COMPRESSIVE CLASSIFICATION FOR THROUGH-THE-WALL RADAR IMAGING. Mark R. Balthasar, Michael Leigsnering, Abdelhak M. Zoubir

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1 20th European Signal Processing Conerence (EUSIPCO 2012) Bucharest, Romania, August 27-31, 2012 COMPRESSIVE CLASSIFICATION FOR THROUGH-THE-WALL RADAR IMAGING Mark R. Balthasar, Michael Leigsnering, Abdelhak M. Zoubir Signal Processing Group Technische Universität Darmstadt Darmstadt, Germany ABSTRACT Through-the-Wall Radar Imaging (TWRI) is an emerging technology, desirable or a variety o military and civilian applications. Some o these applications require the ability to make a rapid decision on the contents o a target scene, rather than reconstruct it perectly. In this work, we address this problem by modiying the smashed ilter proposed by Davenport et al., which is based on compressive sensing (CS) theory. We introduce two compressive classiication methods, which allow or an eicient hardware implementation as well as deliver accurate classiication results in the considered scenarios using only a raction o the available data. Index Terms Through-the-Wall, radar imaging, compressive sensing, compressive classiication 1. INTRODUCTION Through-the-Wall Radar Imaging (TWRI) is a very promising ield o research with a variety o applications including military, police, and ire brigade missions as well as search and rescue in the atermath o natural disasters [1, 2]. By delivering inormation about obscured areas, which cannot be observed by other means, it provides a way to estimate the layout o an observed scene as well as localize, identiy, and classiy possible obscured targets. Conventional TWRI systems usually orm an image rom the gathered data and apply target detection beore classiying on the obtained eatures [3, 4]. Compressive sensing (CS) has been shown to tremendously decrease the amount o necessary data or high-resolution TWRI reconstruction [5]. Aspects in sparse reconstruction o extended targets have been studied in [6], however, it is unclear i these images can be used or classiication as the image statistics dier strongly rom beamormed images. Apart rom that, there are many applications, such as search and rescue missions, in which it is crucial to rapidly decide on the scene under observation rather than reconstruct its layout perectly. This work seeks to provide a solution to this problem by bringing CS theory to the classiication domain. Our aim is to achieve reliable classiication using only a small amount o measurements, or which no imaging would be possible. To this end, we will resort to the smashed ilter, which was developed by Davenport et al. in [7]. This algorithm oers a way to apply compressive classiication (CC) in image target classiication by using a maximum likelihood classiier (MLC) directly on compressed measurements. It is even able to deal with arbitrary perturbations, such as a shit or rotation, by prepending a maximum likelihood estimator (MLE). We propose two adaptations o the smashed ilter or the application in TWRI, namely Fixed Frequency Classiication (FFC) and Fixed Antenna Element Classiication (FAEC), which randomly combine measurements in the space or requency domain, respectively, beore applying the smashed ilter. Both methods can be eiciently implemented and deliver close to perect classiication in the considered scenarios using less than 0.1 % o the available data. In Sections 2 and 3, we will briely revisit the smashed ilter and introduce the TWRI signal model used in our experiments, respectively. In Section 4 FFC and FAEC are presented. Section 5 compares their perormance to a standard method in a simulated as well as a real TWRI scenario. 2. THE SMASHED FILTER The smashed ilter is a two-step algorithm, consisting o a MLE, i.e. a least-squares estimator (LSE) in the case o additive white Gaussian noise (AGWN), ollowed by a MLC. It was initially developed to classiy compressed images o objects that are taken using a single-pixel camera, a detailed description o which can be ound in [8]. Compressive measurements o a signal x R N in a noisy image can mathematically be expressed as y = Φ(x + w), (1) where Φ R M N, M N is a pseudorandom orthoprojector and w denotes AWGN with variance σ 2 [7]. Note, that Φ can either be illed with zeros and ones, which corresponds directly to the single-pixel camera imaging process, or with ±1. For each class C i, i = 1,, P, containing the signal s i, the smashed ilter requires access to a set o training data, EURASIP, ISSN

2 i.e. a maniold M i, containing all possible transormations x = i (θ, s i ) o the signal, parametrized by θ. Note that the unction i could represent scaling, rotation, or translation o the object or the whole scene in an imaging scenario. I i is unknown, we can resort to a data-based representation using training data. The smashed ilter [7] irst estimates the closest point on each maniold M i to the observed signal y using a LSE, which is equal to inding the most probable transormation or each class. Mathematically, this is given by ˆθ i = arg min y Φ i (θ, s i ) 2 2. (2) θ Θ i In the second step, MLC selects the maniold and thereby the class or which the distance between y and the closest point on the maniold is minimal, as C(y) = arg max p(y ˆθ i, C i ), (3) i=1,,p where the likelihood p(y ˆθ i, C i ) is approximated by p(y ˆθ i, C i ) = 1 1 e 2σ y Φi( ˆθ i,s i ) 2 2 (2πσ) M/2. (4) Note that this expression holds exactly, i Φ is an orthoprojector [7]. 3. THROUGH-THE-WALL RADAR IMAGING SIGNAL MODEL Consider a monostatic uniorm linear array o A antenna elements t a placed at a distance z o rom the observed target area. The stepped-requency approach is used to synthesize an ultrawideband (UWB) pulse. This is done by transmitting short continuous wave segments at U dierent requencies u or each antenna element, exploiting the Fourier equivalency in the process [1]. In the requency domain, the received signal at array element t a and requency u can be regarded as a superposition o P point targets [1]: r[u, a] = P 1 p=0 σ p e j2πuτpa, (5) where σ p denotes the target relectivity o target p and τ pa is the two-way propagation delay between target p and antenna element t a. I a wall is present, the EM waves are reracted according to Snell s law and τ pa has to be calculated accordingly. In the absence o a wall, however, τ pa can simply be computed using the Euclidean distance d pa between target p and antenna element a: τ pa = 2 d pa c 0 (6) where c 0 denotes the speed o light. Eventually, we obtain an U A measurement matrix, representing the target area in the requency-space domain. Conventional imaging and classiication algorithms would apply beamorming, or example, employing Direct Frequency Domain Image Formation [9]. Since we are not interested in the exact layout o the target scene and, thus, want to skip the imaging step, the smashed ilter shall be applied directly to the N = U A measurements. As these do not correspond to actual pixel locations, the algorithm has to be adapted in order to work in a TWRI scenario. In the ollowing, we will explore two such adaptations. 4. COMPRESSIVE CLASSIFICATION FOR THROUGH-THE-WALL RADAR IMAGING As mentioned earlier, many TWRI applications require rapid classiication decisions on the scene under observation. Due to portability reasons, it is, urthermore, desirable to make reliable decisions with as ew measurements as possible. FFC and FAEC are two approaches at bringing the smashed ilter to the classiication domain. By linearly combining random measurements in the space and requency domain, respectively, they orm compressive measurement vectors to which the smashed ilter is applied Fixed Frequency Classiication The irst and most intuitive adaptation o the smashed ilter is FFC. For each compressive measurement a single requency u is picked at random and the received signals are linearly combined according to the pseudorandom measurement matrix. Let u m be a random sampling index with u m [1,, U]. In analogy to Eq. (1), FFC can be mathematically expressed as y m = φ T mx m, x m = (r[u m, 0], r[u m, 1],, r[u m, A 1]) T (7) where φ m is an A 1 vector. We only consider antipodal patterns, which corresponds to simply inverting some o the received signals beore the summing step, as we ound them to generally yield better results. Fig. 1 shows the concept o taking one o M compressive measurements using FFC. Note that white elements are zero or simply unconsidered, while grey elements are valued with +1 and black ones with 1. Clearly, considering only ±1 as elements o the measurement matrix, which corresponds to a 0 /1 phase shit, allows or a simple hardware implementation. Thereore, a single radio requency rontend would be suicient, assuming access to a real aperture antenna. FFC can be urther simpliied by considering the same requency or every compressive measurement. Although this leads to the total loss o downrange resolution, it can seriously 2289

3 t t t A 1 t A 1 t A 2 t A 2 t 3 t 3 t 2 t 2 t 1 t 1 t U 2 U 1 t U 2 U 1 Fig. 1. Fixed Frequency Classiication, taking 1 o M compressive measurements Fig. 2. Fixed Antenna Element Classiication, taking 1 o M measurements decrease hardware costs since a ixed requency transceiver setup can be employed Fixed Antenna Element Classiication Considering only one requency but all antenna elements is not necessarily cost-eective, as additional requencies are cheap, while additional sensors are usually expensive. Thereore, we introduce FAEC, which randomly chooses an antenna element t a and linearly combines the values at that element or each requency to obtain one compressive measurement. Let a m be a random sampling index with a m [1,, A]. In matrix notation, FAEC can be expressed as y m = φ T mx m, x m = (r[0, a m ], r[1, a m ],, r[u 1, a m ]) T (8) where φ m is an U 1 vector. Using an antipodal pattern again corresponds to inverting some values beore the summing step. The concept o taking one o M compressive measurements using FAEC is depicted in Fig. 2. FAEC, too, can be urther simpliied by choosing the same antenna element or every compressive measurement. This allows or a natural and eicient implementation o the smashed ilter in real lie. I we design a waveorm [10] with the appropriate power spectral density to relect the necessary requencies, only one pulse has to be transmitted. However, the consideration o only one sensor leads to the total loss o crossrange resolution. In other words, there will be ambiguities between shited objects, which are located at the same radius rom t a, as they produce the same propagation delay τ pa Template Matching Template Matching () is the simplest non-compressive alternative to FFC and FAEC, which we will use to assess the perormance o these two methods. In this case, the measurement vector y is simply composed o M non-compressive measurements, which are randomly picked rom the original U A measurement matrix, which we introduced in Section 3. Subsequently, the smashed ilter is applied to y as previously described. 5. EXPERIMENTAL RESULTS In the ollowing experiments we evaluate the perormance o FFC, FAEC and in simulated as well as real TWRI scenarios. To this end, we will estimate the classiication rate using a Monte Carlo simulation. Hence, every experiment is executed 0 times or dierent values o M and correct classiications are counted. The classiication rate, thus, denotes the number o correct classiications in percent. Apart rom that, we employ leave-one-out testing, i.e. randomly remove one entry rom the training data and use it as the test class in each Monte Carlo run. For simplicity, we only consider a single class to be present in the observed scene. While the transition to multi-class scenarios is straightorward, the classiication process becomes increasingly more complex with each additional class. In order to test every possible combination o classes, we would need to have access to training data or all possible constellations o arbitrarily and individually transormed classes Simulated Data Simulation Setup Consider a square target area with a side length o approximately 3.6 m, mapped to a grid o pixels. The linear transceiver antenna array consists o A = 57 antenna elements equally spaced at 2.2 cm and is placed at a distance z o = 1.05 m rom the target area. At each element 22

4 Classiication Rate, % Classiication Rate, % Classiication Rate, % FFC (multi req.) FFC (single req.) (a) FFC, σ 2 = FFC (multi req.) FFC (single req.) (c) FFC, σ 2 = 0.1 FFC (multi req.) FFC (single req.) (e) FFC, real data Classiication Rate, % Classiication Rate, % Classiication Rate, % FAEC (multi req.) FAEC (single req.) (b) FAEC, σ 2 = FAEC (multi req.) FAEC (single req.) (d) FAEC, σ 2 = 0.1 FAEC (multi req.) FAEC (single req.) () FAEC, real data Fig. 3. Classiication rates or FFC, FAEC and using simulated data under dierent SNRs (a-d) and real data (e,) Class No./No. o targets Pos. [cross-,downrange] in m 1 [0, 3.5] 2 [-1.25, 3.0] [1.25, 3.0] 3 [-0.75, 2.0] [0, 5.0] [0.75, 2.0] Table 1. Three point target layouts as simulation classes position, U = 1 measurements are generated, representing multiple requency steps between 0 MHz and 3.1 GHz. Table 1 details the point target layouts, which will serve as classes in the ollowing experiments. The training data is generated by successively shiting each point target layout by 1 cm in cross- or downrange. The maximum shit is restricted to 1 m. At this point, we assume ree space propagation, i.e. no wall is present Results on Simulated Data A comparison o using FFC, FAEC and with dierent levels o additive noise is given in Fig. 3 (a-d). Considering a high SNR, i.e. σ 2 = 0.001, leads to a very good classiication rate, reaching % or M 20, which corresponds to 0.04 % o the available data. FFC and FAEC yield even better results, reaching % already at about M = 10, i.e. using 0.02 % o the data. In the low SNR regime, i.e. σ 2 = 0.1, the perormance o all three algorithms degrades. While needs an M >, i.e % o the data, to achieve perect classiication, FFC and FAEC only require M 42 and M 30, i.e % and 0.06 % o the data, respectively. I % correct classiication is suicient, we can resort to the simpliied version o FFC, which uses only one requency, and thereby decrease hardware costs. All in all, compressive classiication outperorms in terms o the amount o necessary data, especially in the ace o a low SNR Real Data Experimental Setup The real measurement setup is similar to the one assumed in the model or simulated data. The 3.6 m 3.6 m target scene is set up in an anechoic chamber, the loor o which is not covered by anechoic material. Hence, relections o the loor will lead to multipath propagation [1]. In contrast to the simulation setup, the transceiving antenna consists o a element planar array. Essentially, this gives us 57 complete sets o measurements or dierent elevation layers. A horn antenna is iteratively moved by 2.2 cm in crossrange and height, 2291

5 taking measurements or all 1 requencies at each point. A concrete wall with a thickness o d = 14.3 cm and a relative permittivity o ε = is placed at a stando-distance z o = 1.05 m rom the antenna. A detailed description o the setup can be ound in [1]. We consider TWRI measurements o 4 single objects a metallic dihedral, a metallic trihedral, a metallic sphere, and a gallon jug o saltwater placed on a 1.2 m high oam column, located at about 1.9 m downrange. They will serve as classes in the ollowing experiments. For simplicity, we will only consider measurements that were taken using horizontal transmitter and receiver polarization Results with Real Data We do not have access to measurements o shited objects and, thus, we cannot translate the experiments rom the previous section to a real TWRI environment. The 57 available sets o measurements, corresponding to dierent measurement heights, however, represent some kind o shit, too, only this time the antenna array is shited instead o the object. Thereore, we will use them as training data or each class. A comparison o using FFC, FAEC and is given in Fig. 3 (e,). All three methods lead to a similar classiication rate, reaching % at about M = 15, which corresponds to 0.03 % o the available data. The simpliied approaches o FFC and FAEC even yield worse results, as their classiication rate saturates slightly below %. Obviously, compressive classiication does not oer much beneits in this scenario. 6. CONCLUSION Two dierent approaches to modiying the smashed ilter or the application in TWRI were presented. They were employed to classiy shited targets in a simulated environment as well as unshited targets recorded by a shited antenna in a radar imaging lab. Their perormance was evaluated in terms o classiication rate and compared to a non-compressive method. While already yields excellent results, the compressive methods are able to urther reduce the number o measurements. For low SNR, FAEC is the best choice since its perormance stays almost constant. FFC and FAEC cannot improve the results o in the real data experiment, with all three methods perorming equally well. However, we have to take into account that the two experiments are not perectly comparable or various reasons. First o all, we consider shited targets in the simulated case, compared to a shited antenna in the real setup. Second, the simulation classes are composed o 1 to 3 point targets with the same radar signature, compared to a single target with a dierent radar signature or each class. Third, the simulated environment lacks a wall and is assumed to be aected by AWGN noise only, while the measurements taken in the radar lab are subject to a huge wall-echo and the noise is not necessarily white. Last, Φ cannot be considered an orthoprojector in a real radar scenario. Due to the unknown eect o these dierences, it is not surprising that the real data results dier rom the simulation one. In order to improve the perormance o FFC and FAEC even urther, both methods can be extended to consider all our possible combinations o horizontal and vertical transceiver polarizations. That way we can take advantage o the act that some classes might dier more when viewed rom a certain angle. ACKNOWLEDGEMENT The authors would like to thank Pro. Dr. Moeness Amin and Dr. Fauzia Ahmad rom the Center o Advanced Communications at Villanova University, Villanova, PA, USA, or providing the experimental data. 7. REFERENCES [1] F. Ahmad and M. Amin, Multi-location wideband synthetic aperture imaging or urban sensing applications, Journal o the Franklin Institute, vol. 345, pp , September [2] E. J. Baranoski, Through-wall imaging: Historical perspective and uture directions, Journal o the Franklin Institute, vol. 345, pp , January [3] C. Debes, J. Hahn, A. Zoubir, and M. Amin, Target discrimination and classiication in through-the-wall radar imaging, IEEE Transactions on Signal Processing, vol. 59, pp , oct [4] G. Smith and B. Mobasseri, Robust through-the-wall radar image classiication using a target-model alignment procedure, IEEE Transactions on Image Processing, vol. 21, pp , eb [5] Y.-S. Yoon and M. Amin, Compressed sensing technique or high-resolution radar imaging, in Proceedings o SPIE, vol. 6968, p A, [6] M. Leigsnering, C. Debes, and A. M. Zoubir, Compressive sensing in through-the-wall radar imaging, in IEEE International Conerence on Acoustics, Speech and Signal Processing (ICASSP), [7] M. A. Davenport, M. F. Duarte, M. B. Wakin, J. N. Laska, D. Takhar, K. F. Kelly, and R. G. Baraniuk, The smashed ilter or compressive classiication and target recognition, Proc. SPIE Computational Imaging, vol. 5, January [8] M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, Single-pixel imaging via compressive sampling, IEEE Signal Processing Magazine, vol. 25, pp , March [9] G. Alli and D. DiFilippo, Beamorming or through-the-wall radar imaging, in Through-the-Wall Radar Imaging (M. G. Amin, ed.), ch. 3, pp , CRC Press, [10] F. Yin, C. Debes, and A. M. Zoubir, Parametric waveorm design or improved target detection, in European Signal Processing Conerence (EUSIPCO), August