Fast 3D Synthetic Aperture Radar Imaging from Polarization-Diverse Measurements
Pierre Minvielle, Pierre Massaloux, Jean-François Giovannelli
11 Fast 3D Synthetic Aperture Radar Imaging fromPolarization-Diverse Measurements
Pierre Minvielle, Pierre Massaloux, and Jean-Franc¸ois Giovannelli
Abstract
An innovative 3-D radar imaging technique is developed for fast and efficient identification and characterizationof radar backscattering components of complex objects, when the collected scattered field is made of polarization-diverse measurements. In this context, all the polarimetric information seems irretrievably mixed. A direct model,derived from a simple but original extension of the widespread “multiple scattering model” leads to a highdimensional linear inverse problem. It is solved by a fast dedicated imaging algorithm that performs to determine ata time three huge 3-D scatterer maps which correspond to HH, VV and HV polarizations at emission and reception.It is applied successfully to various mock-ups and data sets collected from an accurate and dedicated 3D sphericalexperimental layout that provides concentric polarization-diverse RCS measurements.
Index Terms
3D radar imaging, polarization, multiple scattering model, linear inverse problem, regularization.
I. I
NTRODUCTION S YNTHETIC Aperture Radar (SAR) imaging is a widespread technique to make a map or image ofthe spatial distribution of reflectivity or backscattering of an object or scene from measurements ofthe scattered electric field. It is of great importance in many areas: land mapping, target recognition,environment monitoring, surveillance, nondestructive testing, etc. Radar imaging can also be used in ananalysis purpose. For instance, to reduce the interference of wind turbine blades with air traffic controlradars [1] or to characterize tree signatures for canopy monitoring [2]. It is then called ”Radar CrossSection analysis” or “RCS analysis”. Regarding the data, it is usually collected from a remote platform,for instance a satellite or an airborne. It can also be collected nearby, e.g. inside an indoor facility, if theobject is small enough, for analysis imaging. Most of the time, radar imaging is one-dimensional (1-D)or two-dimensional (2-D), leading respectively to 1-D backscattering profiles or 2-D backscattering maps.Three-dimensional (3-D) radar images are known to be far more complicated and challenging to process[3], beyond the required extensive signal processing. Indeed, if a two-dimensional (2D) backscatteringimage can be formed by synthesizing an 1-D aperture with a wide-band radar, a three-dimensional (3D)backscattering image needs an entire 2D aperture. More precisely, in an indoor anechoic chamber, typical2D aperture geometries are planar, spherical and cylindrical [4]. Their physical implementations requiremany illumination viewpoints around the object. It requires to combine many object and/or antennarotations, a time-consuming task while the measurement conditions may derive. Once the backscatterfield data is recorded, it is processed in order to get 3D images of the target radar backscattering spatialdistribution.Many techniques have been developed for 3D radar image formation, both in SAR where the radarplatform is moving while the target stays motionless and ISAR (Inverse SAR) where the target is movingwhile the radar platform radar is stationary. High resolution (HR) radar images are commonly obtainedby processing coherently the backscattered fields as a function of the frequency and the angle (i.e. objectattitude relatively to the radar). Among 3D HR radar imaging techniques emphasized in [4], there are thePolar Format Algorithm (PFA), also known as Range-Doppler, the Range Migration Algorithm (RMA),
P. Massaloux and P. Minvielle are with CEA, DAM, CESTA, F-33114 Le Barp, France.J.-F. Giovannelli is with Univ. Bordeaux, IMS, UMR 5218, F-33400 Talence, France. a r X i v : . [ c s . I T ] J un its 2D version being also known as k - ω algorithm, and the Chirp Scaling Algorithm. Based on the polarnature of the frequency-domain backscatter data, 3D PFA is extensively used. Taking advantage of theprocessing that reduces in far-field condition to a Fourier synthesis problem, it is practically achievedvia an interpolation, that reformats the data in the spatial frequency domain, and an inverse fast Fouriertransform (FFT). Refer to [3] or [5] for implementation details on 3D PFA, including the polar reformattingmapping technique. Regarding RMA, it comes from seismic engineering and geophysics. Based on 1-DStolt interpolation and again FFT through an approximation with the method of stationary phase (MSP),RMA is able to compensate completely a potential wavefront curvature. The last one, i.e. the Chirp ScalingAlgorithm, is widely used in airborne SAR. Besides, let us mention space-time domain methods, suchas the 3D Backprojection (BP) algorithm. Also known as Time-Domain Correlation, this tomographicreconstruction is achieved by coherent summation and stems from the projection-slice theorem; see forexample [6].Back to the far-field condition of PFA, it can be relaxed even if it must be stressed that 3D near-fieldradar imaging remains computationally expensive [2], [3]. Let us mention for example [7] that developsa near-field focusing function that accounts for wavefront curvature and propagation loss; it is employedin [2] for RCS analysis of large trees inside an anechoic chamber. Another condition, generally implicitlyrequired in radar imaging, is the so-called ”small bandwidth small angle” [5]. This assumption means thatthe data is collected for a bounded excursion of the observation angles and frequency where the natureof the wave-object interaction is unchanged. It is closely related to the validity domain of the standard”multiple scatterer model” where the object is represented by a collection of coherently illuminated pointscatterers [8], the properties of which do not vary in the limited frequency and angle excursion band.In this article, we present a fast and efficient 3D non conventional radar imaging technique for identifica-tion and characterization of radar reflectivity/backscattering components of complex objects, in the contextof RCS or signature analysis. It goes far beyond the previously proposed ad hoc localization processingtechnique, the limitations of which are described in [9]. Unlike [2], our spherical measurement set-up,designed and developed at CEA, is especially appropriate for measuring small targets of lower RCS. Thespecific feature is that the collected scattered field data is made of polarization-diverse measurements: theelectric field varies both at emission and reception during the acquisition, forming concentric circles witha singularity in the main direction. In a way, the issue is related to polarimetric imaging and to what isoften called imaging radar polarimeter [10]. SAR polarimetry is a widely used technique for measuringand identifying polarimetric properties of scatterers [11], [12]. Many works have been achieved in HRpolarimetric characterization or decomposition, especially for automatic target recognition [13], [14], inorder to determine the full scatterer matrix of a scatterer and get a relevant information on the shape,orientation or dielectric properties. Here, our singular set-up implies that all the polarimetric informationis mixed. To overcome the issue, an original approach is proposed and developed. It is based on a suitableextension of the multiple scattering model that leads to a high dimensional linear direct problem. Thisinverse problem is solved by a fast regularization algorithm that manages to determine at a time threehuge 3D scatterer maps.The article is organized as follows. First, in section II, a general description of the polarization-diversemeasurement problem is made. In section III is dedicated to the 3D radar imaging approach. After abrief introduction to classical radar imaging and the multiple scatterer model, an extension is developed.Inspired by HR polarimetric characterization, it leads to the expression of the direct model and to theproposition of a fast regularized inversion method. In section IV, the 3D spherical RCS set-up is presented:it is able to accurately acquire polarization-diverse measurements. After a few details on the associatedspecific 3D radar image processing, results are presented and discussed for various mock-ups. Finally,conclusions are summarized in section V. II. P
ROBLEM FORMULATION
Standard RCS Synthetic Aperture Radar
A monostatic radar illuminates an object with a quasi-planar monochromatic continuous wave (CW) ofgiven frequency f . The incident electric field is vector E I of complex amplitude E I . The object backscattersa CW with the same frequency. The scattered electric field is vector E S , of complex amplitude E S . Thecomplex scattering coefficient S quantifies the whole object-EM wave interaction; it indicates the wavechange in amplitude and phase. Note that it is directly linked to the Radar Cross Section. They can bedefined in far field condition by (see [15] for more details): S = lim R →∞ √ πR E S E I , RCS = |S| (1)where R is the radar distance at which scattering is observed. The complex scattering coefficient S can bemeasured with an appropriate instrumentation system (antenna, network analyzers, etc.) and a calibrationprocess. The measurements can be repeated for different wave frequencies, which is usually called”Stepped Frequency Continuous Wave” (SFCW) acquisition mode. They can be repeated also for differentincidence angles, by rotating the object or/and the antenna. Finally, it leads to a sequence of M measuredcomplex scattering coefficients {S , S , · · · , S M } . Considering all these successive measurements, it isgenerally assumed in HR radar imaging that the wave-object interaction nature is unaltered. In particular,the polarization configuration remains stable, i.e. the electric field orientation relatively to the target isunchanged. M O ˆ x ˆ y ˆ z measurement S measurement S measurement S M … Fig. 1. Polarization-diverse radar imaging acquisition at successive wave frequencies { f , f , · · · , f M } Polarization-diverse measurements
Far from above standard RCS SAR, we consider afterwards the singular but relevant problem of amonostatic acquisition composed of M polarization-diverse measurements. It is represented in figure 1.Our goal is to analysis the scattering from O ˆ z viewpoint. In this singular set-up, each measurement isassociated with a different linear polarization, i.e. a different electric field direction. Let us focus onthe i -th measurement at wave frequency f i . It is described by the wave vector k i , i.e. its direction (unitvector ˆ k i ) and its modulus or frequency wavenumber k i = 2 πf i /c (where c denotes the light speed), by thepolarization direction at emission (unit Jones vector ˆ e Ei ) and by the polarization direction at reception (unitJones vector ˆ e Ri ). The incident electric field E Ii is collinear to ˆ e Ei while ˆ k i ∧ ˆ e Ei indicates the direction of the incident magnetic field. Finally, the polarization-diverse measurement is defined, i.e. both microwaveemission and reception, by the successive frequencies { f , f , · · · , f M } and the successive correspondingdirectional triplets { (ˆ k , ˆ e E , ˆ e R ) , (ˆ k , ˆ e E , ˆ e R ) , · · · , (ˆ k M , ˆ e EM , ˆ e RM ) } .Consequently, the polarization-diverse measurement 3D radar imaging problem consists in determining3D scatterer maps from O ˆ z radar viewpoint, based on this sequence of polarization-dependent measure-ments. Compared to classical radar imaging, the main issue is linked with the variation of polarization, i.e.the rotation of both the electric and magnetic fields. Indeed, the classical multiple isotropic scatterer modelon which radar imaging usually relies is no longer valid. Whereas the backscatter data is usually recordedseparately for each polarization at emission and reception (i.e. HH, VV and HV), here all the informationis mixed up. Besides, the huge number of unknown quantities needs to be processed efficiently.III. F AST RADAR IMAGING WITH POLARIZATION - DIVERSE MEASUREMENTS
A. Overview of classical 3D HR radar imaging and the multiple scatterer model
Let us briefly introduce classical HR radar imaging [3] and first the so-called ubiquitous ”multiplescatterer” model on which it is based. The multiple scatterer (MS) model lies on the interpretation ofthe target-electromagnetic wave interaction by a collection of coherent illuminated and localized pointscatterers [8]. Related to the high frequency behavior of EM scattering from geometrically complex bodies,the MS model can be derived from Maxwell’s equations by high frequency approximations [8]. It mustbe stressed that it requires several restrictive conditions and does not explicitly account for various EMphenomena: multiple scattering, creeping waves, etc. Even so, the MS model is practically and intensivelyused in radar interpretation and analysis, far beyond the validity of its assumptions. It works pretty wellin the far field as long as there is a critical look at its output where possible artifacts may occur [8],mainly related to the above mentioned EM phenomena.
MS model:
In a monostatic radar context with far field/planar monochromatic wave (with wave vector k i ), the MS model considers a target made of N elementary isotropic scatterers. They are located at points r n ( n = 1 , · · · , N ), of coordinates ( x n , y n , z n ) in the target coordinate system ( O ˆ x ˆ y ˆ z ) . Each scatterer isdefined by its complex scattering coefficient or power s n ( n = 1 , · · · , N ), that quantifies the relationshipbetween the incident and scattered field amplitudes ( E I and E Sn ) , resulting from the elementary wave-scatterer interaction. Considering a large enough radar-target distance R , it is possible to derive from(1): E Sn ≈ √ π | R · ˆ u R + r n | ( s n · e − j k i · r n ) · E I (2)where ˆ u R is the unit vector corresponding to radar- O direction in ( O ˆ x ˆ y ˆ z ) , so that | R · ˆ u R + r n | is thedistance between the radar and the current scatterer n . In the phase shift, the term k i · r n accounts forthe back and forth delays.In the MS model, E S = (cid:80) Nn =1 E Sn , i.e. the received echo or scattered signal is ideally formulated as acoherent superposition of elementary echoes [8], neglecting multiple scattering terms. That leads to thefollowing scattering coefficient: S i = 2 √ πR E S E I = N (cid:88) n =1 R | R · ˆ u R + r n | s n · e − j k i · r n (3)Since the ratio R | R · ˆ u R + r n | = | ˆ u R + r n /R | ≈ , the MS model reduces to: S i = N (cid:88) n =1 s n · e − j k i · r n (4)The N complex scattering coefficients s n associated to the multiple scatterers determine the systemresponse. They do not depend on the wave vector k i , i.e. its amplitude and direction. It is assumed that the incident scattered field amplitude E I does not depend on n . Typical application:
Consider the typical acquisition of Fig. 2 that can be encountered in spherical3D radar imaging. It requires a sequence of M viewpoints around O ˆ z . Considering that the i-th wavedirection is defined by the standard spherical angles ( ϑ i , Φ i ) , the successive wave vector directions ˆ k i aregiven by: ˆ k i = [ − cos ϑ i sin Φ i − sin ϑ i − cos ϑ i cos Φ i ] t It must be noticed that in such an acquisition the electric field varies slightly as long as the variationsof ( ϑ i , Φ i ) are limited. This is also called the ”small angle” condition. Moreover, the electric field stays inthe same plane, as shown in Fig. 2, and the polarization remains more or less constant. Another conditionis required: it is called the ”small bandwidth” condition. It implies that the target-wave interaction doesnot significantly vary within the limited frequency variation domain. If these conditions are fulfilled, eachcomplex scattering coefficient s n is considered constant for all the measurements. The MS model (4) canbe formulated for each measurement i (for i = 1 , · · · , M ) as: S i = N (cid:88) n =1 s n · e j k i · (sin ϑ i cos Φ i x n +sin ϑ i sin Φ i y n +cos ϑ i z n ) O … ˆ x ˆ y ˆ z Fig. 2. Classical spherical radar imaging with ˆ e Ei = ˆ e Ri = ˆ e i Remark that the MS model can be formulated as: S i = (cid:90) I ( r ) · e − j k i · r · d r + ε (5)where I ( r ) = (cid:80) Nn =1 s n δ ( r − r n ) is the spatial density of scatterers and δ ( · ) is the Dirac function. Theadditive term ε accounts for noise and uncertainty, due to interfering echoes generated by the environmentand MS model limitations. It is generally considered as white and Gaussian (centered, complex andcircular). Radar imaging:
Radar imaging processing is an inverse problem. For a given radar viewpoint (e.g. O ˆ z ), it consists of the determination the complex 3D image or map I , i.e. the complex amplitudes s i atpoints r i . The above mentioned ”small bandwidth small angle” condition is assumed. It must be stressedthat it requires that the wave polarization remains constant so that the target-wave interaction does notvary from one measurement to another.The Polar Format Algorithm (PFA), formerly introduced, is based on (5) which is the discrete Fouriertransform (DFT) of the sought image I . The computation of I is achieved via the inverse Fourier transform of the measured hologram, S j (for j = 1 , · · · , M ). In actual practice, the limited and discrete acquisition inangles and in frequencies must be taken into account. For a regular acquisition in angle and frequency, aninterpolating pre-processing step of regridding is performed before an inverse 3D Fast Fourier transform. B. Extension to the multiple scatterer model
Considering the polarization-diverse acquisition of Fig. 1, the MS model is no longer valid and as aconsequence conventional radar imaging processing can not be applied. Actually, the complex scatteringcoefficients s n ( n = 1 , · · · , N ) can no longer be considered constant. The electromagnetic interactionstrongly depends on the electric field direction; consider for example the scattering with the edges of anobject. To overcome this issue, we next develop an adaptation of the MS model in this varying polarizationcontext that will latter support the imaging inversion process. General formulation:
Remind that the purpose is to achieve radar imaging analysis from O ˆ z radarviewpoint. Let us consider the i-th acquisition of Fig. 1, around O ˆ z . Then, the basic idea is to considerthat each elementary isotropic scatterer is associated with not only a complex scattering coefficient s n butwith a full polarization scattering matrix S n : S n = (cid:20) s xx n s xy n s xy n s yy n (cid:21) (6)More exactly known as the radar cross section matrix [16], S n is defined in the target reference frame O ˆ x ˆ y , associated with an illumination direction collinear to vector ˆ z . It quantifies the amplitude andpolarization of the scattered wave for an arbitrary polarization of the incident wave. Note that s xy n = s yx n due to the reciprocity theorem for a monostatic case [12], so that the scattering matrix S n is completelydefined by the three complex coefficients: s xx n , s yy n and s xy n . Next, we derive an extension to the MS model.For each illumination, it introduces the appropriate combination of scattering matrix terms.Introducing polarization, the former general definition (1) of the complex scattering coefficient can bereformulated by: S = lim R →∞ √ πR T S E I (7)where ˆ e R is a unit vector aligned along the electric polarization at reception and the scalar product T S = ˆ e R · E S corresponds to the transmitted electric field. Refer to [15] for more details.The polarization configuration of the n-th scatterer is represented in Fig. 3. For the i -th acquisition, thescattering field vector E Sn resulting from the interaction between the incident wave and the n-th elementaryisotropic scatterer is given in ( O ˆ x (cid:48) ˆ y (cid:48) ) by (for a large target-radar distance R ): E Sn (ˆ x (cid:48) , ˆ y (cid:48) ) ≈ e − j k i · r n √ π | R · ˆ u R + r n | ( S (cid:48) n ( i ) · E Ii (ˆ x (cid:48) , ˆ y (cid:48) ) )= e − j k i · r n √ π | R · ˆ u R + r n | (cid:18) S (cid:48) n ( i ) · (cid:20) E Ii · ˆ x (cid:48) E Ii · ˆ y (cid:48) (cid:21)(cid:19) where S (cid:48) n ( i ) is the scattering matrix in the reference frame O ˆ x (cid:48) ˆ y (cid:48) , normal to the current direction ofpropagation ˆ k i . Note that there is no component of E Sn and E Ii in the direction of ˆ k i .Let denote ˆ e Ei the unit Jones vector at emission and ˆ e Ri the unit Jones vector associated with thereceiving linear polarized antenna. The radar received voltage is proportional to the transmitted electricfield: T Sn = [ˆ e Ri (ˆ x (cid:48) , ˆ y (cid:48) ) ] t · E Sn (ˆ x (cid:48) , ˆ y (cid:48) ) , so that: T Sn = e − j k i · r n √ π | R · ˆ u R + r n | (cid:20) ˆ e Ri · ˆ x (cid:48) ˆ e Ri · ˆ y (cid:48) (cid:21) t · S (cid:48) n ( i ) · (cid:20) E Ii · ˆ x (cid:48) E Ii · ˆ y (cid:48) (cid:21) It comes to: T Sn = E I e − j k · r n √ π | R · ˆ u R + r n | (cid:20) ˆ e Ri · ˆ x (cid:48) ˆ e Ri · ˆ y (cid:48) (cid:21) t · S (cid:48) n ( i ) · (cid:20) ˆ e Ei · ˆ x (cid:48) ˆ e Ei · ˆ y (cid:48) (cid:21) i i i r n ˆ x ˆ y ˆ z ˆ x’ ˆ y’ ˆ z’ i i Fig. 3. Interaction of an EM wave with a point scatterer
Similarly to the standard MS model, the received echo T S can be simply formulated as a coherentsuperposition of the elementary echoes: T S = (cid:80) Nn =1 T Sn . Thus, the complex scattering coefficient scatteringcoefficient is: S i = 2 √ πR T S E I = 2 √ πR (cid:80) Nn =1 T Sn E I (8)In the far field ( r n /R ≈ ), it is then straightforward to derive the following extended MS model forpolarization-diverse measurement, incorporating noise: S i = N (cid:88) n =1 s (cid:63)n ( i ) · e − j k i · r n + ε i (9)with s (cid:63)n ( i ) = [ˆ e Ri · ˆ x (cid:48) ˆ e Ri · ˆ y (cid:48) ] · S (cid:48) n ( i ) · (cid:20) ˆ e Ei · ˆ x (cid:48) ˆ e Ei · ˆ y (cid:48) (cid:21) (10)As previously with the MS model about the scattering coefficients s n , a ”small bandwidth small angle”will be further assumed about the scattering matrices. Due to a limited acquisition domain, in angle(around O ˆ z ) and frequency, the scattering matrices are supposed to be nearly constant: S (cid:48) n ( i ) ≈ S n for i = 1 , · · · , M . Former expression (10) leads to: s (cid:63)n ( i ) ≈ [ˆ e Ri · ˆ x (cid:48) ˆ e Ri · ˆ y (cid:48) ] · S n · (cid:20) ˆ e Ei · ˆ x (cid:48) ˆ e Ei · ˆ y (cid:48) (cid:21) (11)Basically, this expression provides the complex scattering coefficients or powers s (cid:63)n of the MS model thatare adapted to the i-th configuration. Note again that it depends on the relative target attitude towardsthe emitting and receiving antenna. In classical radar imaging, the directions of ˆ e Ei and ˆ e Ri remain stable,selecting constantly a term of the scattering matrix S n or a fixed linear combination of them. In thisvarying polarization context, they are all ”mixed”: (9) and (11) show that each acquisition is to select adifferent combination of the scattering matrix S n . Next, the extended MS model is detailed for a specificset-up. Specific form for concentric polarization-diverse measurements:
Let us consider the concentric acqui-sition, with the successive wave and electric field vectors represented in Fig. 4. It stems from a spherical3D RCS set-up that will be further detailed in section IV. Firstly, note the following sort of singularity:the vectors spin around axis ˆ z and there are various acquisitions from wave vectors colinear to ˆ z whileelectric fields differ. Note that it induces different scattered fields, it goes against the classical MS modelwith scalars s n . O … ˆ x ˆ y ˆ z Fig. 4. Concentric Polarization-Diverse Radar Imaging Acquisition
The i-th measurement is given by ( θ i , ϕ i , f i ) , for i = 1 , · · · , M . The azimuth θ i and the roll ϕ i correspond to two rotation angles (the roll rotation is defined around ˆ z , see section IV for details), thatdefines the directions of the wave vector direction ˆ k i as well as the emitting/receiving linear polarization: ˆ k i = − sin θ i cos ϕ i − sin θ i sin ϕ i − cos θ i (ˆ x , ˆ y , ˆ z ) ˆ e Ei = − cos θ i cos ϕ i − cos θ i sin ϕ i sin θ i (ˆ x , ˆ y , ˆ z ) or − sin ϕ i cos ϕ i (ˆ x , ˆ y , ˆ z ) depending on the emitting and receiving linear polarized mode of the antenna (respectively H or V), andso for ˆ e Ri .Afterwards, we determine the specific forms of (ˆ e Ri · ˆ x (cid:48) , ˆ e Ri · ˆ y (cid:48) ) and (ˆ e Ei · ˆ x (cid:48) , ˆ e Ei · ˆ y (cid:48) ) in order to express s (cid:63)n ( i ) for concentric acquisition. First of all, let us define the intermediate coordinate system (ˆ x (cid:48) , ˆ y (cid:48) , ˆ z (cid:48) ) : ˆ z (cid:48) = − ˆ k i (12) ˆ y (cid:48) = ˆ z (cid:48) ∧ ˆ z (cid:107) ˆ z (cid:48) ∧ ˆ z (cid:107) = − sin ϕ i cos ϕ i (ˆ x , ˆ y , ˆ z ) (13) ˆ x (cid:48) = ˆ y (cid:48) ∧ ˆ z (cid:48) = cos θ i cos ϕ i cos θ i sin ϕ i − sin θ i (ˆ x , ˆ y , ˆ z ) (14)The unit Jones vector at emission ˆ e Ei , and so at reception ˆ e Ri , can be expressed in (ˆ x (cid:48) , ˆ y (cid:48) ) as: ˆ e Ei =[ − t(ˆ x (cid:48) , ˆ y (cid:48) ) (H mode) or [0 1] t(ˆ x (cid:48) , ˆ y (cid:48) ) (V mode). The antenna coordinate system (ˆ x (cid:48) , ˆ y (cid:48) , ˆ z (cid:48) ) is definedby: ˆ z (cid:48) = ˆ z (cid:48) = − ˆ k i , ˆ x (cid:48) is collinear to the projection of ˆ x on the orthogonal plane (ˆ x (cid:48) , ˆ y (cid:48) ) and ˆ y (cid:48) = ˆ z (cid:48) ∧ ˆ x (cid:48) . ˆ x (cid:48) = cos θ i cos ϕ i ˆ x (cid:48) − sin ϕ i ˆ y (cid:48) (cid:112) cos θ i cos ϕ i + sin ϕ i (15) ˆ y (cid:48) = sin ϕ i ˆ x (cid:48) + cos θ i cos ϕ i ˆ y (cid:48) (cid:112) cos θ i cos ϕ i + sin ϕ i (16) For the H mode, it results in the following expressions for ˆ e Ei · ˆ x (cid:48) and ˆ e Ei · ˆ y (cid:48) (and so ˆ e Ri ): e I · ˆ x (cid:48) = − cos θ i cos ϕ i (cid:112) cos θ i cos ϕ i + sin ϕ i (17) e I · ˆ y (cid:48) = − sin ϕ i (cid:112) cos θ i cos ϕ i + sin ϕ i (18)Let us sum up the complex scattering coefficient in this concentric polarization diverse acquisition. Forthe i -th polarization acquisition, the complex coefficient s (cid:63)n ( i ) is given by:1) HH acquisition mode s (cid:63)n ( i ) ≈ [ˆ e Ri · ˆ x (cid:48) ˆ e Ri · ˆ y (cid:48) ] · S n · (cid:20) ˆ e Ei · ˆ x (cid:48) ˆ e Ei · ˆ y (cid:48) (cid:21) ≈ K i [cos θ i cos ϕ i · s xx n + sin ϕ i · s yy n + cos θ i sin 2 ϕ i · s xy n ] (19)2) VV acquisition mode s (cid:63)n ( i ) ≈ K i [sin ϕ i · s xx n + cos θ i cos ϕ i · s yy n − cos θ i sin 2 ϕ i · s xy n ] (20)3) HV acquisition mode s (cid:63)n ( i ) ≈ K i [cos θ sin 2 ϕ · s xx n − cos θ sin 2 ϕ · s yy n − (cos θ cos ϕ − sin ϕ ) · s xy n ] (21)with: K i = cos θ i cos ϕ i + sin ϕ i .Afterwards, we consider the whole measurements, i.e. the M observed complex scattering coefficients,in the associated acquisition conditions defined by the successive frequencies f i , azimuth θ i and roll ϕ i for i ∈ , · · · , M . They are stacked in the following complex vector S : S = [ S S · · · S M ] t (22) C. Derived direct matrix model
We now derive the observation model (forward model) from the previous extended MS model. Letus stress again that the ”small bandwidth small angle” is assumed: the scattering matrices are supposedto be nearly constant for each elementary scatterer. The unknown is made up of the scattering matricesassociated to the N discretized scatter points r n ( n = 1 , · · · , N ) of a 3D grid, the coordinates of whichare ( x n , y n , z n ) in the target coordinate system ( O ˆ x ˆ y ˆ z ) . The scattering matrices can be separated inthree 3D maps s xx , s yy et s xy (in the target reference frame), respectively for HH, VV and HV antennapolarization mode (defined at the initial reference rotation, for θ = 0 ◦ and ϕ = 0 ◦ ). In vector form,they can be expressed respectively by: s xx = [ s xx s xx · · · s xx N ] t , s yy = [ s yy s yy · · · s yy N ] t and s xy = [ s xy s xy · · · s xy N ] t . From (9), the complex scattering coefficients S i are clearly related to theFourier transform of the s (cid:63)n coefficients. Furthermore, it is obvious from (19) and (20) that the scatteringcoefficients are linear combinations of components s k ( k ∈ { xx , yy , xy } ). Therefore, the direct model (9)can be rewritten as (with noise vector ε ): S = (cid:88) k ∈{ xx,yy,xy } T W k F s k + ε (23) where F is a DFT (Discrete Fourier Transform), that can be computed by FFT (Fast Fourier Transform) al-gorithm. Finally, T is the truncation matrix related to the position of the observed points in Fourier domain.The three matrices W k are diagonal weight matrices that affect Fourier coefficients ( k ∈ { xx , yy , xy } ): W k = diag (w k (1) , w k (2) , · · · , w k ( M )) (24)Depending on the polarization acquisition mode (HH, VV or HV), the weights W k are given respectivelyby (19), (20) and (21). They define the elements or combination of elements of the N MS scatteringmatrices (i.e. s xx n , s yy n or s xy n , for n = 1 , · · · , N ) which determine the scattering coefficient observation S i .When the acquisition mode is HH, the weights are given by (for i = 1 , · · · , M ): w xx ( i ) = cos θ i cos ϕ i / K i w yy ( i ) = sin ϕ i / K i w xy ( i ) = cos θ i sin 2 ϕ i / K i (25)They are shown in Fig. 5, for various acquisition angles θ and ϕ . Typical acquisition angles ( θ ≤ ◦ and ◦ ≤ ϕ ≤ ◦ ) are represented by black points. Notice that the weights w xx , w yy and w xy dependmainly on the value of ϕ . For ϕ is close to ◦ , only the s xx n terms matter. Conversely, when ϕ close to ± ◦ , only the s yy n matter and for that reason are observable. Notice that it is true as long as θ ≤ ◦ . Forlarger θ , the scattering terms s xx n no longer affect the coherent superposition echo. Anyway, the ”smallbandwidth small angle” assumption of constant S n will be violated for common complex targets. w xx w yy |w xy | Fig. 5. Weigths w xx , w yy and | w xy | in function of θ and ϕ , with typical acquisition angles (black points). Finally, merging the 3 maps in an unique column vector s = [ s xx ; s yy ; s xy ] , the final direct model reads: S = As + ε (26)where A is a large block matrix of size M × N : A = (cid:2) T W xx F T W yy F T W xy F (cid:3) (27)The observation matrix A defines the deterministic part of the relation between the observations and theunknown state vector made of the three vectorized 3D map. It must be noticed that its dimensions canbe huge, e.g., N ≈ and M = 15 · . And yet, the model (26) is still linear (linear transformsand additive noise). Moreover, it relies on simple and fast transform (FFT, weight, truncation). Again,remark that a close parallel can be made with polarimetric SAR [10], [12], with a linear and additivenoise model and a polarimetric measurement matrix which, similarly to A , is determined by the successivepolarizations at emission and reception. D. Regularized inversion and Minimum Norm Least Squares
Regarding the inverse problem, a common approach relies on a discrepancy between measurementsand model outputs. A standard discrepancy is based on a quadratic norm and yields the so called LeastSquares (LS) criterion: J LS ( s ) = (cid:107) S − As (cid:107) . (28)A potential solution minimizes J LS , i.e., it is the one that best fits the data: (cid:98) s LS = arg min s ∈ C N J LS ( s ) . (29)The minimizer (cid:98) s LS can be found by setting the gradient of J LS ( s ) to zero and it leads to a linear system: A † As = A † S . (30)Unfortunately, it cannot be solved since A † A is rank-deficient (the number of unknowns is larger thanthe number of measurements) and, as a consequence, an infinity of objects minimizes (and nullifies) theLS criterion. In other words, an infinity of backscattered maps is exactly consistent with the data. Amongthese solutions, a possible simple approach selects the one with minimum norm: (cid:98) s MNLS = arg min s ∈ C N (cid:107) s (cid:107) s.t. S − As = 0 (31) i.e., the Minimum Norm Least Squares (MNLS) solution. There is a very large literature on variousmethods to deal with this kind of situations and build other solutions. The interested reader can, forinstance, refer to books on inverse problems [17]–[21]. Coming back to our simple MNLS solution, itcan be shown that (cid:98) s MNLS = A † ( AA † ) − S . (32)A possible proof is based on Lagrange multipliers and it is given in Appendix. The estimated map (cid:98) s MNLS is a linear transform of the data, nevertheless the matrix AA † is huge and its inverse cannot be computed.Fortunately, it can be inverted analytically: given the specificity of (27), we have AA † = T ( W xx + W yy + W xy ) T t that is a diagonal matrix and hence easily invertible. Finally, relation (32) becomes (cid:98) s MNLS = F † W xx T t F † W yy T t F † W xy T t (cid:2) T ( W xx + W yy + W xy ) T t (cid:3) − S that jointly determines the three 3D backscattered maps associated with the xx, yy and xy polarizations.In addition, each individual map (cid:98) s k MNLS is obtained separately by: (cid:98) s k MNLS = F † W k T t (cid:2) T ( W xx + W yy + W xy ) T t (cid:3) − S from the data set S . It can be seen that they are obtained by IFFT of the zero-padded weighted measure-ments. The involved weights are: π k ( m ) = w k ( m ) w xx ( m ) + w yy ( m ) + w xy ( m ) (33)based on direct weight (25), that depends on the polarization mode and the acquisition angles (azimuthand roll).The resulting fast 3D radar imaging algorithm is composed of two steps as described in Fig. 6. The firststep consists in processing the data by applying the weights of Eq. (33), for each polarization xx, yy and xy,and for each associated weights. Afterwards, in the second step, three 3D backscattered maps, associatedwith xx, yy and xy (in the target reference frame), are separately reconstructed by a 3D PFA method[8], [22]. It includes regridding and inverse 3D IFFT steps. It is finally a very efficiently implementation,based on weight, zero-padding and IFFT. regridddingIFFT zero-paddingPolarizationweighting z P o l a r Fo r m a t A l go r i t h m D data3D map Measurement angles (azimuth,roll)
For k ∈∈∈∈ {HH, VV, HV} (polarization) ππππ k Fig. 6. Fast regularized inversion
IV. A
PPLICATION
Indoor 3D spherical near field facilities are dedicated to various tasks. Beyond scattering analysis with2D or 3D ISAR imaging, they are encountered for RCS measurement, microwave absorber measurement,nondestructive testing, material characterization, antenna measurement and diagnosis, research in near-field measurement techniques, etc. These facilities are generally based on a mobile arch that providesvarious illumination viewpoints. Most of them are dedicated to antenna measurements; let us mention forexample the commercial systems ”G-DualScan” and ”700S-90”, respectively developed by Satimo andNSI. Others are specialized in RCS measurement. See, for instance, the system of the French researchinstitute Fresnel at the Microwave Common Resources Centre (MCRC), equipped with five positionersthat allows spherical diffraction patterns [23], as well as the bistatic anechoic chamber BIANCHA [24]of the Italian institute INTA. It provides a bistatic, spherical field scanner based on a dual-axis azimuthturntable and two elevated scanning arms. Related to these 3D spherical facilities, our 3D spherical RCSset-up, designed and developed at CEA, is specifically dedicated to RCS analysis of small targets of lowerRCS. After a short description of the facility and its microwave instrumentation, we present 3D radarimaging results on various mock-ups.
A. The 3D RCS CEA facility
The concentric polarization-diverse measurements of previous section III-B are achieved by the exper-imental layout of Fig. 7. Dedicated to the RCS characterization of small targets (typically ≤ ft), it iscomposed of a 4 meters radius motorized rotating arch (around horizontal axis) holding the measurementantennas while the target is located on a polystyrene mast mounted on a rotating positioning system(around vertical axis). The combination of the two rotation capabilities leads to a full 3D near fieldmonostatic RCS characterization.The microwave instrumentation is made of two bipolarization monostatic RF transmitting and receivingantennas that are driven by a fast network analyzer. A phased array antenna is used for the 0.8 -1.8GHz frequency band. It is optimized in order to reduce spurious signals originated from interactions withthe arch metallic structure. For higher frequencies from 2 to 12 GHz, a wideband standard gain horn,equipped with a lens, is preferred (see Fig. 7). The target under test is located vertically on a polystyrenemast below the measurement antenna. The target rotation around vertical axis is determined by the rollangle ϕ . The arch rotation around horizontal axis is determined by the azimuth angle θ . A full set ofdata S is therefore composed of successive frequency, azimuth, and roll sweeps. The arch is built ofaluminum, weights about 200 kg, measures approximately 5 m × network analyser horn target mast roll( j ) azimuth ( q ) Fig. 7. Near-field 3D RCS measurement facility potential mechanical deformations and to hold the measurement antenna over a total travel range of ± o . The θ rotation is achieved using a direct drive positioner including a brushless motor with a highaccuracy encoder directly coupled to the motor for accuracy and repeatability concerns. The ϕ rotationuses another positioner also including a brushless motor with a high accuracy encoder directly coupledto the motor. The polystyrene mast [25], [26] supporting the target under test is located on this rotatingpositioning system (vertical axis). Let us emphasize the precision of the whole positioning system. Thetwo major errors are the residual radial error and the repositioning (repeatability) error of the arch. Thefirst one is about ± . o and is equivalent to a maximum error of ± . mm on the vertical axis. Thesecond one is about ± . mm maximum, once again on the vertical axis. Refer to [27] for details onthe laser tracker characterization.Concerning the measurement process, it is composed of successive stages. First, a calibration bysubstitution [16] is performed: it consists in replacing the test target with a calibration standard whoseecho is well known in order to determine the inverse transfer function K of the entire RCS measurementsystem. This enables to convert indirectly the received transmitted electric field T out to a complex scatteringcoefficient value: S = K .T out . Basically, the substitution method establishes a phase reference relatively tothe rotation center and normalizes out the dispersive frequency response of the RCS measurement system.Note that for large objects, it can be enhanced by a multi-calibration approach [28] which overcomes mostof near-field effects, such as the power decrease or the EM wave sphericity. The second stage correspondsto background subtraction [16]. It is commonly introduced when the clutter is too high, compared to thetest target and calibration signals. It removes coherently the background echoes. Eventually, an adaptiverange filter is applied to eliminate the residual interferences, e.g. interactions with walls and floor that can affect the useful signal. B. Results
Let us first illustrate our 3D radar imaging method by an application to the following synthetic RCSdata. We consider the metallic ogival-shaped object from the EM benchmark [29], with polarization-diverse measurements at θ ∈ [0 ◦ : 2 ◦ : 20 ◦ ] , ϕ ∈ [0 ◦ : 5 ◦ : 360 ◦ [ and f ∈ [1 GHz : 10
MHz : 3
GHz ] . Thesynthetic data is computed by an efficient parallelized harmonic Maxwell solver . Fig. 8 shows the 3Dscatterer map (cid:98) s xx MNLS , corresponding to xx polarization. The 3D map corresponds to the indicated aboveradar viewpoint, along O ˆ z . Here, the 3D volume is represented by several transparent isosurfaces, thatare linked to different scatterer levels. In this basic situation, the main EM scatterers are located: thediffraction on the above tip and, at a higher level, the diffraction on the below tip. It comes with sidelobesthat derives from the limited acquisition sampling. Fig. 8. Ogival shape 3D radar map (cid:98) s xx MNLS [ N = 64 × × , M ≈ · ] Next, it is shown how the 3D radar imaging approach can support RCS analysis in more complexsituations from real polarization-diverse measurement data. Various targets have been characterized inorder to evaluate the 3D radar imaging method: a metallic cone with patches, a metallic arrow and aglider. Each of them enables to test the capacity to deal with various target-wave interaction phenomena:localized scatterers with the metallic cone, multiple diffractions as well as different polarization responseswith the metallic arrow and near field effects due to the large wingspread of the glider mock-up (regardingto the wave number). All the scattering measurements have been acquired for a ◦ roll sweep ( ϕ ) anda ± ◦ azimuth sweep ( θ ), from . to . GHz frequency.
1) Localized scatterers:
Fig. 9 shows the 3D radar map (cid:98) s xx MNLS in polarization xx of a metallic cone(height: 60cm) where 3 metallic patches (1cm × z = 250 mm ( − ◦ roll), z = 400 mm ( ◦ roll) and z = 112 mm ( ◦ roll). For convenience, the 3D volume is represented byseveral isosurfaces, superimposed upon a volume section relatively to the target shape. The main scatterersare perfectly located: the tip, the diffraction of the rear edge and each metallic patch. Notice that, sincethe frequency band is here larger than for the previous ogival-shaped object, the resolution is improvedalong O ˆ z . The scatterers corresponding to the 3 metallic patches are correctly located. It combines a volume finite element method and integral equation technique, taking benefit from the axisymmetrical geometry of theshape [30] Fig. 9. Metallic cone with 3 metallic patches (left) - 3D radar map (cid:98) s xx MNLS [ N = 256 × × , M ≈ ] (right) It is confirmed in the above view of Fig. 12, where the image can be compared to the photo, as well asin the exploded image of Fig. 9. There, the true patch locations are represented by black pellets. Noticethat although the creeping waves are not shown in Fig. 9, they would appear, under the cone, in otherrepresentations.
Fig. 10. Exploded image of the 3D radar map (cid:98) s xx MNLS [ N = 256 × × , M ≈ ] Fig. 11. Above view of the metallic cone with 3 metallic patchesFig. 12. Above view of the 3D radar map (cid:98) s xx MNLS (shape section) [ N = 256 × × , M ≈ ] Next, the metallic patches are replaced by 3 RAM (Radar Absorbing Material) rectangular patches(1cm ×
2) Multiple diffraction & polarization dependence:
The second target is a metallic arrow of 472 mmlong and 150 mm large (it belongs to Airbus Group Innovation, see [31] for details). It is representedin Fig. 13. With such a target, diffraction edges are known to differ according to the polarization of theincident wave. Moreover, multiple diffractions are likely to appear between the back of the arrow and itsbase.Fig. 14 provides a comparison of 3D radar maps of the arrow in polarization xx. Radar imaging fromreal data is compared to radar imaging from simulated data, computed with a parallelized 3D EM solver(see [32] for details). It can be checked that both maps are very close one another. The sensitivity to wavepolarization is emphasized in Fig. 14 and 15. The xx polarization map of Fig. 14 is again represented inthe shape section of Fig. 16. Depending on the polarization (i.e. xx, yy or xy), the diffraction on the edgesis more or less important, as well as the multiple reflections of the electromagnetic wave between the rearof the arrow and its base. Located below the target, they are much more important in (cid:98) s xx MNLS and (cid:98) s xy MNLS , dueto the higher resonance with the base. Let us stress that the 3D Radar imaging (cid:98) s xy MNLS enables to localize notonly 3D scatterers associated to diffraction and reflexion but also scatterers that depolarize the incidentwave, like the corners of the target. So the simultaneous determination of the three 3D scatterer maps isparticularly helpful for RCS analysis: it permits to localize and characterize each 3D backscatterer. Fig. 13. Metallic cone with 3 RAM patches - 3D radar map (cid:98) s xx MNLS (shape section) [ N = 256 × × , M ≈ ]Fig. 14. 3D radar image ( (cid:98) s xx MNLS ) of the arrow, from real data (left) and simulated data (right) [ N = 40 × × , M ≈ · ]
3) Near field effects:
They can be shown with the full aluminum glider-shaped mock-up of Fig. 7, andits large wingspread. Its main dimensions are approximately 3 ft × × θ and ϕ , in the context of the extended wingspread inthe transversal dimension. It also partly results from the spherical RCS data and the anisotropic radiationpattern of the antenna [33]. These defaults can be corrected by improving the angular sampling and byintroducing a 3D near field/far field correction in the radar imaging algorithm.V. C ONCLUSION
A 3D radar imaging technique has been presented. It is able to process a collected scattered field datamade of polarization-diverse measurements, where the electric field direction varies during the backscatter Fig. 15. 3D radar image of the arrow (from real data): (cid:98) s yy MNLS (left) and (cid:98) s xy MNLS (right) [ N = 40 × × , M ≈ · ]Fig. 16. 3D radar image of the arrow: (cid:98) s xx MNLS (shape section) [ N = 40 × × , M ≈ · ] data acquisition. It is based on fast and efficient regularized inversion that reconstruct three huge 3-Dscatterer maps at a time. The approach rests on a simple but original extension of the standard multiplescatterer point model, closely related to HR polarimetric characterization. It is applied successfully tosynthetic and real data, collected from a 3D spherical experimental layout dedicated to accurate RCScharacterization.To go further, the resolution could be enhanced by introducing non-quadratic approaches (L2-L1) [34],[35] or considering none differentiable criteria. From [36], constraints could be introduced, e.g. from theknown shape, in order to improve the scatter inference and remove ambiguities [3]. Besides, similarly to[2], [37], it could be possible to take into account the spherical near field illumination of the target as Fig. 17. 3D radar image of the glider: (cid:98) s xx MNLS [ N = 128 × × , M ≈ . · ]Fig. 18. 3D radar image of the glider: (cid:98) s yy MNLS [ N = 128 × × , M ≈ . · ] well as the radiation pattern of the antenna. R EFERENCES [1] J. Pinto, J. Matthews, and G. Sarno, “Stealth technology for wind turbines,”
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PPENDIX
Consider the general quadratic problem in C N with P linear equality constraints: (cid:98) x = arg min x ∈ C N (cid:107) x (cid:107) Q s.t. c − Ax = 0 (34)where • Q is a N × N positive-definite matrix that defines the criterion to be minimized • while the constraint is decribed through the vector c ∈ C P and the P × N matrix A . This matrix isassumed to be of full rank.A standard solution to such a problem relies on Lagrange theory: Lagrange multipliers, duality and saddlepoint, as follows. The Lagrangian of the problem (34) reads: L ( x , u ) = x † Qx + u † ( c − Ax ) (35)where u ∈ C P is the Lagrange multiplier, also referred to as the dual variable ( x ∈ C N is referred to asthe primal variable).Let minimize L with respect to x (for a fixed u ) by setting to zero the gradient of the Lagrangian: ∂ L ∂ x ( ¯ x , u ) = 2 Q ¯ x − A † u = 0 The solution is directly given by: ¯ x = 12 Q − A † u . (36)Note that the Hessian is equal to Q and hence is strictly positive. Consequently, ¯ x corresponds to aminimum. Then, the introduction of its value in the expression of L given by (35) provides the dualfunction ¯ L : ¯ L ( u ) = inf x L ( x , u ) = L ( ¯ x , u ) = − u † AQ − A † u + u † c . Next, let maximize ¯ L ( u ) relatively to u by setting to zero also the gradient with respect to the dualvariable: ∂ ¯ L ∂ u ( ¯ u ) = − AQ − A † ¯ u + c The solution is directly given by: ¯ u = 2( AQ − A † ) − c . (37)Note that the Hessian is equal to − ( AQA † ) / and hence is strictly negative; ¯ u corresponds to a maximum.Let insert the expression (37) of the dual optimizer into the expression (36) of the primal optimizer. Itleads to: (cid:98) x = Q − A † ( AQ − A † ) − c (38)which is the Minimum Norm Least Squares (MNLS) solution. When Q is the identity matrix, it resultsin the expression (32): (cid:98) x = A † ( AA † ) − c . Note that, if N = P , A is invertible and (cid:98) x = A − cc