Translational motion compensation for ISAR imaging under low SNR by minimum entropy
© Zhang et al.; licensee Springer. 2013
Received: 13 June 2012
Accepted: 8 January 2013
Published: 22 February 2013
In general, conventional error correction for inverse synthetic aperture radarimaging consists of range alignment and phase adjustment, which compensate range shift and phase error, respectively. Minimum entropy-based methods have been proposed to realize range alignment and phase adjustment. However, it becomes challenging to align high-resolution profiles when strong noise presents, even using entropy minimization. Consequently, the subsequent phase adjustment fails to correct phase errors. In this article, we propose a novel method for translational motion correction, where entropy minimization is utilized to achieve range alignment and phase adjustment jointly. And, a coordinate descent algorithm is proposed to solve the optimization implemented by quasi-Newton algorithm. Moreover, a method for coarse motion estimation is also proposed for initialization in solving the optimization. Both simulated and real-measured datasets are used to confirm the effectiveness of the joint motion correction in low signal-to-noise ratio situations.
Inverse synthetic aperture radar (ISAR) imaging has been a widely addressed topic in last few decades [1–3]. In order to achieve high-resolution both in range and cross-range of target imagery, the ISAR imaging technique exploits both wideband characteristics of radar waveform and the diversity of viewing aspect angle from radar to the target. In general, the range resolution is proportional to bandwidth of waveform, and the cross-range resolution is dependent on both the coherent processing interval (CPI) and the target rotational motion from variation of radar viewing angles. Therefore, CPI should be long enough to achieve high cross-range resolution by Doppler analysis. In ISAR scenarios, the target is often engaged in complicated maneuvers and the translational motion should be compensated before performing imaging processing. Translational motion introduces range misalignment and high-order phase error. For ISAR imaging of a non-cooperative target, a data-driven compensation procedure must be accounted, which generally consists of range alignment and phase adjustment.
Range alignment is to compensate the range shifts of profiles. Without priorknowledge available about the range shifts, range alignment is usually realized based on the similarity of high-resolution range profiles (HRRPs). Typical methods for range alignment can be sorted into three groups. The first class is based on a maximum correlation between adjacent profiles . The dominant scatter method tracks a prominent scatter and estimates the range shift. And, the maximum correlation method  aligns each HRRP by using the principle that the envelope correlation of two adjacent profiles reaches a maximum when they are aligned. Actually, the principle of maximum correlation of two adjacent profiles can be regarded as a local optimization of alignment. Because the estimation of range shifts between every two adjacent envelopes is independent of each other. As a result, this method is sensitive to target scintillation. And, it is likely to fail when strong noise presents, as the coherence of HRRPs is contaminated seriously. The other group is optimization-based methods [5–9]. It is widely accepted that global optimization methods are more robust to reflectivity scintillation and additive noise than maximum correlation method, in which the problem of ISAR range alignment is formulated by using some global metrics. Minimum entropy or maximum contrast of the synthetic profile, such as the average range profile (ARP) of the aligned profiles, is used as the criterion to evaluate the performance of range alignment. The synthetic profile usually has the highest sharpness when the profiles are aligned perfectly. Or else, the sharpness of synthetic envelope reduces. It is reasonable to evaluate the sharpness of the synthetic profile by contrast or entropy and establish an optimization for range alignment. In general, the synthetic profile is calculated as the mean of magnitude of the aligned HRRPs. Therefore, the synthetic profile can be viewed as a non-coherent energy accumulation of aligned profiles, and it can overcome the noise interference in some degree. However, in the situations of low signal-to-noise ratio (SNR), the SNR gain from the non-coherent integrant is not enough to overcome the interference from noise on the synthetic profile, leading to that the consistency between the sharpness of the synthetic profile and the quality of range alignment is broken. In this sense, it is still challenging to achieve optimal range alignment in presence of strong noise.
If range alignment is done well, afterwards, phase adjustment is carried out to remove the error phase. There are many different schemes to perform phase adjustment, which can be sorted into different groups. The first is based on tracking a phase of dominant scattering centers. Phase gradient autofocus algorithm , the multiple dominant scatters algorithm , and the weighted least square phase estimation  can be classified into this group. These methods usually perform well when dominant scattering centers can be extracted from HRRPs. However, presence of strong noise brings inherent difficulty to precise phase tracking through several dominant scatters. Another group numerically optimizes the phase error correction to improve a global metric consistent with image focus, in which image contrast (IC) [13–16] and entropy [17–22] are utilized as the cost function to optimize the phase error. Image metric-based approaches are usually able to obtain an optimal solution even in the presence of strong background noise and clutter. However, successful phase adjustment can only be ensured when perfect range alignment is obtained, while if the profiles are misaligned in some cases, such as strong noise situation, even the image metric-based methods fail to correct phase errors. To overcome this problem, in , a joint correction scheme for simultaneous range alignment and phase adjustment was proposed based on a two-order polynomial model of the translational range history, and a novel coarse estimation of both velocity and acceleration was also developed to accelerate the motion estimation. In the issue of ISAR motion compensation, adaptive joint time-frequency algorithm (AJTFA) also plays a significant role in ISAR motion compensation and imaging processing [11, 23, 24]. Especially, AJTFA is inherently suitable to dealing with maneuvering targets by projecting the one-dimensional signal onto the two-dimensional (2D) time-frequency plane, and the motion parameters can be extracted from the time-frequency spectrum of dominant scatters. However, strong noise can easily submerge the time-frequency spectrum degrading its performance in low SNR scenarios.
In radar remote sensing, strong noise usually presents, due to the signal decay from long range and absorption of transmit medium. The SNR problem is among the most significant challenges that ISAR imaging systems frequently face. In the presence of strong noise, motion compensation for ISAR imaging inherently encounters some difficulties. Therefore, the technique of motion compensation under low SNR is important, which may furnish imaging capability and improve the effective operating range of some ISAR systems with low power. Based on the fact that both range shift and phase error are directly related to the quality of the focused image, in this article, we present a novel entropy-based approach to joint range alignment and phase adjustment. It should be emphasized that the idea of joint correction of range shift and phase error has been proposed in , which motivates this study. In the joint correction, instead of separating motion compensation into the two dependent steps, it estimates the range shift and phase error simultaneously. Therefore, high SNR gain from 2D coherent integrant is benefited by both range alignment and phase adjustment.
In this article, the joint correction with entropy minimization models the translational motion as a high-order polynomial function , and 2D image entropy is minimized to optimize the polynomial coefficients. A novel coordinate descent algorithm is proposed to solve the minimum entropy optimization. The coordinate descent algorithm is implemented by the quasi-Newton algorithm, yielding fast convergence. For an optimization problem, the initialization is usually important to the efficiency and precision of the solutions. In this article, we also propose a method to estimate a coarse motion, which is effective to obtain the coarse coefficients efficiently. They can be applied as additional information to accelerate and promote the coordinate descent estimation. By using real datasets, the effectiveness of the joint motion compensation is validated.
The organization of this article is as follows. In “Signal model and minimum entropy compensation methods” section, we introduce the signal model and recall some existing minimum entropy-based methods for translational motion correction. In “Joint range alignment and phase adjustment by minimum entropy” section, algorithm for joint range alignment and phase correction by minimum entropy is illustrated in detail. Experimental results from both synthetic and real data processing are given in “Performance analysis” section to illuminate the effectiveness of the proposal. In the last section, some conclusions are drawn and some future works are viewed.
2. Signal model and minimum entropy compensation methods
2.1 Signal model
2.2 Related works
We focus on motion compensation for ISAR imaging under low SNR, which has been gained noticeable interests till now. To overcome the interference of strong noise and clutter, current methods are usually following a similar principle, in which, estimation of errors is converted into a problem of minimizing (or maximizing) a cost function. And, the cost function is selected such that when it is minimized (or maximized), optimal focusing is achieved. Entropy is one of the most primary function in error compensation for ISAR imaging. Entropy of 2D image (or one-dimensional synthetic range profile) represents its sharpness, and generally the “sharpest” image is corresponding to the fully focused image (or aligned HRRPs). Before presenting our method, herein, we introduce two typical entropy-based methods for range alignment and phase adjustment, respectively.
2.2.1 Minimum entropy for range alignment (MERA)
In order to solve the optimization problem in (16), many searching approaches have been proposed by adopting exhaustive search or iterative search.
2.2.2 Minimum entropy for phase adjustment (MEPA)
Many searching approaches have been proposed to solve the minimization problem in Equation (21). Especially, three schemes are introduced in  in detail. Although they have different searching schemes and efficiencies, their performances are similar as the underlying principle is the same. In general, minimum entropy phase adjustment is implemented by an iterative solution, the image entropy decreases with the increase of iteration number until the estimate reaches an acceptable optimal convergence.
3. Joint range alignment and phase adjustment by minimum entropy
3.1 Algorithm description
To solve this optimization, many standard algorithms are available, such as gradient-based algorithms, genetic algorithms, and gold section search. Due to the random characteristics of noise, it contributes little to the variance of entropy during the motion estimation [19, 22]. And, some dominant scatters will exceed the noise in amplitude distinctively due to energy accumulation when optimal motion correction is achieved, enhancing the sharpness of image significantly. Therefore, minimizing the entropy of 2D image for error correction is reasonable. Due to the strong noise interference, the cost function is usually not convex along with a lot of local minimums. Straightforwardly, utilizing the standard optimization algorithms encounters high risk that the iteration is trapped into a local minimum that far away from the true solution. In this article, we propose a coordinate descent algorithm implemented by quasi-Newton algorithm. The coordinate descent algorithm sequentially minimizes the objective function with respect to a single parameter while holding the remaining parameters constant. By this, coordinate decent algorithms have monotone convergence in the objective function and trend to have the ability of achieving the global optimization .
, where ρ is a parameter small enough for the pre-determined threshold or p reaches a pre-determined maximum iteration number P. For computational efficiency, fast Fourier transform should be used in the implementation of the calculation above.
3.2 Coarse motion estimation for initialization
In this section, real datasets are used to evaluate the performance of the joint range alignment and phase adjustment for ISAR imaging. Contrast, entropy and peak value of the focused RD images are used as the evaluation criterion of the image quality. First, the Yak-42 data are utilized to analyze the performance of the proposed method. High-order polynomial motion and different noise are added into the data, and the motion compensation is performed by both the conventional minimum entropy-based methods and the proposal. Comparisons are also provided to illuminate the improvement of our method. A ship data with strong sea clutter is also used to confirm the effectiveness of our method. All results illuminate that the enhanced quality is achieved by the joint correction.
MERA + MEPA
MERA + MEPA
Peak value evaluation
MERA + MEPA
In this article, we focus on the error correction under low SNR for ISAR imaging. It is widely accepted that minimum entropy-based phase correction is very robust to noise due to the high-SNR gain from the 2D coherent integral in image generation. Inspired by this, a novel scheme is established by using the 2D image entropy as the penalty function to jointly optimize the range shift and phase error. Implemented by quasi-Newton algorithm, a coordinate descent algorithm is developed to estimate the high-order coefficients of the translational motion. To avoid being trapped into a local minimum far away from the ideal solution, a novel method for coarse motion estimation is also proposed. Coarse motion usually provides solution close to the global optimization, leading to significant iteration number decrease in the fine estimation. Utilizing a parametric model on the range history, the proposed method performs well under strong noise. However, this model restricts the translational motion model, leading to limitations on specific scenarios. For example, severe vibration of the target or radar platform could induces random range shifts and phase errors. In this case, random translational motion compensation for ISAR imaging under low SNR is still open.
The authors thank the anonymous reviewers for their valuable comments to improve the paper quality. This study was supported by “973” program of China under grant 2010CB731903 and National Natural Science Foundation of China under grant JJ0200122201 and 61179010. Lei Zhang's work was also supported by the Fundamental Research Funds for the Central Universities under grant K5051302001 and K5051302038.
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