IT IS of practical significance to detect, locate, characterize,

Transcription

1 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 56, NO. 4, APRIL Active Microwave Imaging II: 3-D System Prototype and Image Reconstruction From Experimental Data Chun Yu, Senior Member, IEEE, Mengqing Yuan, John Stang, Elan Bresslour, Rhett T. George, Gary A. Ybarra, Senior Member, IEEE, William T. Joines, Fellow, IEEE, and Qing Huo Liu, Fellow, IEEE Abstract A 3-D microwave imaging system prototype and an inverse scattering algorithm are developed to demonstrate the feasibility of 3-D microwave imaging for medical applications such as breast cancer detection with measured data. In this experimental prototype, the transmitting and receiving antennas are placed in a rectangular tub containing a fluid. The microwave scattering data are acquired by mechanically scanning a single transmit antenna and a single receive antenna, thus avoiding the mutual coupling that occurs when an array is used. Careful design and construction of the system has yielded accurate measurements of scattered fields so that even the weak scattered signals at 21 = 90 db (or 30 db below the background fields) can be measured accurately. Measurements are performed in the frequency domain at several discrete frequencies. The collected 3-D experimental data in fluid are processed by a 3-D nonlinear inverse scattering algorithm to unravel the complicated multiple scattering effects and produce high-resolution 3-D digital images of the dielectric constant and conductivity of the imaging domain. Dielectric objects as small as 5 mm in size have been imaged effectively at 1.74 GHz. Index Terms Born iterative method, breast cancer detection, diagonal tensor approximation, distorted Born iterative method, inverse scattering, microwave imaging, nonlinear inverse scattering algorithm, stabilized biconjugate-gradient fast Fourier transform (FFT) algorithm. I. INTRODUCTION IT IS of practical significance to detect, locate, characterize, and image tumors in healthy tissue of the breast. Over the last two decades, intensive investigations have been conducted for early breast cancer detection using microwaves (e.g., [1] [23]). These studies include confocal microwave imaging [6], [7], [12] [14], 2-D microwave topographic imaging [8], [9], [11], and 3-D active microwave tomographic imaging [15] [17]. Microwave imaging has been proposed for breast detection [8], [9], [6], [7] because of its potentially high specificity for breast cancer diagnosis due to the high contrast in electrical properties between normal and malignant human breast tissues. The electrical properties of normal and malignant human breast tissues have been a subject of great interest over the last two decades. It has been reported that a significant contrast in electrical properties at microwave frequencies exists Manuscript received June 9, 2007; revised November 1, This work was supported by the National Institutes of Health (NIH) under Grant 5R01CA The authors are with the Department of Electrical and Computer Engineering, Duke University, Durham, NC USA ( qhliu@ee.duke.edu). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TMTT between normal and malignant human breast tissues [3]. This high contrast is due to significantly different sodium concentrations, fluid contents, and electrochemical properties. A recent research on the variability of normal breast tissue properties [21] reveals that the breast can, in fact, be quite heterogeneous, and this may pose a significant challenge for microwave breast imaging, although actual clinical tests in [22] and [23] have demonstrated the significant potential and new advances of microwave breast imaging from previous simulations and phantom studies. Previously, microwave imaging has been reported primarily based on numerical simulation data and on 2-D reconstruction. Only recently has imaging from 3-D experimental data been reported based on space time beamforming [20] and tomographic reconstruction [15]. For a more thorough review, the reader is referred to [12], [18], and [19]. Our previous work, presented in [11], [16], [17], and [19], focused on theoretical modeling and numerical investigation of active microwave imaging to demonstrate the feasibility of 3-D microwave imaging for breast cancer detection using fast volume integral-equation solvers and nonlinear inverse scattering algorithms. In this study, we further demonstrate the feasibility of a high-resolution 3-D microwave imaging system with accurate experimental data. We will discuss the design of a 3-D experimental prototype and utilize some recently improved simulation and reconstruction algorithms for 3-D microwave imaging simulation and image reconstruction. The 3-D microwave imaging system prototype consists of one transmitting and one receiving dipole antenna that are placed in a rectangular tub containing fluid. To avoid mutual coupling, we do not use an array of antennas in this study; instead, we scan the receiving antenna using a positioner (stepper motor). Measurements are performed in continuous wave (CW) mode for both background (without objects) and with anomalous objects in the fluid, with the corresponding fields called the incident and total fields, respectively. The scattered field is obtained by subtracting the incident field from the total field. The collected 3-D scattered field is then processed by a newly developed 3-D hybrid nonlinear inverse scattering method based on the combination of several algorithms, which are: (1) the diagonal tensor approximation [27] [29] combined with the Born iterative method to produce an approximate image rapidly and (2) the stabilized biconjugate-gradient fast Fourier transform (FFT) method [25], [26], [30] [32] combined with the distorted Born iterative method to solve the full 3-D nonlinear inverse problem. To test the capability of the microwave imaging system with the proposed 3-D nonlinear inverse scattering algorithm, reconstructions from experimentally measured data have been /$ IEEE

2 992 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 56, NO. 4, APRIL 2008 Fig. 1. Experimental setup of microwave imaging system prototype. Two dipole antennas are placed in a rectangular tub containing a lossy fluid. The dimension of the tub is cm. The distance of the antennas from the tub walls is at least 15 cm during scanning so that the reflection from the tub walls is negligible. performed. High-resolution reconstructed results are presented for multiple objects in fluid. Here we note that the proposed algorithm and system has been tested against accurate experimental data, while the experimental model is based on a simplified scenario, rather than a real breast. We are currently studying microwave imaging for more realistic breast phantoms. This paper is organized as follows. Section II presents a description of the experimental 3-D microwave tomographic imaging system, including the system prototype, the design and performance of the transmitting and receiving dipole antennas, and the measurement procedure for 3-D data acquisition. Section III summarizes the inverse scattering algorithm used in this study. Reconstructed results from experimental data and discussions are presented in Section IV. Finally, conclusions are given in Section V. II. 3-D MICROWAVE IMAGING SYSTEM A. System Overview As shown in Fig. 1, the experimental 3-D microwave imaging system consists of two dipole antennas (one acting as a transmitter and the other as a receiver), a rectangular tub as an imaging chamber, one automatic positioner (two-axis stage controller: SHOT-602), and an HP 8753E vector network analyzer. The tub is filled with either a fluid with electrical properties matching those of normal breast tissue or water. The automatic positioner, serving as a mechanical scanning system, moves the transmitting and/or receiving antennas in the fluid through a series of locations with high precision. As a result, the 3-D data acquisition is implemented with flexible multiview transmitters/receivers on given measuring surfaces around the object to be imaged. This setup avoids the issue of mutual coupling in an actual array. The HP network analyzer is used to collect both the incident field (i.e., the field when no anomalous objects are in the chamber) and the total field (i.e., the field in the presence of anomalous objects in the chamber) in the form of scattering parameters ( -parameters). In particular, measurements with and without objects in the imaging domain inside the chamber (corresponding to the total field and incident field ) are recorded (these parameters Fig. 2. (a) Geometry of the dipole antenna with a balun to reduce the radiation from the outer conductor of the semirigid coaxial cable. (b) Measured return loss S of two dipole antennas in water. will be defined later). The positioner, its scanning speed and location, and the collection of data by the network analyzer are controlled by a desktop computer. The collection time needed for each receiver position is 3 s. There are several advantages of this microwave imaging system prototype, which are: (1) the coupling between multiple transmitting antennas and receiving antennas in a typical high-resolution 3-D microwave imaging system is minimized due to the use of scanning antennas, thus simplifying the design; (2) the antenna positions are known to a high precision (3 m); and (3) the system does not require switching devices as in systems containing multiple antenna array elements, thus reducing the additional noise and loss from these switching devices. The main disadvantage of this system is its relatively long acquisition time, but this is not a critical issue during prototyping and will be improved in the clinical version. B. Design and Performance of Antennas The data acquisition of the 3-D microwave imaging system requires accurate and sensitive measurements of the scattered electromagnetic waves. To achieve this aim, a linearly polarized dipole antenna has been designed to function as both the transmitting antenna and the receiving antenna. Fig. 2(a) shows the geometry of the dipole antenna. The two arms of the linearly polarized dipole antenna were made from the inner and outer conductors of a semirigid coaxial cable. The total length of the two arms is. A matching balun of length is formed at the front-end of the antenna to reduce the radiation from the outer

3 YU et al.: ACTIVE MICROWAVE IMAGING II: 3-D SYSTEM PROTOTYPE AND IMAGE RECONSTRUCTION FROM EXPERIMENTAL DATA 993 respectively. If the transmitting and receiving dipole antennas are oriented in the and directions, the signal measured at the receiver is for the incident field when only the background medium exists (i.e., no objects are in the imaging domain), and (1) Fig. 3. Example of transmitting dipole locations (on a plane at x = 04:1 cm) and receiver locations (on a plane at x =4:1 cm) along five dotted lines separated by 1 cm in the z-direction, each with 21 points (increments of 0.4 cm in the y-direction). The total number of receiver points is = 105 for each transmitter location. conductor of the coaxial cable. Antennas of various dimensions have been designed and fabricated for matching fluid (, S m) and water in the chamber. In particular, for the dipole in water, we chose mm. Fig. 2(b) shows the measured for two fabricated dipole antennas in water. The return loss of these two antennas is below 11 db at 1.74 GHz, and they are well matched to work at a common optimal frequency with a bandwidth of approximately 200 MHz. The difference between the values of the two antennas is due to a small difference between the antennas in size. These fabricated dipole antennas were also characterized experimentally for radiation into a matching fluid with a relative permittivity of 20 and a conductivity of 0.16 S/m. For this matching fluid, the measured return loss is below 15 db at the optimal frequency of 2.7 GHz for both transmitting and receiving antennas. C. 3-D Data Acquisition In microwave breast cancer detection, tomographic imaging is achieved by transmitting a sequence of electromagnetic waves through the breast and measuring the scattered fields around the breast. For the experimental prototype shown in Fig. 1, the transmitting antenna sends out electromagnetic waves to illuminate a breast phantom placed in a fluid background that matches the breast in terms of electrical parameters; for each transmitting antenna position, the receiving antenna collects the fields at different locations controlled by the positioner with a locating accuracy of approximately 3 m. The transmitting antenna is not scanned automatically in this experimental setup. An example of the transmitting dipole location distribution (on a plane at cm) and the receiver-scanning pattern (on a plane at cm) is shown in Fig. 3. The multiple views of the receiving antenna are obtained by scanning on five lines (increments of 1 cm in the -direction) with 21 points along each line (increments of 0.4 cm in the -direction), resulting in a total number of receiver points of. The microwave signals at the receiving antenna are measured in the form of scattering parameters (or -parameters) through the HP vector network analyzer with its two ports connected to the transmitter and receiver. Consider just two antennas in the system, one for transmitting and the other for receiving. The complex scattering parameters are,, and, where subscripts 1 and 2 refer to the transmitter and receiver, for the total field when there are objects in the imaging domain. In the above, the incident field and the total field at the receiver depend on the orientation of the transmitting dipole antenna and the dyadic Green s function. is a network analyzer calibration constant such that the -parameter is equal to one when ports 1 and 2 are directly connected. From the incident and total field measurements, we can obtain the scattered signal where is the scattered field at the receiver location due to the presence of anomalous objects in the imaging domain. In the general case, there are transmitter locations, each with receiver locations. The scattering parameter is measured for each transmitter receiver combination. Therefore, there are complex data points for, corresponding to the th receiver signal due to the th transmitter excitation. From this set of data, the complex permittivity distribution in the imaging domain can be found using an image reconstruction algorithm. III. IMAGE RECONSTRUCTION ALGORITHM The objective of the image reconstruction algorithm is to infer the complex permittivity distribution in the imaging domain, where and are the unknown dielectric constant and electrical conductivity, respectively. Consider a general 3-D inhomogeneous object with in a background medium with. The integral equation governing the inverse problem [33] is where the magnetic permeability in the target is assumed to be the same as in the background medium, is the transmitter location, is an electric dyadic Green s function due to an electric current source, and is the contrast function defined as Methods for solving the above inverse problem include the Born iterative method, the distorted Born iterative method, and the contrast source inversion (see [34] [39] for more details). In (2) (3) (4) (5)

4 994 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 56, NO. 4, APRIL 2008 this study, we utilize a hybrid inversion method, as summarized below. A. Forward Solver A forward solver is necessary in an inversion method such as the Born iterative method or the distorted Born iterative method to calculate the predicted measured data and the gradient information. The first task of the forward solver is to calculate the electric field inside the imaging domain (i.e., ) in (4). The scattered field at any receiver location on a surface can then be calculated as (6) Equation (6) is called the data equation, as it defines the scattered field at the observation point. In contrast, the integral equation (4) for is called the object equation, and can be solved by using the full-wave stabilized biconjugate-gradient FFT method, or by using approximation methods such as the diagonal tensor approximation. In this paper, we use a hybrid technique [40] by combining the diagonal tensor approximation and the stabilized biconjugate-gradient FFT method where the diagonal tensor approximation acts as a preconditioner for the stabilized biconjugate-gradient FFT method. Such a hybrid method has the advantage that, for low-to-moderate contrasts, the diagonal tensor approximation alone will give accurate results without going through the stabilized biconjugate-gradient FFT iterations; for high contrasts, the diagonal tensor approximation gives a good preconditioner so that the stabilized biconjugate-gradient FFT iterations will converge rapidly. Thus, both low- and high-contrast problems can be solved efficiently and accurately by this hybrid method. B. Nonlinear Inverse Scattering Method In the inverse scattering problem, the total number of data points collected is. Suppose the imaging domain is discretized into small voxels. By using the trapezoidal rule, the data equation with the unknown contrast function can be discretized as follows: where is an -dimensional data column vector whose elements are the measured scattered electric field collected by the receiver, is the volume of each voxel, and and denote the indices for the receiver and transmitter, respectively. For measurements and discretized voxels, (7) can be written compactly as where represents the simulated scattering fields, is an -dimensional column vector of the contrast function, and is an matrix whose elements are given by where and. (7) (8) (9) Since the total field within the objects (and in the distorted Born iterative method) is an unknown function of the material contrast function, (8) is a nonlinear equation in. Moreover, the limited amount of information makes the problem ill posed. The Born iterative method [34] or the distorted Born iterative method [35] [37] can be used to solve the above nonlinear inverse scattering equation. In this study, we use an algorithm combining the diagonal tensor approximation and the Born iterative method to obtain an initial reconstruction for a fast estimate of the image, and then use an algorithm combining the stabilized biconjugate-gradient FFT and the distorted Born iterative method to provide an accurate and fast inversion. In iterative inversion algorithm, since the actual location of the objects to be imaged is not given, we can only rely on the data misfit information to determine whether the inversion has converged. Thus, we define the relative data residual error RES as (10) where and are the th measured and simulated scattered field, respectively. When this data residual reduces to a predetermined criterion (e.g., 0.1%), or when the residual error changes very little between subsequent iterations (e.g., 0.1%), or when a maximum iteration number is reached, the inversion iteration is terminated. In the first-step inversion from experimental data, the stopping criteria is related to the above relative data residual error and a maximum iteration number. When the data error is less than the small number (e.g., 0.1%), or when the iteration exceeds the maximum iteration number (10), the first-step inversion is terminated and the inversion is switched into the secondstep inversion using the stabilized biconjugate-gradient FFT and the distorted Born iterative method. The CPU time on an AMD Opteron Processor 250 is approximately 2 min/iteration for this first-step inversion. Finally the iteration in the second-step inversion stops with a given number of iteration (e.g., 20). The flowchart of the computer program for the algorithm combining the diagonal tensor approximation and the Born iterative method for the microwave imaging system is shown in Fig. 4. The flowchart of the algorithm combining the stabilized biconjugate-gradient FFT and the distorted Born iterative method is the same as shown in Fig. 4, except that Updating Diagonal Tensor and Solving Expected Scattered Field by DTA in the box should be replaced with Updating Green s Function and Solving Expected Scattered Field by BCGS-FFT. The CPU time on an AMD Opteron Processor 250 is approximately 10 min/iteration for this second-step inversion. Note that the above performance of the codes has not been optimized; furthermore, the codes have been developed for the more general case of objects embedded in a layered medium, thus there is an overhead for the computation of Green s functions. IV. MICROWAVE IMAGING FROM MEASURED DATA In order to avoid the environmental effects from the reflections caused by an outer boundary of the fluid container, we

5 YU et al.: ACTIVE MICROWAVE IMAGING II: 3-D SYSTEM PROTOTYPE AND IMAGE RECONSTRUCTION FROM EXPERIMENTAL DATA 995 Fig. 5. Measured js j versus frequency for a pair of dipole antennas separated by 8.2 cm in water. Solid line: S without object; heavy dotted line: S with an clay ball of 5-mm diameter located at the center of the two antennas; lighter dotted line: S with two metal balls of 5-mm diameter (the locations of two metal balls are the same as the example in Section IV-C). Dashed line: the scattered field S for a clay ball. Dashed dotted line: the scattered field S for two metal balls. Fig. 4. Flowchart of the algorithm combining the diagonal tensor approximation (DTA) and the Born iterative method (BIM) for microwave imaging. use a large container and fill it with water (measured, S m). The following examples use the microwave imaging system shown in Fig. 1 to collect the scattered data. In our experiments, dielectric spheres made of clay of various diameters have been used. The clay material has been measured to have a dielectric constant of approximately 5 and lossless; however, when immersed in water, it is likely that the dielectric constant and conductivity increase because the material is slightly porous. Measurements have been made to determine the dielectric constant and conductivity changes during immersion in water, and the results have shown that there is a small change less than 15% for both dielectric constant and conductivity before and after immersion. Such objects have large contrasts with the background medium, and thus large multiple scattering effects are expected. The transmission coefficient ( ) between the transmitter and receiver, both polarized in the -direction and separated by 8.2 cm in water, is measured as a function of frequency, as shown in Fig. 5. The five curves in Fig. 5 represent the background without object (solid line), the total with a clay ball of 0.5-cm diameter placed at the center of the two antennas (heavy dotted curve), the total with two metallic spheres of 0.5-cm diameter (lighter dotted curve), the scattered obtained from (3) for a clay ball (dashed curve), and for two metallic spheres (dashed dotted curve), respectively. It is noted that at 1.74 GHz, corresponding to the common frequency for the -parameters shown in Fig. 2, the background is approximately 42 db, while the scattered is approximately 67 db, or 25 db lower than the background field for the clay ball placed at the center of the two antennas, and approximately 59 db, or 17 db lower than the background field for the two metallic spheres. The use of a lower frequency would yield a weaker scattered field, as well as lower resolution in image reconstruction. The following examples will use 1.74-GHz microwaves to image small objects in water. Although the objects and water in our initial experiments are not exactly equivalent to tumors and normal breast tissue, it serves as a good test for our 3-D inverse scattering algorithm and 3-D microwave imaging experimental setup. Current research is investigating more realistic phantoms and imaging by array antennas. Since experimental data sets are measured by linearly polarized resonant dipole antennas, a calibration procedure is used in the image reconstruction algorithm with an infinitesimal electric dipole having the same polarization as the finite-length dipole. In fact, the radiation field patterns from a short dipole and a resonant dipole are very similar within the viewing angles of the given antenna array. The difference between the measured data and simulated data can be calibrated out with a scale factor or a simple normalization procedure. In the following imaging examples, we use a normalization procedure in which the normalization factor is the measured background field and the simulated background field, respectively, at the same receiver location. A. Microwave Imaging With One Dielectric Ball The first example utilizes a small dielectric sphere of 0.5-cm diameter placed at the center of the chamber (also the center of the imaging domain and the origin), as shown in Fig. 6. To produce multiple source excitations, the dipole transmitting antenna is placed at nine locations on the plane cm with,, where cm, and cm. As shown in Fig. 3, the receiving antenna collects multiview microwave field information by scanning automatically at 105 locations on the opposite side with cm and,, where, cm, and, cm. Only the -polarized field is collected since the two antennas are oriented in the same vertical direction. Measurements are performed in the frequency domain

6 996 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 56, NO. 4, APRIL 2008 Fig. 6. Imaging of a dielectric sphere in water using nine transmitter and 105 receiver locations. The transmitting antenna is placed on the plane x = 04:1 cm. The receiving antenna scans on the plane x =4:1 cm. Fig. 8. Convergence curve of the data error as a function of the iteration number of the stabilized biconjugate gradient FFT-distorted Born iterative method for one dielectric sphere in water. Fig. 9. Imaging of two clay balls in water. The two spheres are placed along y-axis and separated center to center by 2.4 cm. The transmitting antenna is placed on the plane x = 04:1 cm. The receiving antenna scans on the plane x = 4:1 cm. Fig. 7. Reconstructed permittivity profile for a 0.5-cm clay ball from measured data in 3-D plots using: (a) the diagonal tensor approximation Born iterative method and (b) the stabilized biconjugate gradient FFT-distorted Born iterative method; the reconstructed permittivity profile on the x =0plane using: (c) the diagonal tensor approximation Born iterative method and (d) the stabilized biconjugate gradient FFT-distorted Born iterative method. at several discrete frequencies. Here, only single-frequency reconstructed results are presented. For the first example, the imaging domain of dimension of 8 8 8cm is divided into voxels so the total number of complex unknowns to be reconstructed is, while the number of measured data is. Both the diagonal tensor approximation Born iterative method and the stabilized biconjugate gradient FFT-distorted Born iterative method are used for inversion. In the second step inversion process, the inverse result produced by the diagonal tensor approximation Born iterative method is used as an initial solution to speed up convergence. Fig. 7 shows reconstructed results of the permittivity at 1.74 GHz using the diagonal tensor approximation Born iterative method and the stabilized biconjugate gradient-distorted Born iterative method, respectively. The reconstructed images indicate highly accurate location of the object. The value of the relative permittivity is substantially higher than 5, perhaps because of the small size of the object. Fig. 8 shows the convergence curve of the data error as a function of the iteration number of the stabilized biconjugate gradient FFT-distorted Born iterative method. The inversion is terminated at the twentieth iteration step. It is observed that, due to a good inverse result of the diagonal tensor approximation Born iterative method being used as an initial solution (data error 20%) for inversion, the data error of the inversion using the stabilized biconjugate gradient FFT-distorted Born iterative method converges quickly to 6% after 20 iterations for the real measurement data. B. Microwave Imaging for Two Dielectric Balls Fig. 9 shows the geometry of the imaging system for two dielectric spheres (0.5-cm diameter) in water. The dipole transmitting antenna is placed at three locations on the plane cm with,, where and cm, to test the capability to image in a different experimental setup. The receiving antenna collects multiview microwave field information by scanning automatically at 105 locations on the opposite side with cm and the receiving points are exactly the same as in the last example. The transmitting and receiving dipoles are all -polarized. The measured data at 1.74 GHz is used for image reconstruction of an imaging domain of size cm divided into voxels. The total number of complex unknowns to be reconstructed is. Fig. 10 presents the measured for all combinations of transmitters for the total and incident fields. The scattered field can be obtained by subtracting the incident field from the total field. It is observed that: 1) the difference is very small between the total field with the objects and incident field without the objects; 2) the magnitude of the scattered field is

7 YU et al.: ACTIVE MICROWAVE IMAGING II: 3-D SYSTEM PROTOTYPE AND IMAGE RECONSTRUCTION FROM EXPERIMENTAL DATA 997 Fig. 10. Measured js j from three transmitters and 105 receivers for two dielectric spheres in water. (a) Magnitude js j for the incident and total fields. (b) Magnitude js j for the incident and scattered fields. Fig. 11. Reconstructed permittivity profile for two clay balls in 3-D plots using: (a) the diagonal tensor approximation Born iterative method and (b) the stabilized biconjugate gradient FFT-distorted Born iterative method. The 2-D cross section at x =0 for the: (c) diagonal tensor approximation Born iterative method and (d) the stabilized biconjugate gradient FFT-distorted Born iterative method. approximately from 70 to 80 db, or approximately from 20 to 30 db lower than the incident field ( 50 db). This means that the signal-to-noise ratio (where the signal refers to the total field) should be better than an average of the ratio of the total field to the scattered field (say, 25 db) to image such small objects. The system has been carefully designed to meet this requirement. Fig. 11 shows the 3-D tomographic results in a 3-D view and a 2-D cross section with the diagonal tensor approximation Born iterative method and the stabilized biconjugate gradient FFTdistorted Born iterative method. The positions and sizes of the spheres are clearly determined. As expected, Fig. 11(a) and (b) shows that the resolution in the -direction is slightly lower than in the other two directions because the scattered field information is collected only on the -plane. The reconstructed conductivity profiles for the two clay balls are presented in Fig. 12 in 3-D view and in a 2-D cross section with the diagonal tensor approximation Born iterative method and the stabilized biconjugate gradient FFT-distorted Born iterative method. It is noted that the positions and sizes of the objects can also be determined by the reconstructed conductivity profiles. Comparison between the experimental scattered field and reconstructed scattered field in Fig. 13 shows excellent agreement. This example demonstrates the capability of our 3-D imaging system and reconstruction method in the reconstruction of multiple small objects from measured data. C. Microwave Imaging for Two Metallic Balls In the above experiments, dielectric spheres with permittivity value smaller than the background have been imaged. To demonstrate the performance of the system and algorithms for objects having a higher complex permittivity value than Fig. 12. Reconstructed conductivity profile for two clay balls in 3-D plots using: (a) the diagonal tensor approximation Born iterative method and (b) the stabilized biconjugate gradient FFT-distorted Born iterative method. The 2-D cross section at x =0for: (c) the diagonal tensor approximation Born iterative method and (d) the stabilized biconjugate gradient FFT-distorted Born iterative method. the background, here we show an imaging example for two metallic spheres. The setup is exactly the same as in the previous experiment in Fig. 9, except that the two metallic spheres are separated center to center by 3.2 cm. Fig. 14 shows the measured for this imaging example. The scattered field is obtained by subtracting the incident field from the total field. It is observed that the difference between the total field and incident field is substantially

8 998 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 56, NO. 4, APRIL 2008 Fig. 13. Comparison of measured and reconstructed scattered field js j for two dielectric spheres. Fig. 15. Conductivity profiles reconstructed from the experimental data for two metallic spheres in 3-D plot using: (a) the diagonal tensor approximation Born iterative method and (b) the stabilized biconjugate gradient FFT-distorted Born iterative method. The 2-D cross section at x =0for the: (c) diagonal tensor approximation Born iterative method and (d) the stabilized biconjugate gradient FFT-distorted Born iterative method. simple dipole antennas in isolation and in a large water chamber. No background heterogeneities have been added. We are currently investigating more realistic imaging environments with array antennas. Fig. 14. Measured js j from three transmitters and 105 receivers for two dielectric spheres in water. (a) Magnitude js j for the incident and total fields. (b) Magnitude js j for the incident and scattered fields. larger than in the previous example of two dielectric spheres; the magnitude of the scattered field is approximately 20 db lower than the incident field. The reconstructed results for the conductivity are shown in Fig. 15. The reconstructed images clearly demonstrate that the two metallic objects can be localized very effectively. D. Discussion The above results have shown that the prototype and hybrid algorithms can successfully image small objects using microwave imaging. For simplicity and for the sake of demonstrating 3-D microwave tomographic imaging, we have used V. CONCLUSION We have demonstrated a 3-D microwave tomographic imaging system. An inversion technique that combines the diagonal tensor approximation, stabilized biconjugate gradient FFT algorithm, Born iterative method, and distorted Born iterative method has been applied to nonlinear image reconstruction in microwave imaging for breast cancer detection. In this system, two dipole antennas are designed to transmit and receive electromagnetic waves, reducing the mutual coupling of antennas and resulting in accurate measurements of scattered near-field waves. The collected 3-D electromagnetic wave data in water are processed by a 3-D nonlinear inverse scattering algorithm to unravel the multiple scattering effects. With the developed 3-D microwave imaging system prototype, dielectric objects of 5-mm diameter have been detected and localized effectively with high resolution, even when the scattered field intensity is approximately 90 db, or approximately 30 db below the incident field. One remaining challenge is the imaging of tumors in the presence of skin and highly heterogeneous structures. Future work includes developing a 3-D microwave tomographic imaging system designed for a more realistic heterogeneous environment and for clinical applications. REFERENCES [1] S. S. Chaudhary, R. K. Mishra, A. Swarup, and J. M. Thomas, Dielectric properties of normal and malignant human breast tissues at radiowave and microwave frequencies, Indian J. Biochem. Biophys., vol. 21, pp , 1984.

9 YU et al.: ACTIVE MICROWAVE IMAGING II: 3-D SYSTEM PROTOTYPE AND IMAGE RECONSTRUCTION FROM EXPERIMENTAL DATA 999 [2] A. J. Surowiec, S. S. Stuchly, J. R. Barr, and A. Swarup, Dielectric properties of breast carcinoma and the surrounding tissues, IEEE Trans. Biomed. Eng., vol. 35, no. 4, pp , Apr [3] W. T. Joines, Y. Zhang, C. Li, and R. L. Jirtle, The measured electrical properties of normal and malignant human tissues from 50 to 900 MHz, Med. Phys. J., vol. 21, no. 4, pp , [4] L. Jofre, M. S. Hawley, A. Broquetas, E. de los Reyes, M. Ferrando, and A. R. Elias-Fuste, Medical imaging with a microwave tomographic scanner, IEEE Trans. Biomed. Eng., vol. 37, no. 3, pp , Mar [5] S. Caorsi, G. L. Gragnani, and M. Pastorino, An electromagnetic imaging approach using a multi-illumination technique, IEEE Trans. Biomed. Eng., vol. 41, no. 4, pp , Apr [6] S. C. Hagness, A. Taflove, and J. E. Bridges, Two-dimensional FDTD analysis of a pulsed microwave confocal system for breast cancer detection: Fixed-focus and antenna-array sensors, IEEE Trans. Biomed. Eng., vol. 45, no. 12, pp , Dec [7] S. C. Hagness, A. Taflove, and J. E. Bridges, Three-dimensional FDTD analysis of a pulsed microwave confocal system for breast cancer detection: Design of an antenna-array element, IEEE Trans. Antennas Propag., vol. 47, no. 5, pp , May [8] P. M. Meaney, K. D. Paulsen, A. Hartov, and R. K. Crane, An active microwave imaging system for reconstruction of 2-D electrical property distributions, IEEE Trans. Biomed. Eng., vol. 42, no. 10, pp , Oct [9] P. M. Meaney, M. W. Fanning, D. Li, P. Poplack, and K. D. Paulsen, A clinical prototype for active microwave imaging of the breast, IEEE Trans. Microw. Theory Tech., vol. 48, no. 11, pp , Nov [10] E. C. Fear and M. A. Stuchly, Microwave detection of breast cancer, IEEE Trans. Microw. Theory Tech., vol. 48, no. 11, pp , Nov [11] Q. H. Liu, Z. Q. Zhang, T. Wang, G. Ybarra, L. W. Nolte, J. A. Bryan, and W. T. Joines, Active microwave imaging I: 2-D forward and inverse scattering methods, IEEE Trans. Microw. Theory Tech., vol. 50, no. 1, pp , Jan [12] E. C. Fear, S. C. Hagness, P. M. Meaney, M. Okoniewski, and M. A. Stuchly, Enhancing breast tumor detection with near-field imaging, IEEE Micro, vol. 3, no. 1, pp , Mar [13] S. K. Davis, H. Tandradinata, S. C. Hagness, and B. D. Van Veen, Ultrawideband microwave breast cancer detection: A detection theoretic approach using the generalized likelihood ratio test, IEEE Trans. Biomed Eng., vol. 52, no. 7, pp , Jul [14] P. Kosmas and C. M. Rappaport, A matched-filter FDTD-based time reversal approach for microwave breast cancer detection, IEEE Trans. Antennas Propag., vol. 54, no. 4, pp , Apr [15] S. Y. Semenov, R. H. Svenson, A. E. Bulyshev, A. E. Souvorov, A. G. Nazarov, Y. E. Siziv, 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, Three-dimensional microwave tomography: Experimental prototype of the system and vector born reconstruction method, IEEE Trans. Biomed. Eng., vol. 46, no. 8, pp , Aug [16] Z. Q. Zhang, Q. H. Liu, C. Xiao, E. Ward, G. Ybarra, and W. T. Joines, Microwave breast imaging: 3-D forward scattering simulation, IEEE Trans. Biomed. Eng., vol. 50, no. 10, pp , Oct [17] Z. Q. Zhang and Q. H. Liu, 3-D nonlinear image reconstruction for microwave biomedical imaging, IEEE Trans. Biomed. Eng., vol. 51, no. 3, pp , Mar [18] K. D. Paulsen, P. M. Meaney, and L. C. Gilman, Alternative Breast Imaging: Four Model-Based Approaches. Berlin, Germany: Springer-Verlag, [19] W. T. Joines, Q. H. Liu, and G. Ybarra, Electromagnetic imaging of biological systems, in CRC Handbook on Biological Effects of Electromagnetic Fields, B. Greenebaum and F. Barnes, Eds. Boca Raton, FL: CRC, [20] X. Li, E. J. Bond, B. D. Van Veen, and S. C. Hagness, An overview of ultra-wideband microwave imaging via space time beamforming for early-stage breast-cancer detection, IEEE Antennas Propag. Mag., vol. 47, no. 1, pp , Feb [21] M. Lazebnik, L. McCartney, D. Popovic, C. B. Watkins, M. J. Lindstrom, J. Harter, S. Sewall, A. Magliocco, J. H. Booske, M. Okoniewski, and S. C. Hagness, A large-scale study of the ultrawideband microwave dielectric properties of normal breast tissue obtained from reduction surgeries, Phys. Med. Biol., vol. 52, pp , [22] S. P. Poplack, T. D. Tosteson, W. A. Wells, B. W. Pogue, P. M. Meaney, A. Hartov, C. A. Kogel, S. K. Soho, J. J. Gibson, and K. D. Paulsen, Electromagnetic breast imaging: Results of a pilot study in women with abnormal mammograms, Radiology, vol. 243, no. 2, pp , [23] P. M. Meaney, M. W. Fanning, T. Raynolds, C. J. Fox, Q. Fang, S. P. Poplack, and K. D. Paulsen, Initial clinical experience with microwave breast imaging in women with normal mammography, Acad. Radiol., vol. 14, pp , [24] C. F. Smith, A. F. Peterson, and R. Mittra, The biconjugate gradient method for electromagnetic scattering, IEEE Trans. Antennas Propag., vol. 38, no. 6, pp , Jun [25] Q. H. Liu, X. M. Xu, B. Tian, and Z. Q. Zhang, Applications of nonuniform fast transform algorithms in numerical solutions of differential and integral equations, IEEE Trans. Geosci. Remote Sens., vol. 38, no. 4, pp , Jul [26] X. Xu, Q. H. Liu, and Z. Q. Zhang, The stabilized biconjugate gradient fast Fourier transform method for electromagnetic scattering, J. Appl. Comput. Electromagn. Soc., vol. 17, no. 1, pp , Mar [27] L. P. Song and Q. H. Liu, Fast three-dimensional electromagnetic nonlinear inversion in layered media with a novel scattering approximation, Inv. Problems, vol. 20, no. 6, pp. S171 S194, [28] L. P. Song and Q. H. Liu, A new approximation to three dimensional electromagnetic scattering, IEEE Geosci. Remote Sensing Lett., vol. 2, no. 2, pp , Apr [29] C. Yu, L. P. Song, and Q. H. Liu, Inversion of multi-frequency experimental data for imaging complex objects by a DTA CSI method, Inv. Problems, vol. 21, pp. S165 S178, [30] M. Xu and Q. H. Liu, The BCGS-FFT method for electromagnetic scattering from inhomogeneous objects in a planarly layered-medium, IEEE Antennas Wireless Propag. Lett., vol. 1, pp , [31] X. Millard and Q. H. Liu, Fast volume integral equation solver for electromagnetic scattering from large inhomogeneous objects in planarly layered-media, IEEE Trans. Antennas Propag., vol. 51, no. 9, pp , Sep [32] B. Wei, E. Simsek, C. Yu, and Q. H. Liu, Three-dimensional electromagnetic nonlinear inversion in layered media by a hybrid diagonal tensor approximation Stabilized biconjugate gradient fast Fourier transform method, Waves in Random Complex Media, vol. 17, no. 2, pp , May [33] W. C. Chew, Waves and Fields in Inhomogeneous Media. Piscataway, NJ: IEEE Press, [34] Y. M. Wang and W. C. Chew, An iterative solution of the two-dimensional electromagnetic inverse scattering problem, Int. J. Imag. Syst. Technol., vol. 1, pp , [35] W. C. Chew and Y. M. Wang, Reconstruction of two-dimensional permittivity distribution using the distorted born iteration method, IEEE Trans. Med. Imag., vol. 9, no. 2, pp , Jun [36] A. Roger, A Newton Kantorovich algorithm applied to an electromagnetic inverse problem, IEEE Trans. Antennas Propag., vol. AP-29, no. 2, pp , Mar [37] R. F. Remis and P. M. van den Berg, On the equivalence of the Newton Kantorovich and distorted born method, Inv. Problems, vol. 16, pp. L1 L4, [38] F. Li, Q. H. Liu, and L. P. Song, Three-dimensional reconstruction of objects buried in layered media using born and distorted born iterative methods, IEEE Geosci. Remote Sensing Lett., vol. 1, no. 2, pp , Apr [39] P. M. van den Berg and R. E. Kleinman, A contrast source inversion method, Inv. Problems, vol. 13, pp , [40] B. Wei, E. Simsek, and Q. H. Liu, Improved diagonal tensor approximation (DTA) and hybrid DTA/BCGS-FFT method for accurate simulation of 3-D inhomogeneous objects in layered media, Waves in Random Complex Media, vol. 17, no. 1, pp , Feb Chun Yu (SM 07) received the Ph.D. degree in electrical engineering from Shanghai University, Shanghai, China, in From 1982 to 1992, he was an RF Design and Research Engineer with the China Research Institute of Radiowave Propagation, Xingxiang, China. From 1997 to 2001, he was a faculty member and an Associate Professor with the Department of Communication Engineering, Shanghai University, Shanghai, China. In March 2001, he joined the Electromagnetics Laboratory, Department of Electrical and Computer Engineering, University of Kentucky, initially as a Post-Doctoral Research Fellow and then as a Research Scientist. Since December 2004, he has been a Research Associate with the Department of Electrical and Computer Engineering, Duke University, Durham, NC. His research interests include computational electromagnetics, electromagnetic scattering and wave propagation, inverse scattering, microwave and biomedical imaging, and antenna analysis and design.

10 1000 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 56, NO. 4, APRIL 2008 Mengqing Yuan, photograph and biography not available at time of publication. John Stang, photograph and biography not available at time of publication. Elan Bresslour, photograph and biography not available at time of publication. Rhett T. George, photograph and biography not available at time of publication. Gary A. Ybarra (S 86 M 86 SM 06) was born in Hampton, VA, on May 13, He received the B.S., M.S., and Ph.D. degrees from North Carolina State University, Raleigh, in 1983, 1986, and 1992, respectively, all in electrical and computer engineering. He is currently a Professor of the Practice and Director of Undergraduate Studies with the Department of Electrical and Computer Engineering, Duke University, Durham, NC. His research interests include radar signal processing and microwave imaging. Qing Huo Liu (S 88 M 89 SM 94 F 05) received the Ph.D. degree in electrical engineering from the University of Illinois at Urbana-Champaign, in From September 1986 to December 1988, he was a Research Assistant with the Electromagnetics Laboratory, University of Illinois at Urbana-Champaign, and from January 1989 to February 1990, he was a Post-Doctoral Research Associate. From 1990 to 1995, he was a Research Scientist and Program Leader with Schlumberger-Doll Research, Ridgefield, CT. From 1996 to May 1999, he was an Associate Professor with New Mexico State University. Since June 1999 he has been with Duke University, Durham, NC, where he is currently a Professor with the Department of Electrical and Computer Engineering. He has authored or coauthored over 350 papers in refereed journals and conference proceedings. He is an Associate Editor for Radio Science. His research interests include computational electromagnetics and acoustics, inverse problems, geophysical subsurface sensing, biomedical imaging, electronic packaging, and the simulation of photonic devices and nanodevices. Dr. Liu is a Fellow of the Acoustical Society of America. He is a member of Phi Kappa Phi, Tau Beta Pi. He is a full member of the U.S. National Committee, URSI Commissions B and F. He is currently an associate editor for the IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING. He was the recipient of the 1996 Presidential Early Career Award for Scientists and Engineers (PECASE) presented by the White House, the 1996 Early Career Research Award presented by the Environmental Protection Agency, and the 1997 CAREER Award presented by the National Science Foundation (NSF). William T. Joines (M 61 SM 94 LSM 97 F 08) was born in Granite Falls, NC. He received the B.S.E.E. degree (with high honors) from North Carolina State University, Raleigh, in 1959, and the M.S. and Ph.D. degrees in electrical engineering from Duke University, Durham, NC, in 1961 and 1964, respectively. From 1959 to 1966, he was a Member of Technical Staff with Bell Telephone Laboratories, Winston-Salem, NC, where he was engaged in research and development of microwave components and systems for military applications. In 1966, he joined the faculty of Duke University, where he is currently a Professor of electrical and computer engineering. His research and teaching interests are in the area of electromagnetic-wave interactions with structures and materials, mainly at microwave and optical frequencies. He has authored or coauthored over 100 technical papers on electromagnetic-wave theory and applications. He holds seven U.S. patents. Dr. Joines was the recipient of the Scientific and Technical Achievement Award presented by the Environmental Protection Agency in 1982, 1985, and 1990.