Space-time versus frequency domain signal processing ... - CiteSeerX

Space-time versus frequency domain signal processing for 3D THz imaging Roel Heremans

Marijke Vandewal

Marc Acheroy

Royal Military Academy Renaissancelaan 30 1000 Brussels, Belgium Email: [email protected],

Royal Military Academy Renaissancelaan 30 1000 Brussels, Belgium [email protected]

Royal Military Academy Renaissancelaan 30 1000 Brussels, Belgium Tel: + 32 2 742 66 60 Fax: + 32 2 742 64 72

Abstract— For the first time, to the best of the authors knowledge, a 3D image reconstruction is developed using wide beam THz radiation. The reconstruction method finds its origins in the domain of radar and sonar where it is known as Synthetic Aperture Radar (SAR) and Synthetic Aperture Sonar (SAS) respectively. The extension to the SAR/SAS reconstruction algorithms on wide beam terahertz radiation results in a highresolution 3D image by combining the depth information (due to the penetration aspect of THz radiation) with a 2D scanning setup. Two 3D reconstruction algorithms have been developed, one in the space-time domain and one in the frequency domain. They both have been validated and analysed using simulated data: the azimuth resolution dependence - on the transmitted frequency and on the opening angle - is compared between the space-time and the frequency domain algorithm as well as their respective computational load. The application of the proposed imaging techniques lays in the domain of non-destructive testing (NDT) in particular for composite aircraft samples.

I. I NTRODUCTION In this paper multidimensional processing is applied on signals simulated in the terahertz (THz) spectrum, which covers a frequency spectrum from the far-IR (Infra-Red) region to the mid-IR region (center frequency between 100 GHz and 10 THz). There are three major factors contributing to the choice of this research domain: (1) THz radiation enables systems to see-through composite materials making them very useful for non-destructive testing (2) THz radiation is nonionizing and poses no health risk to the systems operator, (3) the use of high frequencies promises potential for high resolution results when applying imaging techniques. Despite the knowledge that synthetic aperture (SA) processing leads to high resolution images, very little research exists in SA using THz radiation [1], [2]. In high-resolution imaging two types of spatial resolution need to be defined for pulsed THz systems. The first one is referred to as the depth (or range) resolution which depends on the spectral bandwidth of the pulses and which will not be altered by SA imaging. The available bandwidth of the pulse is limited by two times the carrier frequency, and from this bandwidth, the achievable pulse width can be determined and thus the range resolution. It can be easily calculated that this resolution is an order of magnitude smaller than the used peak wavelength. The second type of resolution is a 2-dimensional (2D) azimuth resolution (measured in the plane (SA surface in Fig. 1)

perpendicular to the pulse direction) which depends fully on the opening angle of the used THz beam and the distance at which the object is placed from the THz source. Since most composite materials have a maximum transparency in the sub THz (typically between 0.1 and 0.8 THz), the minimum wavelength and thus the minimum opening angle one can use is limited and therefore the achievable azimuth resolution. A set of optics is responsible for keeping the opening angle small and the positioning and fine tuning of these optics need to be done with care [3], [4]. It should be noted that this resolution is only reached for a certain layer within the illuminated object given the so-called beam focusing depth inherent to the used optics. If the requirements on this beam width could be relaxed without deterioration of the azimuth resolution, the hardware would be a lot less complex and the focusing would not be limited to a certain layer of the test object. This justifies a measurement technique with a wide angle beam, working in the sub THz band, for which the necessary resolution is reached through the application of an appropriate signal processing, in this case the synthetic aperture processing. As indicated above the SA principle is based on the use of a wide antenna beam. In order to achieve a high resolution 3D image, this wide beam has to scan the object to gather redundant information which has to be combined intelligently by the SA processing. Indeed, the antenna moves in two dimensions along the SA surface of Fig. 1 to gather redundant information of a point target at position (ut , vt , zt ) and to produce the equivalent of a large antenna. The size of the synthesized antenna -called the synthetic aperture- is determined by the amount of time the point target is covered by the antenna beam and is several orders of magnitude longer than the real aperture size. If the start time and phase of each transmitted and received pulse along tohe synthetic aperture are carefully controlled, then the phase of the received echoes (called THz raw data) will vary with respect to azimuth time in a predictable manner: applying a 2D scanning motion the variation of the phase will be proportional to the variation of the distance between the target and the sensor position, the latter being illustrated by the hyperboloid in Fig. 2. This phase variation can be matched, compressing the signals, hence reducing the recorded hyperboloids to an imaged point target after application of the SA processing. The obtainable

Fig. 1.

Geometry: scanning plane and reconstruction pixel.

azimuth resolution is approximated by half the length of the real antenna aperture [5], [6]. For this intensive signal processing a compromise will have to be found between image accuracy and computational load (and thus imaging speed) which will be illustrated by two processing algorithms applied on simulated data: one working in the space-time domain and one working in the 3D frequency domain. The following methodology has been applied in this paper: at first the concept of the simulation of the THz raw data obtained by a 2D scanning along the SA surface using a wide beam antenna is explained. This simulator allows verification and validation of the developed SA processing algorithm presented next. Two very different types of 3D SA processors are demonstrated and tried out on the simulated data. The performances (in terms of resolution) of these 3D imaging techniques are evaluated for varying SA parameters. The paper concludes with the analyzed results and achievements, and the future research work. II. R AW DATA SIMULATOR Following the measurement principle illustrated in Fig. 1 one can see that the response of a transmitted THz pulse1 can be simulated for each sensor position (us , vs , zs ), with a delay time characterized by the delay distance from the sensor to the target position (ut , vt , zt ). The THz sensor, using a wide beam antenna, is scanned in two dimensions along the SA surface. The delay distance for a chosen zs = 0 is given by q R(us , vs , ut , vt , zt ) = (ut − us )2 + (vt − vs )2 + zt2 (1) Fig. 2 illustrates the range variation for two point targets during the 2D scanning. The simulator takes a rectangular antenna footprint into account characterized by the footprint size Wu and Wv in respectively the u- and v-direction. Wu and Wv are determined by the antenna opening angles θu and θv and considering a reference depth zref as (see Fig. 1): Wu = 2zref tan(θu /2)

(2)

1 Here a rectangular pulse is considered as the input pulse with a length of τ seconds.

Fig. 2. 3D representation of the delay-time as a function of the sensorposition for 2 targets placed at (ut , vt , zt ) = (−0.1, 0.05, 0.2) and (0.07, 0.2, 0.2) where the three coordinates are expressed in meters. f0 V C

= 160/300/800/1000 GHz = 1 m/s = 299792458 m/s

θu,v OSF zref

= 6/10/20/30 deg = 1.4 = 0.2 m

TABLE I THz parameters used in the raw data simulation.

The same expression holds for the footprint size Wv by replacing θu by θv in equation (2). The opening angles θu and θv can vary, however chosen to be equal in order to obtain comparable azimuth resolutions in the u- and v-direction (see section IV-B). The system parameters that have been used in the simulator are summarized in Table I, and are explained in the text bellow. (Nu , Nv ) samples have been generated in the u-and vdirection with a bin spacing ∆u = ∆v = P VRF , with V the velocity of the sensor displacement and PRF the pulse repetition frequency. Those bin spacings have been calculated with respect to the SA constraints as mentioned in [7]. The PRF is chosen to respect Nyquist regarding the expected bandwidth BWaz in azimuth directions u and v [6],  P RF = OSF × BWaz (3) BWaz ≈ 4V tan(θλu /2) where OSF stands for over sampling factor. Since the signal bandwidth in azimuth direction depends mainly on the carrier frequency f0 (through λ, the carrier wavelength) and on the antenna opening angles, the P RF has been chosen to be slightly bigger than the expected azimuth bandwidth, so that the simulation is computational efficient. Fig. 3 shows as a function of the antenna opening angle for three carrier frequencies (f0 = 160, 300 and 800 GHz) the variation in P RF and thus in (Nu , Nv ). The bin size in range is defined as:  ∆r = Cτ = C2 BW1 ra 2 (4) 2f0 BWra = OSF with C the speed of the electromagnetic wave in vacuum,

THz raw data for which the range hyperboloid of Fig. 2 is not too curved as will be explained in paragraph IV-B. A. Time domain processing algorithm

Fig. 3. PRF values (up) as a function of antenna opening angle and the number of samples in the synthetic aperture for three carrier frequencies (f0 = 160, 300 and 800 GHz).

BWra the bandwidth of the signal in range direction and τ the pulse length adapted to the carrier frequency f0 . The output of the simulator is a 3D matrix d(., ., .) (see equation (5)), containing Nt range samples for each sensor position (us , vs ); for a given sensor position each range sample corresponds with a specific range containing the impulse response amplitude and phase information of the point targets at that range from the sensor as given by, d(us , vs , R) =

N X

sn (ut , vt , zt ) exp (−i

n=1

4π R) λ

(5)

where the sum is taken over the total number of targets N and where sn represents the reflectifity of target n. III. R ECONSTRUCTION A SA processing algorithm can be defined as a range dependent filter operation on the received raw data. They generally differ from each other in the approximations they use to combat problems as large data sets, range migration effects (demonstrated by the hyperboloids of Fig. 2), and others. One way of classifying them is by looking at the domains they mainly perform the image reconstruction in. Time domain algorithms are also referred to as space-time domain algorithms where typically all calculations are performed in the time domain for the 3 dimensions. The major advantage is that the range dependence of the raw data is easily taken into account since a new matched filter is calculated per range and the image is built up pixel by pixel. The major drawback is the low overall computational efficiency as a direct result of the above characteristic. Frequency domain algorithms are the typical wavenumber domain algorithms or three-dimensional frequency domain algorithms. They use a 3D transfer function in the frequency domain to compress the raw data. The implementation of this filter can be (almost) exact (as in the range migration algorithm explained below) which results in a good accuracy and an enhanced imaging speed. However, these performances are only guaranteed for

The time domain processing algorithm consists of two main steps for each image reconstruction pixel. On one hand, one has to determine the exact distance R (see equation (1)) between the sensor position in the SA sum under study and the reconstruction position. Due to the fact that this exact distance does not necessarily correspond to a position in the range vector of d(., ., .), an interpolation of the range data has to be performed. On the other hand, one has to sum the signal coherently by multiplying this interpolated range data with the term exp(i 4π λ R) which can be considered as a phase correction term. For one reconstruction point one has thus to sum the result of the interpolation step and the phase correction for each term in the synthetic aperture sum along the u and the v directions. Indeed, the backscattered synthetic aperture signal for the reconstructed grid point at position (ur , vr , zr ) is given by; I(ur , vr , zr ) = Pur +Wu /2 Pvr +Wv /2 k=ur −Wu /2

l=vr −Wv /2

d(k, l, R) exp(i

4π λ

R)

(6) where R determines the distance between the reconstruction grid point (ur , vr , zr ) and the scanning position under consideration varying as a function of the synthetic aperture sum indices k and l; p R = (ur − us (k))2 + (vr − vs (l))2 + zr2 (7) and d(., ., .) indicates the 3D simulated raw data matrix. This procedure has to be repeated for each pixel in the reconstruction grid. The time domain processing algorithm is schematically shown in the blockdiagramma of Fig. 5. The result of the time domain processing algorithm is a 3D image, shown in Fig. 4 using 14 point targets. Isosurfaces are shown at reconstruction intensities corresponding to values equal to 50 percent of the maximum intensity present in the 3D reconstruction cube. The simulated point targets are indicated by the crosses. The size of the isosurfaces around the crosses are an indication of the obtained resolutions.

Fig. 4. Time domain processing algorithm for a carrier frequencies f0 = 160 GHz. A isosurface plot is shown at surface values corresponding to a value equal to 50 percent of the maximum value in the 3D cube.

B. Frequency processing algorithm Assume a point scatterer target located at (ut , vt , zt ) with reflectivity s(ut , vt , zt ) and that any position in the scanning plane can be given by (us , vs , zs ) with zs = 0, than the measured backscatter at position (ur , vr , zr ) associated with this point scatterer can be expressed from [8] starting from equation (5), as ZZ Z s(ur , vr , zr ) = dus dvs dkr d(us , vs , kr ) A kr

p × exp (i2kr (ur − us )2 + (vr − vs )2 + zr2 )

(8)

where A denotes the synthetic aperture scanning surface as indicated in Fig. 1. However to facilitate the Fourier analysis that follows, the integrand d(us , vs , R) is already Fourier transformed over the range variable R into d(us , vs , kr ) in the previous equation (with kr = 2πfr /C as the one-way wavenumber and fr the range frequency corresponding to R). The surface integral in equation (8) can be represented by a 2D convolution between d and e as Z s(ur , vr , zr ) = d(ur , vr , kr ) ⊗u,v e(ur , vr , zr ) dkr kr

Z =

F −1 {F(d(ur , vr , kr )).F(e(ur , vr , zr ))} dkr

(9)

kr

with e(ur , vr , zr ) = exp (i2kr

p u2r + vr2 + zr2 ),

(10)

and where F and F −1 represents respectively the 2D Fourier transform and the corresponding inverse Fourier transform with variables (ur ,vr ) and (ku , kv ) = 2π V (fu , fv ). fu and fv correspond to the respective azimuth frequencies of u and v. The Fourier transform of e(ur , vr , zr ) can be calculated using the method of stationary phase and can be expressed as, ZZ p F(e(u, v, z)) = exp (i2kr u2 + v 2 + z 2 )

'

× exp (−iku u − ikv v) du dv 4πkr exp (−ikz z) ikz2

with the Stolt transformation [9], p kz = 4 kr2 − ku2 − kv2 .

(11)

(12)

(13)

Substituting equation (11) into (9), substituting the frequency wavenumber kr by kz and applying the definition of the inverse 2D Fourier transformation one obtains; ZZZ 2π × s(ur , vr , zr ) = D(ku , kv , kz ) × ikz exp[i(ku u + kv v − kz z)] dku dkv dkz

that the reflectivity image can simply be obtained through a 3D invers Fourier transform of the product of the resampled wavenumber domain backscatter data by a complex exponential. The latter is also known as matched filter. The algorithm implementation to reconstruct the reflectivity image consists of four sequential steps (see Fig. 5), i.e. a 3D FFT, a matched filter, a Stolt interpolation and a 3D IFFT. It is the matched filter step that corrects for the range curvature of all scatterers at the same depth as the scene center. The phase corresponding to the matched filter is space invariant and depends only on the distance to the center of the scene, the carrier frequency and the scanning plane wavenumbers ku and kv . The Stolt interpolation compensates the range curvature of all scatterers by an appropriate warping of the backscattered data. This warping is performed to obtain a uniformly kz sampled dataset in order to be able to perform the 3D IFFT. IV. P ERFORMANCE ANALYSIS

The asymptotic expansion of the method of stationary phase is only valid in the region where 4 kr2 ≥ ku2 + kv2 .

Fig. 5. Blockdiagramma for the time domain (left) and frequency domain (right) reconstruction algorithm.

(14)

Remark that the term D(ku , kv , kz ) has been modified due to the change of variables kr into kz . Equation (14) indicates

A. Computational load The time domain processing algorithm has a very high computing load and is by consequence extremely slow with respect to the frequency domain processing algorithm. Analysing the number of operations for the time domain reconstruction algorithm for an opening angle of 10o in u and v and for a carrier frequency of 160 GHz, leads to a total number of 131075 pixels (N t, N u, N v) = (108, 68, 68). For each of those pixels a dubbel sum has to be evaluated which can become an even bigger slowdown factor, mainly for high opening angles in combination with a high carrier frequency. For θu = 10o and f0 = 800 GHz, Fig. 3 mentions a sum over 132 terms for the central depth zref = 0.2 m, where also the number of reconstruction pixels is higher (here 28.106 pixels) than for the case of lower opening angles or lower carrier frequencies. Taking into account the dubble sum, the reconstruction algorithm has to perform 5.1011 interpolations. Tabel II compares for three opening angles (θu =6o , 10o and

20o ) and three carrier frequencies (f0 =160 GHz, 300 GHz and 800 GHz) the total computation time. θ 6o 10o 20o 30o

f0 = 160 GHz (49) [1.8] (4275) [2.2] (17.105 ) [7.1] [16.8]

300 GHz (1.104 ) [4.0] (8.105 ) [9.2] (1.109 ) [41] [71]

800 GHz (6.105 ) [33] (4.108 ) [/] (4.101 2) [/] [/]

TABLE II Total processing time in seconds to perform the time domain processing (left number indicated between (.)), and to perform the frequency domain processing (right number indicated between [.]) as function of the opening angles (θ=6o , 10o , 20o and 30o ) and as function of the carrier frequencies (f0 =160, 300 and 800 GHz). [/] means that the processing was not feasable due to out of memory problems.

B. Obtained resolution: opening angle, object size The three graphs in Fig. 6 (from top to bottom corresponding to f0 =160, 300 and 800 GHz) show the resolution representation for the time domain algorithm and the frequency domain algorithm, and this for different opening angles. A first observation is that the validity of the frequency domain is restricted to limited values of the opening angle. The maximum opening angle for which the algorithm is still valid, depends on the value of the carrier frequency. The valid opening angle for carrier frequencies f0 =160 and 300 GHz is around θ = 10o , for f0 = 300 GHz. The falldown of the algorithm at θ = 20o is more prominent for f0 =300 GHz than for f0 =160 GHz. For f0 =800 GHz even an opening angle of θ = 10o is performing worse than for θ = 6o . This indicates that a valid opening angle for a carrier frequency of f0 = 800 GHz is somehere between 6 and 10 degrees. A second observation is that the time domain algorithm performs better in terms of resolution for higher opening angles and higher carrier frequencies. The performance as function of carrier frequency for simulated data with an opening angle of θ = 20o goes from 2.2 mm over 1.38 mm to 0.49 mm for respective freqencuies f0 =160, 300 and 800 GHz, which are in good agreement with the theoretical expectation as mentioned in [2]. The time domain algorithm is however computationally far more expensive. For instance for the reconstruction of the simulated data with f0 = 800 GHz and an opening angle θu,v = 6o the total processing time corresponds to 7.28 days. Parallelizing the algorithm reduced the total processing time to aproximately 8 hours. The frequency domain reconstruction needs only 33 seconds as shown in Tabel II. C. Validity of the predicted SAT performance Up till now predictions of the SAT (synthetic aperture THz) performance have been formulated based on the synthetic aperture theory and simulated data. It is clear that this performance will be somewhat degraded when applying SAT on real data. These measurements are planned in the near future and an

Fig. 6. Obtained resolution for both reconstruction algorithm for the different carrier frequencies f0 =160 GHz, 300 GHz and 800 GHz.

ISAT or Inverse SAT setup will be used: The THz system will be static and the testobject will move on a 2D-scanner. The validity of the predicted SAT performance, when extrapolating to real data, will depend on: •

•

•

•

Source and detector coherence A necessary condition to allow synthetic aperture processing is a phase stable source and detector. Otherwise the coherency between the data within the synthetic aperture length cannot be guaranteed. Scanning precision It is obvious that when the spatial position of the source and receiver with respect to the testobject is unaccurately determined during the data recording, an incorrect delay time will be measured. This will lead to an imprecise synthetic aperture summation and will consequently degrade the obtained resolution. Considering that the absolute spatial precision has to obey ∆d < λ/8 [5], for a 1 THz wave (worst case) a spatial precision for the THz system position with respect to the testobject has to be less than 0.3 mm. Using a setup where the source and receiver device are fixed and where the testobject is placed on a moveable platform one should be able to fullfill this requirement in practice taking into account the specifications of existing scanning systems. Phase errors due to a varying propagation medium Another origin of phase errors can be due to the fact that the speed of light C with which the THz wave travels in vacuum slows down to a speed Cm for different materials. The lower speed depends on the material’s relative permittivity r , and its relative permeability µr √ as Cm = C/n where n = r µr indicates the refractive index. Size of the opening angle and the signal to noise ratio Imaging with a high opening angle looks promising, however one has to consider that in the case of real data the dispersed signal due to the wide beam opening angle can fall below the noise level. The optimal opening angle will depend on the dissipated power of the source and the sensitivity of the detector used for the test measurements. V. C ONCLUSION

The reconstruction result on simulated THz data sets has been shown for two reconstruction algorithms. One algorithm has been performed in the time domain whereas the second algorithm has been performed in the frequency domain. The dependency of the obtained resolutions with respect to different values of opening angles and different values of carrier frequencies have been shown. The time domain processing algorithm looks a promising reconstruction choice, given its high resolution capability. However the computing time is rather high so that the technique can not be used in real time situations. The frequency domain algorithm is on the other hand much faster and real time imaging can be possible. However here the algorithm will not be applicable for opening angles bigger than 10o due to the fact that the range migration is performed for a reference

distance mostly chosen in the center of the 3D reconstruction box. In order to be able to deliver a real time processing reconstruction in the future, one has to investigate and improve the frequency performance for large opening angles. R EFERENCES [1] R. Heremans, M. Vandewal, and M. Acheroy, “Synthetic aperture imaging extended towards thz sensors,” in Proceedings of the IEEE Sensors 2008, Lecce, Italy, Oct. 2008. [2] M. Vandewal, R. Heremans, and M. Acheroy, “Synthetic aperture signal processing for high resolution 3d image reconstruction in the thz-domain,” in In proceeding of the 17th European Signal Processing Conference (EUSIPCO), Glasgow, Scotland, Aug. 2009. [3] A. Redo-Sanchez, N. Karpowicz, J. Xu, and X.-C. Zhang, “Damage and defect inspection with terahertz waves,” N. Dartmouth, MA, 2006. [4] R. F. Anastasi and E. I. Mandaras, “Terahertz nde for metallic surface roughness evaluation,” in The 4th international workshop on ultrasonic and advanced methods for nondestructive testing and material characterization, 2006, pp. 57–62. [5] P. Lacomme, Air and Spaceborne Radar Systems, An introduction. William Andrew Publishing, New York, 2001. [6] I. G. Cumming and F. H. Wong, Digital processing of synthetic aperture radar data. Boston: Artech House, 2005. [7] M. P. Bruce, “A processing requirement and resolution capability comparison of side-scan and synthetic-aperture sonars,” IEEE Journal of oceanic engeneering, vol. 17, no. 1, pp. 106–117, 1992. [8] J. M. Lopez-Sanchez and J. Fortuny-Guasch, “3d radar imaging using range migration techniques,” IEEE Transactions on antennas and propagation, vol. 48, no. 5, pp. 728–737, 2000. [9] R. Stolt, “Migration by forier transform techniques,” Geophys, no. 43, pp. 49–76, 1978.