IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 6, NO. 3, JULY 2009
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Cross-Range Scaling for ISAR Based on Image Rotation Correlation Chun-Mao Yeh, Jia Xu, Ying-Ning Peng, and Xiu-Tan Wang
Abstract—For better understanding of the imaging result provided by inverse synthetic aperture radar (ISAR), it is more desirable to present it from the conventional range–Doppler domain to the homogeneous range–cross-range domain. To accomplish the process of cross-range scaling, the rotating velocity (RV) of the target should be estimated first. For a uniformly planar rotating object, a novel method based on a concept of rotation correlation is proposed to obtain the RV in this letter, and neither prominent scattering centers nor phase coefficient extraction is required. Furthermore, a three-step fast-Fourier-transform-based convolution interpolation scheme is proposed to realize the above correlation, which makes the proposed method more efficient for implementation. Finally, experimental results with both simulated and real ISAR data are provided to demonstrate the effectiveness of the proposed method for different targets. Index Terms—Cross-range scaling, inverse synthetic aperture radar (ISAR), rotation correlation.
I. I NTRODUCTION
I
NVERSE synthetic aperture radar (ISAR) is a kind of a microwave imaging system that exploits signal processing techniques to provide 2-D imaging results for a noncooperative moving target and can be used for both military and civilian purposes [1], [2]. Generally speaking, an ISAR system may obtain the range-resolving ability by pulse compression of wideband echoes, and it may further obtain high cross-range resolution by coherently accumulating a number of received echoes from different aspects. After translational motion compensation, the target to be imaged can be modeled as a rotating object [1], [2]. Furthermore, for a nonmaneuvering target in a limited observation interval, this target can be assumed as a uniformly planar rotating object, and the computationally efficient range–Doppler (RD) algorithm can be used to provide well-focused imaging results. For better image understanding, it is more preferable to rescale the RD image and display it in the homogeneous range–cross-range domain [3]–[7]. However, different from the range scaling factor (SF), which is determined by the known system parameters, the cross-range SF is related to the rotating
Manuscript received December 9, 2008; revised February 18, 2009. First published June 2, 2009; current version published July 4, 2009. This work was supported in part by the Coalition for National Science Funding under Grant 60502012, by the China Ministry Research Foundation under Grant 9140A07020106JW0103, and by the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China (706004). The authors are with the Department of Electronic Engineering, Tsinghua University, Beijing 100084, China (e-mail:
[email protected]). Digital Object Identifier 10.1109/LGRS.2009.2021990
velocity (RV) of the object, and it is usually unknown for a noncooperative moving target. In recent years, many methods have been proposed to estimate this RV either by auxiliary configurations [4] or directly from the received data [5]–[11]. The received-data-based methods may be roughly categorized into three kinds. The first is to search the RV based on iterative imaging with elaborated algorithms such as polar format algorithm (PFA) [5] or convolution back-projection algorithm, and the right RV may be obtained with the best imaging quality. The second kind is to obtain the motion information by extracting high-order phase coefficients from the received echoes such as the wellknown prominent point-processing scheme [6]. Also, some time–frequency analysis techniques can be used to extract the phase information for scattering centers that may not be prominent [7]–[10]. The third kind of method exploits an optical flow concept and extracts the motion information through subpixellevel position extraction of prominent scattering centers on a series of RD images [11]. Recently, Prodi [12] has also presented an RV estimation method using a correlation-based functional, which depends on careful selection of two range cells. In a word, all these methods may be effective to some extent, as illustrated by numerical simulations or experiments on some real data. However, they may computationally be inefficient for the iterative imaging process or less robust to a wide variety of targets with different scattering properties. Because the conventional RD imaging algorithm linearly maps the scattering centers from the physical plane to the image plane, the pose of a target on the RD image plane may also change according to the radar line of sight. By exploiting this basic effect, a new method is proposed in this letter to search the RV based on a concept of rotation correlation, where neither prominent scattering centers nor phase coefficient extraction is required. Furthermore, the rotation of an RD image can efficiently be implemented with a three-step fast-Fourier-transform (FFT)-based convolution interpolation process, which makes the proposed method very computationally efficient. Finally, experimental results are presented to demonstrate the effectiveness of the proposed method. II. RV E STIMATION A. RD Imaging of a Rotating Object Suppose that the translational motion compensation has been ideally implemented for the received echoes [1], [2]. Then, the uniformly rotating object imaging model may be assumed for a nonmaneuvering target in a limited observation interval, as
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IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 6, NO. 3, JULY 2009
where S = diag(1/ηr , 1/ηa ) and cos(ωo tm ) − sin(ωo tm ) R(tm ) = sin(ωo tm ) cos(ωo tm )
(8)
are the scaling and aspect matrices, respectively. B. RV Estimation Based on Rotation Correlation
Fig. 1. Rotating object imaging geometry.
shown in Fig. 1. Thus, the slant range from a given scattering center P (xo , yo ) on the rotating object to the radar varies with the pulse sampling time tm as 1/2 r(tm ) = ro2 + ra2 − 2ro ra cos(θo + ωo tm ) (1) where ra is the constant distance between the radar and the rotating center O, ωo is the interested and unknown RV, and (ro , θo ) is the polar coordinate for scattering center P . Under the far-field condition, which means ra is far larger than the size of the target being imaged, this distance can be approximated as r(tm ) ≈ ra − [xo cos(ωo tm ) − yo sin(ωo tm )] .
(2)
For an RD image formed around tm , this scattering center is mapped on the discrete image domain according to the time delay and the Doppler as 2r(tm ) c = Xo + [xo cos(ωo tm ) − yo sin(ωo tm )] /ηr
X(tm ) = − fs
Y (tm ) =
M 2 dr(tm ) xo sin(ωo tm ) + yo cos(ωo tm ) = fr λ dtm ηa
(3) (4)
where XO = −ra /ηr is a constant term, fs is the sampling frequency, fr is the pulse repetition frequency, c is the velocity of light, λ is the wavelength, M is the number of accumulated pulses for this RD image, and ηr and ηa are, respectively, the range and cross-range SFs, i.e., ηr = c/(2fs ) ηa = λfr /(2M ωo ).
(5)
Also, for the ISAR using the dechirp method for pulse compression, the mapping in (3) still holds, and only the range SF should be rewritten as ηr = cfs /(2N γ) = cfs Tp /(2N B)
(6)
where N denotes the number of range sampling points for a given echo, B is the bandwidth, Tp is the pulse duration, and γ is the frequency modulation rate. Rewrite the above RD mapping relation in the matrix form as x X(tm ) − Xo Xc (tm ) (7) = = SR(tm ) o Y (tm ) yo Yc (tm )
Suppose that two RD images are formed at tm1 and tm2 by equally dividing all the received echoes from the uniformly planar rotating target. Then, from (7), we have Xc (tm2 ) Xc (tm1 ) = SR(tm2 )R−1 (tm1 )S−1 Yc (tm2 ) Yc (tm1 ) Xc (tm1 ) = H(θd ) (9) Yc (tm1 ) where H(θd ) is a kind of a rotation matrix that relates the two RD images, and θd = ωo (tm2 − tm1 ) = ωo tm is the corresponding aspect angle difference between them. With (5) and (8) H(θd ) = SR(tm2 )R−1 (tm1 )S−1 cos(θd ) −(ηa /ηr ) sin(θd ) = . (10) (ηr /ηa ) sin(θd ) cos(θd ) Thus, the RD image formed at tm2 can be rotated to present the same appearance as the RD image formed at tm1 . Although the two variables, i.e., ηa and θd , are unknown in the above rotation, they are all determined by the RV for a uniformly rotating object. A fast processing scheme is proposed in [13] to rotate the video image that has the same SF in both dimensions. For an RD image that has different SFs between the range and cross-range dimensions, this scheme can be generalized by factorizing H(θd ) as follows: H(θd ) = UR (θd , ω)DA (θd , ω)UR (θd , ω)
(11)
where UR and DA denote the signal shifting in the range and cross-range directions, respectively, i.e., 1 −(ηa /ηr ) tan(θd /2) UR (θd , ω) = 0 1 1 − (λfr /(2M ωηr )) tan(ωtm /2) = (12) 0 1 1 0 DA (θd , ω) = (ηr /ηa ) sin(θd ) 1 1 0 = . (13) (2M ωηr /(λfr )) sin(ωtm ) 1 With the above matrix factorization, the whole image rotation process can be decomposed in an appropriate sequence of 1-D signal translations that can all be implemented via simple
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TABLE I COMPUTATIONAL LOAD COMPARISON
Fig. 2.
Convolution implementation in the spatial domain.
convolutions. Furthermore, the rotation correlation coefficient function versus the searched RV ω ˜ can be defined as F1 (X, Y )f2 (X, Y ) x y S(˜ ω ) = (14) F12 (X, Y ) f22 (X, Y ) x
y
x
y
where f2 (X, Y ) denotes the RD image formed at tm2 , and F1 (X, Y ) is a rotated version of the RD image f1 (X1 , Y1 ) formed at tm1 , i.e., F1 (X, Y ) = H(˜ ω )f1 (X1 , Y1 ) X X1 = H(˜ ω) . Y Y1
(15) (16)
Obviously, S(˜ ω ) is a function for the similarity measurement between two images, and it is supposed that this function reaches its maximum only with the right RV, i.e., ω ˆ = max (S(˜ ω )) . ω ˜
(17)
Consequently, with (5), the RD image can be cross-range scaled. C. Rotation Realization and Computational Load Analysis Obviously, image rotation occupies the most computational load for the proposed method. Conventional image rotation often requires 2-D interpolation operations, which are computationally inefficient and complex for implementation. With the rotating matrix factorization in (11), the rotation of an RD image can be realized with an appropriate sequence of 1-D signal translations that can all be implemented via simple convolutions. Furthermore, this convolution can often be implemented in the spatial domain as shown in Fig. 2. Thus, to rotate an RD image that accumulates Ny pulses and has Nx range cells, the computational load is mainly composed of 4Ny times of Nx -point FFTs for signal shifting in the range direction and 2Nx times of Ny -point FFTs for signal shifting in the Doppler direction. Thus, such RD image rotation requires Nx Ny (2 log2 Nx + log2 Ny ) complex multiplications (mc) and 2Nx Ny (2 log2 Nx + log2 Ny ) complex additions (ac), which also means that the computational load is mainly determined by the size of the RD image. For comparison, the computational load of the iterative PFA-imaging-based RV estimation method [5] is considered. For each PFA imaging, a polar-to-rectangular interpolation is required before the final image forming with 2-D FFT, and this 2-D interpolation can approximately be implemented with two 1-D interpolations [14].
While the PFA imaging requires the involvement of all range cells and all pulses, the proposed method can be made more computationally efficient by getting rid of some marginal areas of the RD images since the target usually occupies only a fractional area in the center of the image. Thus, the main computational load of the above two methods for each possible RV may be listed in Table I. In Table I, Mx and My denote the remained Doppler and range cells after getting rid of the marginal areas of the RD images. According to the table, for an ISAR image that accumulates 1 024 pulses and has 256 range cells, each PFA imaging requires more than 7 077 888 mc operations and 14 155 776 ac operations. Typically, suppose that 512 Doppler cells and 128 range cells are remained. Then, the rotation of a tailored RD image requires only 1 507 328 mc operations and 3 014 656 ac operations. III. E XPERIMENTAL R ESULTS Experiments with both simulated and real ISAR data are presented in this part to show the effectiveness of the proposed method to different types of targets, and an iterative PFAbased method is applied to illustrate the effectiveness of the proposed method. According to the above discussion, for a uniformly rotating object, all the received echoes are equally divided to form two required RD images. Then, for a given RV, one of the RD images is rotated according to (11). The rotated RD image is then correlated with the other RD image to provide the rotation correlation function that changes with the searched RV. The right RV may then be obtained according to the maximum of the rotation correlation function. With the estimated RV, the original RD image can be cross-range-scaled as (5). A. Numerical Simulations In this numerical experiment, the simulated ISAR system transmits 300 chirp signals with 300-MHz bandwidth per second. The carrier frequency is 5.52 GHz, and the echoes are coherently demodulated and inphase/quadrature phase channel (I/Q)-sampled with a rate of 400 MHz for pulse compression. The targets are set to rotate with a constant RV of 0.015 rad/s. A total of 2048 echoes are used, corresponding to an aspect angle change of about 5.867◦ . Two kinds of target are used in this experiment. The first kind of target is made of isolated scattering centers, as shown in Fig. 3(a), and the second kind of target can be seen from Fig. 3(b), in which the isolated scattering centers can hardly
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Fig. 3. Simulated target. (a) Target with isolated scattering centers. (b) Target without isolated scattering centers.
Fig. 5. Experimental results for the target without isolated scattering centers. (a) Rotation coefficient versus searched RV with the proposed method. (b) Image contrast versus searched RV with the iterative PFA-based method. (c) Imaging result in the RD domain. (d) Image result in the range–cross-range domain.
Fig. 4. Experimental results for the target with isolated scattering centers. (a) Rotation coefficient versus searched RV with the proposed method. (b) Image contrast versus searched RV with the iterative PFA-based method. (c) Imaging result in the RD domain. (d) Image result in the range–cross-range domain.
be extracted with the above system parameters. Applying the proposed method, the RV is searched in a scope span from 0.001 to 0.02 rad/s. The experimental results for the first target are given in Fig. 4, and the RD image is rescaled according to the searched RV of about 0.01497 rad/s, as shown in Fig. 4(d). Furthermore, the iterative PFA-imaging-based RV estimation is about 0.01680 rad/s for this target, as can be seen from Fig. 4(b). Also, the experimental results for the second target are given in Fig. 5, and the RD image is scaled according to the searched RV of about 0.01497 rad/s provided by the proposed method, as shown in Fig. 5(d). Also, the RV estimation provided by the iterative PFA-imaging-based method is about 0.01710 rad/s, as can be seen from Fig. 5(b). It is shown that the proposed method may be effective for both targets with and without prominent scattering centers and may, thus, be robust to adapt to the scattering properties of different targets.
Fig. 6.
Graph of Yak-42.
B. Experiments With Some Real Data Experiments with some real data of a Yak-42 airplane (see Fig. 6) recorded by a C-band (5.52 GHz) ISAR experimental system are also used to demonstrate the effectiveness of the proposed method. This system transmits 400-MHz chirp signals with 25.6-μs pulsewidth, and the target’s echoes are dechirped and I/Q-sampled for pulse compression at 10 MHz. Thus, the range SF is 0.375 m/pixel. A total of 2048 pulses are extracted for the proposed method, which are also equally divided to form two RD images. At 200 Hz, the time interval between the two images is 5.12 s. The RV is searched in a scope span from 0.001 to 0.02 rad/s with the proposed method, and the maximum rotational correlation coefficient is obtained with the right RV of about 0.0103 rad/s, as can be seen in Fig. 7(a). With this estimated RV, the conventional RD image in Fig. 7(c) can be rescaled in the homogeneous range–cross-range domain, as shown in Fig. 7(d), which is very similar to the appearance of the Yak airplane shown in Fig. 6. Also, the RV estimation provided by the PFA-based method is about 0.0078 rad/s.
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ACKNOWLEDGMENT The authors would like to thank Prof. X.-G. Xia from the University of Delaware, Newark, for his valuable suggestions and Dr. Z. G. Su, Dr. X. Z. Dai, Dr. M. C. Yu, the anonymous reviewers, and the associate editor for their valuable help. R EFERENCES
Fig. 7. Experimental results for the Yak-42 airplane. (a) Rotation coefficient versus searched RV with the proposed method. (b) Image contrast versus searched RV with the iterative PFA-based method. (c) Imaging result in the RD domain. (d) Image result in the range–cross-range domain.
IV. C ONCLUSION In this letter, a method has been proposed to search the RV of a uniformly rotating object based on a concept of rotation correlation of two RD images, and a three-step FFT-based convolution interpolation scheme has been applied to effectively rotate an RD image. With the searched RV and the known system parameters, the cross-range SF may be obtained, and the conventional RD image can be cross-range-scaled to facilitate image understanding. In addition, both experimental results with some simulated and real ISAR data have been provided to demonstrate the effectiveness of the proposed method for different targets.
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