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To present focused ISAR imaging results in the homogenous range and cross-range domain, an integrated scheme is proposed to estimate both the targets equivalent rotational velocity (RV) and rotational center (RC). The RV estimation is improved by radial projection combined with keystone processing, and then the RC is estimated through image entropy minimization. Finally, delicate imaging results may be obtained for wide-angle scenarios. Experiment results are provided to demonstrate the effectiveness of the proposed method.

Inverse synthetic aperture radar (ISAR) may provide high-resolution images for noncooperative moving targets [

When the integrated aspect angle interval is small, ISAR images may be formed using the efficient range-Doppler (RD) algorithm. For better image understanding and target feature extraction, it is more preferable to rescale the RD image into the homogeneous range-cross-range domain. However, the cross-range scaling factor (SF) is related to the rotational velocity (RV) of the target, which is usually unknown, and it should be estimated for a noncooperative moving target. Furthermore, when echoes are collected from a relatively large aspect angle interval, more sophisticated imaging algorithms such as convolution back projection (CBP) algorithm or the polar format algorithm (PFA) should be used to compensate the scattering centers’ migration through resolution cells (MTRC). In these cases, both the RV and the equivalent rotational center (RC) should be known, which are necessary for high-resolution ISAR image forming and understanding.

In recent years, a number of methods have been proposed for the RV estimation from collected wideband echoes. These data-driven methods may be roughly categorized into three classes. The first is to search the RV which provides the best image focusing quality using the PFA [

In this paper, a method combining the 2D FT with radial projection is proposed to estimate the RV. The radial projection is used to convert the 2D maximum correlation into a one-dimensional (1D) polar curve matching problem. Then, a representation of the polar curves using the cosine series is introduced to avoid the interpolation operation. With an easy initialization, the 1D maximization problem can be solved in a few iterations more efficiently.

There have been few reports about the estimation of the target’s RC in open literatures. Here, on the basis of effective RV estimation, the RC may be found with the minimum entropy criterion. With all the rotational parameters, the delicate CBP imaging algorithm is then used to generate size-scaled ISAR images with high resolution.

The paper is organized as follows. The existing RV estimation methods are introduced in Section

After careful imaging interval selection and effective TMC, an ISAR target may be considered as a uniformly planar rotating object as shown in Figure

Rotating target imaging geometry.

Suppose that two RD images are formed at

There is a displacement term

Equation (

Let

The normalized power spectrum is defined as follows:

Then, in the polar coordinate system,

Here,

The keystone transform is introduced in the preprocessing step to get better focused subaperture images.

The radial projection is introduced for the estimation of RV, and the time cost 2D image interpolation and correlation operations are avoided.

With the estimated RV, a method is proposed to search the RC based on minimum entropy criterion, and both the RV and RC may be provided for focused image forming with large aspect angle.

A block diagram of the whole procedure for rotation estimation is illustrated in Figure

Proposed method for rotation estimation.

Before the RD image formation of two subapertures, the keystone processing [

To make the computation more efficient, we introduce the radial projection for the estimation of RV. Radial projection is proved to be an efficient and robust algorithm for the estimation of 2D affine transformations in frequency domain [

According to (

In (

By combining (

In the total procedure for the estimation of RV, the 2D interpolation is done only once in the polar coordinate system, and the 2D maximum correlation problem is converted to a 1D polar curve matching problem according to (

When echoes are collected from a relatively large aspect angle interval in wideband radar imaging system, the target’s MTRC cannot be ignored. In order to apply the delicate imaging algorithm like CBP for high-resolution imaging, not only the RV but also the RC should be known. The commonly used autofocusing methods are not able to locate the target’s equivalent RC in the RD images. Generally speaking, the range bias of RC is caused by the range alignment and the unknown reference point. It is usually difficult to estimate the RC directly. However, on the basis of effective RV estimation, the range bias of RC may be obtained by 1D searching directly.

After range compression and TMC, the radar echoes related to a single scatter

Taking the second-order approximation of (

The range position of RC is then estimated by the following:

With all the rotation parameters known, the CBP algorithm is then used to generate a high-resolution and rescaled image. The traditional CBP can be accelerated using GPU parallelization [

This section demonstrates the effectiveness and robustness of the proposed algorithm using both simulated and some collected ISAR data, and the algorithm proposed by [

In the numerical simulation, a 3D target which contains 330 isolated scattering points is used. The 3D model of the target and its three projected views are shown in Figure

Three-dimensional model of the target, consisting of 330 points. (a) The three-dimensional distribution of the scattering points. (b) Front view. (c) Side view. (d) Top view.

The simulated raw data and the TMC results. (a) The simulated raw data with line motion. (b) The received data after motion compensation.

By dividing the received data equally into two parts and taking the keystone transform, the RD images are formed as in Figures

Experiments with simulated data (I). (a) RD image of 1–2048 echoes (dB). (b) RD image of 2049–4096 echoes (dB). (c) Squared FT magnitude relative to (a) (dB). (d) Squared FT magnitude relative to (b) (dB). (e) Radial projections. (f)

The RV is initialized to be 0.0124 rad/s using (

Experiments with simulated data (II). (a) The iteration steps of Nelder-Mead algorithm. (b) The entropy of RD images when searching for RC. (c) RD image of total 4096 echoes (dB). (d) CBP imaging result (dB).

Since the estimation of RC is directly related to the result of RV estimation, this section only illustrates the effectiveness of the proposed RV estimation. The root mean square error (RMSE) of the proposed RV estimation method against the signal-to-noise ratio (SNR) is presented in Figure

The RMSE of RV estimation versus SNR.

Then, the performance of proposed RV estimation method is compared with the method in [^{−5} rad/s^{2} to 10^{−4} rad/s^{2}, is shown in Figure

Robustness analysis in presence of nonuniform rotating. (a) The RMSE of RV estimation with different rotational acceleration. (b) The RMSE of RV estimation in presence of 3D rotation.

Experiments with some real data of a Yak-42 airplane (see Figure

Graph of Yak-42.

Experiments with real ISAR data (I). (a) RD image of 1–2048 echoes (dB). (b) RD image of 2049–4096 echoes (dB). (c) Squared FT magnitude relative to (a) (dB). (d) Squared FT magnitude relative to (b) (dB).

Experiments with real ISAR data (II). (a) Radial projections. (b)

An improved method for estimating the target rotation parameters has been proposed in this paper. For the RV estimation, the keystone transform was used to compensate the linear range migration through resolution cells in subaperture. Then, the two adjacent RD images were processed by 2D FT to eliminate the influence of unknown RC, and the radial projection was applied to avoid the time-consuming process of 2D interpolation and image correlation. In addition, the RV was found by solving a 1D polar curve matching problem in a few iterations efficiently. Then, the influence of unknown RC on wideband ISAR imaging was studied in this paper. Based on the RV estimation, the target’s RC was estimated by a line search using minimum entropy criterion. With all the rotation parameters, the parallelized CBP algorithm was used to obtain the high-resolution and rescaled image.

In a simulation performed with known RV using 3D point scattering model, the target was properly rescaled and the scatters’ MTRC were compensated very well. And the robustness of the proposed method has been validated with numerous experiments in presence of noise or nonuniform rotation. Furthermore, experiments with some collected data also demonstrated the effectiveness of the proposed algorithm, the CBP image of Yak-42 was well focused, and the contour of the target on image was very close to the real aircraft.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was in part supported by the NSFC (no. 61271417), in part supported by the major research plan of the NSFC (no. 61490693), and in part supported by the Research Foundation of Tsinghua University.