1108
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 18, NO. 11, NOVEMBER 1999
Correspondence Microwave Imaging Using the Finite-Element Method and a Sensitivity Analysis Approach Ioannis T. Rekanos, Stavros M. Panas, and Theodoros D. Tsiboukis*
Abstract—A method for reconstructing the constitutive parameters of two-dimensional (2-D) penetrable scatterers from scattered field measurements is presented. This method is based on the differential formulation of the forward scattering problem, which is solved by applying the finiteelement method (FEM). Given a set of scattered field measurements, the objective is to minimize a cost function which consists of two terms. The first is the standard error term, which is related to the measurements and their estimates, while the second term, which is related to the Tikhonov regularization, is used to heal the ill posedness of the inverse problem. The iterative Polak–Ribi`ere nonlinear conjugate gradient algorithm is applied to the minimization of the cost function. During each iteration of the algorithm, the direction of correction is computed by using a sensitivity analysis approach, which is carried out by an elaborate finiteelement scheme. The adoption of the finite-element method results in sparse systems of equations, while the computational burden is further reduced by applying the adjoint state vector methodology. Finally, a microwave medical imaging application, which is related to the detection of proliferated bone marrow, is examined, while the robustness of the proposed technique in the presence of noise and for different regularization levels is investigated. Index Terms—Inverse scattering, microwave imaging, sensitivity analysis, regularization.
I. INTRODUCTION Medical imaging techniques such as X rays, ultrasounds, positron emission, nuclear magnetic resonance and microwave imaging are undoubtedly powerful tools for detecting and monitoring injuries and diseases of biological tissues. Each one of these imaging techniques exploits possible differences between the values of various properties of tissues; for example, X-ray tomography is based on density differentials. Microwave imaging seems to be very attractive in cases where the differences in the permittivity and conductivity of adjacent tissues prevail [1]–[4]: the detection of proliferated bone marrow belongs to this class of problems [5]. The basic objective of microwave imaging is to reconstruct the spatial distributions of permittivity and conductivity of the body that is investigated. The body is illuminated by electromagnetic waves from various directions at microwave frequencies. Usually, these waves are generated by electric or magnetic dipoles placed around the domain of investigation, or they are synthetic plane waves. Microwave radiations provide a rather safe exploration of tissues, because they are not ionizing. The reconstruction is based on scattered electromagnetic field measurements which are produced by the interaction between the incident wave and the inhomogeneity Manuscript received March 11, 1998; revised September 1, 1999. This work was supported in part by the Greek General Secretariat of Research and Technology under the PENED’94 Research Project. The Associate Editor responsible for coordinating the review of this paper and recommending its publication was V. Johnson. Asterisk indicates corresponding author. I. T. Rekanos, S. M. Panas, and *T. D. Tsiboukis are with the Division of Telecommunications, Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, Thessaloniki 54006, Greece (e-mail:
[email protected];
[email protected]). Publisher Item Identifier S 0278-0062(99)09624-X.
introduced by the presence of the body. Since scattered field is related to the constitutive parameters of the object, the spatial distributions of the material properties of the body can be reconstructed by postprocessing scattered data. Although this technique can lead to valuable imaging results, it introduces some difficulties. First, its resolution is of the order of the wavelength inside the body, while high nonlinearities and difficulties in the reconstruction process arise by increasing the excitation frequency. Second, the energy transmitted into the body should be maximized by selecting a host medium that has a permittivity value of the same order with the body. Third, in case of high dielectric contrast of tissues, imaging algorithms require a great deal of computation time and may diverge. The development of microwave imaging methods passes through the description of the forward scattering problem. This problem is usually described by the electric field volume integral formulation, resulting in the Lippmann–Schwinger equation [6]. In this case, the inverse scattering problem suffers from nonlinearity and ill posedness in the Hadamard’s concept [7] where the existence, uniqueness, and stability of the solution are not ensured. Much effort has been made toward the investigation of existence and uniqueness conditions, for two-dimensional (2-D) [8] and 3-D (3-D) [9] inverse scattering. Furthermore, the instability of the inverse scattering solution, owing to the compactness of the forward scattering operator, is treated by means of regularization techniques [10] that tend to reduce the dimension of the space of solution. Moreover, any a priori knowledge about the geometric characteristics and the range of values of the permittivity and conductivity can also play the role of regularization. The nonlinearity is bypassed in various ways. Linearization techniques, which are closely related to the conventional diffraction tomography and are based on the Born and the Rytov approximations [4], [11], give qualitative results. Unfortunately, these results are inadequate when strong scatterers are examined. Spatial iterative algorithms such as the Newton–Kantorovich or its equivalent, the distorted Born iterative method, result in quantitative scatterer reconstruction [12]–[14]. Other techniques are associated with an iterative minimization of the distance between measured and estimated values of the scattered field data [15]–[17]. Finally, nonlinear quadratic inversion techniques are very promising [18]. Unlike the previous inversion approaches, which are based on the integral formulation of the forward problem and the method of moments (MoM), the inverse scattering technique proposed in this paper is oriented to the differential formulation of the problem [19]. The differential formulation allows the application of the finiteelement method (FEM) to the solution of the forward scattering problem. Two basic advantages arise from the finite-element analysis. First, in the 2-D case examined here, the FEM fulfills the preservation of the continuity of the tangential component of the unknown field along the edges of the elements. This physical requirement is not satisfied by the MoM. Thus, the field inside the scatterer domain is modeled more precisely by the FEM compared to the MoM. The second advantage is that the FEM results in the solution of sparse systems of equations, whereas the MoM is associated with dense systems. Furthermore, the sparsity achieved by the FEM is preserved by applying a local-type absorbing boundary condition [20]. The objective of the proposed method is to estimate the spatial distribution of the constitutive parameters (permittivity and conduc-
0278–0062/99$10.00 1999 IEEE
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 18, NO. 11, NOVEMBER 1999
tivity) of a scatterer by minimizing an appropriate total error function that consists of two terms: a standard error term and a Tikhonov regularization term. The first term is related to the difference between the scattered far field measurements and their estimated values, which are computed using the FEM and the Helmholtz–Kirchhoff integral theorem (HKIT) [20]. The regularization term is independent of the measurements and is related only to the shape of the function that describes the constitutive parameters of the scatterer. This can be viewed as a penalty term that demands the smoothness of the reconstructed profile. During the inversion, the relative complex permittivity distribution is first initialized and afterwards is iteratively updated until the total error function reaches a lower bound. The correction of the scatterer constitutive parameters is made towards a direction, which is obtained by the Polak–Ribi`ere nonlinear conjugate gradient algorithm [21]. The gradient of the error function is computed by a sensitivity analysis approach [22], which is performed by means of an elaborate FEM scheme and the assistance of the adjoint state vector methodology [23]. This methodology prevents the extensive computation of the Jacobian matrix of the field solution with respect to the unknown scatterer profile. Finally, the proposed method is applied to the reconstruction of 2-D scatterers’ profiles. A very interesting application which is related to the detection of proliferated bone marrow in the case of leukemia is presented, while the robustness of the method in the presence of noise is examined. In addition, this paper investigates the case of estimating the location and shape of a scatterer that has known constitutive parameters [24], [25]. The paper is organized as follows. In Section II, the mathematical formulation for the solution of the forward 2-D scattering problem is presented. In Section III, the proposed inverse scattering technique is described. Numerical results are presented and discussed in Section IV. Finally, conclusions are drawn in Section V. II. THE FORWARD SCATTERING PROBLEM Let us consider an infinitely long nonmagnetic isotropic inhomogeneous lossy cylindrical scatterer of bounded arbitrary cross section. The scatterer is embedded in a lossy homogeneous surrounding medium and is parallel to the z axis of a Cartesian system of coordinates (Fig. 1). It is assumed that the constitutive parameters of the scatterer do not vary along the z axis. Thus, the properties of the scatterer are represented by the relative complex permittivity (RCP)
"~r (x; y ) = ["(x; y) 0 j(x; y)=!]="~s
(1)
where j 2 = 01; " and are the scatterer dielectric permittivity and conductivity, respectively, ! is the excitation frequency, "~s = "s 0 js =! is the constant complex permittivity of the surrounding medium, and (x; y ) denotes a point that lies within the scatterer domain, D. Finally, the scatterer is illuminated by a single frequency TM incident field,having the electric field polarized in the z direction. Under these assumptions, the scattered field, E , is derived from the solution of the Helmholtz differential equation r2 E (x; y) + k2 "~ E = 0k2 (~" 0 1)Einc (2) s r
s
r
where E inc represents the incident electric field, and ks = (! 2 0 "s 0 j!0 s )1=2 is the complex wave number [19]. The solution of (2) is obtained by applying the FEM to the Galerkin formulation [20], given by
rE 0 1 rE 0 ks2 "~r E 0 E d
= ks2
(~"r 0 1)E 0 E inc d +
@
E 0 @E=@n dl
(3)
1109
Fig. 1. Geometric configuration of the 2-D scattering problem.
where E 0 is the weighting function, is the 2-D area that contains the scatterer domain (D ), and n denotes the outward pointing direction normal to the boundary @ . If @ is circular, then n coincides with the radial direction. In this case, the derivative of the scattered field with respect to the radial coordinate r, which is associated with the closed line integral in (3), is approximated by the scalar second-order absorbing boundary condition (ABC) [26] (4) @E=@r = a(r)E + b(r)@ 2 E=@ 2 where
a(r) = 0jks 0 (2r)01 + (8jks r2 )01 + 8ks2 r3 01 b(r) = (2jks r2 )01 + 2ks2 r3 01
(5) (6)
and denotes the azimuthal coordinate. Substituting (4) in (3) and applying the FEM, we get the system of equations S(")E = b("Einc ) (7) where E is the vector of the scattered field values at the nodes of the mesh, the vector Einc represents the corresponding incident field values, S and b are the matrices obtained by the FEM, and " is a complex vector that represents the unknown constitutive parameters (" = [~"r1 "~r2 1 1 1 "~rM ]T ). The dimension M of " is equal to the number of finite elements used for the discretization of the scatterer domain, while its components are equal to the RCP of the corresponding elements. It is obvious that both S and b depend on ", while b depends also on the incident field. To reduce the memory demands and the computation time, @ is placed as close as possible to the scatterer domain. The distance of @ from the scatterer surface usually varies from 0:2 to 0:4 where is the wavelength of the incident field. It should be noted that the application of FEM and the adoption of this type of ABC produces a sparse matrix S, a result that is advantageous from the computational point of view. After the computation of the scattered field in the bounded region
, the far field at an arbitrary point, r(x; y) outside is evaluated by means of the HKIT given by
E (r) =
0
[E (r0 )@G(r; r0 )=@n0 0 G(r; r0 )@E (r0 )=@n0 ] dl0 :
(8)
In (8), 00 is any closed curve that lies entirely in the domain (it could coincide with the boundary @ ), r0 is a point on 00 ; n0 denotes the outward pointing direction normal to 00 , and G(r; r0 ) is the free space Green’s function for 2-D problems. The latter is given by G(r; r0 ) = 00:25jH0(2) (ks jr 0 r0 j) (9)
(2) where H0 is the Hankel function of the second kind and zero order.
1110
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 18, NO. 11, NOVEMBER 1999
Finally, if we approximate the line integral (8) by a numerical integration technique, the far field F at a set of K distinct points is given by the matrix equation
E
EF =
1 1 1 EKF T 1 1 1 EK ]T = QE
E1F E2F = Q[E1 E2
where E1 ; E2 ; . . . ; EK are the nodes of the FEM mesh and on the Green’s function and its on the field values at the nodes matrix is sparse.
(10)
values of the scattered field at the
Q is a K 2 N matrixFwhich depends derivative. Since E depends only 0
Q
that are related to the curve
0 , the
III. THE INVERSION The proposed inverse scattering technique is based on scattered far field measurements that are obtained at various positions around the scatterer domain which are illuminated from various directions of incidence. Thus, we obtain a set of vectors
fi = [fi
fi2
1
1 1 1 fiK ]T ;
(i = 1; 2; . . . ; I)
E
= E
E
F i1
F i2
111 E
F T iK
:
(12)
A. Description of the Inverse Scattering Method The scatterer is reconstructed by minimizing the total error function given by
F (") = FS (") + FR (") =
I i=1
kfi k0 kfi 0 EiF (")k + k="k : 2
2
In (13) FS is the standard error term, FR is the Tikhonov’s regularization term, = is a real matrix that approximates a spatial differential operator, and is a positive scalar parameter called regularization factor [10]. Substituting (10) in (13), the total error is written as
F (") =
i=1
wi Fi (") + k="k
2
(14)
2 where wi = k i k02 and Fi (") = k i 0 i (")k . The minimization of the first term of (14) satisfies the obvious demand that the scattered field produced by an estimate of the scatterer profile should match, as closely as possible, the measurement data. In addition, the error terms of each individual incidence Fi are weighted by wi in order to normalize error terms that are related to different incidences. On the other hand, the regularization term stabilizes the solution and forces it to be smooth. It actually penalizes the RCP distribution whenever it tends to be oscillatory. These oscillations arise due to the presence of noise in the measurements. In this work, the differential operator that we consider is the gradient of the RCP distribution (first-order Tikhonov’s regularization scheme) [23], which is approximated by first differences. The factor tunes the influence of the regularization to the whole minimization process. It is obvious that the selection of is closely related to the level of noise. In general, the higher the noise level is, the higher the value of to be selected. However, a high regularization factor decreases the spatial resolution of the RCP. As a consequence, the choice of the regularization factor should insure a convenient stability and spatial resolution compromise. Although some very interesting attempts for selecting optimum values of the
f
f QE
(15)
d
where the superscripts denote the iterations, is the direction of correction, and g is the size of the movement along . Since (14) is nonlinear with respect to ", the Polak–Ribi`ere nonlinear conjugate gradient algorithm has been implemented [21]. The vector is directly related to the gradient of (14) with respect to the vectors "R and "I that denote the real and the imaginary part of ", respectively. For convenience, this gradient is represented by the notation
d
d
D" F (") =
@ + j @ F ("): @"R @"I
(16)
We will refer to (16) as the complex gradient of the real-range function F ("), with respect to its complex variable vector, ". Following this notation, the complex gradient of the regularization term in (14) with respect to " is given by
D" k="k2 = 2=T =":
(17)
The calculation of the complex gradient of the standard error term needs more discussion. We will present the procedure of evaluating the complex gradient of Fi ("). The same procedure is valid for every incidence. It can be shown that the complex gradient of Fi (") fulfills the property
(D" Fi )3 = ZiT DE Fi 3
2
(13)
I
"(n+1) = "(n) + g(n) d(n)
(11)
where fik denotes the scattered field measurement at the kth position for the ith incidence. On the other hand, if we have an estimate " of the scatterer profile, the scattered far field can be evaluated via the application of the FEM and the HKIT, resulting in a, similar to (11), set of estimates F i
regularization factor in zero-order Tikhonov’s regularization schemes (= is the identity matrix) have been carried out [14], [28], it is recognized as a very difficult problem from a mathematical point of view and it is usually dealt with by numerical experimentation [17], [27]. A gradient optimization method is applied to the minimization of (14). The distribution of the RCP of the scatterer domain is iteratively updated according to the line search scheme
(18)
where the asterisk denotes the complex conjugate and 2 M Jacobian matrix
N
Zi = [@Ei =@" 1
@Ei2 =@"
1 1 1 @EiN =@"]T
Zi
is the (19)
which contains the partial derivatives of the field solution with respect to ". In addition, the complex gradient of Fi (") with respect to the FEM solution i is given by
E
DE Fi = 02QH (fi 0 QEi )
(20)
where the superscript H denotes the Hermitian transpose operator. Each column of i represents the sensitivity of the field solution with respect to the corresponding unknown RCP in the scatterer domain. We can evaluate the columns of i by differentiating the FEM system of equations (7) [22]. Hence, we obtain M systems of equations
Z
Z
SZim = @ bi =@~"rm 0 (@ S=@~"rm )Ei ; 1 m M (21) where Zim represents the mth column of Zi , @ bi =@ "~rm is an N 2 1 and @ S=@ "~rm a symmetric N 2 N matrix that have as components the partial derivatives of the components of bi and S with respect to "~rm . The systems of (21) are analogous to (7) because the matrix S is the same. This property is exploited by decomposing S only once during each iteration of the Polak–Ribi`ere algorithm. It is clear that the computation of Zi is a time consuming task; it requires the solution of M systems. By applying the adjoint state vector methodology [23], we can overcome this difficulty. According to this method, we define the adjoint state vector, i , which is given by the solution of the system
v
Svi = Ci
(22)
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 18, NO. 11, NOVEMBER 1999
1111
Fig. 2. The case of the lossy discontinuous scatterer profile. RMSE of the reconstruction after 100 iterations for various values of the SNR of the measurements and different values of the regularization factor .
where
C
T i =
02(fi 0 QE Q: H i)
(23)
If we solve (22) and combine (18), (20) and (23), we obtain the complex gradient from
D" Fi )3 = [@ bi =@" 0 (@ S=@")Ei]T vi
(
(24)
where
@ bi =@" = [@ bi =@ "~r1 @ bi =@ "~r2 1 1 1 @ bi =@ "~rM ] @ S=@")Ei = [(@ S=@ " ~r1 )Ei
(
@ S=@ "~r2 )Ei 1 1 1
(
@ S=@ "~rM )Ei ]:
(
(25)
"~rm = 1 + (~"r0 0 1) (tm ) (26)
Therefore, the introduction of the adjoint state vector reduces the computational burden, since (24) is evaluated by means of just one solution of the system (22). Finally, the weighted sum of the D" Fi terms over all the incidences gives the complex gradient of the standard error term. The remaining part of the line search process (15) is to estimate the step size g . By means of the Taylor expansion of the total error function, we estimate the step size by
a n Tq n a n TH n a n R; n T I; n T T [(d ) (d ) ] ; qn I; n T T
g(n) = 0 where
D" F
an R; n T ( )
=
( )
( )
( )
( ) ( )
( )
(27)
( )
=
) ] , (the superscripts R and I (n) denote the real and the imaginary part, respectively), and is a diagonal matrix estimate of the Hessian matrix in the Taylor expansion and is obtained by forward differences of gradients. [(
( )
)
D" F
( )
(
( )
a homogeneous surrounding medium). This could be applicable to medical imaging cases where we know a priori from previous measurements, the values of the constitutive parameters of healthy and diseased tissues and we intend to identify their presence and location. This could also be valuable when we already have previous images of the body under investigation, and we want to validate the progress of a therapy. Since the RCP of the scatterer is known, the total error function (13) is not differentiable with respect to ". In order to overcome this limitation, we express the RCP of each element in the form
H
where "~r0 is the known RCP of the scatterer [25]. The function is given by
t
=
0tm =T )]
( m ) = 1 [1 + exp(
(28)
1
()
(29)
where T is a positive parameter (temperature) and tm varies from to 1. Since (1) is a monotonic function having an upper bound equal to one and a lower bound equal to zero, we conclude that the mth element is occupied by the scatterer when (tm ) is close to one. Usually, (29) is called function of support due to the fact that when it reaches its highest value it supports the presence of the scatterer. Substituting (28) in (13), the cost function becomes differentiable with respect to the real vector = [t1 t2 1 1 1 tM ]T and is minimized by applying the Polak–Ribi`ere algorithm. Details about the role and the selection of the parameter T can be found in [25]. The a priori knowledge of the RCP value reduces the degrees of freedom of the unknown spatial distribution of "~r , resulting in a kind of regularization. This aspect is examined during the inversion by setting the regularization factor in (13) equal to zero.
01
t
B. Binary Inverse Scattering Finally, a very interesting inverse scattering problem is to identify the shape and the location of scatterers that have known constitutive parameters. The simplest case that belongs to this class of problems is the binary one where we are dealing with a single complex permittivity inhomogeneity (a homogeneous scatterer embedded in
IV. NUMERICAL RESULTS In this section, numerical results of a number of 2-D RCP reconstruction problems are presented. The measurements of the scattered field are obtained numerically by the solution of the forward scattering problem using the FEM and HKIT. Then, these noiseless data are
1112
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 18, NO. 11, NOVEMBER 1999
Fig. 3. Detection of proliferated bone marrow. The original and the reconstructed profiles of the RCP’s of the leg tissues and the surrounding water. The reconstructed profiles are after 100 iterations for various values of the SNR and the regularization factor .
corrupted by adding Gaussian white noise at various signal-to-noise ratios (SNR’s), defined as
S=N
kf k
0 2
= 10 log10
2 2IKn
(dB)
(30)
where f 0 denotes the vector of all the noiseless data (for I incidences and K measurement points), and n is the standard deviation of the additive noise. To quantify the reconstruction results we define the relative mean square error (RMSE) of the reconstruction. If " is the original scatterer profile and "(n) is its estimate after the nth iteration of the Polak–Ribi`ere algorithm, then the RMSE of the reconstruction is given by
Err(n) = " 0 "(n)
k"k:
(31)
A. Reconstruction of a Lossy Discontinuous Scatterer In the first example, we examine the reconstruction of a lossy discontinuous scatterer profile. The scatterer lies within a square region of side equal to 2 and is surrounded by air. Inside a 2 region centered at the scatterer domain the RCP is constant and equal to 1:6 0 j 0:2, whereas in the remaining domain it is equal to 1:3 0 j 0:4. The scatterer is discretized via a 24 2 24 grid where each subsection is assumed to have constant RCP. In addition, due to the application of triangular first-order elements in the FEM, each square subsection is divided into two triangles. The scatterer region is illuminated by TM plane waves from 24 directions, (I = 24),
which are uniformly distributed around the scatterer. For each angle of incidence, a set of 24 far field measurements, (K = 24), are obtained at uniformly distributed positions on a circle of radius equal to 7. Then, these noiseless measurements are corrupted by additive gaussian noise at various SNR’s (40 dB, 35 dB, 30 dB, 25 dB, 20 dB, and 15 dB). During the reconstruction procedure, five different values of the regularization factor, ; (0:0; 0:005; 0:01; 0:05; 0:1), have been used, while the initial guess of the RCP of the scatterer is chosen to be that of the air. In Fig. 2, the RMSE of the reconstruction after 100 iterations of the Polak–Ribi`ere algorithm and for the above values of SNR and is illustrated. After 100 iterations we observed that no essential further reduction of the standard error term was achieved. It is clear that the regularization heals the ill posedness of the inverse scattering problem. In addition, we conclude that for a low SNR we have to use a high regularization factor, whereas for high SNR’s a lower factor should be applied. B. Detection of Proliferated Bone Marrow In the second example, we investigate the applicability of the proposed method to the detection of proliferated marrow inside the lower part of a leg. The leg is considered as a cylinder of radius equal to 0.05 m, immersed in water. The tissues that compose the leg are assumed isotropic, while the locations of fat, muscle, bone, and marrow are known. We also ignore the presence of arteries and veins. The objective is to reconstruct the RCP profile of the region occupied by marrow (inside the bone) from scattered field measurements.
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 18, NO. 11, NOVEMBER 1999
1113
Fig. 4. The original and the reconstructed RCP profiles of the leg by applying the binary inverse scattering approach to both configurations. The reconstructed profiles are after 100 iterations for various values of the SNR and the regularization factor . TABLE I. RELATIVE COMPLEX PERMITTIVITY (RCP) WITH RESPECT TO THE WATER AT 800 MHz
Synthetic measurements are obtained by illuminating the original structure from 20 uniformly distributed directions by means of TM plane waves at 800 MHz. The RCP’s of the tissues with respect to the water are given in Table I [29]. The square domain (10 2 10cm2 ) that covers the leg and part of the water is divided into 32 2 32 square subdivisions. For each angle of incidence, 20 scattered field measurements are computed at uniformly distributed positions on a circle of radius equal to 7. Gaussian noise is added to these noiseless data at various SNR’s (20 dB, 30 dB, and 40 dB). The initial guess of the scatterer profile is that of the absence of the proliferated marrow. During the inversion, only
the RCP inside the bone is updated, while in the remaining regions it is kept constant and equal to the value of the corresponding type of tissue. Thus, the problem is strictly constrained to the detection of proliferated marrow. The original and the reconstructed RCP profiles for different values of the regularization factor are illustrated in Fig. 3. By observing the reconstructed profiles, we conclude that the introduction of the regularization term in the total error cost function is of great importance. Actually, it results in acceptable reconstruction even at an SNR equal to 30 dB. We also observe that by applying a high regularization factor, we obtain a blurred reconstruction without edge preservation. This is due to the smoothness introduced by the regularization and it could be overcome by applying a powerful edge preservation technique introduced by Lobel et al. [17]. On the other hand, when the SNR is low, a low results in a noisy RCP reconstruction. Finally, in the last example we deal with the same problem of detecting proliferated marrow from the binary inverse scattering point of view. Two different configurations of the original profile of the leg are examined, shown in Fig. 4. The regularization parameter is set equal to zero, while the a priori knowledge of the values of the constitutive parameters of the normal and the proliferated marrow are taken into account. The same number of incidences and measurement positions as in the previous example is used. By applying the support function methodology described in (28) and (29), we obtain excellent reconstruction results (Fig. 4), even at low SNR’s. As can be seen, the
1114
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 18, NO. 11, NOVEMBER 1999
a priori knowledge of the RCP values acts as a regularization process by reducing the degrees of freedom of the reconstructed RCP. V. CONCLUSION A new spatial inversion method for microwave medical imaging applications was presented. Unlike the previously proposed inverse scattering methodologies, which were related to the Lippmann–Schwinger integral equation, the proposed method was totally oriented toward the differential formulation of the scattering problem. This approach allowed the implementation of the FEM, a powerful tool for the analysis of electromagnetic scattering problems. During the inversion, which was performed by minimizing a total error function by means of the Polak–Ribi`ere algorithm, a sensitivity analysis technique was developed. This analysis was implemented via an elaborate FEM scheme, while the introduction of the adjoint state vector reduced the computational burden dramatically. The method was successfully applied to the detection of proliferated marrow inside a leg. To overcome the ill-posed character of the inverse problem, a regularization term was introduced to the total error function. This term, which described the variation of the reconstructed RCP, was proven very efficient even when the SNR was close to 30 dB. Furthermore, the influence of the regularization factor was the expected one. A high value of the regularization factor resulted in a poor edge preservation, whereas a low one was inadequate for low SNR. Finally, when we exploited the a priori knowledge about the RCP value by means of a binary inverse scattering technique, the reconstruction results were excellent, even without any regularization scheme. ACKNOWLEDGMENT The authors acknowledge the helpful comments of Dr. A. Papagiannakis, Dr. P. Hagouel, and Dr. T. Yioultsis. REFERENCES [1] L. E. Larsen and J. H. Jacobi Medical Applications of Microwave Imaging. Piscataway, NJ: IEEE Press, 1986. [2] C. Pichot, L. Jofre, G. Peronnet, and J. C. Bolomey, “Active microwave imaging of inhomogeneous bodies,” IEEE Trans. Antennas Propagat., vol. 33, pp. 416–425, Apr. 1985. [3] S. Caorsi, G. L. Gragnani, and M. Pastorino, “Reconstruction of dielectric permittivity distributions in arbitrary 2-D inhomogeneous biological bodies by a multiview microwave numerical method,” IEEE Trans. Med. Imag., vol. 12, pp. 232–239, June, 1993. [4] S. Y. Semenov, R. H. Svenson, A. E. Bulyshev, A. E. Souvorov, A. G. Nazarov, Y. E. Sizov, A. V. Pavlovsky, V. Y. Borisov, B. A. Voinov, G. I. Simonova, A. N. Starostin, V. G. Posukh, G. P. Tatsis, and V. Y. Baranov, “3-D microwave tomography: Experimental prototype of the system and vector Born reconstruction method,” IEEE Trans. Biomed. Eng., vol. 48, pp. 937–946, Aug. 1999. [5] D. Colton and P. Monk, “The detection and monitoring of leukemia using electromagnetic waves: Numerical analysis,” Inv. Problems, vol. 11, pp. 329–342, 1995. [6] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory. New York: Springer-Verlag, 1992.
[7] J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations. New Haven, CT: Yale Univ. Press, 1923. [8] D. Colton and B. D. Sleeman, “Uniqueness theorems for inverse problem of acoustic scattering,” IMA J. Appl. Math., vol. 31, pp. 253–259, 1983. [9] D. Colton and L. P¨aiv¨arinta, “The uniqueness of a solution to an inverse scattering problem for electromagnetic waves,” Arch. Rational Mech. Anal., vol. 119, pp. 59–70, 1992. [10] A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems. Washington, D.C.: Winston, 1977. [11] M. Slaney, A. C. Kak, and L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech., vol. 32, pp. 860–874, Aug. 1984. [12] A. Roger, “Newton-Kantorovich algorithm applied to an electromagnetic inverse problem,” IEEE Trans. Antennas Propagat., vol. 29, pp. 232–238, Mar. 1981. [13] W. C. Chew and Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imag., vol. 9, pp. 218–225, June 1990. [14] A. Franchois and C. Pichot, “Microwave imaging—Complex permittivity reconstruction with a Levenberg-Marquardt method,” IEEE Trans. Antennas Propagat., vol. 45, pp. 203–215, Feb. 1997. [15] R. E. Kleinman and P. M. van den Berg, “A modified gradient method for two-dimensional problems in tomography,” J. Comp. Appl. Math., vol. 42, pp. 17–35, 1992. [16] H. Harada, D. J. N. Wall, T. Takenaka, and M. Tanaka, “Conjugate gradient method applied for two-dimensional problems for electromagnetic imaging,” IEEE Trans. Antennas Propagat., vol. 43, pp. 784–792, Aug. 1995. [17] P. Lobel, L. Blanc-F´eraud, C. Pichot, and M. Barlaud, “A new regularization scheme for inverse scattering,” Inv. Problems, vol. 13, pp. 403–410, 1997. [18] T. Isernia, V. Pascazio, and R. Pierri, “A nonlinear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sensing, vol. 35, pp. 910–923, July 1997. [19] A. F. Peterson and S. P. Castillo, “A frequency-domain differential equation formulation for electromagnetic scattering from inhomogeneous cylinders,” IEEE Trans. Antennas Propagat., vol. 37, pp. 601–607, July 1989. [20] J. Jin, The Finite Element Method in Electromagnetics. New York: Wiley, 1993. [21] D. G. Luenberger, Linear and Nonlinear Programming, Reading, MA: Addison-Wesley, 1989. [22] S. Gitosusastro, J. L. Coulomb, and J. C. Sabonnadiere, “Performance derivative calculations and optimization process,” IEEE Trans. Magn., vol. 25, pp. 2834–2839, 1989. [23] P. Neittaanm¨aki, M. Rudnicki, and A. Savini, Inverse Problems and Optimal Design in Electricity and Magnetism (Monographs in Electrical and Electronic Engineering 35). Oxford, U.K.: Clarendon, 1996. [24] R. E. Kleinman and P. M. van den Berg, “Two-dimensional location and shape reconstruction,” Radio Sci., vol. 29, pp. 1157–1169, 1994. [25] L. Souriau, B. Duchˆene, D. Lesselier, and R. E. Kleinman, “Modified gradient approach to inverse scattering for binary objects in stratified media,” Inv. Problems, vol. 12, pp. 463–481, 1996. [26] A. Bayliss and E. Turkel, “Radiation boundary conditions for wavelike equations,” Comm. Pure Appl. Math., vol. 33, pp. 707–725, 1980. [27] P. M. van den Berg and R. E. Kleinman, “A total variation enhanced modified gradient algorithm for profile reconstruction,” Inv. Problems, vol. 11, pp. L5–L10, 1995. [28] G. Wahba, “Practical approximate solutions to linear operator equations when the data are noisy,” SIAM J. Numer. Anal., vol. 14, pp. 651–667, 1977. [29] J. W. Penn and E. L. Bell, “Electrical parameter values of some human tissues in the radio frequency radiation range,” USAF School of Aerospace Medicine, Brooks Air Force Base, San Antonio, TX, Tech. Rep. SAM-TR-78-38, 1978.
