Contrast maximisation bad technique for 2-D ISAR autofocusing
M.Martorella,F.Berizzi and B.Haywood
Abstract:An image contrast bad algorithm for 2-D ISAR image autofocusing is propod.The problem of ISAR image autofocusing is formulated analytically by defining geometry and dynamics of the radar-target system and by assuming a mathematical model for the received signal.The image focusing is then achieved by estimating the model parameters through the maximisation of the image contrast.The problem of the maximum arch is solved numerically by means of an iterative arch method.An algorithm able to produce an accurate initial guess is also developed by using the radon transform.The good accuracy of the initial guess guarantees the convergence of the optimisation problem solution to the global maximum.The performance of the propod autofocusing technique is tested by comparing it to the point prominent processing (PPP)algorithm,the pha gradient algorithm (PGA)and the image entropy bad technique (IEBT),through the u of real data.Results confirm the effectiveness of the propod algorithm.
1Introduction
ISAR is a well-known method for high-resolution target image reconstruction.High bandwidth puls ar
e trans-mitted and target echoes,received at different aspect angles,are procesd coherently to form the image.One of the critical and fundamental steps of the ISAR technique is the ‘image focusing’or ‘motion compensation’[1].The received signal is multiplied by a time-varying pha term related to the movement of a ‘focusing’point O belonging to the target (e Fig.1).The pha term associated with the focusing point is modelled by means of a polynomial [2,3].The polynomial order must be chon as a trade-off between the image focusing accuracy and the computational load required by the autofocusing algorithm.When spatial resolutions of the order of one metre are required and the target motion is sufficiently regular,the pha term can be approximated effectively by means of a cond order polynomial.In this ca,the computational time needed for the ISAR image focusing is reasonably low and real-time applications can be considered.The first and cond order coefficients of the polynomial pha term will now be referred to as ‘focusing parameters’.The aims of this paper are:
(i)Propo a two-dimensional autofocusing technique bad on image contrast maximisation for estimating the polynomial coefficients of the focusing point pha term.Such a technique is an extension of the technique propod in [2]to full range–Doppler ISAR imaging (for the sake of
clarity,in [2]only single frequency signals where considered for cross-range imaging).
含辛茹苦造句
(ii)Develop an initialisation technique bad on the u of the radon transform (RT)to provide an accurate initial guess for the iterative arch of the image contrast maximisation.(iii)Define and test a stand-alone reconstruction technique compod of three steps:(i)preliminary estimation of the focusing parameters by means of the initialisation tech-nique;(ii)estimation refinement by means of image contrast maximisation;(iii)reconstruction of the ISAR image through the 2-D-Fourier transform (2-D FT)of the motion compensated data.In the following,such an image reconstruction technique will be referred to as image contrast bad technique (ICBT).
The ICBT reconstructs the image by maximising the image contrast (IC),which is an indicator of the image quality.This characteristic makes such an algorithm basically different from other classical techniques,such as the PPP [4]and the PGA [5]algorithms.It has to be acknowledged that similar approaches,which make u of the concept of entropy,have been developed,such as the IEBT [6,7].Nevertheless,in this paper the authors consider the straightforward concept of image quality being defined by image contrast.Moreover,a stand-alone ISAR image reconstruction technique is fully developed and tested.Modern fast computation abilities also allow the ICBT to be implemented in real-time systems.It is worth mentioning that a non-parametric technique has been propod that makes u of the image entropy without using a parametric model for the pha history [8].Neverthel
ess,in this paper the convenience of using parametric or non-parametric models for the reprentation of the pha history is not discusd.The authors simply consider cas where the target radial motion is sufficiently regular and therefore the propod algorithm is compared with a parametric implementation of the entropy bad technique.2Mathematical aspects of signal modelling and ISAR processing
Let the system geometry be reprented by Fig.1where the radar is located at ð0;0;h r Þin the system of co-ordinates
q Australian Crown Copyright 2005IEE Proceedings online no.20045123doi:10.1049/ip-rsn:20045123
M.Martorella and F.Berizzi are with Dept.of Information Engineering,University of Pisa,Via Diotisalvi 2,56126Pisa,Italy
B.Haywood is with the Defence Science &Technology Organisation,PO Box 1500,Edinburgh,SA 5111,Australia E-mail:m.martorella@iet.unipi.it
Paper first received 7th November 2004and in revid form 16th March 2005
ðw 1;w 2;w 3Þ.The reference system ðz 1;z 2;z 3Þis embedded on the target,which is assumed to be moving along an arbitrary trajectory and undergoing angular motions.The back-scattering properties of the target are described by the complex reflectivity function z ðz Þ,where z is the vector that locates a generic scatterer position in the ðz 1;z 2;z 3Þcoordinate system.From the Appendix,the received signal,in free space conditions,can be written in a time–frequency domain,as follows:
S R ðf ;t Þ¼W ½f ;t exp Àj
4p f
c R 0ðt Þ&'Z
V
z ðz ÞÂexp Àj 4p f c z T
Ái R 0ðt ÞÂÃ&'
d z ð1Þwhere
W ðf ;t Þ¼rect t T obs
rect f Àf 0
B
;
f 0reprents the carrier frequency,B the transmitted signal
bandwidth,T obs the obrvation time,R 0ðt Þthe modulus of vector R 0ðt Þ,which locates the position of focusing point O,i ðz Þ
R 0ðt Þthe unit vector of R 0ðt Þand V is the spatial domain where the reflectivity function z ðz Þis defined.The function rect(x )is equal to 1for j x j <0:5,otherwi 0.
Motion compensation consists of removing the pha term exp fÀj ð4p f =c ÞR 0ðt Þg owing to the radial movement of the focusing point O.The received signal after motion compensation can be written as follows:
S Rc ðf ;t Þ¼W ½f ;t Z
V z ðz Þexp Àj 4p f c z T i ðz ÞR 0ðt Þh i
&'d z ð2ÞIn order to reconstruct the ISAR image we consider the
simple approach of the range–Doppler (RD)technique.The RD makes u of a two-dimensional Fourier transform (2-D FT)of S Rc ðf ;t Þto reconstruct the complex image in the time lag –Doppler frequency domain.If we consider the target model as the composition of K ideal and independent
z ðz Þ¼
X
K k ¼1
a k d ðz Àz k Þ;the ISAR complex image of such a target can be written as
follows:
I ðt ;n Þ¼FT À2W ½f ;t X K k ¼1
a k exp Àj 4p f c ðz Àz k ÞT i ðz Þ
R 0ðt Þh i &'()¼T obs B
X K k ¼1
a k sinc ½T obs ðn Àn k Þ sinc ½B ðt Àt k Þ e Àj 2p f 0ðt Àt k Þ
ð3Þ
where a k is the complex reflectivity of the k th scatterer,d ðz Àz k Þis the Dirac function centred at the position of the k th scatterer and n k and t k reprent the coordinates of the scatterers on the image plane in the time lag –Doppler frequency domain.As shown in (3),the ISAR complex image of a t of ideal and independent scatterers is compod of a sum of sinc-like shaped terms,which is a well know result in the literature.The time lag t and the Doppler frequency n are related to the range r and cross-range c r by the following equations [9]:
n ¼
c r
R c r T obs
ð4Þt ¼
r
R r B
ð5Þ
新老生交流会where R c r ¼c =2f 0O eff T obs is the cross-range resolution,O eff is the modulus of the effective rotating vector and R r ¼c =2B is the range resolution [1].
When the required cross-range resolution is of the order of one metre and the relative radar-target motion is sufficiently regular,the distance R 0ðt Þcan be approximated around the central time instant t ¼0by its cond order Taylor polynomial,as follows:
R 0ðt Þffi~R
0ðt Þ¼a þb t þg 2
t 2ð6Þwhere a ¼R 0ð0Þ,b ¼_R
0ð0Þand g ¼€R 0ð0Þ.Therefore,the image focusing problem reduces to the estimation of three parameters.
The received signal,compensated by means of the
approximated term exp Àj ð4p f =c Þ~R 0ðt ÞÈÉ,can be written
as follows:
~S
Rc ðf ;t Þ¼W ½f ;t exp Àj 4p f c D R 0ðt Þ&'
ÂZ
V z ðz Þexp Àj
4p f c z T i ðz ÞR 0ðt Þh i
&'d z ð7Þwhere
D R 0ðt Þ¼R 0ðt ÞÀ~R
0ðt Þð8Þ
is the distance error.When the approximation in (6)holds,
the error D R 0ðt Þis very clo to zero and the blurring effect in the reconstructed ISAR image is negligible.
Becau the last step of the image formation is the 2-D FT,the compensation of the pha term component exp fÀj ð4p f =c Þa g can be avoided.In fact,it only provokes a shift along the range coordinate,without giving image distortion.To demonstrate this effect,let the signal reprented in (1)be compensated by the pha term exp fÀj ð4p f =c ÞR 00ðt Þg ,where R 00ðt Þ¼b t þðg =2Þt 2.The reconstructed image of K scattering centres,neglecting the distance error D R 0ðt Þ,becomes:
I ðt ;n Þ¼FT À2W ½f ;t X K k ¼1
a k exp Àj 4p f c ðz Àz k ÞT i ðz Þ
R 0ðt Þh i &'(Âexp Àj 4p f
c a
&''
¼T obs B X
K k ¼1
a k sinc ½T obs ðn Àn k Þ
Âsinc ½B ðt Àt k Þ e Àj 2p f 0ðt Àt k Þ d ðn Þd ðt Àa Þ
ð9Þ
where the symbol reprents the two-dimensional
convolution operation in the ðt ;n Þdomain and d ðÁÞ
is
Fig.1
Geometry of system
the Dirac function.The shift of all the target scatterers along the range coordinate does not provoke any defocusing of the image.Therefore,the compensation of parameter a can be avoided.The coefficients b and g physically reprent the radial velocity and acceleration of the focusing point,respectively.3
Image contrast bad technique (ICBT)
In this Section we prent and describe the ICBT for ISAR image reconstruction.Such a technique is implemented in two steps:(i)preliminary estimation of the focusing parameters,which are provided by an initialisation technique that makes u of the radon transform (RT)and of a mi-exhaustive arch;(ii)fine estimation,which is obtained by solving an optimisation problem where the function to be maximid is the image contrast (IC).
3.1Initialisation technique
This technique is ud to obtain two rough estimates of b and g .Its effectiveness will be shown in Section 5,where the ICBT is applied to real data.
3.1.1Rough estimation of the target radial velocity (b ðin Þ):In real applications the radar transmits
a pul every T R conds,hence,the received signal must be written in its sampled form S R ðf ;kT R Þwhere k ¼ÀM =2;ÀM =2þ1;...;M =2À1and M is an even number of sweeps in the obrvation time T obs ¼MT R .By taking the
one-dimensional Fourier transform (1D-FT)of S R ðf ;kT R Þwith respect to f ,we obtain the target range profile S R ðt ;kT R Þfor each sweep.In Fig.2a a range profile time history S R ðt ;kT R Þis plotted.The data is from an experimental data acquisition of an airplane (the data t will be detailed in Section 4).It is worth noting that the t axis is scaled,by means of (5),in order to obtain the range coordinate r .We can easily note that the stripes,owing to the main scatterers’range migration,are almost linear.Each stripe reprents the trace of the time history of a generic scatterer distance R s ðkT R Þ.
In order to estimate the value of b we assume that:
A1)to a first approximation,the distance R si ðkT R Þof the i th scatterer varies linearly with slope equal to b ,i.e.R si ðkT R Þ%R si ð0Þþb ÁðkT R Þ;
A2)the focusing point distance R 0ðkT R Þhas roughly the same quasi-linear behaviour of each R 0ðkT R Þ%R 0ð0Þþb ÁðkT R Þ.It is worth noting that in general the focusing point does not need to be coincident with any scatterers.
If conditions A1)and A2)are roughly satisfied,a preliminary estimation of b can be obtained by calculating the mean slope of the scatterer distance traces.Let b ¼tg ðf Þ,where the angle f ,given by the scatterers trace and the abscissa axis,can be estimated by means of the radon transform (RT)[10]of S R ðt ;kT R Þas follows:
^f ¼arg max f ½RT S R
ðr ;f Þ &'
Àp 2
ð10
Þ250
150
100500–50–100–150
200
150
100
50
0–1.5
–1.0
–0.5
0time, s
a
b
c d
水稳碎石
angle, deg
0.5
1.0
1.5
–1.5
–1.0
–0.5
0time, s
0.5
cad面积测量1.0
1.5
20
40
60
80
100120
140160r a n g e , m
250
200黑板上的记忆作文
150
100
50
0r a n g e , m
r a d i a l c o o r d i n a t e
150
100500–50–100–150
angle, deg
20
40
60
80
100120
140160
r a d i a l c o o r d i n a t e
ˆ(in )f
ˆ(in )f
Fig.2Radial velocity estimation by means of radon transform
a
Range profile time history
b Radon transform of the range profile time history
c Maske
d rang
e profile time history
d
Radon transform of masked range profile time history
where RT S
R
ðr;fÞis the RT of S Rðt;kT RÞ.Hence,the estimate^bðinÞis obtained by equating^bðinÞ¼tgð^fÞ.
The RT of S Rðt;kT RÞ(e Fig.2a)is shown in Fig.2b. The position of the peak of the RT is quite evident and corresponds to the angle^fffi140 .In weak signal-to-noi ratio(SNR)conditions it is convenient to mask the range profile time history S Rðt;kT RÞwith a threshold and t to zero all the values that are below it.The result of the masking is plotted in Fig.2c and the relative RT is in Fig.2d. In this ca the threshold is equal to80%of the peak value of S Rðt;kT RÞ,hence the distance traces of the main dominant scatterers are lected.
3.1.2Rough estimation of the target radial acceleration(g):Let iðt;n;~b;~gÞbe the absolute value of the complex image obtained by compensating the received signal by using two initial valuesð~b;~gÞ.The image contrast is defined as follows
ICð~b;~gÞ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A½iðt;n;~b;~gÞÀA f iðt;n;~b;~gÞg 2
ÈÉ
q
A f iðt;n;~b;~gÞg
ð11Þ
where the operator AðÁÞreprents the image spatial mean over the coordinatesðt;nÞ.The function ICð~b;~gÞreprents the normalid effective power of the image intensity iðt;n;~b;~gÞand gives a measure of the image focusing.In fact,when the image is focud correctly,it is compod of veral pronounced peaks(one for each scatterer),which enhance the contrast.When the image is defocud,the image intensity levels are concentrated around the mean value and the contrast is low.Thefinal estimation of the focusing parameters b and g is obtained by maximising the contrast.Therefore,the following optimis-ation problem must be solved:
ð^b;^gÞ¼arg max
~b;~g ½ICð~b;~gÞ
ð12Þ
A preliminary estimate^g inðÞof g is obtained by means of an exhaustive linear arch,over the variable g,of the maximum of the image contrast ICð^bðinÞ;~gÞin a pre-defined interval g0min;g0max
½ ,where^bðinÞis obtained by means of(10), written as:
^gðinÞ¼arg max
~g h
ICð^bðinÞ;~gÞ
i
ð13Þ
The initial guess for the iterative numerical arch that solves the problem of(13)is obtained by means of an exhaustive arch within the pret interval g0min;g0max
½ .If a strong acceleration of the target occurs and the value is found to be clo to one of the boundaries,a new arch interval is defined in order to investigate further values of g. The value of the arch step D g is chon in order to satisfy the rule D g T2ob R he error produced by the quantisation step multiplied by the squared value of the obrvation time must not exceed the range cell resolution. The choice of both the arch interval and the arch step is heuristic and has been tested on aircraft and ship data with good results.
Without strongly affecting the accuracy,the number of frequency steps in addition to the number of radar sweeps integrated coherently can be reduced in order to speed up the estimation process.In our tests the received signal matrix was reduced to a32Â32matrix(32frequency steps and32sweeps).The reduction can be done simply by lecting the central32frequencies and the central32sweeps of the received signal data t.It is worth noting that suitable data t reductions must be considered according to the data quality.
The accuracy of the estimation of^gðinÞalso depends on the accuracy of the estimation of^bðinÞ.Ev
en if this method does not reprent an optimum method for estimating the couple (b,g),it is sufficiently accurate to produce a good initial guess for the estimation refinement,as will be detailed in the next Section.Moreover,in Section4,an idea of the accuracy of the preliminary estimation will be given by using real data.
For the sake of clarity the algorithm that provides the initial guess of g is summarid below:
Step1.The raw data is reduced to a N0ÂN0data t (N0¼32in our tests);
Step2.Afirst range g0min;g0max
冬奥会冰墩墩½ is lected heuristically. Typically,it is a symmetrical bound,such asÀg0max;g0max
½ ; Step3.An exhaustive linear arch of the IC maximum is performed by tting b¼^bðinÞand arch along g;
Step4.If the value of^g0that maximis the IC is clo to one of the boundaries the arch is repeated for further values of g; Step5.When a value of^g0far enough from the boundaries is found,it is ud to initiali the iterative linear arch to find the solution of(13),which reprents the desired value of gðinÞ.
3.2Estimation refinement
A refinement of the preliminary estimatesð^bðinÞ;^gðinÞÞis obtained by maximising the image contrast by using classic optimisation algorithms.In our work,we u the optimis-ation algorithm of Nelder and Mead[11],initialid with the guess valuesð^bðinÞ;^gðinÞÞ.The convergence of the algorithm to the global maximum depends on the initial guess.The IC shows a good convexity near to the global maximumð^b;^gÞand a strong multimodal behaviour far from it.An example of IC is provided in Fig.3and the ctions along b and g, corresponding to the global maximum,are shown in Fig.4. It is worth noting that in this particular ca,corresponding to a real data analysis,the IC shows a pronounced peak and a quite regular behaviour around it.A more detailed look at the two ctions shows a general multimodal characteristic along b and a quite regular behaviour along g,within an interval around the global maximum position.
The convexity of the IC will not be proven mathema-tically.The convergence of the algorithm to the global maximum will be shown in Section4by means of real data. For the sake of clarity,aflow chart of the ICBT is reported in Fig.5
.
Fig.3Image contrast
4Application to real data
In this Section we u real data to test the effectiveness of the ICBT.The results obtained by means of the ICBT are compared to the results obtained by using the PPP,PGA and IEBT algorithms.We refer to the techniques becau they are well consolidated and already implemented in most of the operating ISAR systems.
4.1Data t
All data ts ud for comparing autofocusing algorithm performance were collected using the same low power instrumented radar system developed by the DSTO.The radar is able to transmit simple frequency stepped waveforms in the band 8.0–18.0GHz.Generally,the radar is houd in a mobile van for ground-bad measurements,such as tho of aircraft taken at Adelaide airport (Australia).On occasion,the system has been deployed in a C-130transport aircraft with the antennas pointed out of the open ramp door,as was the ca for the ship measurements ud in this paper.
Tables 1and 2give the relevant radar parameters ud to collect the aircraft and ship datats,respe
ctively.All datats were taken using horizontally polarid transmit and receive antennas mounted on a pedestal that was pointed manually to track the targets.Aircraft image data was gathered soon after take-off from Adelaide airport at ranges between 1.5and 3.0km.The bulk loader ship (e Fig.6)data was gathered as the C-130aircraft containing the radar flew away from the vesl at ranges between 1.0and 6.0km.While the speed of the bulk loader was unknown,the large vesls typically travel in the vicinity of 12–15knots.The estimated a condition for the measurements was a state 3(significant wave height of between 1.0and 1.5
m).
Fig.5Block diagram of the ICB
technique
梯田风景Fig.4Image contrast ctions along b and g in correspondence of the peak
a b b g
Table 1:Radar parameters (aircraft)
No.of sweeps
256No.of transmitted frequencies 128Lowest frequency 9.26GHz Frequency step 1.5MHz Range resolution 0.78m Radar height (h s )ground level Target type Boeing 737PRF =sweep rate王者荣耀胜率排行
20kHz =156.25Hz
Table 2:Radar parameters (ship)
No.of sweeps
256No.of transmitted frequencies 256Lowest frequency 9.16GHz Frequency step 0.6MHz Range resolution 0.97m Radar height (h s )305m Target type bulk loader PRF =sweep rate
20kHz =78.13
Hz
Fig.6Bulk loader photograph
