving antenna measures the total and the incident field from 72 different locations distributed equidistantly around the object. Due to physical limitations there is a minimal angle between the transmitting and receiving antenna such that for each illumination angle the field is measured for 49 out of the 72 receiver angles. Then, for each source position we have 49 data points per frequency. We use the Maxwell model for the frequency dependent constitutive parameters of the object. We have extended the CSI method handling frequency diversity data in a similar way as done by Bloemenkamp et al. [2001].

[25] As the first experimental dataset (twodielTM-8f.exp), we consider the one with respect to two dielectric cylinders with circular cross-section of radius 15 mm, see Figure 7a. These cylinders are placed at about 30 mm with respect to the center of the experimental setup. The permittivity of this twin dielectric cylinder is supposed to be ε/ε0 = 3 ± 0.3 (χ = 2 ± 0.3). Eight frequencies are measured from 1–8 GHz.

image

Figure 7. The geometry of two dielectric circular cylinders (a) and the reconstructed profiles from CSI with L1-norm TV (b) and CSI with weighted L2-norm TV (c).

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[26] In the inversion the square test domain D is discretized into 63 × 63 subsquares of 2.7 mm by 2.7 mm. Then, at the highest frequency of 8 GHz, the dimensions of the test domain D is equal to 4.5λ × 4.5λ, where λ is the wavelength in free-space. No a priori information was used that the contrast of the objects is real valued. Although we perform a reconstruction process taking into account all eight frequencies simultaneously, in Figure 7 we present the reconstruction results of the real and the imaginary part of the contrast at highest frequency (the real part is the same for all frequencies). The inversion results after 512 iterations are given in Figure 7b using the CSI method with L1-norm factor and in Figure 7c using the CSI method with weighted L2-norm factor. The top value of the reconstructed profile of the CSI method using both regularization factors approximately amounts to χ = 1.8. Because the results of inversion using the non-regularized CSI and CSI with L2-norm factor method are inferior to the other regularization factors, we do not show them in this paper. Anyway, from Figure 7 we conclude that with the weighted L2-norm factor the circular forms of the twin dielectrics are reconstructed very well.

[27] As second experimental dataset (uTM-shaped.exp), we consider the one with respect to the U-shaped metallic cylinder as given in Figure 8a. Eight frequencies are measured from 2–16 GHz. The same test domain and discretization parameters as in the previous experimental data are used for this dataset. Then, at the highest frequency of 16 GHz, the dimensions of the test domain D is 9λ × 9λ. The inversion results after 512 iterations are given in Figure 8b using the CSI method with L1-norm factor and in Figure 8c using the CSI method with weighted L2-norm factor. The top value of the reconstructed profile of the CSI method using the L1-norm factor approximately amounts to χ = i0.8 while using the weighted L2-norm factor amount to χ = i2.0. The results using the weighted L2-norm factor given in Figure 8c clearly indicate that we are dealing with an object which is purely conductive (real part is small with respect to imaginary part).

image

Figure 8. The geometry of U-shaped metallic cylinder (a) and the reconstructed profiles from CSI with L1-norm TV (b) and CSI with weighted L2-norm TV (c).

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5. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Inversion Problem
  5. 3. Inversion Algorithm
  6. 4. Numerical Test Examples
  7. 5. Conclusions
  8. Acknowledgments
  9. References

[28] The multiplicative regularization reduces substantially the use of the a priori knowledge about the material composition and shape of the unknown object. An artificial tuning process, with a weighting parameter of the regularization to obtain the “cosmetically best” result, seems superfluous. By a slight change of the contrast updating scheme in the multiplicative regularized CSI method, we are able to show that even by multiplying the cost functional with a regularization factor the convexity of the total cost functional is not destroyed. Three different regularization factors, namely the so-called L1-, L2- and weighted L2-norm total variation factors have been compared. Numerical results using synthetic and experimental data show that CSI method using the weighted L2-norm total variation factor produces the ‘best’ reconstructed profiles. In the reconstruction results a clear distinction between the real and imaginary part of the dielectric and conductive scatterers have been obtained.

[29] We have treated in detail the two-dimensional problem. The extension of the presented method to a full 3D electromagnetic problem will be published elsewhere.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Inversion Problem
  5. 3. Inversion Algorithm
  6. 4. Numerical Test Examples
  7. 5. Conclusions
  8. Acknowledgments
  9. References

[30] The authors wish to thank M. Saillard and K. Belkebir for making their experimental data available as an objective test of our inversion algorithm.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Inversion Problem
  5. 3. Inversion Algorithm
  6. 4. Numerical Test Examples
  7. 5. Conclusions
  8. Acknowledgments
  9. References
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