Compressed Sensing mm-wave SAR for Non-Destructive Testing Applications using Side Information

Transcription

1 Compressed Sensing mm-wave SAR for Non-Destrutive Testing Appliations using Side Information Mathias Bequaert 1,2, Edison Cristofani 1,2, Gokarna Pandey 2, Marijke Vandewal 1, Johan Stiens 2,3 and Nikos Deligiannis 2,4 1 CISS Department, Royal Military Aademy, 3 Av. de la Renaissane, B-1 Brussels 2 ETRO Department, Vrije Universiteit Brussel, Pleinlaan 2, B-15 Brussels, Belgium 3 SSET-department, Ime, Kapeldreef 75, B-31 Leuven, Belgium 4 iminds, B-95 Ghent, Belgium mathias.bequaert@rma.a.be, edison.ristofani@ele.rma.a.be, gpandey@etro.vub.a.be, marijke.vandewal@rma.a.be, jstiens@etro.vub.a.be, ndeligia@etro.vub.a.be Abstrat This paper evaluates the appliability of an innovative strategy for applying Compressed Sensing (CS) on Syntheti Aperture Radar (SAR) imaging, in the mm-wave range, using prior or strutural side information. The studied tehnique adds the side information to the onventional CS minimization problem using an approah, allowing for lower sub- Nyquist sampling than standard CS predits. The appliability of this strategy on ultra-wideband SAR measurements is tested through simulations and real Non-Destrutive Testing (NDT) experiments on a 3D-printed polymer objet. Index Terms Compressed sensing (CS), Syntheti Aperture Radar (SAR), Non-Destrutive Testing (NDT) I. INTRODUCTION Three-dimensional printing has beome an extremely popular tehnology in many domains. This tehnology is widely used for prototyping, opying and produing omplex objets [1]. Non-Destrutive Testing (NDT) tehniques are needed to image the inner struture of these objets in order to identify defets due to, for example, high temperature differenes, alibration problems or unadapted printing speeds [2]. In this paper a SAR sensor is presented for imaging the interior of 3D-printed polymer objets. Compressed Sensing (CS) was initially introdued to Syntheti Aperture Radar (SAR) appliations for reduing the hardware omplexity and storage needs [3], [4]. One of the requirements of ompressed sensing is that the reonstruted image must be sparse in a given domain. In many SAR appliations, this requirement is not met [5], thereby limiting the derease of the undersampling bound. For 3D-printed objets additional side information an be added in the CS minimization equation, in order to overome this problem: the exat physial dimensions and material properties of the designed objet are known. The printed objet an thus be ompared with a similar printed objet or with a simulated referene objet. In this paper it is shown that adding the minimization of the differene between referene and real objets an severely derease the number of needed samples. The rest of the paper is organized as follows: Setion 2 introdues briefly the theory of CS and the SAR sensor used in this work. In Setion 3 a simulated sensor is presented and the impat of adding side information into the CS minimizaton equation is studied through simulations. Those results are then validated by means of real measurements on a 3D-printed objet. Finally, onlusions are drawn and possible approahes for future work are presented. A. Compressed Sensing II. BACKGROUND Suppose an unknown sparse signal x C N with k N non-zero elements is sensed by applying n linear measurements: y = Ax, where A is a sensing matrix of size n N, satisfying the Restrited Isometry Property (RIP) [6]. Conventional ompressed sensing [7] theory ensures an exat reonstrution of the signal x in the ase of severe undersampling (n N) with high probability, by solving the following basis pursuit problem: ɛ, (1) where 1, 2 denote the l 1 -norm and l 2 -norm, respetively, and ɛ is an auray threshold. The CS sub-sampling bound an be dereased even further if side information is available. Suppose x C N to be a good estimate of x. The similarity between the side information and the unknown signal an be added to the minimization problem and equation (1) beomes: + β g(x x ) ɛ, (2) with β > and with g a funtion expressing the similarity between x and x. Popular hoies for g in literature are the l 1 - and the l 2 -norm, known as and l1- l2 minimization respetively. In [8] and [9] both approahes are ompared and the performane bounds are determined

2 for Gaussian random sensing matries. The authors onlude that, in general, as long as the similarity is signifiant, l1- l1 minimization performs better than lassi CS or the l1- l2 minimization. The authors in [8] and [9] also prove, both theoretially and experimentally, that the sharpest bound is obtained for β = 1. Taking those results into aount equation (2) beomes: + x x 1 ɛ. (3) TABLE I SPECIFICATIONS OF THE SENSOR USED FOR THE SIMULATED AND REAL EXPERIMENTS NDT Sensor parameters Value Starting frequeny f 45 GHz Bandwidth 3 GHz Frequeny step f MHz Sanning distane.3 m Number of azimuth measurements 3 Number of frequenies 41 Aperture angle (-3 db) 15 In literature, has been applied in medial imaging: X-ray Computed Tomography (CT) [1] and Magneti Resonane Imaging (MRI) [11], and for ompressive video bakground subtration [12] [13] lassi CS l1-l2 minmization B. Stepped FMCW SAR An ultra-wideband stepped Frequeny-Modulated- Continuous Wave (FMCW) SAR was used for performing measurements. The parameters of this sensor were used to proess both experimental and simulated data. The signals emitted by this type of sensor are expressed by: s(t) = L exp(j2πf l t), (4) l=1 with f l = f + (l 1) f being the disrete frequenies, f being equal to B/L and f equal to the starting frequeny. B stands for the total emitted bandwidth and L equals the number of disrete frequenies. At reeption, the ehoes from K satterers, with refletion oeffiients a k,l and at distanes equal to r k from the sensor, are demodulated, using a homodyne demodulation sheme and resulting in a beat signal: s b (t) = K L k=1 l=1 ( a k,l exp 2r ) k j2πf l, (5) where is the propagation speed of the eletromagneti waves through the medium ( = µɛ ). A satisfying resolution in range (ross-trak) and in azimuth (along-trak) an be obtained, after applying a SAR ompression algorithm (e.g.: [14] [15]) on this beat signal, due to the ultra-wide bandwidth and the syntheti aperture reated by the relative motion between sensor and objet. The range and azimuth resolutions an be estimated by equations (6) and (7), respetively: r ra = r az = 2B, (6) 2f sin(θ), (7) where θ is the opening angle of the antenna. Examples of the use of this type of sensor for NDT appliations an be found in [16] Change rate (= p/n) with Sparsity = Change rate (= p/n) with Sparsity = Fig. 1. Experimental results obtained through simulation. 64 random referene senes with a sparsity equal to.5 (top figure) and equal to (seond figure) were built. Eah of them was hanged with an inreasing hange rate going from.4 to 5. For eah hange rate the mean minimum number of samples (undersampling rate δ) was determined for obtaining a orret reonstrution. This was done for the three different reonstrution approahes: (1) lassi CS (in blue); (2) (in red) and (3) l1-l2 minimization (in green). III. SIMULATIONS The parameters of the sensor used for the results obtained through simulations are idential to the real sensor and are listed in Table 1. These parameters are used to build the sensing matrix A: eah of the N olumns of A orresponds to the simulated responses, and thus the beat signals, of a sene with a single point satterer. These vetorized responses of length N are randomly sampled by seleting n rows of the sensing matrix. This orresponds to a random sampling over all the rows and olumns of the raw data. In order to experimentally evaluate the appliability of the approah on SAR data, random senes of 123 pixels are simulated. The refletion oeffiients of those senes are the side information x, used for the reonstrution. Two sparsity rates are onsidered, namely, k/n =.5 and k/n =. The refletion oeffiients of p random pixels of these referene senes are then substituted by different random refletion oeffiients. The simulated reeived signals from the adapted senes are randomly sub-sampled with a varying undersampling rate (δ n = n/n (, 1)). Fig. 1 shows the mean minimum undersampling rate neessary for a good reonstrution using the onventional CS approah, the l1-l2 and the approahes for a varying hange rate (= p/n). A reonstrution is onsidered to be

3 suessful if the Peak Signal-to-Noise Ratio (PSNR) between the reonstruted sene and the adapted original sene x is less than an arbitrarily hosen value of 2 db. Three reonstrution strategies are ompared: (1) Conventional Compressed Sensing: (2) l1-l2 minimization: ɛ (8) x x 2 2 ɛ (9) (3) : + x x 1 ɛ (1) The results presented in Fig. 1 show that performs drastially better than the lassi CS as long as the similarity between the referene sene and the reonstruted sene is signifiant. The also performs slightly better than the l1-l2 minimization. As an be expeted intuitively, the positive impat of the use of side information beomes more important when the sparsity of the sene dereases. If the sene is already sparse (top graph of Fig. 1), the impat of the side information dereases rapidly for inreasing hanging rates. For a lower sparsity rate of the referene image (bottom graph of Fig. 1), the impat stays more important for inreasing hanging rates. Further simulations are performed in order to evaluate the robustness against noise of the side-information CS algorithms ompared to the onventional CS strategy. 64 random referene senes with a sparsity rate equal to are simulated and adapted with a hange rate of. Noise is added to the measurements of the adapted senes to obtain a varying Signal-to- Noise Ratio (SNR) from to 28 db. Fig. 2 depits the mean minimum sub-sampling rate for a suessful reonstrution for inreasing SNR. A positive impat on the robustness against noise an be seen: the side-information algorithms allow for sub-sampling at lower SNR and the noiseless mean undersampling rates (the horizontal lines) are also reahed at lower SNR ompared to the onventional CS. IV. EXPERIMENTAL RESULTS ON REAL DATA Only the results obtained with the algorithm will be ompared to the onventional CS approah, given the better performane of the, as disussed in [8] and verified with the simulations presented in this doument. The objet under test (Fig. 2) is a 3D-printed retangular blok (dimensions: 18 m in length x 5 m in depth x 1 m in height) made out of ABS (Arylonitrile-Butadiene- Styrene) polymer, with three ylindrial holes (radius: 5 mm) reahing from top to bottom. This objet is ompared to an undamaged massive referene objet made out of the same material and possessing the same dimensions. The blok is plaed on a platform fixed on a sanning stage at a standoff distane of 27 m from the antenna. Between eah range measurement, the blok is moved 1 m and one omplete measurement is omposed of 3 azimuth lassi CS l1-l2 minmization SNR [db] Fig. 2. Experimental results obtained through simulation. 64 random referene senes with a sparsity equal to were built. Eah of them was hanged with a hange rate equal to. The mean minimum number of samples (undersampling rate δ) was determined for obtaining a orret reonstrution for inreasing SNR (from to 28 db) of the measurements. This was done for the three different reonstrution approahes: (1) lassi CS (in blue); (2) (in red) and (3) l1-l2 minimization (in green). The horizontal dashed lines represent the mean value of the undersampling rate for a suessful reonstrution using noiseless measurements. Fig. 3. Objet under test: Blok of ABS (dimensions: 18 m x 5 m x 1 m) with three vertial and ylindrial (radius: 5mm) defets through the blok at different depths. measurements. Fig. 3 shows the ompressed images in range and azimuth obtained from a measurement sampled at the Nyquist rate of the referene blok (on the left) and the blok with three defets (on the right). Further reonstrutions will be ompared with these images and expressed in PSNR ([db]) in order to evaluate the reonstrution performane. Fig. 4 depits a visual omparison between the lassi CS reonstruted images and the l1-l1 reonstrution for four different undersampling rates (8%, 6%, 4% and 2%). The l1-l1 performs remarkably better than the onventional reonstrution: with only 4% of the samples, the defets an still be deteted, whereas even the front of the blok annot be identified with the lassi approah. These results are of ourse subjeted to the stohasti behavior of the results, due to the random sampling. Fig. 5 gives a statistial overview of the PSNR for 32 random sampling exeutions for both approahes and for the same four undersampling rates. One again, a remarkable gain in reonstrution performane an be identified for the l1-l1 reonstrution ompared to the lassi CS reonstrution. The results obtained with the onventional CS approah are notieably more spread out around the mean value, resulting in larger boxplots and differenes between the extrema (Fig. 5).

4 Azimuth [m] Referene objet Azimuth [m] Objet with defets PSNR [db] CS with prior knowledge Standard CS Range [m] Range [m] Subsampling [%] Fig. 4. Reonstruted images from the fully sampled data. The left image is the image obtained for the referene blok. The two interfaes (the front of the blok at 7 m and the bak of the blok at.33 m) are learly visible. The right image is obtained for the blok with defets. The three defets are learly visible and highlighted in green. Fig. 6. Statistial result for the reonstrution from 32 different random sampling exeutions with a subsampling rate equal to 2%, 4%, 6% and 8% of the Nyquist sampling rate. The (light blue) learly outperforms the onventional CS (in dark blue) approah. The quality of the reonstrution is ompared to the reonstrution using 1% of the samples and expressed in PSNR [db]. V. CONCLUSIONS AND FUTURE WORK 8% 6% This paper evaluates the appliability of adding side information to the CS minimization equation using an l1-l1 minimization for SAR imaging. Simulations and tests on real data prove the benefit by dereasing the number of samples, needed to obtain an aurate reonstrution, for this approah ompared to onventional CS as long as the similarity with the referene image is aeptable. Simulations show that the l1- l1 minimization performs better than the l1-l2 minimization. Further sub-sampling would be possible if the non-sparse data were expressed in a sparser domain (e.g. wavelets) and with the use of a blok sparsity algorithm. REFERENCES 4% 2% CS Fig. 5. Visual omparison of the reonstruted images obtained by solving the lassi CS minimization (left olumn) and obtained with the l1-l1 minimization approah (right olumn), for different sub-sampling rates (from top to bottom: 8%, 6%, 4%, 2%). [1] T. Wohlers, additive manufaturing and 3d printing state of the industry, Wohlers Assoiates, Fort Collins, CO, 213. [2] troubleshooting pitorial guide. [3] S. Bhattaharya, T. Blumensath, B. Mulgrew, and M. Davies, Fast enoding of syntheti aperture radar raw data using ompressed sensing, in Statistial Signal Proessing, 27. SSP 7. IEEE/SP 14th Workshop on. IEEE, 27, pp [4] L. Anitori, M. Otten, and P. Hoogeboom, Detetion performane of ompressive sensing applied to radar, in Radar Conferene (RADAR), 211 IEEE. IEEE, 211, pp [5] G. Rilling, M. Davies, and B. Mulgrew, Compressed sensing based ompression of sar raw data, in SPARS 9-Signal Proessing with Adaptive Sparse Strutured Representations, 29. [6] E. J. Candes and T. Tao, Deoding by linear programming, Information Theory, IEEE Transations on, vol. 51, no. 12, pp , 25. [7] D. Donoho, Compressed sensing, IEEE Transations on Information Theory, vol. 52, no. 4, pp , 26. [8] J. F. Mota, N. Deligiannis, and M. R. Rodrigues, Compressed sensing with side information: Geometrial interpretation and performane bounds, in Signal and Information Proessing (GlobalSIP), 214 IEEE Global Conferene on. IEEE, 214, pp [9], Compressed sensing with prior information: Optimal strategies, geometry, and bounds, arxiv preprint arxiv: , 214. [1] G.-H. Chen, J. Tang, and S. Leng, Prior image onstrained ompressed sensing (pis): a method to aurately reonstrut dynami t images from highly undersampled projetion data sets, Medial physis, vol. 35, no. 2, pp , 28.

5 [11] L. Weizman, Y. C. Eldar, D. B. Bashat, D. Nguyen, L.-N. Tran, P. Pirinen, M. Latva-aho, A. Mansoor, V. Patsekin, D. Sherl et al., Compressed sensing for longitudinal mri: An adaptive-weighted approah, arxiv preprint arxiv: , 214. [12] J. Mota, N. Deligiannis, A. C. Sankaranarayanan, V. Cevher, and M. Rodrigues, Dynami sparse state estimation using 1-1 minimization: Adaptive-rate measurement bounds, algorithms and appliations, in 4th IEEE International Conferene on Aoustis, Speeh and Signal Proessing (ICASSP) 215, no. EPFL-CONF-2577, 215. [13] J. F. Mota, N. Deligiannis, A. C. Sankaranarayanan, V. Cevher, and M. R. Rodrigues, Adaptive-rate reonstrution of time-varying signals with appliation in ompressive foreground extration, IEEE Transations on Signal Proessing, 216. [14] R. Bamler, A omparison of range-doppler and wavenumber domain sar fousing algorithms, IEEE Transations on Geosiene and Remote Sensing, vol. 3, no. 4, pp , [15] A. Ribalta, Time-domain reonstrution algorithms for fmw-sar, IEEE Geosiene and Remote Sensing Letters, vol. 8, no. 3, pp , 211. [16] E. Cristofani, M. Vandewal, C. Matheis, and J. Jonusheit, In-depth high-resolution sar imaging using omega-k applied to fmw systems, in Radar Conferene (RADAR), 212 IEEE. IEEE, 212, pp