Electromagnetic Time-Reversal-Based Target Detection in Uncertain Media

Transcription

1 Electromagnetic Time-Reversal-Based Target Detection in Uncertain Media Dehong Liu, Jeffrey roli and Lawrence Carin Department of Electrical and Computer Engineering Due University Durham, NC Abstract An experimental study is performed on imaging targets that are situated in a highly scattering environment, employing electromagnetic time-reversal (ETR) methods. A particular focus is placed on performance when the electrical properties of the bacground environment (medium) are uncertain. It is assumed that the (unnown) medium characteristic of the scattered fields represents one sample from an underlying random process., with thisthis random process representing represents our uncertainty in the media properties associated with the scattering measurement (for example, based on a set of Green s function measurements performed for similar media). While the specific Green s function associated with the scattered fields is unnown, we assume access to an ensemble of Green s functions sampled from the aforementioned distribution. This ensemble of Green s functions may be used in several ways to mitigate uncertainty in the true Green s function. Specifically, when performing time-reversal imaging we consider a Green s function representative of the average of the ensemble, as well as Green s functions based on a principal components analysis (PCA) of the ensemble. We also develop a wideband Minimum Variance beamformer with Environment Perturbation Constraints (MV-EPC), in which the unnown Green s function is constrained to reside in a subspace spanned by the Green s-function ensemble. These algorithms are examined using electromagnetic scattering data measured in a canonical set of laboratory experiments. Qualitative performance of the different techniques is presented in the form of images, with quantitative results presented in the form of receiver operating characteristic (ROC) performance.

2 I. Introduction Time reversal is a technique that exploits the multipath characteristic of a complex media. Specifically, assume that a source emits a pulse of energy into a complex, multiscattering environment, and that propagated energy is observed on a finite set of receiving antennas. The multi-path phenomenon causes a temporally localized excitation to be temporally extended as viewed at the receiver antennas, with the signals arriving later in time having traveled a longer distance than the corresponding early arrivals. If the time-domain signals are time reversed, and coherently re-radiated from the antenna array, the signals arrive coherently at the original source location, approximately reconstructing the original source [-3]. In the above discussion the time-reversed fields were assumed to be physically reradiated into the original domain, but this process may also be performed computationally, with an approximate medium Green s function, thereby providing the opportunity to perform imaging. In the wor presented here we are interested in the problem of time-reversal imaging, particularly for cases in which the exact Green s function is uncertain. Moreover, the above discussion considered imaging a localized source; in the wor presented here we are interested in imaging a target based on scattered fields, for which the problem complexity is heightened by the need to consider the two-way Green s function (from the source to the target and from the target to the receivers). Time-reversal has important implications for imaging and detecting targets situated in a complex medium. or example, in conventional imaging the focusing resolution is limited by the size of the sensor-array aperture. owever, time-reversal imaging may have an effective aperture [4-6] much larger than that of the physical array, especially in a complex, highly scattering medium. Interestingly, the more complex the media the more propagation paths are manifested from a source to the receiver array, improving focusing quality. rom this viewpoint, when we try to image targets situated in a complex environment, for which conventional methods lie synthetic aperture radar (SAR) may 2

3 yield poor imaging quality due to the highly scattering medium, time reversal may achieve super-resolved images even with a small physical array [7]. owever, this superresolution property is based on the assumption that the Green s function used in time-reversal imaging is the same as that associated with the scattered fields, with this assumption difficult to meet in practice. Specifically, when performing time-reversal imaging, for which time reversal of the measured scattered fields is performed with a numerical model, one must address the inevitable differences between the true Green s function characteristic of the scattered fields and the simulated Green s function employed when imaging. The refocusing performance of the time-reversal technique degrades when the medium through which time-reversal imaging is performed is different from that corresponding to the actual physical measurement [8,9]. Wor addressing target detection in changing or uncertain environments has been carried out experimentally and theoretically with acoustic waves in random ocean channels for underwater target detection, termed matched-field processing (MP) [-3]. In particular, the Minimum Variance beamformer with Environment Perturbation Constraints (MV-EPC) [] is a well-nown algorithm that sees to improve imaging robustness in the presence of an uncertain sound-speed profile. This algorithm has been demonstrated to be effective in multiple experiments in imaging underwater targets using a large aperture array with narrowband acoustic waves. The analogy between MP and time reversal provides useful guidance for robust timereversal imaging in a changing environment. or example, in [4-8] time-reversal refocusing quality is examined in an ocean channel, as the acoustic medium changed with time. Our objective in this paper is to image and detect targets situated in a highly scattering environment using electromagnetic time-reversal (ETR) based techniques. A particular focus is placed on performance when a small antenna aperture is employed; in addition, we consider cases for which the true media characteristic of the scattered fields is uncertain. If one does not now the precise Green s function of the media under test, other information must be exploited to obtain useful imaging results. We treat the 3

4 bacground Green s function as a random process, with the true (unnown) Green s function representing one sample from the random process. While the true Green s function is unnown, we assume access to an ensemble of Green s functions, these representing multiple realizations of the underlying random process. The ensemble of Green s functions may now be exploited when performing inversion. or example, one may use the average of the ensemble as the Green s function in numerical time-reversal imaging, or alternatively one may employ a principal components analysis (PCA) [9], with the principal eigen Green s functions used in time-reversal imaging. We consider these approaches, as well as a wideband extension of minimum variance with environmental perturbation constraints (MV-EPC) []. The MV-EPC algorithm performs imaging based on a constrained optimization, with the constraint based on the idea that the actual Green s function must reside in the subspace spanned by the aforementioned ensemble of Green s functions. We extend the MV-EPC to situations of interest in wideband time-reversal imaging (previous MV-EPC wor assumed large apertures and narrowband signals []). The imaging algorithms are examined based on a controlled set of electromagnetic scattering measurements. Specifically, we consider quasi-two-dimensional experiments, with the complex propagation media composed of a closely paced set of 68 randomly distributed dielectric rods. The different realizations of the random media are constituted by redistributing the positions of the rods. The measured scattered fields are performed for a target placed within one such media, with the ensemble of Green s functions corresponding to distinct (but statistically related) rod placements. The remainder of the paper is organized as follows. A brief review of time-reversal basics is provided in Sec. II, with a discussion of the literature for time reversal applied to situations of uncertain (or changing) media. In Sec. III techniques are described for performing time-reversal imaging in uncertain environments, and in Sec. IV we describe the experimental system used for algorithm evaluation. Example imaging results are presented in Sec. V, followed by conclusions in Sec. VI. 4

5 II. Theory of Time Reversal in ixed and Changing Media A. Superresolution A ey property of time-reversal sensing is that multiple scattering manifested by a complex (cluttered) environment may be used to constitute an effective aperture that is larger than the actual physical aperture [4-6]. In ig. we consider a two dimensional problem with a line-source radiator, with the fields observed along a linear aperture. In ig. (a) the source radiates in a homogeneous medium, while in (a) ig. (b) the source radiates in a medium characterized by multi-path (clutter). Because of the homogeneous medium in which the fields radiate, in ig. (a) the range of observed angle-dependent wavefronts is dictated by the size of the linear (b) aperture and the distance of the source to the igure. Wavefronts emitted from a aperture (these parameters, as well as the source, observed at a linear antenna array (at left). (a) Source in a wavelength, dictate the well-nown imaging homogeneous medium, (b) in a highly multi-pathed environment. resolution [4-6]). By contrast, in ig. (b) a richer set of waves emitted from the source arrive at the linear receiver aperture, as a result of multi-path. rom ig. we therefore note that the number of different source wavefronts (emitted at different angles) observed by the receiver increases with increasing multi-path. When the data observed on the linear aperture is time reversed and synchronously reradiated, the multiple angle-dependent wavefronts emitted from the source are bacpropagated, coalescing at the original source. Since the diversity of different angledependent waves arriving at the focus point increases with increasing clutter, the resolution (tightness) of the focus increases commensurately. This phenomenon has been examined in detail theoretically [4], and has been demonstrated in previous experiments [-3]; it has been referred to as superresolution, because in a cluttered environment the focusing is tighter than would be expected in a homogeneous medium for the same physical aperture size. 5

6 or simplicity the above discussion assumed a point radiator. In the wor of interest here we consider time-reversal based on scattered fields, and therefore one must account for the two-way Green s function [7]. In this case the Green s function from the source antenna(s) to the target constitutes multiple wavefronts, impinging on the target from multiple angles. The scattered field may be represented as waves emitted from equivalent sources on the target [7], and in the asymptotic limit (wavelength small with respect to the target dimensions) localized scattering centers dominate. Each of these localized scattering-center equivalent sources is analogous to the single source considered in ig.. The number of scattering centers illuminated is dictated by multi-path propagation from the source antenna(s) to the target; superresolution image quality is a consequence of the multipath from the scattering centers (equivalent sources) to the receiving antenna(s). B. Correlations When there is a mismatch between the Green s function employed for time-reversal imaging and that characteristic of the scattered fields, one must address the deterioration in imaging quality. Researchers have theoretically investigated time-reversal imaging quality when there is a mismatch between the true Green s function characteristic of the scattered fields and the Green s function used when performing time reversal. In this previous research [2] it was assumed that the observed fields were manifested from a source (i.e., the scattering problem wasn t considered); we review this literature, to motivate the imaging techniques considered for the scattering problem, when the true Green s function is uncertain. Time reversal of a source in the frequency domain is the product of two Green s function propagators: the first one sends the signal from the source to the array of detectors, and the second one bacpropagates the signal from the detector array. or the case of different Green s functions in the forward and inverse phases, the product of operators has a ernel represented by the correlation function of the two corresponding Green s functions. In the Lippman-Schwinger representation of the wave equation [2], the 6

7 correlation of the two Green s functions is proportional to the power spectrum of the random heterogeneities of the underlying heterogeneous media. If we denote by δc (x) the fluctuations in the propagation speed of the underling medium, relative to a reference homogeneous medium, the correlation function as a function of position x is defined by R ( x ) = δc( y) δc( x + y), which is independent of y when the fluctuations are statistically independent by translation. The power spectrum R ˆ( ) is defined as the ourier transform of the above correlation. When the underlying medium does not change between the two phases of the time reversal experiment, it is well nown that the correlation of the two Green s function solves a radiative transfer equation in the phase space [2]. When the two media are different, then the radiative transfer equation must be modified. The salient feature of this modification is the following: the scattering coefficient in the radiative transfer equation, which was proportional to the power spectrum of the heterogeneities, is now proportional to the cross-correlation of the two sets of heterogeneous fluctuations; see [2]. When δc j (x) are the speed fluctuations during states j = (forward stage) and j = 2 (bacward stage) of the time reversal experiment, the cross-correlation is given by R x ) = δc ( y) δc ( x + ). 2( 2 y The analysis in [22] shows that the intensity of the time reversed signal is all the more important when the medium is scattering, i.e., heterogeneous. Consequently, when the two random media become less correlated, the scattering term in the radiative transfer equations decreases and so does the intensity of the time reversed signal; such decay was validated by numerical studies in [2]. The above discussion underscores that the time-reversal image quality degrades with decorrelation of the actual Green s function and that used when performing time-reversal imaging. This suggests that imaging quality may be maintained even if the exact media (Green s function) is unnown, as long as there is sufficient correlation between the actual and simulated media. In the wor that follows we assume that the actual Green s 7

8 function is unnown, and that it represents one sample from an underlying random process (the random process captures our understanding of the range of Green s functions from which the actual Green s function is sampled). Moreover, while the true Green s function is unnown, we assume access to an ensemble of Green s functions, sampled from the aforementioned random process. The above review suggests that if the ensemble of Green s functions (or the associated subspace) is sufficiently correlated to the true Green s function, then it should be possible to obtain useful imaging quality. III. Imaging in Changing Environments In this section we examine electromagnetic time-reversal imaging in changing environments. In particular, we mae the assumption that the precise Green s function used in the forward process (associated with the measured scattered fields) is unnown; however, we assume access to an ensemble of Green s functions (, rn = ω, with this ensemble motivated by the discussions in Secs. I and II. We first review the basic time-reversal imaging technique, followed by a discussion of how the ensemble of Green s functions (, rn = ω may be employed. A. Time-reversal imaging We consider a two-dimensional problem, with a localized (small relative to wavelength) target situated at r t ; below we consider the case of multiple such localized targets, as would be the case for multiple scattering centers on an extended target. We assume that the source is located at r n, and the nth receiving antenna at r n, and a set of N receiver antennas are employed. The scattered field measured by the nth antenna due to this target is S ω, r, r ). An electromagnetic time-reversal (ETR) space-time image is computed ( n n using Green s function as [7] I N * ( t, rn ) = dω S ( ω, rn, rn ) G ( ω, r, rn ) G ( ω, rn, r)exp( jωt n= ωbw r ), () 8

9 where the ourier integral is performed over the system bandwidth ω BW complex conjugate., and * denotes Assume that the Green s function characteristic of the scattered fields is represented as G ( ω,, ) ; the corresponding scattered field S ω, r, r ) can be expressed as r n r ( n n S ω, r, r ) = B( ω) G ( ω, r, r ) G ( ω, r, r ), with the antenna response embedded in the ( n n n t t n Green s functions and B (ω) representing the spectrum of the excitation. Refocusing at r = is expected at time t = for the case in which G ( ω,, r) matches G ( ω,, ) [9], r t although we here assume that all members of the ensemble from G ( ω,, ). r n r r n (, rn = r n r ω are distinct Typically the scattering data and Green s function are sampled at discrete frequencies, and the image at t = is expressed as I N ( t =, r r ) = s ( n, n ) g ( n, r, n, (2) n n= ) where s n, n ) is an N vector representing the frequency dependence of S ω, r, r ) ( f ( n n at N chosen frequencies, g ( n, r, n) is an N vector representing the frequency f f dependence of the product g ω, rn, r, rn ) = G ( ω, r, rn ) G ( ω, rn, ) at those N f ( r frequencies, and the superscript represents complex transpose. If each antenna sequentially plays the role of the source, with scattering data measured at all N antennas, we obtain a time-reversal image by averaging across excitations I N N ( t =, r ) = s ( n, n ) g ( n, r, n). (3) N n = n= In the above discussion we have used one of the mismatched members of the ensemble ω r r to yield a corresponding time-reversal image. There has been (, n, )} = theoretical and experimental research [8,9,2-22] that demonstrates that the time-reversal quality may deteriorate precipitously as the degree of mismatch between G ( ω,, r) and r n 9

10 r n r G ( ω,, ) increases as discussed in Sec. II, as the media underlying G ( ω,, r) and r n r G ( ω,, ) become increasingly decorrelated. r n Since we assume access to the entire ensemble r r ( ω, n, )} =, rather than producing an image based on a single Green s function, one may consider using all of the information in ( ω, rn =, through simultaneous use of all Green s functions. Below we consider several alternatives for this strategy, with performance on measured data considered in Sec. V. B. Time-reversal imaging with average and PCA Green s function The simplest approach one might consider would be simply to use the average Green s function yielding a time-reversal image I avg g ( n r, (4) avg, r, n) = g ( n,, n) = N N ( t =, r ) = s ( n, n ) g avg ( n, r, n), (5) N n = n= where g ( n, r, n) is an N column vector representing the frequency dependence of avg the average Green s function. f A similar use of the ensemble of Green s functions is considered via an eigen analysis, analogous to principal components analysis (PCA) [9]. We perform an eigen decomposition of the N N matrix f f, r, n) = g ( n, r, n) g ( n,, n) = G ( n r. (6) Let w ( n, r, n) represent the lth eigenvector of (6) corresponding to the lth largest eigen l value; note that the eigen analysis is performed at all potential imaging points r.

11 Considering that the dominant eigenvectors have arbitrary phasing, we incoherently sum over elements. A PCA-based imaging result is represented as I PCA N N LP ( t =, r ) = s ( n, n) w l ( n, r, n). (7) N n = n= l= In the example results (see Sec. V) we consider L = principal component. p C. Wideband MV-EPC To enhance the imaging quality and robustness, we developed a wideband MV-EPC algorithm, this representing an extension of a related technique applied previously to problems involving narrowband signals and large array apertures []. We first review the narrowband MV-EPC algorithm, followed by its extension to the wideband, smallaperture case of interest here. The narrowband MV-EPC beamformer weight w is estimated by minimizing the beamformer output power Z = w Rw, subject to the environment perturbation constraints G w = G h, where R is the cross-spectral density matrix (CSDM) of array outputs of dimension N N, G is a N constraint matrix accounting for the environmental perturbations (captured by the representative Green s functions in the ensemble), and h is the left singular vector corresponding to the largest singular value decomposition of G []. The columns of G correspond to the Green s functions in the ensemble of representative Green s functions, and these Green s functions are a function of the N antennas at which the narrowband data are observed (the nth row of G defines the Green s function variation as viewed by the nth receiver antenna, as a function of the changing media across the ensemble). The array output cross spectral density matrix R is computed at frequency ω by considering N measurements at the N array antennas; the different realizations required for averaging, to compute R, are manifested by sequentially considering excitation at

12 each of the N antennas (this yields N different types of excitation on the scatterer, with R averaged across this variation). The constraint G w = G h implies that the weight vector w should be the same average distance from the ensemble of Green s functions represented by G as the dominant left singular vector h is from this same ensemble. This implies that the variation among the ensemble (, rn = ω is assumed to be wea enough such that there is one dominant left singular vector h (this is discussed further in Sec. IIID). To avoid over-constraining the MV-based beamformer, a singular value decomposition (SVD) is employed on G to get D ( D N and D ) dominant left singular vectors that span the column space of G. The ran- D approximation to G is given by where is a G = Λ Γ, (8) N D matrix whose columns are the left singular vectors of G, Λ is a D D diagonal matrix of the singular values of G, and Γ is a D matrix whose columns are the right singular vectors of G. The weight given by MV-EPC is [] where e = [,...,] T. w ω =, (9) ( ) R ( R ) e It is important to note that the weight in (9) is computed at each sampled frequency, with an incoherent sum over multiple frequencies if multiple frequencies are employed (coherent phase information is lost across frequency). The time-reversal technique is explicitly a wideband approach, and a coherent sum over frequency is desired. urther, we desire high imaging resolution ( superresolution ) with a small physical aperture. Since a coherent frequency analysis is so important for time-reversal imaging, an alternative idea to computing a robust Green s function for time-reversal imaging is to retain the frequency dependence when utilizing the MV beamformer. We therefore 2

13 develop a wideband MV-EPC algorithm, representing an extension of time reversal imaging. As discussed below, the superresolution properties of time-reversal imaging are exploited explicitly, yielding an algorithm in which small apertures are employed, with the multiple frequencies analyzed coherently. The wideband MV-EPC framewor is similar to the aforementioned narrowband MV- EPC, but the CSDM and the constraint matrix have different forms. The wideband MV- EPC beamformer weight w is estimated by minimizing the beamformer output power Z = w R w, subject to the frequency perturbation constraints w G h G =, from which we can deduce the equivalent constraint w = e. To distinguish the parameters used in the narrow band MV-EPC, we use subscript to denote that the MV algorithm now employs frequency-dependent perturbation constraints. The CSDM dimension N N is computed as f f 2 N n = n= R of N N R = s( n, n ) s ( n, n ), () where s n, n ) is an N column vector representing the frequency dependence of ( f S ω, r, r ). Note that s n, n ) is identical with the expression in (2) employed within the ( n n ( time-reversal imager. We are employing the frequency dependence to effect high-quality images and assume that the N antennas are proximate to one another relative to the dominant wavelength (details are discussed further in Sec. V, for the particular data considered here). The constraint matrix G, for antennas at r n and r n, represents different media realizations and is represented as G n, r, ) = [ g,..., g ], () ( n where g n, r, n ) is an N vector representing the frequency dependence of product g ( f ω, r, r, r ) = G ( ω, r, r ) G ( ω, r, ). We perform an SVD decomposition on the ( n r n n n N f constraint matrix G to obtain the dominant constraint space of ran- D 3

14 where is a is a G ( n, r, n ) = Λ Γ, (2) N f D matrix whose columns are the left singular vectors of D D diagonal matrix of the singular values of whose columns are the right singular vectors of D =. G, and Γ is a G, Λ D matrix G. In the example results we consider The left dominant singular vectors can be demonstrated theoretically to be equivalent to the PCA vectors. A PCA-based adaptive time reversal imaging can be performed using the left singular vector corresponding to the largest singular value of w PCA The wideband MV-EPC weight is given by w ) = G [4] ( n, r, n e. (3) (, r, n ) = R ( R ) e n (4) and a wideband MV-EPC based time-reversal image is represented as I PC N N ( t =, r ) = s ( n, n ) w ( n, r, n ). (5) N n = n= rom expressions ()-(5) we observe that wideband MV-EPC maintains the frequency dependence of the Green s function throughout the algorithm, which benefits timereversal imaging. D. Discussion We now attempt to provide a linage between the above MV-EPC algorithm and the theory of time reversal for changing or uncertain media, as elucidated in Sec. II. Theory suggests that the degree to which time reversal degrades with mismatched Green s functions is dictated by the correlation between the true media characteristic of the scattering data, and the media associated with the Green s function used for imaging. The tighter this correlation, the better quality is expected for the time-reversal image. In the MV-EPC algorithm it is assumed that an ensemble of Green s functions (, rn = ω captures the uncertainty in the medium. The tighter the space spanned 4

15 by ( ω, rn =, the more liely that the weights w will correspond to an effective medium that is relatively correlated with the true underlying media (assuming, of course, that the true Green s function resides in a subspace spanned by the (, rn = ω ). This naturally suggests that the more confident we are in the underlying media (tighter distribution on ( ω, rn = ), the better results we expect when imaging. As we become less certain in the underlying media, the space spanned by ( ω, rn = grows, and therefore it is more liely that w will correspond to an effective media decorrelated from the true underlying media yielding diminished imaging quality. We observe, therefore, that the MV-EPC algorithm is a vehicle for extending timereversal for cases in which the underlying media or Green s function is uncertain. If our understanding about the underlying media is accurate and the ensemble ( ω, rn = relatively tight (implying a low-dimensional left-singular-vector-space, i.e., a few principal singular values), then good imaging performance is expected. This therefore represents an extension of time-reversal to relatively small levels of uncertainty in the underlying media. Even if our understanding of the span (, rn = ω is correct, we expect relatively poor performance (due to the potential of a high level of decorrelation between w and the true Green s function) if the uncertainty reflected in ( ω, rn = is high. Nevertheless, as demonstrated in Sec. V, this relatively limited extension of MV- EPC vis-à-vis simple time reversal (with mismatched Green s functions) yields encouraging performance for problems of interest to remote sensing. IV. Experimental Configuration A. Measurement setup The ideas outlined in Secs. II and III are examined based upon controlled electromagnetic scattering measurements. In these experiments we extend many previous measurements performed with acoustic and ultrasonic systems [2,3], as well as recent electromagnetic 5

16 studies [7,9]. As discussed in Sec. II, time reversal is most appropriate in media characterized by a high level of scattering and multi-path. We have therefore constituted a highly scattering random medium by employing 68 low-loss dielectric acrylic rods, as shown in ig. 2. The rods are of.25 diameter,.6 m length, and with approximate dielectric constant ε r = 2.5 ; the rods were distributed in a random manner (with no particular distribution, although the average inter-rod spacing was 4.5, between rod axes). The rods are situated in a domain. m long and.2 m wide, embedded at the bottom in Styrofoam ( ε r ). The..2 m2 rod domain is composed of five distinct and contiguous rod sections (see ig. 3), each 2 long and.2 m wide (there are five distinct Styrofoam sections). Different realizations of the environment are manifested by interchanging the relative position of these five subsections of rods. A target is constituted by wrapping a rod with aluminum foil, with these targets placed to the right of the random medium (see ig. 3). A linear N-element sensor array ( N = 3 in our case) is placed to the left of the Antenna random media (see ig. 3), with interelement spacing = 4.. Antenna array The Target antennas are placed at a height that bisects the midpoint of the rods, and the measurements are approximately two- igure 2. Dielectric-rod setup used in experimental time-reversal studies. The right antenna is moved using a precision stepper motor, in the absence of the target, to measure the Green s function. The size of the dielectric-rod domain is. m by.2 m. dimensional. The elements of the array are identical Vivaldi antennas over the.5.5 Gz band, with the electric fields vertically polarized (electric fields parallel to the rod axes). The measurements are performed with a vector networ analyzer. The three antennas to the left of the medium are fixed. We require a means of obtaining the ensemble of Green s functions {G (ω, rn, r )} =, where r corresponds to an imaging position to the right of the domain of interest (in which the target may exist; see ig. 3). To obtain these Green s functions a receiving antenna is placed on a stepper motor, at 6

17 position r, and it is used to sequentially measure the transmitted fields from the nth antenna at the left, located at r n. The stepper motor is deployed at a sampling rate of x = y =.5 in two dimensions (over a domain of size 3 42, and it is within this domain that targets are placed and imaging considered). The different media realizations are manifested by changing the relative positions of the five rod regions (see ig. 3), and in the examples that follow all five regions are placed differently relative to the actual media for which the scattering measurements are performed (i.e., the ensemble of Green s functions (, rn = ω is characteristic of different media realizations, as implemented in ig. 3, and each of the five media regions is distinct from the media corresponding to G ( ω,, r) and the forward o r n measurements). Meter It is desirable to compare the algorithms discussed in Sec. III to more-traditional approaches. Synthetic aperture radar (SAR) is a common technique for imaging a target in a complex scattering environment. In SAR we do not require the ensemble of Green s functions ( ω, rn =, since Meter igure 3. Top schematic view of the rods in ig. 2. The rods are decomposed into five regions, with the rods in each region held together at the bottom via Styrofoam. Different media instantiations are implemented by interchanging the positions of the five rod regions. The mars to the left hand side show the antenna positions of SAR imaging. Three covered by dots represent the physical array used for time reversal imaging. The right grid domain is the imaging domain of size 3 42, with each grid point scanned by an antenna. typically one simply employs the freespace Green s function. owever, in SAR one obtains imaging resolution by utilizing a large (synthetic) aperture. or comparison to SAR, we also consider measurements in which a single antenna is employed at left (see ig. 3) to perform the scattering measurements, with this antenna placed on a stepper motor. The stepper motor is displaced along a linear path with a sampling rate of 4. 7

18 B. Change detection The media considered in these experiments is highly scattering, yielding a very strong clutter signature, in addition to the relatively wea scattering from the target (when present). If one attempts target imaging based on the scattered fields alone, very poor results are obtained, because the scattered fields are dominated by the clutter (rather than the target). Therefore, in the results that follow all imaging is performed based on the signal manifested by change detection. Specifically, it is assumed that scattering data are obtained in the absence of any targets (i.e., a bacground scattered signal). A second measurement is then performed with the same rod medium, but now with the addition of a target. The data with which imaging is performed is based on the difference between these two signals. This is termed change detection [7], and it is widely employed in SAR processing in highly scattering environments. Note that the ensemble of Green s functions is obtained with propagation measurements, in the absence of targets, and therefore change detection is not relevant for measuring the ensemble of Green s functions. V. Example Imaging Results and Analysis In the results that follow the target was composed of a single rod of the type used to constitute the highly scattering media; the rod was wrapped with aluminum foil to constitute a relatively strong scatterer (although, in the absence of change detection, the signature of the target is buried in the clutter induced by the 68 dielectric rods). The target was situated in the imaging region (see ig. 3) over which the Green s functions were measured (although not at a specific discrete point at which the Green s function was measured). The target was placed in 24 distinct places within the imaging region, to constitute different realizations of the scattering data, for statistical characterization of the imaging and detection algorithms. The Green s functions within the imaging domain were measured separately for 5 distinct arrangements of the five dielectric-rod regions (see ig. 3); across the 5 different 8

19 media realizations, there was no overlap in the placement of any of the five dielectric regions, i.e., there was a complete mismatch in the rod locations when comparing any one of the 5 realizations to another. In addition, for each of the 5 distinct media arrangements, we measured separate target-scattering data for the aforementioned 24 different target positions (a) Average (b) PCA (c) MV-EPC - igure 4. Representative imaging results using full-band scattering data, with Green s functions estimated by the (a) average, (b) PCA, and (c) MV-EPC techniques. Although the particular media characteristic of the scattering data is assumed unnown, several of the imaging techniques discussed in Sec. III employ an ensemble of Green s functions, with this implemented as follows. Assume that scattering data were collected for media associated with one of the 5 media realizations; when imaging we consider = 4, with ( ω, rn = composed of the remaining set of Green s functions (from the distinct media arrangements). 9

20 As indicated above, the measurements were performed in the frequency domain over the.5-.5 Gz bandwidth. We consider imaging results when all frequency-domain data are used, as well as results based on a carefully selected (discussed below) subband of frequencies. In addition to presenting a set of qualitative results in the form of images, we present quantitative results, in the form of the receiver operating characteristic. A. Imagery based on full-band data We first imaging based on the full-band data measured from.5-.5 Gz, with (a) Matched TRM (b) Mis-matched TRM (c) ree-space TRM uniform frequency samples. All imaging results are plotted with a db dynamic range, represented with a color bar, and the pea value in each image is normalized to db. The target position is indicated by the white circle in each image. igures 4(a), 4(b) and 4(c) (d) SAR igure 5. Representative imaging using (a) time-reversal mirror (TRM) with a matched Green s function, (b) TRM with a single mismatched Green s function, (c) TRM with the free-space Green s function, and (d) SAR imaging

21 present typical results for I avg ( t =, r), I PCA ( t =, r) and ( t =, r), respectively, for which good imaging quality is observed. We observe that the wideband MV-EPC suppresses more sidelobes than the averaging and PCA techniques, yielding the best result among the three techniques. Using the full-band data (.5-.5 Gz) the measured cross-range resolution is approximately 6, with approximately 3 resolution in down-range. or a homogeneous medium, the anticipated cross-range resolution is R c = λl / a, where λ is the wavelength, a is the real linear aperture length, and L is the distance from the aperture center to the imaging point. If we consider a free-space medium, and the center frequency 5.5 Gz, a real aperture a = 8, and length L =.2 m, the optimal free-space cross-range resolution is R = 8. 8 ; this demonstrates the significant improvement in cross-range resolution manifested by the rod-induced multipath. Regarding computation time, it requires roughly 3.5 minutes of CPU in Matlab TM on a Pentium IV PC with.73gz CPU for the wideband MV-EPC algorithm to synthesize a 2 29-pixel image, versus 7 seconds for the averaging method and 2 seconds for the PCA technique. I PC c As reference, in ig. 5 we show representative imaging results for time-reversal imaging, using a single Green s function in the imaging phase. In ig. 5(a) the Green s function used in time reversal method corresponds to the actual media considered in the scattering data, while in ig. 5(b) one of the 4 mismatched Green s functions is utilized when performing time reversal. In ig. 5(a) a very tight focus is observed at the target, while ig. 5(b) there are more sidelobes. When considering multiple results of the form in ig. 5(b) time reversal with mismatched Green s function we observed the predicted instability [4] in the image quality and in focusing at the proper target location. In ig. 5(c) we show the time reversal result with the free-space Green s function, for which no cross-range resolution is observed due to the small aperture size. In ig. 5(cd) we present synthetic aperture radar (SAR) imaging results, for which the free-space Green s function is applied, and the linear aperture (with target in the center of the aperture) is 4, with 4 sampling. Poor SAR imaging performance is observed, even with the large aperture size (five times the size of the aperture used in the other imaging techniques), with this 2

22 manifested by the highly scattering environment and hence mismatch to the free-space Green s function. The results in igs. 4 and 5 represent our experience with the techniques considered here. Specifically, direct time-reversal with a mismatched Green s function yields unstable results, meaning that the performance is dependent on the particular scattering data and the particular mismatched Green s function considered. This is consistent with theoretical predictions. Considering the analysis in Sec. IIID, we note that this variability or instability in the time-reversal imaging quality with mismatched Green s function is indicative of a decorrelation between the media used in the forward scattering measurement and the inverse time-reversal imaging. owever, the results in ig. 4 indicate that by utilizing nowledge of the subspace in which the Green s function lives, defined by the ensemble ( ω, rn =, it is possible to recover imaging quality despite the fact that the specific Green s function is unnown. These qualitative observations in imaging quality will be made quantitative below in Sec. V(c). B. Example results of sub-band imaging The results discussed above were.5 based on utilization of the full scattering bandwidth. owever, as discussed in Sec. IIID the quality of.5 the imaging results is anticipated to be lined to the correlation between -.5 the Green s function used in the forward measurement and in the requency (Gz) inverse imaging phase. The igure 6. Logarithm of the 4 singular values of the sensitivity of the Green s function windowed constraint matrix, as a function of the center frequency of the sliding window, which is of width Gz on the specific media realization is (2 frequency samples). The blac curve represents the lined to the media variation relative spectrum of the largest singular value. to wavelength. Therefore, one would anticipate that the imaging quality, and sensitivity log (λ) 22

23 to Green s function uncertainty will be dictated by the frequencies considered in the measurements. We address this question in the following analysis (a) Average (b) PCA (c) MV-EPC igure 7. Time reversal with -6 Gz subband, with Green s functions addressed via (a) averaging, (b) PCA, and (c) MV-EPC. Recall that we have measured frequency-dependent Green s functions G n, r, ) = [ g,..., g ], where g ω, r, r, r ) = G ( ω, r, r ) G ( ω, r, ) accounts for ( n the two-way propagation from a source antenna at ( n r n n n r n to point r in the imaging domain, and from r to the receiver antenna at r n. To explore the singular value spectrum across frequency, we apply a sliding window of width Gz (2 frequency samples) on the constraint matrix G, and in ig. 6 we show the singular value spectrum for = 4 distinct media, as a function of the center frequency of the sliding window. We note there is a single dominant singular value in the frequency subband of -6 Gz, implying that in this frequency range the different media are highly correlated (wavelength relatively 23

24 large with respect to inter-rod spacing, and therefore less sensitivity to media variation). Above 6 Gz there is no longer a clear dominant singular value, suggesting a greater degree of wave mixing in the highly scattering media, and therefore anticipated greater requirements for nowledge of the true Green s function when performing imaging (a) Average (b) PCA (c) MV-EPC - igure 8. Imaging with the 5- Gz subband, with Green s functions addressed via (a) averaging, (b) PCA, and (c) MV-EPC We consider the performance of the average, PCA and MV-EPC algorithms in the two frequency-dependent domains indicated above: (a) from of -6 Gz and (b) from 5- Gz, with results for these cases shown in igs. 7 and 8, respectively. The results in ig. 7 are of relatively high quality (good focusing about the target location), while the 5- Gz band alone (ig. 8) yields poor results. It is of interest to compare the results in ig. 7 to the full-band results in ig. 4 for which the same imaging techniques were applied. While the high-frequency data alone yield relatively poor imaging performance (ig. 8), we note when comparing igs. 4 and 7 that the high-frequencies, when combined with 24

25 the -6 Gz band, improve the tightness of the focus around the target. The quantitative analysis that follows is based on the full-band scattering data. C. Quantitative analysis of imaging results As indicated above, a large number of images with different imaging techniques have been generated, by varying the placement of the target (24 target locations); space limitations prohibit showing all imagery, with the results presented above being representative. It is therefore of interest to perform a quantitative analysis of all such images, where here this is quantified in terms of detection performance. To achieve this, a square box is defined corresponding to a contiguous 5 5 pixel (or ) window in image space, and the test statistic l is the pea amplitude within this box. Recall that the images presented above were composed of a total of 2 29 pixels. If a portion of the box encompasses the target than the target is assumed present, and if not there is no target. A false alarm is defined when a target is declared within a region for which the target is not located, while a target is missed if a target is not declared when it is in fact located within the 5 5 pixel region. Based on this definition, the receiver operating characteristic (ROC) of images is investigated. Specifically, to compute the ROC, we set a threshold T and if l T a target is declared. By.9 varying the threshold T from zero to P d Wideband MVEPC PCA Average Mismatched SAR P fa l max (the largest value of l within a given image), an ROC is obtained, which represents the probability of detection as a function of the probability of false alarm. In ig. 9 we plot the ROC curves for igure 9. Receiver operating characteristic (ROC) curves for the different imaging techniques, based on the all techniques considered here. We full-band imaging results. note that the wideband MV-EPC (equation (5)) yields the best performance, and the performance of the average and the 25

26 PCA techniques (equations (5) and (7), respectively) are comparable. These results are averaged across all 5 different media and across all 24 target locations. or the case of time-reversal with mismatched Green s function, the results are averaged across all possible combinations of mismatched Green s functions. The results in ig. 9 considered full-band data, and in ig. we present results for the - 6 Gz subband, over which there was a dominant singular value (see ig. 6). We note that the ROC performance of the various algorithms is improved when using the subband over which there is a dominant singular value. This indicates that, while the focusing quality of the full-band data is superior to that of the subband data (comparing igs. 4 and 7), the subband data is less beset by spurious sidelobes, which undermine ROC performance. This suggests that detection may best be performed by using the lowerfrequency bands, over which there is a dominant singular value, with classification (for which resolution is important) based on the full-band data. VI. Conclusions Time-reversal imaging has been examined experimentally for electromagnetic target detection in highly scattering uncertain media. We addressed time-reversal behavior for the case in which the media employed in the forward and inverse steps are mismatched. P d Wideband MVEPC PCA Average Mismatched SAR P fa We consider techniques for the case in igure. Receiver operating characteristic (ROC) curves for different imaging techniques which the media employed is uncertain based on the -6 Gz subband imaging results. represented by an ensemble of Green s functions. Besides straightforward methods such as averaging and PCA, we extend a wideband Minimum Variance beamformer with Environment Perturbation Constraints (MV-EPC), in which the unnown Green s function is constrained to reside in a subspace 26

27 spanned by the Green s-function ensemble. The imaging results show that the wideband MV-EPC suppresses more sidelobe than the average and PCA techniques, yielding the best result among the three techniques. We compare and analyze full band imaging and subband imaging results. The images and ROCs show that subband imaging is preferable for stable imaging and better detection performance in uncertain media, although the full-band data provided a better focus around the target. The subband is determined by seeing the singular value spectrum of the constraint matrix of different media realizations. The frequencies where there is a dominant singular value constitute the frequency subband for time reversal imaging. It is demonstrated by comparison between time reversal imaging and SAR imaging that time reversal can achieve super-resolution quality while dramatically reducing the needed physical aperture size. Given statistical characteristics of the uncertain media, time reversal imaging is preferred in highly scattering bacground. It has been demonstrated that access to an ensemble of Green s functions (, rn = ω can improve time-reversal imaging quality, in the absence of the true Green s function, assuming we are operating in a regime (frequencies) for which there is still a dominant propagation mechanism; this implies that the ensemble of Green s functions ( ω, rn = reside within a low-dimensional subspace. There is still the practical problem of how this may be achieved in a realistic sensing scenario, i.e., how one might practically measure the ensemble ( ω, rn =. We provide one example here. Assume there is interest in imaging vehicles on roads, but the roads are buried under a foliage canopy, and it s undesirable to use an array of cameras (e.g., there is a desire for covert surveillance). A distant transmitting antenna(s) may be employed, and vehicles with antennas may be utilized to measure the Green s function from the source antenna(s) to the road(s) of interest. These Green s function measurements may be performed periodically (daily, weely, monthly, etc.) to constitute an ensemble, accounting for variation in the propagation (foliage) properties. The ensemble of 27