A Matched-Filter FDTD-Based Time Reversal Approach ... - IEEE Xplore

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 54, NO. 4, APRIL 2006

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A Matched-Filter FDTD-Based Time Reversal Approach for Microwave Breast Cancer Detection Panagiotis Kosmas, Member, IEEE, and Carey M. Rappaport, Fellow, IEEE

Abstract—Based on the finite-difference time-domain (FDTD) method, a numerical time-reversal (TR) algorithm for microwave breast cancer detection, already presented in previous work [1], [2], is further examined. In [2], we assumed that the exact field scattered from the tumor-like anomaly is available for backpropagation, and it was shown that the time reversal process is robust to breast inhomogeneities and uncertainties of the skin thickness or electric properties. In this paper, we use the same time reversal mirror (TRM) and two-dimensional (2-D) breast model based on magnetic resonance imaging (MRI) data, but examine the realistic situation where the target response is not known and can only be estimated from the total signal, which is dominated by clutter. A matched-filter approach to solve this signal processing problem is proposed and applied to the TRM data. Detection and localization is achieved for different target locations, and the ability of the time reversal algorithm to avoid false alarms is demonstrated. Index Terms—Breast cancer detection, finite-difference time-domain (FDTD), matched-filter, microwave imaging, time reversal.

I. INTRODUCTION

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ICROWAVE imaging for breast cancer detection has recently attracted the interest of many researchers, and several different techniques have been proposed for tumor detection and localization. In active microwave imaging [3], [4], Meaney et al. have developed near-field tomographic image reconstruction algorithms which aim to recover the breast electrical properties [4]. Confocal microwave imaging (CMI) [5], [6] uses ultrawideband backscattered data collected by an antenna array and principles from confocal microscopy. A cylindrical CMI system has also been presented and analyzed with simulation data [7]. More recently, a method based on spacetime beamforming (MIST), which overcomes some of the shortcomings of previous CMI algorithms was also proposed [8], and encouraging experimental results with breast and tumor tissue phantoms have been presented [9]. Some experimental results for the cylindrical CMI system have also been reported [10], while the tomographic approach has also been clinically tested [11]. In previous work [1], [2], we examined the possibility of tumor detection and localization using the principles of time

Manuscript received March 22, 2005; September 28, 2005. This work was supported by the Center for Subsurface Sensing and Imaging Systems (CenSSIS) of Northeastern University, under the Engineering Research Centers Program of the National Science Foundation (Award Number EEC-9986821). P. Kosmas was with the Wireless Communications Research Group, Loughborough University, Leicestershire LE11 3TU, U.K. He is now with the Hellenic Navy, Athens, Greece (e-mail: [email protected]). C. M. Rapapport is with the Center for Subsurface Sensing and Imaging Systems, Northeastern University, Boston, MA 02115 USA. Digital Object Identifier 10.1109/TAP.2006.872670

reversal [12]. In [1], FDTD time reversal was tested for detecting a 3-mm diameter tumor-like anomaly in a semiellipsoidal, homogenous two-dimensional (2-D) breast model. The method’s robustness to using an approximation of the background medium was examined in [2]. The main assumption in that work was that the exact signal scattered from the scatterer is available for every element of the TRM, which is used for the data acquisition and virtual backpropagation of our imaging algorithm. In this paper, we present a complete analysis of the microwave breast cancer detection problem, in a realistic situation where the tumor response is obscured by clutter. The paper is structured as follows. In Section II, we review the properties of the FDTD TR algorithm, and present some additional simulations that show the robustness of the method to the background medium’s uncertainties for the microwave breast cancer detection problem. A matched filter technique based on maximum likelihood estimation, combined with algorithms developed to suppress clutter, is presented in Section III. Results of using the output of this technique as the estimated tumor response to be backpropagated with the TRM are presented and discussed in Section IV, and our findings are summarized in the conclusions section. II. TIME REVERSAL IMAGING FOR TUMOR DETECTION A. Mathematical Background Time reversal can be cast in the general context of the distorted-wave Born approximation (DWBA) [13]. The DWBA approach is based on the idea of expressing the scattering potential as a sum of a known term, and an unknown part which is assumed to be a small perturbation around the first dominant term [14]. The resulting scattered field in the frequency domain is given by [15] (1) In (1), is the object function to be imaged, which is the difference in the wavenumber of the scatterer in relation to the background medium . This object function is in general a function of position, and depends on the elecof the scatterer and the background. The intrical properties , and tegration is performed over the volume supporting the function is the (background) dyadic Green function, which is the solution to (2) , Notice that in (1), the total field inside the integral has been replaced by the field due to the background only

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. We note that the above definition of the object function is employed for this general mathematical description of linear inverse scattering problems, such as time reversal, but does not represent accurately the reconstruction objective of the TR process. In particular, the TR algorithm aims to localize the scatterer by treating it as a point source, which radiates the scattered field. No information on the size or shape of the tumor can be recovered from a simple application of the TR is not possible via process, and therefore full recovery of time reversal. In this context, the high contrast of the electrical properties of the tumor relative to the breast tissue that is assumed in the forward model does not break the validity of the DWBA approximation, since the scatterer is treated as a single in (1) is nonzero point, and therefore the object function only for an arbitrarily small region. B. Review of the FDTD TR Algorithm The properties and implementation aspects of our developed FDTD TR algorithm have been presented in previous works [1], [2], [16]. Here we just review some of its main aspects for completeness and clarity purposes. Our approach to time reversal is based on the regular FDTD update equations, modified to account for backpropagation. An important feature of the algorithm is its ability to compensate for the medium’s losses, by introducing gain via a change in the conductivity sign. This modification fully accounts for the medium’s dissipative properties, provided that they are not frequency-dependent [2], and it has been shown that it does not introduce instabilities in the presence of noise [16], [17]. In a dispersive medium, using the same FDTD TR update equations for the electric field partly compensates for the losses and dispersion effects [16]. In these update equations, a nominal value at some central frequency is of the nondispersive TR chosen for the electric properties field medium. As an example, the update equation for the component is given by

(3) This approach is different from inverse filter-type methods that have also been proposed to compensate for losses and dispersion effects, in acoustics [18], and recently in elecromagnetics [19]. The input signal for backpropagation at each receiver can be expressed as [2], (4) where is the time instant when the peak of the received waveoccurs for a short temporal pulse excitation, form and is the taper parameter, which determines the width of the impulse temporal window function. After backpropagation, the time-reversed field reaches its peak at the optimal time instant when it focuses back to the source. This is due to the spatio-temporal matched filtering achieved by the TR process [20]. In sit-

uations where this maximum criterion may lead to a false image due to limited data and background information, an additional minimum entropy criterion, based on the inverse varimax norm, is used [2]

(5) are where is the time step of the FDTD TR algorithm, the grid cell coordinates, and summation is over the portion of the grid that represents the breast. This approach guarantees a focused image, but may not correspond to the maximum power at the focus, which yields the optimal image in the ideal case. Therefore, it may lead to small localization errors and should be used as an additional criterion in the case where the simple maximum criterion fails. C. Forward Model for Data Acquisition The 2-D MRI breast model and system geometry used to generate synthetic data for our reconstructions is identical to that of [2]. In this paper, however, we consider fully dispersive tissues, and a higher variability in the breast and fibroglandular tissue, resulting to a lower contrast between the breast and tumor properties. The proposed TR system is comprised of an array of 23 receivers located at a small distance from the top of the breast, and placed every , with mm. The middle element is also used as a transmitter. This source location ensures that reflections from the chest wall do not obscure detection of tumors distant from the wall [21]. The transmitter is a point source as( geometry), excited by a differsigned an electric field entiated Gaussian pulse of 50 ps width, and each receiver corresponds to an observation point. Dispersive models for normal breast fat and tumor tissue, based on a single-pole conductivity model and an average permittivity value, have been derived in [21], based on published data [22]. To examine the performance of our proposed algorithm in cases of high tissue variability and dense fibroglandular regions, we considered a 40% upper bound of variability in the breast fat regions, and a 33% variability in the fibroglandular regions, which were assigned three to five times higher permittivity and conductivity values than those of normal breast fat. The dispersive models for all the different tissues were derived from the existing dispersive breast fat model, with a linear perturbation of its coefficients, and were mapped to the denser regions of the MRI measured data. The mapped MRI breast model with sample values of different tissues is shown in Fig. 1. To test our algorithms, we have also artificially introduced into S/m) of thickness the model a skin layer ( varying with position from 1.5 to 2 mm, and a 3-mm tumor-like anomaly, in various positions. The surrounding medium is a , which lossless dielectric with dielectric constant is relatively close to that of breast fat in the frequency range of interest. D. Robustness to Skin and Tissue Variability Simulation studies on the effect of the skin and the breast inhomogeneities on TR focusing were first performed in [2].

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Fig. 1. 2-D MRI-based breast model for the 2-D FDTD grid, used to generate the data for input to the TR system. The different regions of the MRI-based data (top) are mapped to various dispersive tissues with different electric properties, shown at the bottom plots for the frequency range of 1–10 GHz. The skin layer and tumor-target, shown in white, are artificially introduced.

Fig. 2. Normalized time-reversed electric field scattered from the target for the breast model of Fig. 1. The breast in the forward model is homogeneous, comprised of nondispersive breast fat tissue and the skin layer. In (a), the skin layer is used in the TR process, while in (b) it is ignored. The target is marked with a black circle.

Here we study further this effect with the help of the previous analysis of Section II-A. In reference to (1), the input of the FDTD TR algorithm at the th TRM element is (the field scattered by the tumor-like target). We note that exact knowledge of in (1) implies that the properties of the medium are fully known. As with most imaging applications, the background medium may not be fully available for the microwave breast cancer detection problem, due to uncertainties related to the tissue inhomogeneities, as well as the skin thickness and electrical properties. Thus, only an approximation of , can be used in the backpropthis medium, denoted as agation process, via (1). Two different approximations in the background medium, the skin and the breast tissue variability, are studied below. 1) Effect of the Skin: The effect of the skin layer in the backpropagation process was first examined in [2], where it was shown that similar focusing can be achieved in a TR model, which ignores the skin, from forward data that may or may not include the skin in the breast model. Here, we examine the effect of the skin in the TR process further, by comparing images with and without the skin present in the TR model, from forward data that includes the skin in the breast model. To compare the present results with those of [2], we consider the forward model of Fig. 1, with a homogeneous breast comprised of the nondispersive normal breast fat tissue, the skin layer and the tumor-target. We obtain the exact field scattered from the tumor-scatterer by running the forward model with and without the tumor and subtracting the results from the two runs at each receiver. We then backpropagate this scattered field in a breast model for two cases: (a) with the skin layer present, and (b) with the skin layer ignored and filled with normal breast fat. The simulation results for the optimal focused image (based on the maximum criterion) are shown in Fig. 2. According to this figure, the quality of the focusing achieved without the skin layer is superior to the case where the skin layer is present in the background model. In other words, the approximate medium leads to a better focused image that the true

medium. This somewhat unexpected result can be attributed to the fact that we time-reverse only a small part of the scattered wave, the part captured by the receiver array. The skin’s interaction with this time-reversed scattered wave will cause a significant part of it to be reflected. Only a fraction of the true scattered field will be transmitted in the breast and focus onto the target. This nonphysical, ghost wave can be observed in the area around the skin in plot (a) of the figure. On the other hand, omitting the skin from the model in (b) may be causing an error in the true propagation velocity inside the skin layer, but this error is negligible due to the small thickness of the layer. Thus, for a limited array TRM configuration where only echoes of the target signal can be recorded, the effect of the ghost wave can be significant for a single iteration of the time reversal process. Ghost waves have also been observed in TR simulations of propagation in a solid-liquid interface [23]. 2) Effect of Tissue Variability: In [2], it was shown that the omission of the inhomogeneities distribution from the TR model will not affect focusing, for a small (16%) upper bound of variability in the breast tissue. In this section, we consider the fully dispersive breast model of Fig. 1, to examine the robustness of the algorithm to breast inhomogeneities and tissue variability for the more challenging present case: as explained in Section II-C and shown in Fig. 1, the tissue variability is higher, and the fibroglandular region is distinct with considerably higher values of electric properties, resulting in a smaller contrast with the properties of the tumor-like target. In addition, the assumed TR medium, which is a nondispersive breast fat simulant with avand S/m, has quite different erage values of properties from the fully dispersive and inhomogeneous forward breast model. The skin is also present in the forward model, as it was shown in the previous section to create no problems in the focusing of the TR process. Results from this simulation are shown in Fig. 3. To counterbalance the effect of the pulse spread due to dispersion, which is not accounted for in the approximate TR medium, a smaller taper parameter was used for the impulse function in (4) than

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and its size, and can, thus, be considered random within a prespecified value range. Writing these two parameters in a vector , we can estimate the random vector by maxform imizing the log-likelihood function [25] (7)

Fig. 3. Same as Fig. 2, but for the full forward model of Fig. 1, and a TR model that considers the breast homogeneous, filled with nondispersive breast fat simulant.

that of Fig. 2. It is clear from the figure that good focusing is achieved despite the various approximations in the TR model, when the exact scattered field from the target is available. Since in a realistic application this signal can only be estimated, the remaining of this paper focuses on techniques and results that include this important detection aspect, i.e., the estimation of the target response at each receiver. III. ESTIMATION OF THE TARGET RESPONSE A. Matched Filter Approach All of the simulations considered so far in the present work and in [2] have assumed that the portion of the received signal due to skin reflections, which are the main source of clutter, can be perfectly removed. This section examines techniques to estimate the tumor signal at each receiver in the case where no a priori information on the skin layer and breast inhomogeneities is available. The method proposed here is similar to a matched filter technique that has been used previously in land mine detection applications [24]. Application of the matched filter algorithm requires that information on the characteristics of the target response is available. In the study of the forward model in [2], it was shown that the shape of the tumor-scatterer response is very similar to the total reflected signal from the skin. This is particulary true for the area of interest around the peak of the target response, where the temporal window is applied. Thus, the input of the matched filter algorithm that corresponds to the target signal at receiver-i will be of the form

where denotes physical logarithm, is the conditional is (posterior) probability density function of given is the marginal the joint density function of and , and density of the observation variable . The maximum likelihood estimate obtained by maximizing (7) is known as the maximum a posteriori probability (MAP) estimate of [25]. Having defined , we now identify the remaining parameters in (7) for our problem. The observation vector is the vector of the total received field (without the tumor response) at receiver-i, which can be written as (8) In (8), is a (deterministic) temporal signal vector, which corresponds to a prediction of the skin-breast contributions at receiver-i. In relation to the analysis in Section II, are approximations of the background and tumor-target fields, respectively ( denotes discrete time). were identical Referring to Fig. 1, if the skin-breast artifact at each receiver, then in (8) would be zero. However, variability in the skin thickness and the breast tissue will lead to a deviation of from the reference signal , represented by , which cannot be taken into account in predicting . If these variations are assumed to be random, then the signal can be modeled as a Gaussian random vector with zero mean and variance equal to the mean square error of around . We note that this approach may not be the optimal way to model , but serves as a good candidate when no additional information on the clutter sources is available. Under the Gaussian distribu, and denotes the tion assumption, we have data samples, the probability normal distribution. Then, for density function in (7) is given by (9)

(6) In (6), denotes the early-time portion of the total reflected field response, which is due to the skin reflections, at the 12th receiver (which is also the transmitter and is used as our reference), is a time delay, which depends on the relative location of receiver-i to the tumor and the transmitter, and is a scaling factor, which is the peaks ratio of the two signals and . We note that other ways of estimating the tumor signal are also possible. To confirm this, we have run a simulation with homogeneous breast fat, no skin, and a target of arbitrary size and location, used the resulting recorded response from the target as , and obtained similar results to those with the estimate of the skin estimate, which are presented later. Estimating the tumor response from the total field is equivalent to estimating the parameters and in (6). These parameters depend on the position of the tumor relative to the receivers

In the same fashion, the total field will follow a normal distribution in the case where a target is present, but this time . Thus, the joint probability density function with mean will be given by

(10) We can then compute the log-likelihood ratio of (7) as

(11)

KOSMAS AND RAPPAPORT: MATCHED-FILTER FDTD-BASED TIME-REVERSAL APPROACH

and expand the squares to obtain (12) From (12), it is clear that is a correlation-type operation where the signal after clutter removal, expressed as is correlated with the tumor signal . In fact, this operation is equivalent to the correlation between the two signals expressed in the form (13) is invariant to where denotes average energy, but while the scaling factor , (12) is a function of both and . The MAP estimate for the parameters and is then found by looking for the values of these two parameters, which maximize the quanin (12). Thus, our algorithm for estimation of the tity tumor-like target response at receivers i=1 to 23 is outlined as follows: 1. Calculate the reference signal, and the residual . 2. Calculate (12) as function of for an initial guess for , and locate the peak corresponding to a tumor signal. 3. Vary , and find the value that maximizes the peak of previous step. as the estimate of the tumor 4. Use response. , We should note that depending on the form of the signal in (12) may correspond to the maximum location of the target response, or to a peak due to clutter. The correlation between the location of a peak with corresponding locations at the neighboring receivers can aid in determining the peak of the target response. More details on the use of the above algorithm will be given in the next section, where simulation results are presented. For now, we conclude this section with a discussion on ways to obtain the reference signal in the first step of this algorithm. B. Estimation of the Skin-Breast Artifact As explained in the previous section, represents a skinbreast reference signal, which can be obtained by any algorithm that aims to estimate this signal from the total data without corrupting the tumor-scatterer response. This latter requirement is essential in order for the matched filter to produce an accurate estimate of the target signal. If clutter from skin and inhomogeneities did not temporally interfere with the target response, that response could be time-gated or the matched filter could be applied directly to the total data . However, observation of the shows that it temporally interferes with and, thus, signal it is not uncorrelated from the tumor response. If estimation of does not affect the target signal, application of the matched filter algorithm will yield a successful extraction of the tumor response, leading to good localization and resolution for the TR process. is an essential step prior to appliAlthough estimation of cation of the matched filter algorithm presented in the previous

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section, we emphasize that the two processes are independent, can be used in and any technique which aims to estimate conjunction with the matched filter algorithm. Two such techniques are applied in this work. Both these techniques are based on a statistical estimation approach that uses the total field at each receiver to estimate and requires no a priori information on the skin or inhomogeneities properties. The first technique is an averaging method which has been successfully applied previously to reduce the roughness effect in GPR land mine detection applications [26]. It consists of aligning and scaling the signal at each receiver in relation to the reference signal, in order to obtain an average. This average is then subtracted from each individual signal at each receiver after it is again appropriately scaled and aligned relative to each signal. The second method that is applied in the present work is an algorithm presented in [8] for estimating the clutter. The algorithm, based on a least mean square (LMS) estimation of the signal at each receiver from the signal at all other receivers, assumes a strong similarity in the early-time (prior to tumor) responses at the receivers. This is true for our configuration once these signals are aligned and scaled properly. Certainly, other approaches to this problem are also applicable. For example, an FDTD inverse-scattering scheme for determination of the electrical properties of the skin was proposed in [27]. This or other similar approaches may be able to predict the desired signal more accurately, but depend on accurate a priori information on some of the skin parameters, as its thickness or electrical properties, and therefore will not be considered here.

IV. RESULTS AND DISCUSSION In this section, we present results of applying the previously presented methods for estimation of the target signals, and then backpropagating these signals via the FDTD algorithm for localizing the tumor-scatterer. We also show how the properties of the TR process aid in avoiding false alarms. For our simulations, we have considered three target locations, at , and cm, which are referred to as the first, second, and third tumor, respectively. To illustrate some of the possible successes and failures of the matched filter algorithm, we first apply it to data after clutter is suppressed using the averaging method discussed in the previous section. Sample of the results for the second target and two receiver positions are shown in Fig. 4. For the first sample , the target signal is not distorted by the clutter removal process, and the matched filter detects this signal with great accuracy. On the other hand, the target signal is obscured can for the receiver-18, and an early peak in the residual lead to false identification of the target signal, which can affect the focusing in the TR process. This false identification can be avoided by comparing the arrival time of the peak in question with corresponding times in other receivers. Then, it is clear that this peak occurs much earlier than anticipated, and thus the output of the matched filter is zero. Receivers 19–23 also fail to identify the tumor signal, and in all these cases the failure

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Fig. 4. Samples of the estimated response for the second tumor-target using the suggested matched filter approach and an averaging technique for suppressing clutter (d denotes the remaining total field after clutter removal, s is the estimated target response based on the matched filter, and t the exact target response, which is plotted for comparison).

is explained by the fact that the target response interferes significantly with the skin reflections, because the tumor-scatterer is located close to the skin. It is therefore possible that for certain tumor locations some of the receivers with strong target responses may not be used in the TR process. We have also applied the matched filter in conjunction with the LMS algorithm. The algorithm is quite efficient in cancelling the early skin reflections, but at the expense of some distortion on the target response. This is particularly true for our receiver geometry, where the portion of the data where skin reflections are dominant depends on the target and receiver location in relation to the transmitter. We have found that it is preferable to choose the algorithm parameters in a way that leads to imperfect clutter removal, but also distorts less the target response. The results we obtain after application of the algorithm, for the same case as in Fig. 4, are shown in Fig. 5. Removal of clutter is successful for both receivers, but some distortion on the target signal leads to a matched filter output that significantly underestimates the target response for receiver-18. A graphic illustration of the steps involved in the matched filter algorithm analyzed in Section III.A is shown in Fig. 6. as function of for the first target and Here we plot is calculated for different values of . The value of , with sampled every two time steps between and . There is a clear peak at , and its value is maximized for . Based on that information, the tumor/target signal is estimated as . The values of the estimated parameter , which represents the signal to clutter (S/C) ratio, are chosen , approximately 60 to 35 within the range dB down from the skin reflections, a range commonly observed in various microwave breast cancer FDTD simulations [6], [21].

Fig. 5.

Same as Fig. 4, when clutter is suppressed using the LMS algorithm.

Fig. 6. Example of the log-likelihood ratio (12) as function of the parameters n and a, for the first target and the 10th receiver. The peak corresponds to a signal centered at the n = 234th time step, and a = 0:0054 is the optimal scaling coefficient.

Results of the FDTD TR algorithm for the three tumor locations are shown in Fig. 7 for the averaging method, and in Fig. 8 for the LMS algorithm. In these cases of imperfect scattered data, the minimum entropy criterion (5) was used for determining the optimally focused image. These figures show that detection and localization is successful for the first two tumor locations, with less accuracy and lower resolution than what was observed when the target signal is fully known [2]. For the third location, estimation of the target response is not sufficiently accurate in order to lead to good localization. This is due to the small S/C ratio for this tumor location, which leads to significant errors in the estimation of the target response. For such locations, detecting the tumor from the overall clutter becomes a difficult task with the present receiver array, and a circular system that encircles the breast may be preferable.

KOSMAS AND RAPPAPORT: MATCHED-FILTER FDTD-BASED TIME-REVERSAL APPROACH

Fig. 7. Normalized rime-reversed focused electric field for the breast model of Fig. 1 and three target locations, after application of the matched filter method, and the averaging technique for suppressing clutter. In all cases, the receivers used in the TR process (nonzero output of the matched filter) are marked with asterisks, and the tumor-target locations are marked with small stars.

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Fig. 9. Illustration of the matched filter TR approach for a tumor-free case. The inverse varimax norm (5), used for determining the optimal image at the time step where a minimum is reached, attains a minimum corresponding to a nonfocused image, shown at the bottom of the figure.

images will have a much lower peak field value at the focus than in the cases where a target is present, and, therefore, a threshold can be set to determine if they correspond to a target or a false estimation of a peak corresponding to clutter. V. SUMMARY AND CONCLUSION

Fig. 8. Same as Fig. 7, after using the LMS algorithm in conjunction with the matched filter technique.

The figures show that the matched filter algorithm in conjunction with FDTD time reversal perform quite well in either case, despite the fact that only approximate cancellation of the clutter effect is achieved. Fundamental properties of the TR mechanism also guarantee that false alarms are avoided to some extent, because outputs of the matched filter technique to tumor-free data will add incoherently. This important advantage of the method is illustrated in Fig. 9. Here we have generated tumor-free data from the forward model and chosen the outputs corresponding to the peaks of (12) (without comparing them from receiver to receiver). As shown in the top plot of Fig. 9, the inverse varimax norm in (5) attains a minimum, which corresponds to an image not focused anywhere inside the breast. We note, however, that if we try to correlate the response from each receiver in the same way that was done for the previous cases with a target present, false alarms may occur due to sufficiently low values of the S/C ratio. The resulting

This paper further discusses our new approach to microwave breast cancer detection, based on numerical time reversal with the FDTD method, which was presented in previous work. A key aspect of this approach is the separation of the problem to two distinct processes: (a) the estimation of tumor response with a signal processing technique at each TRM receiver, and (b) the backpropagation of the (temporally windowed) estimated response via the FDTD TR algorithm. The robustness of the second process to inhomogeneities and uncertainties in the skin thickness and properties implies that the overall performance depends on the success of the first step, i.e., extracting the tumor response accurately from the total received field, which is dominated by clutter. Various methods based on statistical or physics-based signal processing techniques can be used to address the tumor response estimation problem. In the present paper, we present a matched filter technique, inspired by similar work on land mine detection applications. The matched filter is applied to a signal obtained after subtraction of an estimated signal, which ideally should remove most of the clutter without distorting the tumor response. Application of the matched filter technique is illustrated with simulations,andour resultsshowthatdetectionandlocalizationofthe target can be achieved, even when the tumor response estimation is only successful for a limited number of receivers. In addition, the properties of the TR system aid in avoiding false alarms. In future work, the 3-D extension of our TR algorithm will be presented. Our results from 3-D simulations show that the TR algorithm performs well, even when only one vector component of the signal is used as input in the TR process, as will possibly be the case in an experiment with realistic antenna elements. The robustness to measurement noise has already been examined [16]. Other aspects of detection in relation to time reversal can also be explored; for example, application of the

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DORT method [28] for selective focusing can aid in the detection of multiple tumors. The present results encourage us to believe that the proposed method is a promising technique for microwave breast cancer detection, which can be applied to different geometries and data acquisition systems. ACKNOWLEDGMENT P. Kosmas would like to thank Professors E. Miller and A. Devaney and graduate students M. K. Miled and G. Boverman at Northeastern University, Boston, MA, for helpful discussions on various components of this paper. The authors would also like to thank Dr. M. Doyley, Darthmouth College, Hannover, NH, for providing the MRI scan of the breast, and their reviewers for their helpful comments and suggestions. REFERENCES [1] P. Kosmas and C. Rappaport, “Use of the FDTD method for time reversal: Application to microwave breast cancer detection,” in SPIE Proc., vol. 5299, San Jose, CA, Jan. 2004, pp. 1–9. [2] P. Kosmas and C. M. Rappaport, “Time reversal with the FDTD method for microwave breast cancer detection,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 7, pp. 2317–2323, Jul. 2005. [3] Z. Q. Zhang and Q. H. Liu, “Three-dimensional nonlinear image reconstruction for microwave biomedical imaging,” IEEE Trans. Biomed. Eng., vol. 51, pp. 544–548, Mar. 2004. [4] D. Li, P. M. Meaney, and K. D. Paulsen, “Confocal microwave imaging for breast cancer detection,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 4, pp. 1179–1186, Apr. 2003. [5] 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, pp. 1470–1479, Dec. 1998. [6] E. C. Fear, X. Li, S. C. Hagness, and M. A. Stuchly, “Confocal microwave imaging for breast tumor detection:localization in three dimensions,” IEEE. Trans. Biomed. Eng., vol. 49, pp. 812–822, Aug. 2002. [7] E. C. Fear and M. A. Stuchly, “Microwave detection of breast cancer,” IEEE Trans. Microw. Theory Tech., vol. 48, pp. 1854–1863, Nov. 2000. [8] E. J. Bond, X. Li, S. C. Hagness, and B. D. Van Veen, “Microwave imaging via space-time beamforming for early detection of breast cancer,” IEEE Trans. Antennas Propag., vol. 51, pp. 1690–1705, Aug. 2003. [9] X. Li, S. K. Davis, S. C. Hagness, D. Van der Weide, and B. D. Van Veen, “Microwave imaging via space-time beamforming: Experimental investigation of tumor detection in multi-layer breast phantoms,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 8, pp. 1856–1865, Aug. 2004. [10] E. C. Fear, J. Sill, and M. A. Stuchly, “Experimental feasibility study of confocal microwave imaging for breast tumor detection,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 887–892, Mar. 2003. [11] P. M. Meaney, M. W. Fanning, D. Li, S. P. Poplack, and K. D. Paulsen, “A clinical prototype for active microwave imaging of the breast,” IEEE Trans. Microw. Theory Tech., vol. 48, pp. 1841–1853, Nov. 2000. [12] M. Fink, D. Cassereau, A. Derode, C. Prada, P. Roux, M. Tanter, J. L. Thomas, and F. Wu, “Time-reversed acoustics,” Rep. Prog. Phys., vol. 63, pp. 1933–1995, 2000. [13] S. Lehman and A. J. Devaney, “Transmission mode time-reversal superresolution imaging,” J. Acoust. Soc. Amer., vol. 113, pp. 2742–2753, 2003. [14] J. R. Taylor, Scattering Theory, FL: R. Krieger Inc., 1983. (reprint). [15] W. C. Chew, Waves and Fields in Inhomogeneous Media. New York: Van Nostrand Reinhold, 1990. [16] P. Kosmas and C. Rappaport, “FDTD-based time reversal for microwave breast cancer detection: Robustness to dispersion and measurement noise,” in Proc. ICONIC 2005, Barcelona, Spain, Jun. 2005, pp. 72–77. [17] P. Kosmas, “FDTD modeling for forward and linear inverse electromagnetic problems in lossy, dispersive media,” Ph.D. dissertation, Northeastern Univ., Boston, MA, Jan. 2005. [18] G. Montaldo, M. Tanter, and M. Fink, “Revisiting iterative time reversal processing: Application to detection of multiple targets,” J. Acoust. Soc. Amer., vol. 115, no. 2, pp. 768–775, Feb. 2004. [19] M. E. Yavuz and F. L. Texeira, “Frequency dispersion compensation in time reversal techniques for UWB electromagnetic waves,” IEEE Geosc. Remote Sens. Lett., vol. 2, pp. 233–237, Apr. 2005.

[20] M. Fink, “Time reversal of ultrasonic fields-Part I: Basic principles,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 39, pp. 555–566, Sep. 1992. [21] P. Kosmas and C. Rappaport, “Modeling with the FDTD method for microwave breast cancer detection,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 8, pp. 1890–1897, Aug. 2004. [22] S. Gabriel, R. W. Lau, and C. Gabriel, “The dielectric properties of biological tissues: II. Measurements on the frequency range 10 Hz to 20 GHz,” Phys. Med. Biol., vol. 41, pp. 2251–2269, Nov. 1996. [23] C. Tsogka and G. Papanicolaou, “Time-reversal through a solid-liquid interface and super-resolution,” Inverse Prob., vol. 18, pp. 1639–1657, 2002. [24] R. Linnehan, “Mitigating ground clutter effects for mine detection with lightweight artificial dielectrics,” M.S. thesis, Northestern Univ., Sep. 2002. [25] L. L. Scharf, Statistical Signal Processing. New York: Addison-Wesley, 1991. [26] C. M. Rappaport, El-Shenawee, and H. Zhan, “Suppressing GPR clutter from randomly rough ground surfaces to enhance nonmetallic mine detection,” Subsurface Sensing Technol. Appl., vol. 4, pp. 311–326, Oct. 2003. [27] M. Popovic and A. Taflove, “Two-dimensional FDTD inverse-scattering scheme for determination of near-surface material properties at microwave frequencies,” IEEE Trans. Antennas Propag., vol. 52, no. 9, pp. 2366–2373, Sep. 2004. [28] C. Prada, “Detection and imaging in complex media with the D.O.R.T method,” in Topics Appl. Phys.. Berlin, Heideberg: Springer-Verlag, 2002, vol. 84, pp. 107–133.

Panagiotis Kosmas (S’03–M’05) received the Diploma in electrical and computer engineering from the National Technical University of Athens, Greece, in 1999, and the M.S. and Ph.D. degrees in electrical engineering from Northeastern University, Boston, MA, in 2002 and 2005, respectively. From January 2000 until February 2005, he worked as a Research Assistant with the Department of Electrical Engineering and the Center for Subsurface Sensing and Imaging Systems, Northeastern University, Boston, MA. From April 2005 to January 2006, he was a Postdoctoral Research Associate at the Wireless Communications Research Group, Loughborough University, U.K. He is currently serving in the Hellenic Navy. His research interests include computational electromagnetics, and the finite-difference time-domain method, in particular, periodic structures, as well as inverse problems and signal processing techniques.

Carey M. Rappaport (S’80–M’87–SM’96–F’05) received the S.B. degree in mathematics, the S.B., S.M., and E.E. degrees in electrical engineering in June 1982, and the Ph.D. degree in electrical engineering in June 1987, all from the Massachusetts Institute of Technology (MIT), Cambridge. He has worked as a teaching and research assistant at MIT from 1981 to 1987, and during the summers at COMSAT Labs, Clarksburg, MD, and The Aerospace Corp., El Segundo, CA. He joined the faculty of Northeastern University, Boston, MA, in 1987, where he has been Professor of Electrical and Computer Engineering since July 2000. During Fall 1995, he was a Visiting Professor of Electrical Engineering at the Electromagnetics Institute of the Technical University of Denmark, Lyngby, as part of the W. Fulbright International Scholar Program. He has consulted for Geo-Centers, Inc., PPG, Inc., and several municipalities on wave propagation and modeling, and microwave heating and safety. He was Principal Investigator of an ARO-sponsored Multidisciplinary University Research Initiative on Demining and Co-Principal Investigator of the NSF sponsored Center for Subsurface Sensing and Imaging Systems (CenSSIS) Engineering Research Center. He has authored more than 200 technical journal and conference papers in the areas of microwave antenna design, electromagnetic wave propagation and scattering computation, and bioelectromagnetics, and has received two reflector antenna patents, two biomedical device patents, and three subsurface sensing device patents. Prof. Rappaport is a member of Sigma Xi and Eta Kappa Nu professional honorary societies. He was awarded the IEEE Antenna and Propagation Society’s H.A. Wheeler Award for Best Applications Paper, as a student in 1986.