high resolution reconstruction of isar images

High Resolution of ISAR Images

Reconstruction

RA0 M. NUTHALAPATI, Member, IEEE California State University Fullerton

A two-dimensional high resolution spectral analysis algorithm with application to an inverse synthetic aperture radar (ISAR) imaging, is presented. This algorithm is based on two-dimelrsional

linear prediction using autoregressive (AR) coefficients. Stability is guaranteed by AR process pole adjustment. An ISAR target Is modeled for a complex scatterer geometry. Computer simulation results are provided for the high resolution reconstruction of ISAR

images.

Manuscript received January 30, 1991; revised April 30 and May 29, 1991.

IEEE Log No. 9104947.

This work was supported by the Office of Naval Research while the author was at the Naval Weapons Center as a 1990 Summer Wculty Research Fellow.

Author's address: Dept. of Electrical Engineering, California State University Fullerton, Fullerton, CA 92634.

0018-9251/92/%3.00 @ 1992 IEEE

1. INTRODUCTION

Synthetic aperture radar (SAR) finds several commercial and military applications including all-weather terrain mapping and target recognition. SAR is associated with a stationary target and moving radar [l]. When the target to be imaged is rotating or moving while the radar is stationary, the configuration is called inverse synthetic aperture radar (ISAR) [2].

target-reflected data along range and cross-range (Doppler) directions and correlating the data in two dimensions to estimate the target reflectivity function in both directions [3]. There are several ways of estimating target reflectivity in range direction [4]. In cross range direction, usually spectrum analysis is applied. A high resolution estimation of ISAR image is presented here.

Thrget reflectivity can be estimated along the range direction by transmitting a stepped-frequency waveform and inverse Fourier transforming the reflected (received) data. This generates a range profile. In the cross range direction, data corresponding to all range profiles is Fourier transformed to generate a two-dimensional (2-D) target reflectivity function or an image of the target. Paditionally, a fast Fourier transform (FFT) algorithm has been used to estimate the 2-D reflectivity function. Application of a high resolution technique such as autoregressive spectral estimation (ARSE) to ISAR imaging is presented here.

ISAR imaging involves collecting all

II. ISAR TARGET MODELING

Brget reflectivity function for an ISAR target is modeled in this section. An ISAR (stationary) can transmit a stepped-frequency waveform to estimate reflectivity function in the range dimension. The phase shift of the electromagnetic wave received by the radar is a function of distance between the radar and the target and the frequency of the transmitted wave. Different points of the scatterer along the range dimension shift the incident wave differently, the composite wave received by the radar contains information associated with the target in the range dimension.

The unambiguous range extent is a function of a frequency step size and range resolution is a function of total bandwidth of the stepped-frequency waveform. By appropriately varying the frequency step size and the total waveform bandwidth, arbitrary range depth and range resolution can be accomplished. By Fourier transforming the received data for any fEed aspect position of the target, a range profile (target reflectivity versus range) can be generated.

A complete stepped-frequency waveform is transmitted for every aspect and the corresponding received waveforms are collected. Between two consecutive transmissions, the position of the target

462 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 28, NO. 2 APRIL 1992

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is changed which in turn causes a phase shift change in the reflected wave. Along the cross range (Doppler) dimension, for thesame range point, several points of the target reflect the incident wave with different phase shifts. This causes the received wave for the same transmitted frequency to be spectrally correlated. The cross-range resolution is a function of the transmit wavelength and the target rotation rate. By doing the inverse Fourier transform, reflectivity distribution of the target along cross range for the same range cell can be estimated.

If & represents a distance between the center of a target and the radar, then the electromagnetic wave undergoes a phase shift $0 of

4r x do = -Ro

where X is a transmit wavelength ,.

where c is the light velocity and f is the transmit frequency. The received signal can be modeled as

L- 1

where i is the cross-range index. j is the range index. k is the scatterer number. A is the constant amplitude. L is the number of scattering elements in the target

geometry. $ j is the phase of the electromagnetic wave for j th

frequency step position. Rik is the range of the kth scattering element for

ith cross-range position. The phase of the reflected wave is a function of

reference phase $0 associated with Ro and differential phase 6@ which is a function of frequency step 6f. 6$ and 6f are related by

4n 66 = -6f C

where 6f is the frequency step bandwidth (Hz),

with X d i k = r(k)COS(8j) + S(k)Sh(&), and Ydik = s(k)cos(Oi) - r(k)sin(8i). Relative positions of the kth scatterer in range and cross range are denoted by r (k) and s(k), respectively.

$ j = $0 + ( j - 1)6$, R i k = { (RO - Xdik)2 + ( y d i k ) 2 } ” 2 ,

e, = (i - l)68

where 8, is the angular position of target for ith measurement, 68 is the incremental angular shift of target, and

total angular shift number of measurements *

de =

The range and Doppler resolutions are related to bandwidth and target rotation rate as follows [4]:

C Range Depth = - 2 6f C

Range Resolution (6r) = - 2Br

X Cross-Range Resolution (a,) = - 2 be

Cross-Range Width = N 6,

where Br is the total bandwidth of stepped-frequency waveform, 6, is the cross-range resolution, and N is the number of measurements.

111. RECONSTRUCTION OF ISAR IMAGES

There are several methods of 2-D spectral estimation. One of the easiest and fastest methods is based on 2-D FFT. The disadvantage of 2-D FFT method is low resolution associated with limited observed data and high sidelobes. High resolution methods such as AR techniques provide better resolution at the expense of lower throughput rate. In this section, both methods are presented for reconstruction of ISAR images. TI compare these two methods on the basis of resolution, it is appropriate at this point to review the concept of resolution.

Resolution

The time-bandwidth product establishes a relation between time width and band width of a signal. For systems with a constant time bandwidth product, one can have a fine spectral resolution (narrow bandwidth) for a proportionately longer time width 17. Generally, reciprocal of time width is considered as frequency resolution. Similarly, reciprocal of bandwidth is termed as time resolution. For smaller observation intervals, resolution is low. By zero-padding, only an interpolated spectrum can be achieved but not a high resolution spectrum. On the other hand, by extrapolating the observed signal beyond the observation interval, we effectively increase the time width of the waveform, thereby achieving increased resolution. In this process, we do not assume unknown samples as zeros. The unknown signal samples are predicted using weighted summation of observed samples. This is the basis for high resolution spectral estimation methods such as ARSE techniques.

One important note is that a signal with a longer observation interval provides better spectral resolution with conventional methods than with ARSE methods. For large enough data records, the conventional spectral estimators are preferable since they do not make any additional assumptions about the data other than wide sense stationarity. Therefore, ARSE

NUTHALAPATI: HIGH RESOLUTION RECONSTRUCTION OF ISAR IMAGES 463

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techniques offer better resolution than FFT methods for a smaller number of observed samples. For a larger observed data set, if there is no need for data extrapolation, ARSE techniques are superfluous. For this case, conventional spectral estimation techniques can offer adequate resolution. Throughout this paper we assume that the observed data set is small in size. Therefore, ARSE techniques are demonstrated to be superior to FFT methods.

FFT Spectral Estimation Method: In this method, zeros are padded in both dimensions to the original data array of size to create an extended data array. Zero-padding does not increase the inherent resolution of the waveform but simply interpolates the spectrum. Original complex data array is assumed as x(i, j ) with data window as

l < i , j < M

where M is the sample size in each dimension and i and j are row and column index, respectively.

i 5 N . x ( i , j ) = 0; (M + 1) 5 j ,

A onedimensional (1-D) FFT algorithm is used on the data corresponding to all rows. Then a 1-D FFT is applied on the row-transformed data of all columns. The resulting 2-D magnitude spectrum gives locations of scatterers and their amplitudes.

resolution [6]. AR techniques based on the Burg method are widely used in 1-D spectral analysis. This method employs linear predictive extrapolation on the observed data set to a desired size before applying conventional FFT on the data. This results in very low sidelobes compared with the FFT approach and improved spectral estimates [4].

There are several ways of estimating 2-D spectrum using AR techniques [7]. We use the Burg algorithm to estimate linear prediction coefficients based on the observed data. Next these linear prediction coefficients are used to extrapolate the data in two dimensions. Finally a 2-D FFT is applied on the extrapolated data to generate a target reflectivity function.

Due to the separability of range processing and Doppler processing, linear predictive extrapolation can be used on range data and Doppler data separately, and a conventional row-column 2-D FFT can be used on the extrapolated data to yield high resolution spectrum. In this section, this approach is used.

A standard Burg subroutine [7] is used to estimate AR coefficients of data of all rows and columns. Using the AR coefficients, new data is generated recursively. Original data is extended to a required size using the method of linear prediction.

ARSE Method: ARSE techniques provide high

Linear Predictive Data Extrapolation in One Direction

In this method, data is extrapolated in forward directions of both row and column dimensions.

Original complex data array is assumed as x ( i , j ) with data window as

l < i < M

I < j < M

where M is the sample size in each dimension and i and j are row and column index, respectively.

observed data x ( i , j ) (1 5 i, j < M} along rows (first dimension) up to N samples, the following linear predictive extrapolation is used.

Data Extrapolation along Rows: To expand the

' ( M + l ) S j < N l < i < M

where P is a model order and a(k), 15 k < P are AR (linear prediction) coefficients.

the data along columns (second dimension) up to N samples, the following linear predictive extrapolation is used.

Data Extrapolation Along Columns: To expand

P x(i , j ) = - C x ( i - k, j)a(k);

( M + l ) < i < N 15 j < N k = l

where P is a model order a@), 1 5 k < P are AR coefficients.

Linear Predictive Data Extrapolation in Both Directions

Data extrapolation along both directions for data of each dimension reduces the risk of instability of linear prediction process and increases the accuracy of spectral estimates compared with data extrapolation in one direction [SI. In general, the extrapolated data obtained with the Burg algorithm is bounded and therefore stable. But the extrapolation using covariance methods does not generate bounded data. Though we use the Burg algorithm for data extrapolation to generalize the ARSE method, here we assume the data is extrapolated in both directions. Therefore, the original data is assumed to be a center part of a 2-D data array with data window as represented below.

where i and j are row and column index, respectively.

observed data x( i , j ) ( 1 5 i , j < M} along rows (first dimension) up to N samples, the following linear predictive extrapolation is used.

Data Extrapolation Along Rows: lb expand the


- ~~ ~ -~ ~~

111 1 7r- - -

( :+F+l) < j < N

where M is the number of data samples, P is the AR order, and E is the estimated white noise variance. The AR order selected is the one for which the FPE is minimum. Generally, for small data sets (5 32 per dimension), an optimal AR order can be selected in the following range to find the minimum FPE [9].

MI10 < P 5 3M/4

where M is the number of observed data samples and P P is the model order.

x ( i , j ) = - C x ( i , - k - j)a*(k);

(y-+1

k-1 IV. COMPUTER SIMULATION RESULTS

N M Random scatterers are used for the computer

simulation. Three scatterer geometries are considered for the simulation. ISAR targets are modeled as point scatterers and spaced differently for each case.

N M

'?he following parameters are used for the computer simulation of ISAR target reflectivity function. where a* (k) is a complex conjugate of a@).

Data Extrapolation Along Columns: To expand the data for second dimension up to N samples, the following linear predictive extrapolation is used. Range resolution (6,) l m

Cross-range resolution lm

( $ + $ + l ) < i < N l < j < N

P x( i , j ) = - x(-k - i, j)a* (k) ;

k = l

Model Orders

If a model order is selected as low, the spectrum shall have all the peaks smoothed. On the other hand, if we select a higher order, the spectrum may contain additional peaks. Thus, model order selection represents for ARSE the classic tradeoff of increased resolution and decreased prediction error variance. One intuitive approach would be to construct AR models of increasing order until the computed prediction error variance reaches a minimum.

Many criteria have been proposed as objective functions for selection of the AR model order. Only Akaike criterion (final prediction error) is considered here. According to this criterion [q, the prediction error variance is the sum of the power in the unpredictable part of the process and a quantity representing the inaccuracies in estimating the AR parameters. The FPE for an AR process is defined as

M + P + 1 F"E(P) = E M - P - 1

(arc) %tal angular rotation 0.05 rad Angular increment (60) 0.03128

Carrier frequency 3 GHz Stepping frequency 150 MHz bandwidth %get range depth 128 m %get cross-range width 128 m

rad = 2.864789O

To achieve the above range and cross range resolutions, 128 x 128 points of data should have been generated from the reflected waveforms. However, to test the effectiveness of the high resolution ARSE algorithm, only 32 x 32 simulated data is generated. Theoretically, point scatterers spaced at a meter apart in both range and Doppler dimensions can be resolved. Therefore, scatterers should be spaced no closer than 4 m in each dimension. One target geometry is considered violating this condition. The simulated data is generated using a computer program. The distance between the scatterers are adjusted to demonstrate the advantages of the ARSE technique.

Complex Gaussian noise of zero mean and an arbitrary variance is generated and added to the observed data. Then an AR method as described in previous sections is used to estimate the spectrum. Data can be multiplied by a 2-D window before taking a 2-D FFT. This is to reduce sidelobes. Rectangular, triangular, Hann, Hamming, and Blackman windows are implemented in the simulation program.

Amplitudes of all point scatterers are assumed to be of same level for target geometries 1, 2, and 3. Two target geometries are considered with different


Fig. 1. ' h e scatterer locations (target geometry 1).

.3/-.31

Fig. 2. Reconstruction using ETT method (target geometry 1).

' I

Fig. 3. Reconstruction using AR method (target geometry 1).

scatterer amplitudes. In all simulations, maximum amplitude is scaled to 255 to represent &bit video. As shown in several three-dimensional (3-D) plots, the Range (m) Cross-Range (m) Amplitude

Target Geometry 1

amplitude data is plotted across range and cross-range 4 4 1.0 axes. Only a 64 x 64 center portion of the 128 x 128 -2 -5 1.0

2 8 1.0 5 -6 1.0 image is plotted because all of the scatterers are

closely placed in the center portion of the image.


31 .3\

Fig. 4. ?fue scatterer locations (target geometry 2).

Fig. 5. Reconstruction using FFT method (target geometry 2).

1 1 1 ' : ~ II I


The four scatterers are shown on a grid of 128 x 128 as illustrated in Fig. 1. The 32 x 32 simulated data are expanded to 128 x 128 matrix by padding zeros in both dimensions. Then a 2-D is used to reconstruct these scatterers. As shown in Fig. 2, high sidelobes in the target area obscure nearby smaller scatterers if any exist. Next, AR method (linear

prediction) is used to estimate prediction coefficients. Using these coefficients, 32 x 32 data is expanded to 128 x 128 data matrix. Fig. 3 illustrates reconstruction of scatterers with row and column model orders of 4 and 20, respectively. These model orders are found using the automatic order finding program given in Appendix A. As shown in Fig. 3, the AR method

461 NUTHALAPATI: HIGH RESOLUTION RECONSTRUCTION OF ISAR IMAGES

‘3J 31

Fig. 7. Reconstruction using FFT method; SNR = 4.8 dB (target geometry 2).

Fig. 8. Reconstruction using AR method; SNR = 4.8 dB (target geometry 2).

offers high resolution compared with zero-padding m.

Target Geometry 2

Range (m) Cross-Range (m) Amplitude

2 2 1.0 2 -2 1.0

-2 2 1.0 -2 -2 1.0

This geometry as shown in Fig. 4 exactly meets theoretical resolution constraint. As shown in Fig. 5, the FFT method cannot resolve all four scatterers. But the AR method with automatic order selection, resolves all four scatterers. The optimal orders are found to be 10 and 21, respectively, for row and column data processing. This is illustrated in Fig. 6. Next, zero-mean Gaussian noise is added to the complex signal and FFT and AR algorithms are applied to process the noisy data. As shown in Fig. 7, the FFT method cannot resolve all peaks. The AR method with a row order of 23 and a column order of 13 clearly resolves all four peaks even at 4.8 dB signal-to-noise ratio (SNR) as shown in Fig. 8.

Tamt Geometry 3


1.5 1.5 1.0 1.5 -1.5 1.0

-1.5 1.5 1.0 -1.5 -1.5 1.0

This geometry as shown in Fig. 9, does not meet the theoretical resolution constraint because the distance between scattering elements is less than 4 m for the 32 x 32 observed data array. As shown in Figs. 10-11, the AR method with a row order of 7 and a column order of 15 resolves all four peaks, whereas the FFT method fails.

Target Geometry 4


4 4 1.00 -2 -5 0.50

2 8 0.75 5 -6 0.25


3,3\

Fig. 9. ltue scatterer locations (target geometry 3).


-3, s\ Fig. 11. Reconstruction using AR method (target geometry 3).

This geometry is the same as target geometry 1 but shown in Figs. 13 and 14, amplitudes are accurately measured with the FlT method. Though the ARSE method resolves all four peaks, amplitudes are not proportionately displayed. One of the reasons for this performance is additive noise.

the scatterer amplitudes are nonuniform as shown in Fig. 12. Gaussian noise of zero mean and 0.2 variance is added to both inphase and quadrature components of observed data. The SNR is calculated as 5.7 dB. As


'4 3 1

Fig. 12 'Itue scatterer locations (target geometry 4).

Rg. 13. Reconstruction using FFT method; SNR = 5.7 dB (target geometry 4).

Fig. 14. Reconstruction using AR method; SNR = 5.7 dB (target geometry 4).

Target Geometry 5


This geometry is the same as target geometry 2 but with nonuniform scatterer amplitudes (Fig. 15). No noise is added to the observed data. As shown in

2 2 1.00 Fig. 16, the FFT method cannot resolve all scatterers. 2 -2 0.50 This is expected because the FFT method failed to

resolve all scatterers of uniform amplitude (target geometry 2). The AR method with optimal model

-2 2 0.75 -2 -2 0.25


I;

Fig. 15. ?tue scatterer locations (target geometry 5).

-3j .3\


3

' !


orders 10 and 24 resolves all scatterers as shown in Fig. 17. It appears that variable amplitudes of the scatterers are proportionately displayed.

V. CONCLUSIONS

has been demonstrated with severa1 simulated ISAR data. One of the critical requirements of using the AR method is the selection of an optimal order. The throughput rate is reduced drastically if automatic order selection is used on every row and column of data. A reasonable assumption is that an optimal order for first row and first column can also be suboptimal for other rows and columns, respectively.

On

A 2-D ARSE is superior to a conventional 2-D FFT method, in achieving a high resolution. This


Therefore, for a given data set, automatic order selection should be used only two times.

Several experiments on simulated data indicate that the AR method outperforms the FFT method in resolving closely spaced scatterers. Additive Gaussian noise, down to 5 dB SNR does not pose a problem to this method, as long as scatterers meet theoretical resolution constraint. Of course, the FFT method even without noise, fails to resolve scatterers that are placed at theoretical resolution. If there is no significant additive noise (SNR > 12 dB), the AR method resolves even nonuniform amplitude scatterers.

APPENDIX

The computer simulation programs for simulating point scatterer data and for finding an optimal AR model order are written in VAX/VMS FORTRAN and are available upon request from the author.

REFERENCES

[l] Hovanessian, S. A. (1980) Introduction to Synthetic Array and Imaging Radars. Dedham, MA: Artech House, 1980.

High Resolution Radar Imaging. Dedham, MA: Artech House, 1980.

[2] Mensa, D. L. (1980)

[3] Walker, J. L. (1980) Range-Doppler imaging of rotating objects. IEEE Pansactwm on Aermpace and Electronic Systems, AES-16, 1 (Jan. 1980), 23-52.

[4] Prickett, M. J. (1980) Prinaples of inverse synthetic aperture radar (ISAR) imaging. Eascon Record, (1980), 340-345.

Chen, C. C., and Andrews, H. C. (1980) Multifmquency imaging of radar turntable data. IEEE Pansactwns on Aermpace and Electronic Systems, AES-16, 1 (Jan. 1980), 15-22.

High resolution spectrum analysis for airborne pulse Doppler radars. Microwave Journal, (Feb. 1990), 113-124.

[7] Maple, S. L. (1987) Digital Spectral Anabsis with 4plicatwm. Englewood Cliffs, NJ: Prentice-Hall, 1987.

High resolution reconstruction of ISAR images. Summer research report, Naval Weapons Center, Aug. 1990.

An improved method of two-dimensional AR spectral estimation using average model orders and spectral data extrapolation. To be published.

[SI

[6] Nuthalapati, R. (1990)

[SI Nuthalapati, R. (1990)

[9] Nuthalapati, R. (1991)

412

Rao M. Nuthalapati (S’82-M84) received the Ph.D. degree in electrical engineering from the University of Ottawa, Ontario, Canada, 1984.

From 1981 to 1984, he was associated with the Canadian Department of Defence and the Canadian Department of Communications on various contracts related to radar signal processing. In 1984, he worked as an electronic warfare systems engineer at Miller Communications Systems Ltd. on the Threat Radar Simulator project. At Wichita State University, during the 1984-1985 academic year, he taught radar courses to the staff of the Boeing Military Airplane Co. From 1985 to 1988, he worked as a senior systems engineer at Honeywell Military Avionics Division, on the F-15 TISS contract. Since, 1988, he has been at California State University Fullerton, where he is an Associate Professor of electrical engineering conducting research on radar signal processing. He participated in the 1990 Navy-ASEE Summer Eculty Research Program at Naval Weapons Center. His research interests are digital signayimage processing techniques for radar systems, high resolution frequency measurements, radar/EW signal analysis and simulation, and mathematical modeling of radar systems.

Dr. Nuthalapati is a registered Professional Engineer in the State of California.

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 28, NO. 2 APRIL 1992

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high resolution reconstruction of isar images

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