Radio Science

A quasi-Newton reconstruction algorithm for a complex microwave imaging scanner environment

Authors


Abstract

[1] An iterative complex permittivity reconstruction technique for two-dimensional near-field imaging is presented. A particular feature of the algorithm is that it takes into account the complicated environment of a circular 434 MHz microwave imaging scanner, which was developed to conduct biomedical imaging experiments. This is accomplished in a computationally efficient way by means of an embedding technique. The reconstruction technique is further based on a quasi-Newton optimization scheme with approximate line search, in which the Hessian matrix is iteratively updated with the Broyden–Fletcher–Goldfarb–Shanno (BFGS) formula. This way, second derivative information is exploited to a large extent. The technique is illustrated with complex permittivity reconstructions of homogeneous and inhomogeneous lossy dielectric cylinders of moderate and high contrast from simulated data.

1. Introduction

[2] A 2D TM quantitative microwave imaging algorithm for complex permittivity reconstructions of lossy dielectric objects in a 434 MHz scanner is presented. The scanner, which was developed at CNRS/Supélec to conduct biomedical imaging experiments [Geffrin, 1995], consists of a water-filled circular cylindrical metal casing containing an array of transmitting/receiving antennas located on a circle with a slightly smaller diameter. The object under test is placed in the center of this circle. This system can be regarded as a complicated environment, since the reflections from the metal casing, and to a lesser extent the antenna interactions, should be accounted for in a reconstruction algorithm. In the past, quantitative microwave reconstruction algorithms for biomedical imaging generally were configured for the object in a homogeneous medium of infinite extent [e.g., Chew and Wang, 1990; Meaney et al., 1995; Franchois et al., 1998; Tijhuis et al., 2001]. In more recent years, Paulsen and Meaney [1999] and Meaney et al. [1999] have incorporated a nonactive antenna compensation model in their reconstruction algorithm, leading to an improved image quality. In the present paper, we take into account the influence of the metal casing in a computationally efficient way by applying an embedding technique to the forward model, as described by Tijhuis et al. [2000]. With this technique, the forward problem is first solved for the object in a homogeneous medium of infinite extent by means of the efficient conjugate gradient FFT method [Peng and Tijhuis, 1993], and a scattering matrix of the object is computed. Next, the casing is introduced and the scattered field on the measurement circle is expressed in terms of “outgoing” and “source-free” cylindrical wave functions, the coefficients of which are determined by imposing the boundary conditions at the casing and the scattering matrix for the object. Finally, the induced currents on the casing are replaced by equivalent current sources on the measurement circle, leading to an elegant expression for the total field in the object in terms of the homogeneous medium solutions. We have implemented this adapted model into a Broyden–Fletcher–Goldfarb–Shanno (BFGS) quasi-Newton optimization algorithm with line search [Tijhuis et al., 2001; Franchois and Tijhuis, 2001]. This method accumulates approximate second derivative information during the iterations. In our case, the optimization operates on a nonlinear least squares cost function, representing the error between the measured field data and the field computed for a parameterized complex permittivity distribution, possibly augmented with a regularization term. Since we use different meshes for the forward modeling and the parameter reconstruction, the ill conditioning also can be alleviated by choosing a sufficiently coarse mesh for the latter.

[3] Since the beginning of the 1990s, several authors have applied Newton-like nonlinear least squares optimization techniques, such as the linearized Gauss–Newton and Levenberg–Marquardt methods, in their 2D microwave reconstruction algorithms [e.g., Chew and Wang, 1990; Joachimowicz et al., 1991; Franchois and Pichot, 1997]. Others have applied conjugate gradient optimization schemes, either to the forementioned cost function [Harada et al., 1995; Lobel et al., 1997; Rekanos et al., 1999; Tijhuis et al., 2001] or to a modified gradient type of cost function, in which the field is also included as a parameter [Kleinman and van den Berg, 1992; Belkebir et al., 1997]. Still others have tried to tackle the problem of local minima by using global optimization techniques such as simulated annealing [Garnero et al., 1991; Caorsi et al., 1994] or genetic algorithms [Caorsi et al., 2001], often at the expense of considerable computational power. All these methods have yielded results with varying success in terms of dynamic range, spatial resolution, sensitivity, and convergence speed, which can be ascribed to the difficulty of the quantitative microwave reconstruction problem at hand, a (very) nonlinear, ill-conditioned large residual optimization problem, and this seems to hold for the BFGS quasi-Newton algorithm as we have implemented it here as well [Fletcher, 1990].

[4] In section 2, we introduce the configuration and formulate the inverse problem. In section 3, we describe the BFGS quasi-Newton algorithm and in section 4, we expose, for the sake of completeness, the embedding procedure applied to the 434 MHz circular scanner configuration. In section 5, we show reconstruction results from simulated data for some piecewise homogeneous lossy dielectric cylinders of moderate and high contrast in the 434 MHz scanner configuration.

2. Problem Formulation

[5] We consider an inhomogeneous, lossy dielectric cylinder with arbitrary cross-sectional shape and with (unknown) relative complex permittivity εr(ρ), where ρ = (x, y) in Cartesian and ρ = (ρ, φ) in polar coordinates, respectively, in an observation domain equation imageO (Figure 1). The excitation is a time-harmonic electric line source (the factor ejωt will be omitted in the following) on a circular contour ∂equation imageO with radius ρO. The contour ∂equation imageO is located in a region a < ρ < b with a known constant complex permittivity ε1r, which is that of the water filling the casing. The “environment” in equation imageO is a perfectly conducting circular casing with radius b. Bearing in mind the 434 MHz scanner, we conduct a multiincidence experiment at a fixed frequency. Therefore, K antennas are equally spaced on ∂equation imageO. Each antenna is excited in turn and the electric field is measured on all others. In the following, source and receiver positions will be denoted with ρS = (ρO, φs) and ρR = (ρO, φr), respectively. Let us yet specify some of the scanner characteristics to fix the ideas: the complex permittivity of deionized water at, for example, 30°C is ε1r = 76.5 − j1.4 [Stogryn, 1971], the corresponding wavelength is λ1 = 7.9 cm; the diameter of the casing is 59.2 cm or about 7.5 wavelengths; the maximum number of antennas is Kmax = 64; they are spaced apart about λ1/3 and at a distance of about λ1/4 from the casing.

Figure 1.

2D model of the 434 MHz circular scanner configuration.

[6] The field caused by a line source in ρS can be identified as a Green's function, i.e., the solution of the Helmholtz equation

equation image

where k0 is the wave number of free space. For the function G(ρ, ρS) the following contrast-source integral relation can be derived:

equation image

where the bars pertain to a known reference configuration, whose permittivity in equation imageO may be chosen arbitrarily. For the computational reconstruction of the complex permittivity we have to introduce a parameterized configuration

equation image

where equation image(ρ) is the parameterized contrast with respect to the background medium, where {ψα(ρ)} is a finite set of known, real-valued expansion functions with support inside equation imageO, and where the parameters {χα = χ′α + jχ″α} are the unknowns of our problem. In our implementation, we assume the contrasting region to be within a rectangular computational domain equation imageequation imageO with sides Lx and Ly, consisting of N1x × N1y square cells with side h1. The N1 = (N1x + 1) × (N1y + 1) grid points of this mesh are located at ρp,q = xpux + yquy, with xp = ph1Lx/2 for p = 0, … ,N1x and yq = qh1Ly/2 for q = 0, …, N1y. We have used two types of expansion functions: bilinear functions and pulse functions. With the bilinear expansion functions, the parameterized contrast is given by

equation image

where Λ(ξ) is a triangular expansion function:

equation image

[7] With the pulse expansion functions, the functions Λ(ξ) in (4) are replaced by pulse functions Π(ξ):

equation image

[8] The unknowns of our problem {χα} are obtained by minimizing a cost function equation image of the form

equation image

where Gcasscat,m(ρR, ρS) is the known (measured) scattered field at receiver position ρ = ρR for a source at ρ = ρS and equation imagecasscat(ρR, ρS) is the corresponding field in the parameterized configuration given by {χα}. We will omit the tildes over the fields in the following. The subscript “cas” stands for the presence of the casing and the superscript “scat” stands for scattered field, which is defined as the difference between the fields with and without the object in place and the use of which will be motivated in section 4. The factor 1/equation image0 in (7) is a normalization constant

equation image

which alleviates the cost function's dependency on the number of sources/receivers K and on the radius ρO of the contour ∂equation imageO. It is well known that a nonlinear least squares cost function (7) may suffer from the introduction of local minima as well as from ill-conditioning due to the ill-posed nature of the original integral relation. In this paper, we try to alleviate these problems by choosing the cell side h1 sufficiently large. For the case of the bilinear expansion functions, we also have tested the cost function (7) augmented with a regularization term as suggested by Tijhuis et al. [2001]:

equation image

where δ is a small parameter. With this term, variations of the first derivatives in the x and y directions are penalized, with the intention that too small cells combine into larger ones. For piecewise homogeneous contrasts, such as biological objects, one should multiply (9) with the cell side h1 in order to make its relative importance in the cost function independent of h1.

3. Quasi-Newton Reconstruction Algorithm

[9] Newton's method for (local) optimization, which approximates a nonlinear function with a quadratic model based on the function's first and second derivatives at the current iterate and which takes the stationary point (minimum) of this model as the next iterate, has the attractive fundamental property of superlinear convergence if the initial guess is close enough to the solution [Fletcher, 1990]. However, when starting further away from the solution, the Newton correction may lead to an increase in the cost function, if at its stationary point the model is no longer a good approximation to the function or if its Hessian matrix is not positive definite. Several strategies to enlarge the convergence domain of Newton's method have been proposed in the literature [Dennis and Schnabel, 1983; Fletcher, 1990]. One such strategy is the trust-region approach, which constrains the minimization of the model to a region where the model is assumed to be a valid approximation to the function. Another, more classical, strategy is to incorporate a line search along the Newton step direction, which is the approach we have adopted in this paper. Let us denote by χ the real vector of contrast parameters

equation image

in which the real and imaginary parts of {χα} are ordered in the first and second half, respectively, and where T stands for transpose. The shorthand notation [χ′α, χ″α]T will be used in the following. This contrast vector is updated from iterate n to iterate n + 1 with a correction Δχ as follows:

equation image
equation image
equation image

[10] In (12), p is the Newton step, which is used as a search direction in (13). The positive real number ξ is chosen by means of a line search, such that it approximately, in order to limit the number of function evaluations, minimizes equation image(χ); g is the gradient of the cost function

equation image

and H is the Hessian matrix

equation image

[11] It is well known that for the special cases of nonlinear least squares or sums of squares, these first and second derivatives take particular forms. This holds for the cost function (7) and for the regularization term (9) as well. Since Gcasscat(ρ, ρ′) is analytic in {χα}, the first derivatives of (7) with respect to χ*α, for example, where * stands for complex conjugate, are given by

equation image

and the gradient can then be written as

equation image

[12] where equation image and equation image stand for the real and imaginary parts, respectively. The explicit availability of at least the gradient is much appreciated when applying a Newton-like method to a problem involving many parameters. From (2), it is possible to derive a closed form expression for the gradient [Roger, 1981; Chew and Wang, 1990; Franchois and Pichot, 1997]. We use the following analytic expression for ∂Gcasscat(ρR, ρS)/∂χα [Tijhuis et al., 2001]:

equation image

[13] If we introduce a complex Jacobian matrix J, such that ∂Gcasscat(ρR, ρS)/∂χα is an element of the αth column of J and if we order the components Gcasscat(ρR, ρS) and Gcasscat,m(ρR, ρS) in column vectors G and Gm, respectively, we also obtain the following expression for the gradient:

equation image

where stands for complex conjugate transpose. The Hessian matrix can be written as

equation image

where the elements of JJ contain products of only first derivatives of Gcasscat(ρR, ρS) and Gcasscat*(ρR, ρS) and where the elements of B are sums of products of the residuals with second derivatives of Gcasscat(ρR, ρS):

equation image

[14] In our application, the computation of the second derivatives ∂2Gcasscat(ρR, ρS)/∂χα∂χβ is expensive and therefore avoided. One option is to neglect the matrix B in (20), which leads to the so-called Gauss–Newton method, the Gauss–Newton step then being given by (12) with this modified Hessian matrix. This method may fail for very nonlinear or large residual problems. When augmenting the Gauss–Newton Hessian matrix with δI, where δ is a positive number and I the identity matrix, one obtains the more robust Levenberg–Marquardt method, which can be cast into the trust-region approach. Both methods have been used in microwave reconstruction techniques in the past [e.g., Chew and Wang, 1990; Joachimowicz et al., 1991; Franchois and Pichot, 1997]. In the present paper, we do not compute the Hessian matrix (20) explicitly, but we approximate it based on gradient information gained in previous iterations. This approach, which attempts to account for the second derivatives in B as well, is known as the quasi-Newton method, which, by the way, is not restricted to nonlinear least squares problems. A number of formulas for updating H or its inverse H−1 are proposed in the literature [Fletcher, 1990]. We use the BFGS formula for updating H−1:

equation image

where δn = Δχn and where γn = gn+1gn. With formula (22), the initial matrix is an arbitrary positive definite matrix. The successive approximations HBFGS,n−1 then are positive definite and eventually become close approximations to Hn−1. The advantage of updating HBFGS−1 is that we do not need to perform a matrix inversion to obtain p in (12). In this paper, we have chosen H1−1 = I, hence the initial search direction corresponds to the steepest descent direction; the Gauss–Newton approximation might be worth trying here.

[15] For the line search, which is an important aspect in the performance of the optimization algorithm, we have adopted the algorithm described by Fletcher [1990], the main principles of which we briefly expose here. The goal of the line search is to find a value for ξn in (13), which approximately minimizes

equation image

along the search direction pn. The value ξ = 0 in (23) corresponds to the current iterate χn. A step ξ is found to be acceptable if it satisfies the following two conditions:

equation image
equation image

where the derivative of equation image with respect to ξ, equation image′(ξ) = g. p, is also referred to as the directional gradient. The sufficient decrease condition (24) requires that the function value equation image(ξ) be lower than the η-line, where 0 < η < 0.5. The second condition (25) expresses that the derivative equation image′(ξ) should be smaller than a fraction of the derivative in ξ = 0, in absolute values, where η < v < 1. In order to limit the number of function and gradient evaluations, hence the number of steps in the line search, we choose v not too small, for example v = 0.5. The algorithm then proceeds as follows. In the bracketing phase, increasingly large steps ξ are taken in order to find a bracket, which contains the minimum. If for a step during this phase, both conditions (24) and (25) are satisfied, the line search is terminated. In the sectioning phase, the size of the bracket is gradually reduced until an acceptable step is found. For choosing new steps in the bracketing and sectioning phases, we take the minimum of a cubic extrapolation and interpolation polynomial, respectively, fitted to the available function and derivative information.

4. The Forward Problem: Embedding Approach

[16] In this section we treat the computation of the scattered field Gcasscat(ρR, ρS) on the measurement circle and of the total field Gcastot(ρR, ρS) in the computational domain equation image, which are needed to compute the cost function (7), the gradient (19) in combination with (18) and the Hessian update in each iteration of the BFGS quasi-Newton algorithm. For the configuration of an object in a homogeneous environment of infinite extent, it is well known that applying the conjugate gradient FFT method combined with the marching-on-in-angle technique [Peng and Tijhuis, 1993; Tijhuis et al., 1997] leads to a highly efficient forward solver. Due to the presence of the casing in our configuration, the convolution symmetry of the Green's function is broken, and operator products can no longer be evaluated with FFT operations. We circumvent this problem by means of an embedding technique [Tijhuis et al., 2000], the four main steps of which are described below.

4.1. The Forward Problem in a Homogeneous Environment

[17] The first step in the embedding procedure consists of solving the forward problem for the object in a homogeneous environment ε1r of infinite extent. We use the method described by Peng and Tijhuis [1993], which introduces a space discretization of the contrast-source integral equation

equation image

where G1 is the Green's function of homogeneous space ε1r, by isolating the logarithmically singular behavior of G1(k1ρρ′∣) as ∣ρρ′∣ ↓ 0 into a second integral over equation image and by approximating suitable parts of the resulting integrands by piecewise-linear approximations. Since the convolutional symmetry of (26) is preserved in the discretized equations, they can be solved with the conjugate gradient FFT method. Let us mention that we use here a finer mesh than with the contrast parameterization (3) in the optimization scheme. The computational domain equation image is now composed of N2x × N2y square cells with side h2 = h1/n, with n an integer depending on the accuracy desired for Ghomtot(ρ, ρS) and Ghomscat (ρR, ρS), the accuracy being of the order equation image(h22). Since these fields are to be successively computed for varying source positions ρS, we reduce the number of conjugate gradient iterations with the marching-on-in-angle technique [Peng and Tijhuis, 1993; Tijhuis et al., 1997] by taking for the initial guess of Ghomtot(ρ, ρS) an optimized linear combination of the field solutions corresponding to the three previous source positions.

4.2. The Scattering Matrix of the Object

[18] The second step in the embedding procedure is the computation of the scattering matrix of the object in a homogeneous environment ε1r. Let us first define the scattering operator. For a general excitation outside equation imageO, we can write the field in a < ρ < ρO in spectral form as:

equation image

with k1 = k0equation image. The coefficients {Am} represent the “source-free” incident field and the coefficients {Bm} the “outgoing” field scattered by the object. Since the object is linearly reacting, these coefficients are linearly related by a scattering operator:

equation image

[19] The scattering operator can be obtained from the Green's function Ghomscat(ρR, ρS) for the object in a homogeneous environment. In this case, the incident field is the field caused by a line source in a homogeneous background. Identifying the coefficients {Am} and using the definition (28) then leads to the following expression for the scattered field:

equation image

[20] The scattering operator S thus can be obtained from Ghomscat(ρR, ρS) by applying Fourier transformations with respect to φR and φS. In practice, we apply discrete FFTs, hence we choose M equally spaced source/receiver positions and we truncate the summations in (29) to M terms:

equation image

with r, s = 1, …, M and where M should be large enough to yield a sufficiently accurate truncated scattering matrix, or Sm+lM,m′+lMHm+lM(2)(k1ρO)Hm′+lM(2)(k1ρO) ≪ Sm,mHm(2)(k1ρO)Hm(2)(k1ρO) for ± l′ = 1, …, ∞. For the objects considered in this paper the choice M = Kmax, for example, proved to be more than sufficient. The accuracy of the scattering matrix then primarily depends on that of Ghomscat(ρR, ρS). Note that K always is an even number in the 434 MHz scanner configuration.

4.3. The Field on the Measurement Circle

[21] The computation of the field on the measurement circle in presence of the casing is the next step in the embedding procedure. The total field in the region a < ρ < b can be written in spectral form as:

equation image

where ρ< = min {ρ, ρO} and ρ> = max {ρ, ρO}. The first sum in (31) consists of the line source contribution in homogeneous space (indicated with number 1 on Figure 1) and of the reflection from the empty (water-filled) casing in absence of the object (nb. 2 on Figure 1). Rm are the reflection coefficients of the casing, relating “source-free” constituents reflected by the casing to “outgoing” constituents incident on the casing, in a way comparable to (28). For a perfectly conducting circular casing, they are obtained by imposing the boundary condition Ez = 0 at ρ = b:

equation image

[22] The first sum in (31) does not depend on the object, hence we will consider it as the incident field Gcasinc(ρ, ρS). The second sum, with unknown coefficients {CmS)} and {DmS)}, is due to the insertion of the object and depends on both the object and the casing (nbs. 3 and 4 on Figure 1). We will consider it as the scattered field Gcasscat(ρ, ρS). The summation corresponding to the empty-casing reflection converges much slower than this second sum, hence it is advantageous not to include the empty-casing reflection implicitly into the scattered field. The coefficients {CmS)} and {DmS)} are computed as follows. For ρO < ρ < b, we treat the situation as reflection by the casing. This leads, with the aid of Rm, to an expression for the coefficients {CmS)} in terms of {DmS)}. For a < ρ < ρO, we treat the situation as scattering by the dielectric cylinder. This leads, with the use of S, to an expression for the coefficients {DmS)} in terms of {CmS)}. Combination of both expressions gives a set of M linear equations in {CmS)}:

equation image

which we solve with a direct inversion for varying φS, i.e., for multiple right-hand sides. The scattered field on the measurement circle is then given by

equation image

and is used to compute the cost function and the gradient. In an imaging experiment, the measured known data Gcasscat(ρ, ρS) are obtained by subtraction of measurements performed with and without the object in place, which furthermore presents the advantage of eliminating certain systematic measurement errors.

4.4. The Field in the Computational Domain

[23] The final step in the embedding procedure consists of computing the total field Gcastot(ρ, ρS) on the computational domain equation image. In the region a < ρ < b, the “source-free” constituents of Gcastot(ρ, ρS) originate from the current at the line source ρS and from the induced surface current on the casing. Alternatively, we can treat the induced constituents, which correspond to the second and third terms of (31), as originating from an equivalent surface current on ∂equation imageO. In our discretized approach, this equivalent surface current is represented as a set of L equally spaced equivalent line sources in ρQ on ∂equation imageO with unknown complex amplitudes. The “source-free” component of the scattered field yields the following expression for the complex amplitude of the equivalent line source at ρ = ρQ:

equation image

and the reflection of the empty casing yields the complex amplitude:

equation image

The coefficients wcas(ρQ, ρS) are computed once in the beginning of the program. The total field Gcastot(ρ, ρS) on the (N2x + 1) × (N2y + 1) = N2 computational grid is then expressed as a weighted combination of field solutions Ghomtot(ρ, ρQ) computed in the first step of the embedding procedure:

equation image

[24] In this paper we have chosen the number of equivalent line sources L equal to the number of transmitters K. The accuracy of the embedding procedure has been checked against analytical solutions and is also of the order equation image(h22), if M and L are well chosen. The accuracy of the elements (18) of the Jacobian matrix J also depends on the cell size h2. Summations containing complex exponentials, such as (35), (36), and (34) and the right-hand side in (33) are computed by means of discrete FFTs. If we choose N2 > K = L = M, then the most time consuming step in the embedding procedure is the solution of (26) in step 1, with equation image(KN2lnN2) operations, followed by the summations (37) in step 4 and the computation of the Jacobian matrix, both with equation image(K2N2) operations. The computation of the gradient (19) requires equation image(K2N1) operations, and that of the BFGS Hessian update (22)equation image(N12) operations. In our numerical code, we have used the public domain FORTRAN software packages LAPACK [Anderson et al., 1999], for the linear system solutions, AMOS [Amos, 1995], for the Bessel function computations, and NMS [Kahaner et al., 1998], for the 2D discrete FFTs.

5. Numerical Results

[25] We show some examples of reconstructions for three lossy dielectric objects. In all examples, the number of sources/receivers is K = Kmax = 64, the scattered field data are simulated and free of measurement noise, and the initial contrast is chosen equal to zero. All computations have been performed on a SUN ULTRA HPC 4000 workstation.

[26] The first object is a homogeneous circular cylinder with a diameter of 17.6 cm (≈2λ1) and a relative permittivity εr = 65 − j2, centered on the axis of the scanner and immersed in (low-loss) water with permittivity ε1r = 76.3 − j0.1. This corresponds to a relatively low contrast in the real part (Figure 2a). For this configuration we were able to generate the scattered field data analytically. For the reconstruction, we have chosen a square computational domain equation image with side 24 cm. We parameterized the contrast on a grid of 8 × 8 cells with bilinear expansion functions. For the forward problem solution, this grid was further subdivided to 128 × 128 cells. The reconstruction result after 36 iterations is shown in Figure 2b and a slice along the axis y = 0 is shown in Figure 2c, where a reconstruction obtained after 40 iterations with pulse expansion functions is shown as well. The cost function was then about 5 × 10−4.

Figure 2.

Real part (left) and imaginary part (right) of the contrast: (a) exact; (b) reconstruction after 36 iterations using a bilinear expansion function parameterization; (c) slices through y = 0 of the exact contrast (a) (- - -), of the reconstruction (b) (–·–), and of a reconstruction using a pulse expansion function parameterization (—).

[27] The second object differs from the first object in that it contains a decentered circular hole with diameter 5 cm, with a complex permittivity equal to that of the surrounding water (Figure 3a). For this inhomogeneous cylinder, the scattered field data were generated by using a fine mesh of 256 × 256 cells. We have performed reconstructions with various contrast parameterizations. In Figure 3b, a reconstruction is shown after 40 iterations using a grid of 16 × 16 cells with bilinear expansion functions. The use of smaller cells leads to more oscillations in the reconstruction. In Figure 3c we repeated this reconstruction with regularization (9), with δ = 10−6, yielding a smoothed reconstruction, as can be seen as well in Figure 3e, which shows slices along the axis y = 0 of the various reconstructions, including a reconstruction after 30 iterations on a grid of 8 × 8 cells. In Figure 3d, a reconstruction is shown after 42 iterations using a grid of 8 × 8 cells with pulse expansion functions with side 3 cm (≈λ1/2.5). For the reconstruction results shown here, the final value of the cost function varied between 6 × 10−4 and 2 × 10−3. Reconstructions in water with higher losses (ε1r = 76.3 − j3.9) led to comparable results.

Figure 3.

Real part (left) and imaginary part (right) of the contrast: (a) exact; (b) reconstruction after 40 iterations using a bilinear expansion function parameterization; (c) reconstruction after 50 iterations with regularization, δ = 10−6; (d) reconstruction after 42 iterations using a pulse expansion function parameterization; (e) slices through y = 0 of the exact contrast (a) (- - -), of the reconstruction (b) (–·–), of the reconstruction (c) (…), of the reconstruction (d) (—), and of a reconstruction after 30 iterations using a bilinear expansion function parameterization (—).

Figure 3.

(continued)

Figure 3a.
Figure 3.

(continued)

Figure 3b.

[28] The third object is a homogeneous circular cylinder of muscle with a diameter of 8.8 cm (≈λ1) and a relative permittivity εr = 54.2 − j38.4, immersed in water with permittivity ε1r = 76.3 − j3.9. This object presents a high contrast in the imaginary part (Figure 4a). We were again able to generate the scattered field data analytically. The side of the computational domain equation image was now 12 cm. We parameterized the contrast on a grid of 8 × 8 cells with pulse expansion functions with side 1.5 cm (≈λ1/5). For the forward problem solution, this grid was further subdivided to 64 × 64 cells. The result after 41 iterations is shown in Figure 4b and a slice along the axis y = 0 is shown in Figure 4c.

Figure 4.

Real part (left) and imaginary part (right) of the contrast: (a) exact; (b) reconstruction after 41 iterations using a pulse expansion function parameterization; (c) slices through y = 0 of the exact contrast (a) (- - -) and of the reconstruction (b) (—).

[29] The normalized least squares error equation image(χ) as a function of the number of forward problem solutions is shown with dots in Figure 5, for the reconstruction of Figure 3b. The circles enclosing these dots indicate when the BFGS Hessian matrix has been updated and correspond to the successive quasi-Newton iterations. The dots without circle thus correspond to the line search iterations. It can be seen that the error decreases like a staircase with stagnations during several iterations. These stagnations coincide with dips in the directional gradient, which is indicated with crosses and which shows an oscillatory behavior. It was also observed that for too small values of the directional gradient, the line search algorithm failed due to numerical rounding errors. The condition number of the Hessian matrix, multiplied by a factor 10−9 and indicated in squares, gradually increases with the iterations. This kind of behavior of the BFGS quasi-Newton algorithm was observed in all the presented examples. Let us mention that proceeding with the iterations beyond the reconstruction results shown here, led to a further reduction of the cost function but not to an improvement of the reconstruction. This may perhaps be explained by the fact that we try to fit a relatively coarse parameterization of the contrast to independent data. Issues for further investigation thus are the selection of an adequate stopping criterion and the acceleration of the convergence speed.

Figure 5.

For the reconstruction of Figure 3b as a function of the number of forward problem solutions: (·) the cost function equation image(χ), BFGS updates being marked with (⊙); (×) the directional gradient; (□) the condition number of the Hessian matrix ×10−9.

6. Conclusion

[30] In this paper, we have presented a quantitative microwave reconstruction algorithm which accounts for the complicated environment of a 434 MHz biomedical imaging scanner by means of an embedding technique. The reconstruction algorithm is based on a classical BFGS quasi-Newton optimization technique with approximate line search. Some examples of rather satisfactory complex permittivity reconstructions have been shown for homogeneous and inhomogeneous lossy dielectric objects of moderate and high contrast.

Acknowledgments

[31] The authors gratefully acknowledge Z. Q. Peng for the forward scattering algorithm implemented in earlier work [Peng and Tijhuis, 1993], E. S. A. M. Lepelaars and A. C. S. Litman for making available a BFGS code, and J.-M. Geffrin for valuable discussions and for his interest in applying the presented reconstruction technique to his 434 MHz microwave scanner.

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