Traffic monitoring with spaceborne SAR—Theory, simulations, and experiments

Computer Vision and Image Understanding 106 (2007) 231–244 www.elsevier.com/locate/cviu

Traffic monitoring with spaceborne SAR—Theory, simulations, and experiments q Stefan Hinz

a,*

, Franz Meyer b, Michael Eineder b, Richard Bamler

a b

a,b

Remote Sensing Technology, Technische Universitaet Muenchen, Germany1 Remote Sensing Technology Institute, German Aerospace Center, Germany2 Received 6 December 2005; accepted 25 September 2006 Available online 29 December 2006 Communicated by James Davis and Riad Hammoud

Abstract This paper reviews the theoretical background for upcoming dual-channel radar satellite missions to monitor traffic from space and exemplifies the potentials and limitations by real data. In general, objects that move during the illumination time of the radar will be imaged differently than stationary objects. If the assumptions incorporated in the focusing process of the synthetic aperture radar (SAR) principle are not met, a moving object will appear both displaced and blurred. To study the impact of these (and related) distortions in focused SAR images, the analytic relations between an arbitrarily moving point scatterer and its conjugate in the SAR image have been reviewed and adapted to dual-channel satellite specifications. Furthermore, a specific detection scheme is proposed that integrates complementary detection and velocity estimation algorithms with knowledge derived from external sources as, e.g., road databases. Results using real SAR data are presented to validate the theory.  2006 Elsevier Inc. All rights reserved. Keywords: Vehicle detection; Velocity estimation; Spaceborne synthetic aperture radar; Along-track radar interferometry

1. Motivation Increasing traffic appears to be one of the major problems in urban and sub-urban areas. Both components, the increase of transport safety and transport efficiency, as well as the reduction of air and noise pollution are the main tasks to solve in the future. Automated traffic monitoring has consequently evolved to an important research issue during the past years. Nowadays, sensors like induction loops, bridge sensors and stationary cameras acquire the traffic flow on some main roads, while traffic on smaller roads, which represent the main part of road networks, is rarely collected. q Significantly extended version of a contribution for OTCBVS’05 workshop held in conjunction with CVPR’05. * Corresponding author. Fax: +49 89 280 9573. E-mail address: [email protected] (S. Hinz). 1 urlwww.RemoteSensing-TUM.de/lmf 2 urlwww.imf.dlr.de

1077-3142/$ - see front matter  2006 Elsevier Inc. All rights reserved. doi:10.1016/j.cviu.2006.09.008

However, when gathering traffic information of the complete network, the drivers could be provided with much more valuable information like, for instance, precise estimates for the current travelling time for various routes from point ‘‘A’’ to point ‘‘B’’ in the road network. Hence, area-wide images of the entire road network are required to complement these selectively acquired data. A number of approaches have recently been developed to automatically detecting vehicles or vehicle rows in optical satellite imagery—mainly pushed by the launch of the new 1-m class optical satellite systems as Ikonos and QuickBird (see e.g. Refs. in [1,2]). Traffic monitoring based on optical satellite systems, however, is only possible at daytime and cloud-free conditions, while spaceborne SAR (Synthetic Aperture Radar) systems are not affected by these limitations. Yet there are other difficulties inherent in the SAR imaging process that must be overcome to design a reasonably good approach for traffic monitoring using spaceborne radar.

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Table 1 Parameters of the TerraSAR-X satellite Orbit height Wavelength Satellite velocity Beam velocity on ground Range (@ 40) FM rate Processed Doppler bandwidth Processed synthetic aperture Pulse repetition frequency Ground sampling distance

h k vsat vB R0 FM PBW TA PRF

515,000 m 0.0311 m 7600 m/s 7105 m/s 670,000 m 5183 Hz/s 3000 Hz 0.5788 s 4000 Hz 1–3 m

Note. The distinction between vsat and vB can be usually neglected for airborne platforms. In the case of space-borne SAR both values differ significantly due to the curved orbit and the influence of the earth rotation.

In this paper, special emphasis is put on a thorough analysis and validation of the effects of moving objects in SAR images. Based on this, a detection strategy is derived that accommodates for the restrictions of civilian SAR satellite systems. To validate the theory, we use the sensor and orbit parameters of the upcoming TerraSAR-X mission (see Table 1). Its high resolution synthetic aperture radar sensor will operate in X-band and deliver images of 1–3 m resolution. The system allows to splitting-up the antenna into two parts on receive so that two high resolution SAR images of the same scene can be acquired within a small time frame. This paper focuses on the theory of moving object detection in these images. It is thus the basis for the follow-on paper [3], where a comprehensive performance analysis for the adaption of the detection approach to TerraSARX can be found. After a brief overview of related work in Section 2 we review the theoretical background of influences caused by moving objects in one- and dual-channel SAR images in Section 3. Then, Section 4 outlines theory and simulations of a detection and velocity estimation algorithm, that integrates (uncertain) a priori knowledge on location and movement of vehicles. Experiments and validations using real images are presented and discussed in Section 5 before concluding with an outlook in Section 6. ‘Real’ images have been acquired during flight campaigns in which the radar instrument has been parameterized such that the resulting data correspond approximately with the expected data of TerraSAR-X. 2. Background and related work In military research the task of detecting moving vehicles with SAR sensors is well known as ground moving target indication (GMTI). GMTI approaches are commonly based on a SAR sensor with at least 3 channels and use space-time adaptive processing (STAP) for target detection, see e.g. [4–6] for more details. However, civilian spaceborne SAR systems with 3 or more channels are currently not available. The upcoming TerraSAR-X mission as well as the Canadian RADARSAT-2 mission will be equipped with

a single channel SAR that can be switched to an experimental mode with only 2 channels. These spaceborne SAR systems are thus by far not optimal for the task of traffic monitoring. Although some investigations are currently under way, whether a further splitting of 1 phased array antenna in 3 or more sub-antennas will support such kind of applications [7], this question cannot be answered clearly up to now. The benefit of reducing clutter by the use of 3 or more sub-antennas might be compensated by the loss of radiometric quality due to the worse signal-tonoise-ratio (SNR) caused by the smaller area of each subantenna. The classical approach for detecting moving points in images of a two-channel system is the displaced phase center array (DPCA) method. When certain assumptions on the statistical distribution of the involved backscattered signals are met, DPCA can be shown to be the optimal detector for two-channel images [8]. While DPCA exploits primarily the phase information of the two images, along track interferometry (ATI) includes also the amplitude information. The issue of detecting moving targets using ATI is discussed, for instance, in [9,10]. In [11], special emphasis is put on the probability density functions associated with this detection, and the influence of vehicle acceleration is discussed in [12,13]. Traffic monitoring from space is quite rare so far. But as shown in [14–16] first endeavors have already been carried out. Although civilian SAR systems are less specialized for moving object detection than military systems, there are other advantages that can be exploited. In contrast to military applications, civilian applications include more constraints regarding the objects to detect. In the traffic monitoring case, we can assume that vehicles travel on roads of a known road-network, which might not be true in military GMTI. Such knowledge provides a priori information that can be effectively used for detection. It is one of the objectives of this paper to develop a detection strategy that incorporates this kind of a priori information. 3. Moving objects in SAR images We briefly review the SAR imaging process for stationary objects in Section 3.1, before deriving the theory for moving objects in SAR images in Section 3.2. Finally, Section 3.3 quantifies the theoretical findings for selected cases. We assume the SAR system operating in strip map mode for all the following derivations. The squint angle of the SAR system is assumed to be zero. 3.1. SAR image formation Let the position of a radar transmitter on board a satellite be given by Psat(t) = [xsat(t), ysat(t), zsat(t)] with x being the along-track direction, y the across-track ground range direction and z being the vertical. A point scatterer is assumed to be at position Pobject = [xobject(t), yobject(t), zobject(t)]. The

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time-dependent range to an arbitrarily moving point target from the radar platform is simply defined by RðtÞ ¼ P sat ðtÞ  P object ðtÞ

ð1Þ

The minimum distance between point and radar is denoted R0 and the time variable t is chosen such that R(t = 0) = R0. The measured echo signal of this point scatterer can consequently be written as us ðtÞ ¼ ah ðhÞ  ab ðbÞ  gðs  2RðtÞ=cÞ   4p  exp j  RðtÞ k

ð2Þ

with g(s  2R(t)/c) being the delayed complex pulse envelope, c being the speed of light, and ah(h), ab(b) being the amplitude of the two-way antenna patterns in elevation and azimuth, respectively [17,18]. To simplify the equations, the amplitude A = ah(h) Æ ab(b) Æ g(s  2R(t)/c) is discarded in the following. We assume in addition that the SAR data is already range compressed and the range cell migration has already been eliminated. Furthermore, one can approximate in strip map mode   4p exp j  RðtÞ  expfjpFMt2 g ð3Þ k with FM ¼ 

2 d2 2 RðtÞ ¼  vsat vB k dt2 kR0

ð4Þ

being the frequency modulation rate of the azimuth chirp, and vB being the beam velocity on ground [19]. Here it is assumed that the synthetic aperture length is short compared to the orbit height of the sensor to keep higher order terms of the series expansion of R(t) insignificant. The remaining chirp function (the so-called azimuth chirp) is focused using the matched filter concept[17,18]. According to this concept the filter must correspond to sðtÞ ¼ expfjpFMt2 g

ð5Þ

so that an optimally focused image is obtained by complexvalued correlation of u(t) and s(t). For efficiency reasons, this operation is commonly done in frequency domain by multiplying the respective spectra U(f) and S(f) T ðf Þ ¼ U ðf Þ  Sðf Þ

ð6Þ

using highly specialized algorithms like the x–j algorithm or the chirp scaling algorithm (see [18] for details). Please note the definition of the matched filter s(t). To construct the filter correctly, the actual range history of each target in the image must be known. For this, a priori information about sensor trajectory, scatterer position and motion is necessary. Usually, the time dependence of the scatterer position is ignored yielding Pobject(t) = Pobject. This concept is commonly referred to as stationary-world matched filter (SWMF). Because of this definition, a SWMF does not correctly represent the phase history of

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a significantly moving object, which eventually results in image deteriorations. 3.2. Theory of object motion effects in SAR data The main effects that result from object motion depend on the respective component of the object’s velocity and acceleration vectors w.r.t. the co-ordinate system x, y (x along-track or azimuth direction, y across-track direction). 3.2.1. Constant along-track motion First, the target is assumed to move with constant velocity vx0 in azimuth direction. The relative velocity of sensor and scatterer is different for the moving object and the surrounding stationary world. Thus, along track motion changes the frequency modulation (FM) rate of the received scatterer response FMmt. This particular FM rate of a target moving in along-track direction with velocity vx0 is given by  rffiffiffiffiffiffiffi 2 2 pffiffiffiffiffiffiffiffiffiffiffi vsat vsat vB  FMmt ¼  vx0 kR0 vB  2 vx0 ¼ FM 1  ð7Þ vB Consequently, the spectrum of the received signal of a target moving in along track can be written as   f2 U mt ðf Þ ¼ exp jp ð8Þ FMmt Focusing the signal umt(t) with the conventional SWMF s(t) in frequency domain yields   f2 T mt ðf Þ ¼ U mt ðf Þ  Sðf Þ ¼ exp jp ð9Þ dFM where rffiffiffiffiffiffiffi 1 vsat vx0  k R0 dFM vB ðvsat vB Þ32

ð10Þ

The phase of the focused signal Tmt(f) is quadratic and causes a spreading of the signal energy in time or space domain depending on dFM. Unfortunately, the fourier transform of Tmt(f) has no exact analytic solution. Nevertheless, considering the stationary phase approximation of the Fourier-transform the width of the focused peak can be approximated by PRF vx0 ¼ 2T A ½s Dt  ð11Þ d FM vB with TA being the aperture time. Interpretation of Eq. (11) shows that a moving vehicle is smeared by twice the distance it moved along-track during the illumination time TA. It has to be kept in mind that the approximation in Eq. (11) only holds if vx0  0. 3.2.2. Constant across-track motion Let the target now move with constant velocity vy0 in across-track direction. This movement causes a change of range history proportional to the projection of the motion

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vector into the line-of-sight direction of the sensor vlos = vy0 Æ sin(h), with h being the local elevation angle. In case of constant motion during illumination the change of range history is linear and causes an additional linear phase trend in the echo signal. The resulting signal of an object moving in line-of-sight direction with velocity vlos is consequently:   4p 2 ð12Þ vlos ðtÞ ¼ expfjpFMt g  exp j vlos t k The spectrum Tlos(f) = Ulos(f) Æ S(f) of the moving target after focusing with a SWMF results in     2 2 2 fv vlos T los ðf Þ ¼ exp j2p exp j2p 2 kFM los k FM ð13Þ As one can see, Tlos(f) is composed of a linear phase (first part) and a constant phase term (second part). Following the laws of Fourier transform the linear phase component corresponds to a time shift tshift in time domain. Fouriertransforming the linear term of Eq. (13) yields: tshift ¼

2vlos ½s kFM

ð14Þ

Some simple transformations of Eq. (14) give the azimuth displacement of a moving object in space domain vlos Dazimuth ¼ R0 ½m ð15Þ vsat where vB is the beam velocity on ground with tshift ¼ Dazimuth vB (see Table 1). Across-track motion consequently results in an along-track displacement of the moving object. It is displaced in flying direction when the object moves towards the sensor (i.e. the range decreases) and reverse to flying direction when the movement is directed away from the sensor (i.e. the range increases). Fig. 1 gives an impression how the two effects of blurring and azimuth displacement are linked. The circular trajectory of a car results in stronger blurring (depicted in vertical direction) the more the car moves in along-track direction. When

the car turns more into across-track direction, blurring gets less but displacement gets larger (shown by the colors turning from blue to red and widening of the surface). 3.2.3. Object accelerations It is assumed in the above derivations that vehicles travel with constant velocity and along a straight path during the illumination by the radar. However, acceleration is commonplace in real traffic and should be considered in any processor or simulation. Acceleration effects do not only appear when drivers physically accelerate or brake but also due to curved roads, since the object’s along-track and across-track velocity components vary during the radar illumination. To model the influence of object accelerations mathematically, we approximate a point scatterer’s range history by a third-order Taylor series. Let the radar transmitter on board of a satellite at the altitude H move with constant velocity vsat in along-track direction (x-axis). The point scatterer is assumed to be at position (0, y0, 0) at azimuth time t = 0 and to move with velocity components vx0 and vy0 and acceleration components ax and ay in along-track and acrosstrack direction, respectively. The vehicle’s height is assumed to be zero over the entire observation period and the vehicle is assumed to be a point scatterer with rotation invariant reflection characteristics. Then, the third order Taylor series expansion of the range to an accelerating point target from the radar platform is given as: y vy0 RðtÞ ¼ R0 þ 0 t R0     1 y 20 2 2 þ ðvx0  vsat Þ þ vy0 1  2 þ y 0 ay0 t2 2R0 R0     2 1 y þ vy0 ay0 1  02 þ ðvx0  vsat Þax0 t3 2R0 R0 " # 2 1 y 0 vy0 ðvx0  vsat Þ þ y 0 v3y0 3  ð16Þ t 2R0 R20

Fig. 1. Illustration of blurring and azimuth displacement for a circular car trajectory: Stronger blurring is indicated along the vertical direction, while colors (from dark to bright gray values) and widening of the surface visualize azimuth displacement.

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Eq. (16) yields that acceleration components appear in the quadratic and the cubic term of the Taylor series expansion. The acceleration in across-track direction (ay) causes a quadratic phase component in the range equation which results in a spreading of the signal energy in time or space domain. Along-track acceleration appears in the cubic phase term and results in an asymmetry of the focused point spread function. A more in-depth discussion of acceleration effects can be found, for instance, in [12,13]. 3.3. Quantification of object motion effects in satellite radardata We exemplify the above-mentioned effects for the case of a spaceborne radar by using the specifications of the upcoming TerraSAR-X satellite. The side-looking synthetic aperture radar is based on active phased array antenna technology and TerraSAR-X’ Dual Receive Antenna Mode will enable along-track interferometry. Based on the formulae derived above, and using the system parameters as specified in Table 1, we estimate the impact of moving objects in TerraSAR-X data. These investigations help to derive the boundary conditions for building up a strategy for detecting ground moving targets in TerraSAR-X images. 3.3.1. Influence of constant along-track motion The effect of blurring of the focused target response due to an along-track component of the target velocity is calculated based on Eq. (11). The peak width of a moving object in a focused TerraSAR-X is shown in Fig. 2 as a function of viewing angle h and along-track velocity vx0. As the backscattered energy of the moving object is now spread over a larger area the peak value of the signal drops 60

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with increasing vx0. Following the laws of fourier transform the signal amplitude at position t = 0 (the position of the signal peak) can be calculated by integrating the signal spectrum. ( rffiffiffiffiffiffiffi )   Z PRF=2 vsat vx0 1 2 hð0; vx0 Þ ¼ exp jpk R0 f df 2 s vB ðvsat vB Þ32 PRF=2 ð17Þ This Fresnel integral is usually approximated by the stationary phase approximation for vx0  0. Hence, the signal amplitude calculates to hð0; vx0 Þ  dFM ¼ B

3.3.2. Influence of constant across-track motion Based on Eq. (15) the azimuth shift of a target moving with velocity vy0 in across-track is calculated. The results are illustrated in Fig. 4. Fig. 4 shows that moving vehicles are significantly displaced from their real position even for small across-track velocities (about 1 km for 50 km/h at 45 incidence angle). This effect strongly hampers the recognition of cars in TerraSAR-X images as their position is not related to semantic information, e.g. streets. Fig. 4 also shows that the azimuth displacement is ambiguous if jDfd j > PRF or jvy0 j > PRFk . 2 4 sin h This effect can be observed in the lower right corner of Fig. 4. There, the across-track velocity of the object rises 1

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stationary phase approximation

|h(0;vx0)| 2

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ð18Þ

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1 vB T A vx0

It is inversely proportional to the smearing of Eq. (11). The decrease of peak power in TerraSAR-X images as a function of the along-track velocity is illustrated in Fig. 3. As can be seen, the effect of along-track movement has significant influence on the peak amplitude in TerraSAR-X images. Strong blurring distributes the backscattered energy and results in a drop of 50% peak power or more if vx0 P 14.4 km/h. Consequently, nearly all ground moving targets will suffer from energy dispersion, which decreases the signal-to-clutter ratio and renders target detection more difficult.

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Fig. 2. Effect of along-track motion in TerraSAR-X SLCs. Blurring of a moving point scatterer as function of h and vx0. Black curves indicate ‘‘isolines’’ of constant blurring.

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Vx0 [km/h] -3dB @ Vx0 = 14.4 km/h

Fig. 3. Decrease of peak power with vx0 in TerraSAR-X images.

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ay = 0 [m/s2 ] ay= 0.5 [m/s2 ] ay= 1 [m/s2 ] a = 1.5 [m/s2 ] y ay= 2 [m/s2 ]

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above the ambiguity limit. As a consequence the object is wrapped around the frequency band and appears at the position of the ambiguity (negative displacement). 3.3.3. Influence of vehicle acceleration The approximate range history of an accelerating point target was derived in Eq. (16). Based on this equation the effects of acceleration components in along-track and in across-track direction on the point spread function of a moving vehicle are estimated in the following. Along-track acceleration ax appears in the cubic term of the range equation and results in an asymmetry of the focused point spread function as shown in Fig. 5. It can be seen that the deformation of the point spread function is very small even for very strong and very unrealistic acceleration values. For typical accelerations in common traffic 1

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Fig. 6. Broadening of the point spread function as a function of ay in TerraSAR-X images.

scenarios ðax < 2 sm2 Þ the effect is almost zero. Hence, the influence of this kind of acceleration can be neglected. As shown in Section 3.2.3 acceleration components ay in across-track result in a blur of a moving vehicle in the focused SAR image. Based on Eq. (16) and considering the TerraSAR-X system parameters given in Table 1 the amount of defocussing in TerraSAR-X images is calculated. Fig. 6 gives an impression of the amount of blurring that has to be expected for accelerations that commonly occur in vehicle traffic on roads or highways ðax ¼ ½0; 2 sm2 Þ. It can be seen from Fig. 6 that significant image degradation occurs also in case of small across-track accelerations. It is also worth noting that in real traffic scenarios the magnitude of blurring caused by across-track acceleration is comparable with the blurring due to along-track motion. Hence, one needs additional information deduced from position and orientation of the corresponding road axis to separate these two superimposed effects. 4. Detection and velocity estimation of moving objects

2

a = 0 [m/s ] x ax= 40 [m/s2 ]

0.9

It is clear that the effects of moving objects described above hinder the detection of cars in SAR images. However, as has been shown, these effects are mainly deterministic and can thus be exploited to not only detect vehicles but also measure their velocity. The main tasks to accomplish this are the estimation of blurring and displacement, and in addition the interferometric phase of two co-registered images. The solution to this typical inverse problem can be facilitated when incorporating a priori knowledge about the appearance, location, and velocity of vehicles.

0.8 0.7 0.6

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[m]

Fig. 4. Effect of across-track motion in TerraSAR-X SLCs. Azimuth displacement of a moving point scatterer as function of h and vy0 (units: [m]). Black curves indicate ‘‘isolines’’ of constant azimuth displacement.

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Fig. 5. Asymmetry of the point spread function as a function of ax in TerraSAR-X images.

Fig. 7 summarizes our scheme for moving object detection. It consists of two major components: a detection and

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237

Fig. 7. Detection scheme for moving objects in SAR data.

an estimation component. Both components make use of a priori knowledge (framed bright boxes in Fig. 7) in form of a road database and expectation values for the aspect-angle dependent radar cross-section of vehicles. The detector exploiting the along-track blurring of moving objects is run first, in order to obtain initial vehicle hypotheses and to better focus the image regions around such candidates. Then, across-track detectors exploiting the interferometric phase information follow. Typically, detection is performed with a so-called ATI (AlongTrack-Interferometry) detector and optionally a DPCA (Displaced Phase Center Array) detector. All detectors deliver hypotheses for position xcar, ycar and velocity vaz, vlos (see boxes with bold letters in Fig. 7). In addition, the found vehicle positions are projected onto the corresponding road axes, from which also estimates for vaz, vlos are derived. Depending on the road orientation complementary or competing observations can be weighted since, for instance, ATI is less reliable for roads oriented alongtrack than roads oriented across-track. This overdetermination of velocity estimation might also be used to estimate the blurring caused by across-track acceleration. However, this has not yet been realized in the current implementation of the system. In the following, we describe the individual components of the detection approach: (1) The integration of a priori knowledge (Section 4.2); (2) an algorithm for estimating the along-track velocity component based on frequency modulation rate variation (Section 4.3); and (3) the constant false alarm rate (CFAR) detection operating on along-track interferometric data (Section 4.4) to estimate the across-track velocity component.

4.2. Integration of a priori knowledge The above derivations show that, given SAR-system and platform parameters, the effects of moving and accelerating vehicles can be predicted for each SAR image acquisition as long as some necessary assumptions like objects being point scatterers are approximately valid. For instance, the displacement effect in the along-track direction due to an object’s across-track motion can be predicted when real position, velocity, and motion direction of the vehicle are known. Because of the functional relation of interferometric phase and object velocity in across-track direction, also the interferometric phase of a displaced moving object can be derived. Assuming the position and orientation of a road is known, e.g. from a road database, the above derivations enable us to calculate the displacement of each pixel from a road in azimuth direction, and based on that, to predict the velocity, phase and defocus a car must have at each location in the image. These types of predictions may be interpreted as a priori knowledge that can be acquired, analyzed and stored independent of image acquisition. Fig. 8 shows an example for the calculation of a displacement and velocity map based on a single polygonal road segment taken from a road database. While this a priori information can be calculated analytically, the expected behaviour of the vehicle’s radar crosssection needs to be modeled numerically. As it is well known, significant variations of radar cross-section exist over different aspect angles of cars. An example of such a curve derived from simulations and experimental measurements is shown in Fig. 9. Such information can also be

Fig. 8. Derivation of a priori knowledge from a road segment: (a) displacement map predicted for this road segment using sensor and orbit parameters and (b) corresponding velocity map. Along-track direction corresponds to the vertical axis.

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Fig. 9. Radar cross-section depending on aspect angle. Numerical simulation of a medium sized car in X-Band (Courtesy of Erich Kemptner, DLR-IHR).

incorporated into the detection scheme with the help of a road database, since—given the sensor and platform parameters—the aspect angle under which a car must have been illuminated by the sensor can be calculated for each road segment. 4.3. FM rate variation detector With the a priori information derived above, the predicted defocus of a vehicle at each position in the image can be calculated. The main reason for this blurring of moving point scatterers is the violation of the stationary world assumption incorporated in the SAR focusing process (cf. Section 3.2.1). However, focusing moving objects is in fact possible when choosing a FM rate that corresponds to the relative velocity of platform and object. Once a moving object is hypothesized, the image can consequently be adaptively focused depending on the predicted velocity at the respective position. In case of significant across-track velocity of a car such hypotheses can be generated, for instance, by detectors exploiting the interferometric phase (see below). However, if a car travels approximately in along-track direction, the phase values are close to zero so that they cannot be used to find reliable car hypotheses. To detect cars that move in along-track, we employed a specific algorithm based on the SAR image’s amplitude information. Our strategy for finding the correct FM rate relies on hypothesizing a series of FM rates and analyzing a pixel’s ‘‘sharpness function’’ over these FM rates. That means, not only one but a stack of images is processed with

varying FM rates, to find the correct FM rate. Since blurring occurs only in azimuth direction, searching the correct FM rate for a given pixel reduces to a 2D-problem (see also Fig. 10). Every FM-azimuth slice picks out a certain azimuth line (e.g. red line in Fig. 10a)) from every image. A slice consisting of these azimuth lines arranged in order of the FM rates may look like Fig. 10b). The corresponding FM rate is eventually found by searching for a focussed energy peak in the FM-azimuth plane, i.e. a FM rate is chosen, which corresponds to the relative velocity between sensor and vehicle yielding the correct estimate of the vehicle’s along-track velocity. For stationary points, the energy peak of the sharpness function would come up along the slice’s central row. The faster an object moves, the more the energy peak is shifted away from this row. This effect can be seen, for instance, in Fig. 10b). The distance from the central row to the found point gives immediately the estimate of the along-track velocity. For extracting the energy peak, an algorithm must be applied to find the peak of the sharpness function even in the presence of disturbances like speckle or bright background clutter. To this end, a blob detection scheme is implemented that analyzes the local curvatures in azimuth- and FM-direction [20]. The curvatures, which are restricted to a negative value, correspond to the directional second derivatives and must be estimated from the discrete data. Integrating the calculation of the derivatives into a smoothing operation is thus advisable. Finally, combining local curvature maxima and energy amplitude by the geometric mean

Fig. 10. 2D-slice along azimuth line through stack of SAR images, focused with different FM rates. (a) Slice through differently focused SAR images. (b) Sharpness function of moving object in slice.

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yields the final decision function, from which the maximum is selected. Details of this algorithm can be found in [20]. Once the along-track velocity has been determined this way, also the across-track velocity can be computed— again with help of the orientation of the corresponding road segment. However, also effects due to acceleration in across-track have to be taken into account since they have the same blurring effect on the imaged vehicle. The crosstalk of the two effects affects the velocity estimation even for small heading angles with a small across-track component. Fig. 11 depicts a quite difficult case of a real scene. The airplane was flying from bottom to the top of the image and recorded the data in a rightlooking geometry. The rectangles in Fig. 11a) mark example azimuth lines, which were extracted of every image of the FM stack to create FM-azimuth slices. The slice corresponding to the red rectangle is shown in Fig. 11b), while Fig. 11c) illustrates the slice corresponding to the cyan rectangle. At the left axis, FM rates are given in (1/s2), while the right axis represents the corresponding azimuth velocity assuming that acceleration effects can be neglected. One may also note that the sharpness function is tilted and fans out because of a mismatch of the linear component of matched filter and echo signal. Energy peaks extracted by the blob detection algorithm are marked with red circles. As can be seen, one dominant peak corresponding to approximately 65 km/h has been found for the slice indicated by the red rectangle and, in the opposite direction, three peaks corresponding to 50, 52 and 35 km/h have been detected. 4.4. ATI constant false alarm rate detector If moving objects show significant velocity in acrosstrack direction, interferometric approaches are the typical means to detect vehicles, when two-channel systems are available. Along-track interferometry (ATI) and the socalled Displaced Phase Center Array (DPCA) method are the most prominent among them. While the ATI technique combines amplitude and phase information through interferogram calculation, the DPCA technique involves basically a subtraction of the two SAR images from each other.

239

In the case of ATI an interferogram I is formed from the original complex data sets I1 and I2 by calculating I ¼ I 1  I 2 ¼ jI 1 jjI 2 j expðjðu1  u2 ÞÞ ¼ g expðjwÞ with u1 ¼ argðI 1 Þ and u2 ¼ argðI 2 Þ

ð19Þ

with w ¼ u1  u2

ð20Þ

being the interferometric phase. This phase measurement can be expressed as range difference DR in line-of-sight direction, which in turn is related to object motion: w¼

4p 4p DR ¼ vlos t k k

ð21Þ

When the time frame t is limited by the satellite motion and the distance Dl of the phase centers of the two antennas, Eq. (21) can be reformulated as w¼

4p Dl vlos k vsat

ð22Þ

Since both, interferometric phase w and azimuth displacement Dazimuth, are only caused by across-track motion, an analytic relation between both measurements can be established: Dazimuth ¼ R0

vlos k ¼ R0 w 4pDl vsat

ð23Þ

For all stationary targets the interferometric phase values w = (u1  u2) will be statistically distributed around the expectation value E[w] = 0. The joint probability density function (PDF) fc(g, w) of amplitude and phase of an interferogram has been derived in [21,22] using the underlying assumption of jointly Gaussian-distributed data in the two images. It is given by: ! ! 2nnþ1 gn 2ngjqj cosðwÞ 2ng  K n1 exp fc ðg; wÞ ¼ 2 2 2 pCðnÞð1  jqj Þ 1  jqj 1  jqj ð24Þ

where n is the number of so-called looks (effectively the amount of averaging), C(•) is the gamma function and

Fig. 11. Moving object detection by FM-rate variation: (a) Image patch with two selected azimuth lines, from which Azimut-FM slices are calculated. (b) Detected peak for red line and (c) detected peaks for cyan line.

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Kn(•) is the modified Bessel function of the nth kind. Multilooking is done by averaging over n pixels assuming stationarity. For medium resolution SAR the jointly Gaussian assumption has been validated in most agricultural and vegetated areas [23]. Based on this PDF a constant false alarm rate (CFAR) detector can be designed that groups all image pixels into two classes. Class 1, called ‘clutter only’, contains all pixels that only carry image information. Class 2, called ‘no clutter’, contains all pixels that are not part of the image PDF. This second class includes pixels that contain moving vehicles but also all sort of outliers. Since the PDF of this second class is not known, it is assumed to be equally distributed over a large area. With this assumption we can compute a likelihood ratio. Classification is done by comparing that likelihood ratio with thresholds a. This approach provides curves of separation between the two classes, which are actually isolines on fc(g, w). Applying a CFAR detector of the given design for detecting vehicles is optimal only in cases when amplitude and phase of a possible moving target in an arbitrary image pixel is uniformly distributed. This holds for many military applications, where vehicles are not bound to roads and can move in any arbitrary direction. In case of public traffic, where a priori information about position, velocity and movement direction of vehicles is available to a certain degree, the use of a simple CFAR detector is sub-optimal. As outlined in Section 4.2 it is possible to derive expectation values for position, velocity (i.e. interferometric phase), and aspect-dependent radar cross-section of vehicles using ancillary data. In the following, we integrate these data into the ATI-CFAR detector. The moving target signal is assumed to have a peak amplitude b and an interferometric phase shift #. The parameter b is proportional to the square root of the radar cross-section r. A new class describing the superposition of moving target signal and clutter, called ‘vehicle & clutter’ can be introduced now. The Class ‘vehicle & clutter’ is a subset of the class ‘no clutter’. Unfortunately a PDF fc+m(g, w) describing the probability density of this class has not been found yet. An approximation, valid for n  1 has been derived in [11]. This approximation fc+m(g, w) is given by:

These lines are not isolines of fc(g, w) anymore, but they separate the class ‘vehicle & clutter’ much better from the class ‘clutter only’. Thus the risk of falsely detecting an outlier is reduced and the probability of false alarms PFA is decreased. Fig. 12 shows an example of the shape of fc(g, w) and fc+m(g, w) and their relative position to each other. It also illustrates the typical behavior of clutter. The interferometric phase is less accurately defined for small amplitudes and, consequently, varies much more than for higher amplitudes. The PDF fc(g, w) is centered on a phase value of w = 0 as expected. Three examples of curves of separation are also given (varying a). The incorporation of a priori information into the vehicle detector improves the amount of detected targets and also reduces the number of false alarms. But, in order to define the ‘vehicle & clutter’ PDF external data sources are indispensable that allow to obtain the necessary a priori information about the vehicles impulse response b and the vehicles interferometric phase #. To evaluate the quality of this approach, we have implemented an ATI CFAR simulator whose output are receiver operator characteristics (ROC) curves. This simulator can be parameterized in such a way that a priori information about the interferometric phase and amplitude can be integrated or omitted. Hence, the benefits of integration a priori information can be quantified. Since it is impossible to derive probability density functions analytically for certain cases (especially for low number of looks and heterogeneous clutter), we implemented a Monte-Carlo simulation, which is split into two parts. The generation of a sufficient number of random samples followed by the evaluation of these random samples. To generate a random sample, the whole process of data acquisition is simulated: SAR-data-acquisition process,

n1 2

fcþm ðg; wÞ ¼

2nnþ1 gððg  dcosðw  #ÞÞ2 þ d2 sinðw  #Þ2 Þ

pCðnÞð1  jqj2 Þ   2nqðg cosðwÞ  d cosð#ÞÞ  exp 1  q2 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2n ðg  d cosðw  #ÞÞ2 þ d2 sinðw  #Þ2 @ A  K n1 1  q2 with d ¼

b g

ð25Þ

Using this approximation as an alternative hypothesis, fc+m(g, w) allows to define a likelihood ratio. Again thresholds can be applied resulting in new curves of separation.

Fig. 12. Joint PDF of the alternative hypotheses and its position relative to the hypotheses ‘‘Clutter only’’. The three lines are examples for curves of separation.

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241

multilooking if required, and the generation of interferograms. Then the pixel of interest is selected and stored in a list of random samples. Two sets of random samples are created: A set with samples from a pixel containing clutter only; and a set of samples from a pixel containing the radar return of a vehicle embedded in clutter. For each set of random samples a histogram is computed substituting the probability density functions. The simulation is based on the following assumptions: • Clutter statistics known (analytical or empirical, homogeneous or inhomogeneous); • Vehicle is modeled as a point-target; • Vehicle is moving in range with a constant velocity; • Transfer-function of SAR and interferometric SAR incl. antenna pattern known. To evaluate the performance of the detectors the threshold is varied and the probability of detection and probability of false alarm are determined for each step of this variation, eventually resulting in ROC curves. Figs. 13 and 14 show the ROC curves of the two detectors for different signal-to-clutter ratios (Fig. 13) and for different number of looks (Fig. 14). The comparison of the two ATI detectors shows a clear advantage for the one using a priori knowledge. Especially in bad detection scenarios the improvements tend to be larger. Once a vehicle has been detected with this algorithm, there exist two complementary ways to estimate its across-track velocity: (1) 2p-multiples of the interferometric phase measured in this particular pixel; and (2) the displacement of this from the corresponding road. The second measure is much more accurate, yet it might be ambiguous

Fig. 14. ROC curves of an ATI-detector using a priori information (solid line) compared to an ATI-detector using no a priori information (dashed line) for different numbers of looks (signal to clutter ratio fixed at 6 db): 1 look, 3 looks, 9 looks as indicated by the Looks-arrow.

in case of a very dense road network when a displaced car can be re-projected onto more than one road. The first measure is less accurate and also ambiguous. However, since the interferometric phase can be transformed into a displacement value, it can be interpreted as a likelihood function indicating the probability of a particular road section being the true corresponding road. Hence, the interferometric phase helps to robustly resolve potential ambiguities in velocity estimation based on displacement. Having established the match of detected vehicle and its corresponding road section, also the along-track velocity component can be computed via simple trigonometry. To justify the validity of the simulations to a large extent, we replaced the assumptions of knowing exactly the clutter statistics and the SAR-image generation transfer function by ‘real observations’. That means, synthetically generated vehicles were impainted into real SAR-images at varying locations with different types of clutter and the respective clutter statistics were derived from these real data. In so doing, it is possible to compare the results of the detectors with reference and to deduce values for completeness and correctness of detection defined as completeness ¼

#ðcorrect detectionsÞ #ðreference carsÞ

ð26Þ

and correctness ¼ Fig. 13. ROC curves of an ATI-detector using a priori information (solid line) compared to an ATI-detector using no a priori information (dashed line) for one look and varying signal to clutter ratios: 3 dB, 0 dB, 3 dB, 6 dB and 10 dB as indicated by the SCR-arrow.

#ðcorrect detectionsÞ #ðall detectionsÞ

ð27Þ

thereby both measures reaching unity for the ideal case and dropping down to zero for worse results. Fig. 15 shows an example of such a scene and the curves for completeness and correctness as well as the corresponding ROC curve.

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Fig. 15. (a) Example SAR-scene with different clutter regions in which vehicles have been impainted. (b) Completeness (red) and correctness (green) for region ‘‘Field’’, calculated for 10 dB2 vehicle brightness and 80 km/h vehicle velocity (axes indicate correctness and completeness [0, 1] for varying detection threshold). (c) Corresponding ROC curve.

Fig. 16. Experiments with airborne SAR: Validation of displacement effects.

Fig. 18. Experiments with airborne SAR: Detection results and velocity estimation for a denser traffic scenario.

5. Experiments and validations

Fig. 17. Experiments with airborne SAR: Velocity estimation using road data, displacement, and phase values.

In order to verify the validity of the theory, flight campaigns have been conducted to compare the theoretical derivations with real data. These tests also enable to discover bottlenecks of the techniques employed and to reveal unforeseen problems. An additional goal is to simulate TerraSAR-X data for predicting the performance of the extraction procedures. To this end, an airborne radar system has been used with a number of modifications, so that the resulting raw data is comparable with the future satellite data. During the first campaign 8 controlled vehicles moved along the runway of an airfield. All vehicles were equipped with a GPS system with a 10 Hz logging frequency for mea-

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suring their position and velocity. Some small vehicles were equipped with corner reflectors to make them visible in the image. Six GMTI experiments with varying angle a, i.e. the angle between the heading of the aircraft and the vehicles, have been flown. The vehicles have been moved with such velocities vT i that they approximately match traffic scenarios as recorded by satellites (see Table 2). Experiments 1, 2, and 6 are pure across-track motion experiments to study the along-track displacement of moving objects. For investigating the effect of blurring, the flight track of Experiment 3 is aligned with the movement of the cars (along-track motion). The flights 4 and 5 investigate more realistic scenarios with significant along- and across-track motion components. In order to verify the theory, the predicted image position of a moving object is derived from the object’s GPS position and its measured velocity. This position is compared to the position measured detected in the image. To get quantitative estimates of the quality of velocity determination from TerraSARX images, the velocity corresponding to the along-track displacement in the SAR images vdisp has been compared tn to the GPS velocity vdisp (see Table 3). Similar experiments tn have been carried out for determining along-track velocity. These delivered comparable quality. Fig. 16 illustrates the validation of the ATI experiments. The positions of displaced vehicles detected in the image (yellow dots) are compared with their true GPS-position (green dots) and the theoretical displacement computed from the GPS-velocities (red dots). As can be seen, yellow and red dots match very well, so that the theoretical background of detection and velocity estimation seems justified. Although there might be some inaccuracies included in the measurements (varying local incidence angle, GPS-time Table 2 Parameters of moving vehicles

Experiment Experiment Experiment Experiment Experiment Experiment

1 2 3 4 5 6

a

vT 4

vT 5

vT 6

vT 7

vT 8

vT 9

vT 10

vT 11

90 90 0 85 45 90

6 20 6 87 6 20

10 97 10 97 10 97

10 10 10 10 10 10

35 35 35 35 35 35

2.5 2.5 2.5 2.5 2.5 2.5

5 5 5 5 5 5

3 3 3 3 3 3

8 8 8 8 8 8

X

X

X

Corner

X

243

synchronization, etc.) the results show a very good match of theory and real measurements. As expected target 7 is not visible in the image. This is due to the low PBW of only 1/10 of the PRF and the targets velocity. The across-track velocity of target 7 shifts the spectrum of the target outside of the PBW which is centered around fdc = 0. This is another indication for the appropriate mathematical modelling in Section 3. The second flight campaign has been conducted with similar instrument settings, but now the images have been acquired over real-life traffic scenarios on highways. To evaluate the results of SAR-based vehicle detection and velocity estimation, time series of aerial photographs have been taken—almost synchronized with the SAR acquisition. First encouraging results have been achieved with the system described above, although we have to admit that too few scenes have been processed up to now to give reliable and statistically confirmed statements about the system’s performance. Typical results are depicted in Figs. 17 and 18. In Fig. 17 the interferometric phases of two detected vehicles (red rectangles) have been measured and the vehicles have been re-projected onto the corresponding road section (red arrows). The velocity derived from the projection yields 133 and 130 km/h, respectively. Both velocities are reasonable for this section of the highway and, moreover, match closely 2p-multiples of their corresponding interferometric phase values. Another result showing the performance for denser traffic is illustrated in Fig. 18. A visual comparison with the aerial photographs yielded a correctness of approximately 90% due to a stringent setting of the detection parameters. A decrease in completeness must be expected—see e.g. Fig. 15 for comparison—since some vehicles move with a velocity corresponding to an interferometric phase close to zero. Hence, an important hint for a moving object gets less discriminative. An in-depth discussion of the GMTI-processor implementation used in this experiment including an evaluation of the result can be found in [24]. Further details on the detection performance and accuracy of velocity measurements are given in [3].

6. Outlook

Velocities are given in km/h.

Table 3 GPS velocities and estimations from target displacement Target #

vGPS (km/h) Tn

vdisp T n (km/h)

Dv (km/h)

4 5 6 7 8 9 10 11

5.22 9.24 10.03 36.92 2.16 4.78 3.00 6.31

5.47 9.14 9.45 (n.v.) 2.33 4.86 3.01 6.28

0.25 0.1 0.58 — 0.17 0.08 0.01 0.03

Although the results shown in this paper can hardly be compared with those of induction loops or bridge sensors, they show nonetheless great potential to support traffic monitoring applications. The big advantage of satellite data is their large coverage, and thus, the possibility to derive traffic data throughout an extended road network. Evidently, this complements the accurate but sparsely sampled measurements of fixed mounted sensors. A natural extension of the presented approach—beside a number of technical improvements—would be an integration of the accurate, sparsely sampled data with the less accurate but area-wide collected spaceborne traffic data. Already exist-

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ing traffic flow models would provide a framework to do this. Acknowledgments This work has been partly founded by the DLR project ‘‘TerraSAR-X Traffic Products’’. The support of our project team, especially Hartmut Runge, Steffen Suchandt, Andreas Laika and Diana Weihing is gratefully acknowledged. References [1] J. Leitloff, S. Hinz, U. Stilla, Automatic vehicle detection in space images supported by digital map data, in: International Archives of Photogrammetry, Remote Sensing, and Spatial Information Sciences, vol. 36-3W/24, 2005, pp. 75–80. [2] J. Leitloff, S. Hinz, U. Stilla, Detection of vehicle queues in quickbird images of city areas, Photogrammetrie–Fernerkundung–Geoinformation (PFG) 4 (2006) 315–325. [3] F. Meyer, S. Hinz, A. Laika, D. Weihing, R. Bamler, Performance analysis of the TerraSAR-X traffic monitoring concept, ISPRS Journal of Photogrammetry and Remote Sensing 61 (3–4) (2006) 225–242. [4] R. Klemm, Space-time Adaptive Processing, The Institute of Electrical Engineers, London, UK, 1998. [5] C.-E. Livingstone, I. Sikaneta, C.-H. Gierull, S. Chiu, A. Beaudoin, J. Campbell, J. Beaudoin, S. Gong, T.-A. Knight, An airborne synthetic aperture radar (SAR) experiment to support RADARSAT-2 ground moving target indication (GMTI), Canadian Journal of Remote Sensing 28 (6) (2002) 794–813. [6] C. Gierull, Statistical analysis of multilook SAR Interferograms for CFAR detection of ground moving targets, IEEE Transactions on Geoscience and Remote Sensing 42 (2004) 691–701. [7] H. Runge, M. Eineder, G. Palubinskas, S. Suchandt, F. Meyer, Traffic monitoring with TerraSAR-X, in: Proceedings of the International Radar Symposium IRS2005, Berlin, 2005. [8] S. Chiu, C. Livingstone, A comparison of displaced phase centre antenna and along-track interferometry techniques for RADARSAT2 ground moving target indication, Canadian Journal of Remote Sensing 31/1 (2005) 37–51. [9] C. Gierull, Statistics of SAR interferograms with application to moving target detection, Technical Report DREO-TR-2001-045, Defence R&D Canada, 2001.

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