An Efficient Preprocessing Approach for Airborne Hybrid SAR and ISAR Imaging of Ship Target Based on Kernel Distribution

The airborne hybrid synthetic aperture radar (SAR) and inverse synthetic aperture radar (ISAR) imaging for the ship target is very important in the field of ocean surveillance, but it suffers from the problem of high computational complexity. In this article, an efficient preprocessing approach for the airborne hybrid SAR and ISAR imaging based on the kernel distribution is proposed, which can reduce the computational complexity and preserve the image quality simultaneously. First, the residual Doppler frequency is estimated. Then, a novel measurement method with good robustness is proposed for evaluating the stationarity of Doppler frequency from the aspect of statistics, which adopts the kernel distribution to estimate the probability distribution function curve for the Doppler frequency. Afterward, an effective preprocessing approach is addressed consequently to eliminate the incomplete illumination time and select the time interval with steady target motion. Results of simulated and actual SAR data verify the effectiveness of the novel algorithm proposed in this article.

Amplitude function of range direction. a a (·) Amplitude function of azimuth direction. K r Frequency modulation radio. s m (t r ) Matching function. s rb (t m , t r ) Echo data after range compensation.

IFFT[·]
IFFT operator. m ns Start pulse order of range profile. s rb (m, n) Discrete form of s rb (t r , t m ). s k rb (n) The kth subdata. k Order of subdata, k = 1, 2, . . . , K. K Number of subdata, K = N a /N a0 . N a Pulse number of range profile. N a0 Pulse number of subdata. s(m 1 , n) Result of conjugated multiplication. m 1 Azimuth order of s(m 1 , n), m 1 = 1, 2, . . . , N a0 − 1. R 0 (n) Correlation function of s(m 1 , n). R b Autocorrelation function of R 0 (n).  Optimal end time of echo data. s opt r (n) Echo data with optimal time. f δ (x) Estimated PDF off I d (k).
Normal distribution function.

I. INTRODUCTION
A IRBORNE synthetic aperture radar (SAR) imaging of ship target plays a significant role in the field of marine monitoring [1]- [3]. Due to the complex motion of ship target, the traditional SAR imaging algorithm will generate a defocused radar image. To produce a well-focused radar image, the hybrid SAR and inverse synthetic aperture radar (ISAR) technique is presented, which can process the motion of radar and target separately [4]- [10]. The idea of hybrid SAR and ISAR imaging was originally proposed and verified via simulated experiment by Ward [11]- [12]. Afterward, Martorella et al. presented a series of hybrid SAR and ISAR imaging algorithms combining with the techniques of Doppler parameter estimation, cross-range scaling, and moving target detection [13]- [16]. Moreover, F. Breizzi proposed an autofocusing algorithm for hybrid SAR and ISAR imaging in [17]. Some constructive conclusions have been obtained for these algorithms. A hybrid imaging algorithm for high-squint SAR was proposed in [18], which focused on the modification of SAR compensation process. In [19], a hybrid imaging algorithm based on the CLEAN technique and crossrange scaling for frequency-modulated continuous-wave SAR was presented by E. Casalini et al., which verified the possibility of radar imaging for the frequency modulation continuous wave in the Ka band. Recently, a novel time-frequency analysis technique is addressed in [20], which can be applied in the rangeinstantaneous Doppler (RID) analysis for the high-resolution imaging. For the high sea condition, the ISAR imaging technique in [21] can achieve the excellent performance, in which a high accuracy parameter estimation method was presented.
Whereas, some of these hybrid SAR and ISAR imaging methods have the disadvantages as follows: 1) High computational complexity: The hybrid SAR and ISAR imaging methods proposed in [10]- [19] suffer from the high computational complexity owing to processing the entire echo data. Some imaging time selection methods are proposed in [22]- [28], in which the Doppler frequency is segmented in time, and the stationarity is evaluated by the indexes such as image contrast [22] and root mean square error [27]. Hence, the selected time window will be affected by the segment length. Moreover, these methods determine the time window during the ISAR imaging, before which the high computational complexity has been produced. Given the above, the optimal processing time should be determined during the SAR processing, which is denoted as the preprocessing approach for the hybrid SAR and ISAR imaging in this article.
2) Influence on the quality of ship radar image: The performance of hybrid SAR and ISAR algorithm is influenced by the long period movement of ship target. Hence, the optimal time interval selection is necessary for enhancing the image quality.
In this article, the computational complexity can be reduced from two aspects, including the determination of optimal time during the SAR processing and the removal of the incomplete illumination time. For the former, the computational complexity will be greatly degraded owing to processing the data with small size in the ISAR processing. For the latter, the ship target is illuminated in a limited time that can be composed of the incomplete illumination time and the complete illumination time, respectively. Here, the complete illumination time is defined as the time interval in which the entire ship target is located in the wave beam. Correspondingly, the incomplete illumination time is defined as the time interval in which the ship target is partially illuminated by the wave beam. Obviously, the radar image with the entire ship target cannot be obtained via the incomplete illumination time, which will also induce the additional computing time. Hence, the more computational complexity can be reduced via selecting the complete illumination time and ignoring the incomplete illumination time.
For this purpose, an efficient preprocessing approach is proposed in this article for the airborne hybrid SAR and ISAR imaging based on the kernel distribution. First, the Doppler frequencies of echo data are estimated via the average cross-correlation coefficient (ACCC) method in [29]. Then, a novel estimation method for the residual Doppler frequency is proposed. The residual Doppler frequency can reflect the illuminated scope and movement of the imaging target. Afterwards, an efficient preprocessing approach is proposed, through which the optimal processing time interval can be selected. For determining the optimal time interval, the stationarity of residual Doppler frequency needs to be measured. In this article, the kernel distribution in [29] is adopted to estimate the probability distribution function (PDF) of residual Doppler frequency and generate a central frequency and bandwidth, through which the optimal time interval can be determined.
The proposed approach has the advantages as follows. First, it can degrade the computational complexity via determining the optimal time interval during the SAR processing. Second, the stationarity of Doppler frequency is measured from the aspect of statistics, which is accurate and efficient compared with the existing methods in [23]- [28]. Third, the image quality can be equal or higher compared with the radar image before the SAR procedure.
The main work of this article is shown in Fig. 1. The main contents are arranged as follows. First, the circumstances of stationary and moving ship target are analyzed in Section II. Then, an efficient preprocessing approach is proposed in Section III, including the residual Doppler frequency estimation and the optimal time interval selection. Afterwards, the results of  simulated and actual SAR data are shown in Section IV. Finally, the conclusion is drawn in Section V.

II. HYBRID SAR AND ISAR IMAGING GEOMETRY MODEL
The circumstances of stationary ship target and moving ship target are discussed in this section. First, the SAR imaging geometry model is established. On this basis, the range history and Doppler frequency produced by the SAR platform are deduced. Second, the hybrid SAR and ISAR imaging geometry is analyzed. Consequently, the range history and Doppler frequency generated by the SAR platform and ISAR imaging target are derived.

A. Circumstance of Stationary Ship Target
The airborne SAR imaging geometry is shown in Fig. 2, in which the imaging scene and target coordinate systems are established, which are denoted as O−X e Y e Z and O − XYZ, respectively. The scene center is denoted as O, the direction of ship's bow is the same as axis X, and the axis Z is vertical to the sea surface.
For simplification, we choose the ship target with the center located at O for the following analysis. When the ship target is stationary, the positions of scatterer P on the ship target in the coordinates of O − XYZ and O−X e Y e Z can be expressed as follows: where θ 0 is the angle between the ship's bow and axis X e and R T (θ 0 ) is the transform matrix from the coordinate of O − XYZ into O−X e Y e Z, and it can be expressed as The aircraft is moving along the direction of axis X e with a velocity of v r . The position of radar in the coordinate of O−X e Y e Z is shown as where t m , H, and γ 0 are the slow time, the flight height of aircraft, and the look angle of radar, respectively. The range between the radar and scatterer P is calculated as where the range R p0 can be expressed as Then, the range history can be rewritten as wheret p m = t m − x p e /v r . The relationship between R p0 and R S p (t p m ) is shown in Fig. 3. Afterwards, taking the derivative of R p (t p m ), the Doppler frequency produced by the SAR platform can be calculated as where ϕ s (t p m ) is the angle between the direction of axis X and the RLOS in Fig. 3. The squint angle ϕ s (t p m ) can be expressed as

B. Circumstance of Moving Ship Target
The ship target on the sea surface has the three-dimensional (3-D) rotation and translational motion. Under this circumstance, the position of scatterer P in the coordinates of O − XYZ and O−X e Y e Z can be calculated as follows: where v t is the forward velocity of ship and R rot (t m ) represents the 3-D rotation matrix. Then, the range between scatterer P and radar can be calculated as where the range R p0 (t m ) can be expressed as The range variation produced via the target motion is denoted as R p0 (t p m ). Since the position variation of ship target is not large, the time variablet p , the Doppler frequency can be computed as where f d (t p m ) is the Doppler frequency generated by the hybrid SAR and ISAR imaging target, f I d (t p m ) is the Doppler frequency produced via the ISAR imaging target, and the expression of Hence, the Doppler frequency can be decomposed into the combination of frequency produced by the SAR and ISAR procedure.

III. PREPROCESSING APPROACH BASED ON KERNEL DISTRIBUTION
In this section, an efficient preprocessing approach is proposed for the airborne hybrid SAR and ISAR imaging based on the kernel distribution. First, the principle of preprocessing approach is elaborated via the analysis of Doppler frequency. Second, the proposed preprocessing approach is introduced, in which the residual Doppler frequency is estimated, and the stationarity is measured with the method in statistics. On this basis, the optimal processing time interval can be determined.

A. Principle of Preprocessing Approach
Actually, the ship target is illuminated by the wave beam only in a limited time. The length of synthetic aperture is larger than the size of ship target, and the illuminated time consists of the complete and incomplete illumination time. Hence, the computational complexity can be reduced via ignoring the incomplete illumination time.
Furthermore, both the smooth motion and maneuvering movement exist in the complete illumination time. The smooth motion will produce the steady Doppler frequency and has little impact on the radar image quality. Whereas, the maneuvering movement will generate the time-varying Doppler frequency and decrease the image quality. Therefore, the image quality can be enhanced via ignoring the time interval with maneuvering motion of target.
Via the analysis above, we can see that the status of illumination and target motion can be reflected in the residual Doppler frequency, which is related to the theoretical Doppler frequency value of SAR platform and the Doppler frequency center of echo. The theoretical Doppler frequency value, Doppler frequency of echo, and residual Doppler frequency can be calculated as follows.
First, the theoretical Doppler frequency value of SAR platform can be calculated by (8).
Second, the Doppler frequency center of echo can be estimated via the ACCC method in [29]. The ACCC method utilizes the correlation of adjacent echoes to estimate the Doppler frequency center [29]. When the target is not completely illuminated, the correlation of adjacent echoes will be degraded, which induces the obvious difference between the estimated value and theoretical value. As seen in (14), the Doppler frequency is generated by the SAR platform and the ISAR imaging target. When the target has maneuvering motion, the Doppler frequency of ISAR imaging target will be large, which will also cause the obvious difference between the estimated value and theoretical value. Here, this difference can be denoted as the residual Doppler frequency.
Third, the residual Doppler frequency can be estimated aŝ To avoid the influence produced by the translational motion of ship target, the residual Doppler frequency in (15) can be calculated as follows: The residual Doppler frequency in (17) can reflect the status of illumination and target motion, by which the optimal time interval can be determined. When the target has the smooth movement in the complete illumination time, the residual Doppler frequency is stable, and it is concentrated on a stationary frequency. Otherwise, the residual Doppler frequencies is time-varying and dispersive.
The proposed calculation method of residual Doppler frequency is different from the other optimal time selection methods, such as [23]- [27]. In the existing methods, the residual Doppler frequency is calculated with the SAR processing, motion compensation, ISAR refocusing, and time-frequency analysis for the echo data with entire pulses. Hence, the calculation of the residual Doppler frequency consumes a heavy computational complexity.
The proposed method can calculate the residual Doppler frequency only with the range compensation and ACCC method in [29], which reduces the computational complexity.
Moreover, the proposed method is easy to implement compared with other optimal time selection methods. In the existing methods, the time-frequency analysis technique is applied to estimate the residual Doppler frequency, and the range bin with the single scatterer is usually required consequently, which does not exist in some actual cases. Although several time-frequency curve extraction methods are proposed in [23]- [27], the processing procedure becomes complicated and the more computational complexity is produced.
The proposed method can accomplish the calculation of the residual Doppler frequency without finding the range bin with the single scatterer or extracting the time-frequency curve of the single scatterer. The procedure of the proposed method can be implemented easily only with the range compensation and ACCC method.

B. Preprocessing Approach Based on Kernel Distribution
As mentioned above, the stationarity of residual Doppler frequency is related to the computational complexity and image quality. In this article, the kernel distribution in the statistics is applied to evaluate the stationarity of residual Doppler frequency. The stationarity of Doppler frequency is depicted as a central frequency and a narrow bandwidth. On this basis, an efficient preprocessing approach is proposed for determining the optimal time interval, which can reduce the computational complexity and ensure the image quality.
The main procedures of the proposed preprocessing approach are illustrated as follows: Step (1): Obtain the range compression result for the echo data.
The range profile can be obtained via the matched filtering approach. The echo signal of scatterer P can be represented as where t r is the fast time, a r (·) and a a (·) are the amplitude functions of range and azimuth direction, and K r is the frequency modulation ratio. The matching function is Then, the echo data after range compression can be obtained as and IFFT[·] are the fast Fourier transform (FFT) and inverse FFT (IFFT) operator, respectively.
Step (2): Extract the range profile and divide it into subdata.
The range profile of one ship target is extracted. The start pulse order of this range profile is denoted as m ns . The discrete form of this range profile is represented as s rb (m, n), where m and n are the azimuth and range order, respectively. Then, the range profile is divided into some subdata with the form of where k = 1, 2, . . . , K is the order of subdata, K = N a /N a0 is the number of subdata, and N a and N a0 are the pulse number of range profile and subdata, respectively.
Step (3): Estimate the Doppler frequency of echo data based on the ACCC method in [29].
The main procedures of Doppler frequency center estimation method are as follows: 1) The matrix s(m 1 , n) is obtained by the conjugated multiplication of two neighboring pules, which can be expressed as where m 1 = 1, 2, . . . , N a0 − 1 is the azimuth order of s(m 1 , n).
2) The correlation function R 0 (n) is generated by calculating the average along the azimuth direction, which can be represented as 3) The autocorrelation function R b is computed by calculating the average along the range direction, which can be expressed as 4) The Doppler frequency center can be estimated as where PRT is the pulse repetition time and angle(·) is the function of extracting phase.
5) The Doppler frequency history of echo data can be generated asf Step (4): Calculate the Doppler frequency of SAR platform with (8), and it is denoted as f S d (k).
Step (5): Estimate the residual Doppler frequency with (15)- (17), and the Doppler frequency of ISAR imaging target is denoted asf I d (k).
Step (6): Determine the optimal central frequency of f p and optimal bandwidth of δ opt .
The optimal central frequency and bandwidth determination approach based on the kernel distribution will be elaborated in Section III-C.
Step (7): Obtain the time interval based on f p and δ opt . The residual Doppler frequency is selected by the central frequency f p and bandwidth δ opt , i.e., |f I d (k) − f p | ≤ δ opt , which can illustrate that the ship target has the smooth motion in the complete illumination time. For better determining the imaging time interval, the Doppler frequencyf I d (k) can be fitted by the polynomial, which can be expressed aŝ where p i and n p are the polynomial coefficient and order, respectively. The start and end pulse order which are satisfied with |f I d (k) − f p | ≤ δ opt are denoted as m s and m e , respectively. Then, the time interval can be obtained as [m ns + m s , m ns + m e ] · PRT, and the corresponding echo data . . .
Here, the procedure of preprocessing approach is accomplished.
Remark 1: In this article, the Doppler frequency center is estimated via (25), which is suitable for the ship ISAR imaging with high SNR [29]. Under the high sea condition, the sea clutter may exist and the SNR will be decreased. Then, the sea clutter suppression techniques in [35] and [36] can be applied, and the more precious Doppler centroid estimation method in [37] can be utilized consequently.

C. Optimal Selection Approach for the Central Frequency and Bandwidth Based on the Kernel Distribution
Since the PDF can describe the probability of appearing at each frequency, the central frequency, and optimal bandwidth in Step (6) can be obtained by estimating the PDF curve of the residual Doppler frequency. In this article, an optimal selection approach for the central frequency and bandwidth based on the kernel distribution is proposed.
The kernel distribution is a nonparametric method for the estimation of PDF, which describes each sample with a kernel function, and sums all the kernel functions to generate a smooth PDF curve. The principle of kernel function method is illustrated as Fig. 5. The shapes of kernel function can be normal, triangle, and box. Considering the good smoothness, the standard normal distribution is selected. Moreover, an appropriate bandwidth is important because it usually controls the smoothness of PDF curve. In this article, the bandwidth is calculated by the theory in [29].
For the estimated PDF, the position of the highest peak is denoted as f p , which reflects that the most amount of Doppler frequencies are concentrated on f p , and the bandwidth can describe the concentration of central frequency.
The main procedures of the proposed approach can be seen in Fig. 6, and the central frequency and optimal bandwidth can be determined as follows.
Step (1): Estimate the PDF of residual Doppler frequency via the kernel distribution. The PDF of residual Doppler frequency is calculated via the kernel function, which can be expressed as where δ is the bandwidth of kernel function, x represents the real values, the kernel function we choose in this article is the standard normal distribution which has a better smoothness, N (·) represents the standard normal distribution, and the expression of N (·) is given as Step (2): Determine the optimal central frequency and bandwidth.
The optimal central frequency can be solved by searching the highest peak position, which reflects the most amount of Doppler frequencies are concentrated on this position.
The optimal bandwidth of δ opt can be calculated by the principle of minimizing the asymptotic mean integral squared error (AMISE) in [29].
The mean of f δ (x) can be calculated by where f (·) is the true density and z is the variable of integration. With the Taylor expansion, the variance of f δ (x) can be approximated as Then, the mean integral squared error can be calculated as With the weak assumption and Taylor expansion, we can obtain the AMISE as Calculating the derivation of AMISE(f δ (x)) and setting it to be equal to zero, the optimal bandwidth can be computed as where σ is the standard deviation of Doppler frequency samples. The optimal bandwidth in (35) describes the frequency range that can be regarded as neighboring the central frequency.
With the optimal central frequency and bandwidth, the residual Doppler frequency in the domain of |f I d (k) − f p | ≤ δ opt can be considered as stable. Afterwards, the preprocessing procedure can be accomplished.
The proposed approach can maintain or enhance the image quality owing to the selection for the stable Doppler frequency, and the computational will be degraded obviously. Whereas, the azimuth resolution will be reduced due to the decrease of pulse number or imaging time.
Compared with the other optimal time selection methods, the proposed approach can reduce more computational complexity.
Reason 1: Processing the echo data with smaller size in the procedure of motion compensation and ISAR refocusing.
In the existing methods, the optimal time interval is determined after the SAR processing, motion compensation, and ISAR refocusing for the echo data with the entire pulses. Hence, a heavy computational complexity is required owing to process the echo data with the entire pulses.
In the proposed approach, the optimal time interval is determined after the range compensation of the SAR processing, and then the motion compensation and ISAR imaging are implemented for the echo data with the few pulses. Therefore, the proposed method can reduce more computational complexity in the process of motion compensation and ISAR imaging.
Reason 2: Without the generation of the image sequence. In some existing methods, the image sequence is generated to determine the optimal time interval, which produces a large computational complexity.
In the proposed approach, the optimal time interval is selected with the stationary Doppler frequency, which is obtained based on the probability estimation. This procedure consumes a tolerable computational complexity compared with the image sequence generation in the other methods.
Moreover, the proposed approach can adapt the case of maneuvering motion of the ship target. Owing to the change of target motion, the estimated PDF curve can show the multiple peaks. For the better image quality, the higher peak can be selected to determine the optimal time interval, because the more amount of Doppler frequencies are concentrated on this peak position.
Furthermore, the proposed approach can be applied in the circumstance of high sea state. If the rotation amplitudes are different in the multiple periods, the multiple peaks will exist in the estimated PDF curve, and the Doppler frequencies are concentrated on the different frequencies. The optimal time interval can be determined by the maximum peak position and bandwidth. If the rotation amplitude is same in the multiple periods, the peak value will be larger compared with the case of single period, and more amount of Doppler frequencies are concentrated on the peak position. With the peak position and bandwidth, multiple time intervals can be determined. Considering the higher imaging resolution, the optimal time interval with the largest imaging time can be selected.

IV. EXPERIMENTAL RESULT
In this section, the results of simulated and actual SAR data are given to verify the effectiveness of the proposed approach. There are multiple ship targets in the imaging scene. First, the proposed preprocessing approach is implemented to produce the optimal time interval. Afterwards, the hybrid SAR and ISAR imaging algorithm in [13] is applied to generate the well-focused ship radar image. Here, the Chirp Scaling (CS) algorithm in [31] and [32] is chosen for the SAR processing, the range alignment method in [33] and the phase compensation method in [34] are used for the motion compensation, and the RID algorithm based on the spectrum in [10] and [38] is used for the ISAR refocusing.

A. Simulated Data
First, the effectiveness of the proposed approach is verified with the results of simulated experiments. The parameters of the radar and imaging scene are listed in Table I. Two ship targets are located in the scene as shown in Fig. 7, and the 3-D rotating parameters are given in Table II according to the values in [39] and [40]. The SAR image via the CS algorithm is shown in Fig. 8. It is obvious that the CS images of two ship targets are defocused in the azimuth without the preprocessing procedure. Hence, the echo data should be preprocessed to reduce the computing time  and improve the image quality. The simulated results of two ship targets are shown as the following Case 1 and Case 2, respectively.
Case 1. Imaging Results of Ship Target 1: After the extraction of range profile, the theoretical value of f S d (k) can be obtained by (8), and the Doppler frequency center of echo can be estimated with the ACCC method in [29], both of which are shown in Fig. 9. Here, the red label "x," blue line and green dotted line in Fig. 9 represent the estimated value off d (k), fitting curve off d (k), and real value of f S d (k), respectively. Obviously, the estimated values off d (k) and the real values of f S d (k) have similar trends in the orange area.
Then, the residual Doppler frequency can be calculated via (15)-(17) as shown in Fig. 10. The stationarity of residual Doppler frequency is expressed as |f I d (k) − f p | ≤ δ opt , through which the optimal processing time interval can be selected.
Afterwards, the PDF curve is estimated via the kernel distribution to determine f p and δ opt . The histogram and estimated PDF off I d (k) are shown in Fig. 11, where the blue line and red line represent the histogram and estimated PDF, respectively. Obviously, the estimated values off I d (k) are concentrated on    Based on f p and δ opt , the optimal time interval can be selected as shown in Fig. 12, where the red label "x," blue line, green and purple dotted line, and orange shadow area representf I d (k), fitting curve off I d (k), start and end imaging moment, and the area of |f I d (k) − f p | ≤ δ opt , respectively. The time interval and pulse number before and after the preprocessing are listed in Table III. Meanwhile, the theoretical complete illumination time interval and pulse number are given. Obviously, the time intervals after the preprocessing are   within the complete illumination time interval. Compared with the result before the preprocessing, the pulse number after the preprocessing is decreased about 1.21 times. Since the size of processing data is decreased, the computational complexity can be degraded greatly.
The imaging results before and after the preprocessing for the first target are displayed in Figs. 13 and 14, where a) is the SAR image via the CS algorithm, and b) is the radar image via the hybrid SAR and ISAR algorithm. Compared with Fig. 13, the radar images in Fig. 14 have better quality. Especially, the SAR image in Fig. 13(a) has severe azimuth defocus, and the multiple imaging projection planes appear. After the SAR preprocessing, the quality of SAR image in Fig. 14(a) is improved obviously, and the imaging projection plane can be fixed. The improved quality is contributed by retaining the more stable Doppler frequency. The image entropies of Figs. 13 and 14 are listed in Table IV, from which we can see that the image entropies in Fig. 14 are lower.   Case 2. Imaging Results of Ship Target 2: First, the Doppler frequency comparison betweenf d (k) and f S d (k) is shown in Fig. 15. Obviously, the curves off d (k) and f S d (k) have similar trends in the orange area of Fig. 15. Then, the estimated values of residual Doppler frequency are displayed in Fig. 16. With the kernel distribution, the histogram and estimated PDF curve are given in Fig. 17. After calculation, the central frequency is f p = −3.51 Hz and the optimal bandwidth is δ opt = 1.63 Hz. Afterward, the estimated values of |f I d (k) − f p | ≤ δ opt are utilized to obtain the optimal time interval as shown in Fig. 18.
In Table V, the time intervals and pulse number before and after the preprocessing are listed, and the time interval and pulse number of complete illumination time are given simultaneously. Obviously, the optimal time interval is totally within the complete illumination time, and the pulse number after the preprocessing is decreased about 1.01 times. Hence, the proposed approach can reduce the computational complexity for the hybrid SAR and ISAR imaging.  The hybrid SAR and ISAR imaging results before and after the preprocessing for the second target are shown in Figs. 19 and 20, where a) is the SAR image via the CS algorithm, and b) is the radar image via the hybrid SAR and ISAR algorithm. Clearly, there are two ship attitudes appear in Fig. 19(a) before the preprocessing, and only one ship attitude remains in Fig. 20(a) after the preprocessing. Hence, the proposed approach can enhance  Table VI, and the radar images in Fig. 20 have lower entropies.
Given above, the proposed approach can reduce the computational complexity and ensure the image quality.
Remark 2: The use of SAR or ISAR imaging can be determined according to the specific cases. In the case of ship target with slight motion, the high-quality radar image can be obtained only with the SAR processing and the optimal time selection. In the case of ship target with maneuvering motion, the quality of SAR image can be improved obviously after the time selection, for example, the SAR image before the optimal time selection in Fig. 19(a) and the SAR image after the optimal time selection in Fig. 20(a). If the demand of the image quality is not strict, the ISAR refocusing technique does not have to be implemented. If the high-quality radar image is required, the ISAR refocusing technique needs to be implemented.

B. Comparison With the Existing Methods
In this article, the maximum contrast based automatic time window selection (MC-ATWS) technique in [22] is adopted for comparing with the proposed approach from the aspects of computational complexity and image quality.
The MC-ATWS technique can determine the optimal time interval by two steps as follows: first, the imaging central moment is located by the maximum image contrast of the image sequence; then, the time window length is gradually increased until the image contrast of the new image sequence stops increasing.
The proposed approach can select the optimal time interval by the residual Doppler frequency estimation and the PDF fitting with the kernel distribution. The peak position and bandwidth of kernel distribution reflect the stationarity of Doppler frequency, through which the optimal time interval is selected.
1) From the Aspect of Image Quality: As mentioned above, the MC-ATWS selects the optimal time interval by the maximum image contrast, which can guarantee the highest image quality. In this article, the proposed approach also considers the image quality and achieves it by selecting the stable Doppler frequency, while the highest image quality cannot be ensured.
2) From the Aspect of Computational Complexity: It is assumed that the MC-ATWS technique and the proposed approach can export the same optimal time interval, and the azimuth number is denoted as N a = m e − m s + 1.
In the MC-ATWS technique, first, the SAR algorithm and ISAR motion compensation technique need to process the echo data with the size of N a × N r . Then, the ISAR imaging algorithm is implemented (M + N ) times for the echo data with  the size about N a × N r to select the optimal time interval. Hence, adopting the MC-ATWS technique requires the large computational complexity.
In the proposed approach, the computational complexity is mainly derived from the range compression for the echo data with the size of N a × N r . After the optimal time interval selections, the SAR algorithm, ISAR motion compensation, and ISAR imaging algorithm are implemented one time for the echo data with the size of N a × N r . Therefore, the computational complexity can be smaller than the MC-ATWS technique. The computational complexity of the proposed approach is listed in Table VII. It is assumed that one time multiplication produces the computational complexity of O (1).
Here, the simulation results are shown for the comparison of the MC-ATWS and the proposed approach. The parameters of radar system are same as Table I, and the ship target has the same motion parameters with the first target in Table II. The scatterer model and SAR image through the CS algorithm are shown in Fig. 21(a) and (b), respectively. Obviously, the azimuth defocus appears in Fig. 21(b) due to the target movement and long imaging time. Here, the MC-ATWS and the proposed approach are performed to determine the optimal time interval for reducing the computing time and enhancing the image quality.
The time interval and pulse number under the different conditions are listed in Table VIII. Without the time selection, the   Combining with the MC-ATWS technique, the radar images can be yielded via the CS algorithm and hybrid SAR and ISAR algorithm as shown in Fig. 22(a) and (b), respectively. Combining with the proposed approach, the radar images in Fig. 23(a) and (b) can be produced. The entropies of different radar images are listed in Table IX. Apparently, the radar image  generated with the MC-ATWS technique has the better image quality.

C. Actual SAR Data
Afterwards, the results of actual SAR data are applied to verify the effectiveness of the proposed approach. The radar parameters are listed in Table X. The SAR image of the imaging scene is shown in Fig. 24, and the results of preprocessing and hybrid SAR and ISAR imaging are illustrated in the Case 1 and Case 2 as follows.
Case 1. Imaging Results of Ship Target 1: Firstly, the Doppler frequency comparison betweenf d (k) and f S d (k) is shown in Fig. 25, and the curves off d (k) and f S d (k) have the similar trends in the orange area. There is a difference value between f d (k) and f S d (k) in the orange area, which is generated by the smooth movement of target, such as the translational motion   and uniform rotational motion. The translational motion can be eliminated through the motion compensation technique, and the rotational motion can be used for the ISAR imaging. Hence, this difference will not affect the imaging quality. Moreover, this difference is denoted as f p in fact, and will not influence the performance of the proposed approach because the Doppler frequency should satisfy the condition of |f I d (k) − f p | ≤ δ opt . Then, the residual Doppler frequency is estimated as shown in Fig. 26. For measuring the stationarity of residual Doppler frequency, the histogram and estimated PDF are yielded in Fig. 27. The optimal central frequency is determined as −7.73 Hz, and the optimal bandwidth is calculated as 4.58 Hz. Afterwards, the imaging time interval can be determined with |f I d (k) − f p | ≤ δ opt , as shown in Fig. 28.
The time intervals and pulse numbers before and after the preprocessing are listed in Table XI, respectively. After the preprocessing, the processed pulse number is decreased about 1.18 times, which verifies that the proposed approach can reduce the computational complexity.    Table XII, which describe that the radar images after the preprocessing have the equal quality with the radar images before the preprocessing.
Case 2. Imaging Results of Ship Target 2: First, the Doppler frequency comparison betweenf d (k) and f S d (k) is shown in Fig. 31. The orange area in Fig. 31 describes that the curves of f d (k) and f S d (k) have similar trends. Then, the residual Doppler    frequency is estimated and shown in Fig. 32. To evaluate its stationarity, the PDF off I d (k) is estimated as shown in Fig. 33, and the central frequency and optimal bandwidth is 1.86 and 4.79 Hz, respectively. Afterward, the optimal time interval can be selected as shown in Fig. 34.
The time intervals and pulse numbers before and after the preprocessing are listed in Table XIII, respectively. Compared with  TABLE XIII  TIME INTERVALS AND PULSE NUMBER OF THE SECOND TARGET  BEFORE AND AFTER    the pulse number before the preprocessing, the pulse number is decreased about 1.13 times. Hence, the proposed approach can reduce the computational complexity.
The imaging results before and after the preprocessing for the second target are shown in Figs. 35 and 36, respectively, where a) is the SAR image via the CS algorithm, and b) is the radar image via the hybrid SAR and ISAR imaging algorithm. The image entropies are listed in Table XIV, which can illustrate that the image qualities before and after the preprocessing are equal.
Given above, the proposed approach can reduce the computational complexity and ensure the image quality.

V. CONCLUSION
In this article, an efficient preprocessing approach is proposed to address the issues of high computational complexity for the hybrid SAR and ISAR imaging procedure. The key of reducing computational complexity is to select the echo segment with the steady residual Doppler frequency after SAR processing.
First, the residual Doppler frequency is analyzed, which can reflect the status of illumination and movement of target. Then, the stationarity of the residual Doppler frequency is evaluated. Considering that the steady Doppler frequencies are concentrating on a stationary frequency, we applied the kernel distribution to determine the frequency range in which they locate. On this basis, the optimal imaging time interval can be selected, and the procedure of preprocessing approach can be accomplished.
The proposed approach can determine the optimal imaging time interval during the SAR processing, which can efficiently reduce the computational complexity and ensure the image quality. Radar imaging results of simulated and actual SAR data are given to illustrate the effectiveness of the novel method proposed in this article.