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Fast factorized backprojection (FFBP) takes advantage of high accuracy of time-domain algorithms while also possessing high efficiency comparable with conventional frequency domain algorithms. When phase errors need to be compensated for high-resolution synthetic aperture radar (SAR) imaging, however, neither polar formatted subimages within FFBP flow nor the final Cartesian image formed by FFBP is suitable for phase gradient autofocus (PGA). This is because these kinds of images are not capable of providing PGA with a clear Fourier transform relationship (FTR) between image domain and range-compressed phase history domain. In this paper, we make some essential modifications to the original FFBP and present a scheme to incorporate overlapped-subaperture frame for an accurate PGA processing. The raw data collected by an airborne high-resolution spotlight SAR are used to demonstrate the performance of this algorithm.

Backprojection (BP) is essentially a method of phased-array beamforming in the time domain. It is capable of handling an arbitrary flight path and producing synthetic aperture radar (SAR) images free of geometric distortions and defocus effects [

During airborne SAR imaging, air turbulence inevitability induces platform deviation from ideal trajectory. However, it is noteworthy that even the most sophisticated motion measurement system may be not sufficient to ensure diffraction-limited SAR images. Therefore, autofocus techniques are required for further processing, which is studied in [

In this paper, we propose a scheme to incorporate PGA into FFBP and achieve accurate autofocus processing for high-resolution spotlight SAR imaging. The main contributions of this work are twofold:

Pseudopolar coordinate system is employed as the imaging plane, and an analytic expression for azimuth IRF in a backprojected image is derived. Actually, a line-of-sight (LOS) pseudopolar coordinate system not only ensures the spread of azimuth IRF along the horizontal dimension but also provides an approximate FTR for the use of PGA.

Subaperture phase errors estimated by PGA cannot be directly applied to correct the range-compressed phase history data, otherwise they will cause relative shifts among subimages. To achieve an accurate phase correction, overlapped-subaperture frame (OSF) is introduced into FFBP to obtain a full-aperture phase error by coherently combining subaperture phase errors. As phase correction is recursively performed, phase estimation tends to converge and the degraded image becomes refocused.

In the original FFBP, both polar formatted subimages and the final Cartesian image are formed, but neither of them can provide PGA with a FTR between image domain and range-compressed phase history domain. In order to establish a FTR, we focus on theoretical analysis and make some significant modifications to the original FFBP in this section.

An ideal imaging geometry of spotlight SAR is shown in Figure

Imaging geometry for spotlight SAR in polar coordinates.

The sensor transmits linear frequency-modulated (LFM) signal with bandwidth

BP is essentially an integral process performed in the time domain. For a given aperture position

Due to the limited Doppler bandwidth of spotlight SAR, azimuth IRF cannot be a Dirac function which is just located at one angular cell. For target

During the derivation of equation (

To derive an analytic expression of azimuth IRF,

By neglecting the

In the following, we analyze the constraint which needs to be fulfilled for the validity of equation (

By simplifying equation (

As equation (

For spotlight SAR, the angular range of pseudopolar coordinates satisfies

For the sake of clarity, let us take the X-band spotlight SAR used in the experiments in Section

Original FFBP splits up BP integral into an infinite sum over finite subapertures, so reconstructing each subimage can be considered as an individual direct BP. Motivated by Section

Based on aperture factorization and recursive fusion, FFBP consists of a series of stages. Assume that there are

Based on the definition of Doppler

The last two subimages reconstructed by modified FFBP. (a) Subimage 1. (b) Subimage 2.

It should be noted that the LOS polar grid used in the modified FFBP is a polar grid in the image domain, while PFA uses a polar grid in the frequency domain. The modified FFBP inherently accommodates wavefront curvature and elevation-induced defocus effects arising from out-of-plane motion of platform [

FTR is what we emphasize in this paper, but it is not an inherence for time-domain algorithms after all. In

In [

Overlapped-subaperture technology was first combined with PFA to correct spatially-variant phase errors in spotlight SAR imaging [

In the first stage of the modified FFBP, the whole aperture is divided into several overlapped subapertures. Assuming that each subaperture contains

To recover a reliable full-aperture phase error function, obviously, we need to guarantee that the overlap ratio of two adjacent subapertures is big enough. In the following, we will present an innovative method of founding OSF within the modified FFBP. To illustrate this method, here we take a four-step processing as an example for analysis, and its critical steps are as follows:

Step 1: like the original FFBP, the modified FFBP performs the identical processing until 8 nonoverlapped subapertures are left, and each one contains

Step 2: perform aperture fusion among any two adjacent subapertures, and then 7 subapertures can be formed from 8 previous subapertures. As a consequence, the subaperture length is extended to

Step 3: employ PGA to extract subaperture phase errors from the 7 subimages formed by Step 2, and the detailed description will be given in the next section. It should be emphasized that, of the 7 new subapertures, only the odd ones, namely, 4 nonoverlapped subapertures will be transported to the next stage for aperture fusion, which are shown as the blue line segments in the second row of Figure

Step 4: repeat Steps 2

Change of the subapertures in the last four stages. (a) The original FFBP. (b) OSF in the modified FFBP.

An advantage of this method is that the overlap ratio being

“

Assume that phase correction is not performed in the modified FFBP until the

Supposing the bulk of motion errors have been effectively corrected by the high-precision motion measurement system, such as global positioning system (GPS) and/or inertial navigation unit (IMU), migration caused by the residual errors is less than one range cell. In this section, the sign

Based on equation (

To recover a full-aperture phase error function, subaperture phase estimates are coherently combined by eliminating the unknown linear difference between the phase errors extracted from the overlapped samples of the neighboring subapertures, which are illustrated in Figure

Full-aperture phase error combination.

Block diagram of OSF in the modified FFBP.

Profit from OSF scheme, PGA processing can be performed in a recursive manner and compatibly blended into the modified FFBP. Undoubtedly, OSF introduces computational burden, but only two or three successive stages are enough for PGA to refocus a full-resolution image up to an acceptable level. It can be regarded as a tradeoff between the image quality and the processing efficiency. Moreover, the “windowing” step of PGA in this paper should be emphasized that the maximum window width becomes small after the first applying PGA, which is of benefit to the rapid convergence of phase estimation.

In this section, the raw data collected by an airborne high-resolution spotlight SAR are used to demonstrate the performance of this algorithm. The main parameters of the SAR system are tabulated in Table

Main parameters of the SAR system.

Wave band | Transmitted bandwidth | Pulse repetition frequency | Closest slant range | Pulse number |
---|---|---|---|---|

X-band | 1.16 | 2100 | 10.5 | 16384 |

The processed results and estimated motion error are presented in Figure

Images formed by the modified FFBP. (a) Uncompensated SAR image; (b) SAR image restored by PGA; (c) radial error extracted form phase error by PGA.

Magnified images of subscenes A–C in Figure

Images and IRFs. (a) Imaging result and azimuth IRF of the reflector in circle “1”; (b) imaging result and range IRF of the reflector in circle “2.”

Two-dimensional IRFs of the reflector in Scene 3. (a) Range IRF. (b) Azimuth IRF.

From Figure

Main parameters of the SAR system.

Wave band | Transmitted bandwidth (GHz) | Pulse repetition frequency (Hz) | Closest slant range (km) | Pulse number |
---|---|---|---|---|

Ku-band | 500 | 2000 | 5 | 8192 |

Images formed by the modified FFBP. (a) Uncompensated SAR image; (b) SAR image restored by PGA; (c) radial error extracted form phase error by PGA.

Magnified images of subscenes A–C in Figure

Two-dimensional IRFs of the reflector in circle “3”. (a) Range IRF. (b) Azimuth IRF.

In this paper, we incorporate PGA into the modified FFBP to achieve an accurate autofocus for high-resolution spotlight SAR imaging and investigate its performance in detail. In this method, LOS pseudopolar subimages are reconstructed to provide PGA with a FTR between image domain and its corresponding range-compressed phase history domain. Based on OSF, full-aperture phase error function can be obtained to retrieve the linear phase belonging to each subaperture phase error. Within the recursion of the modified FFBP, phase estimation and correction can be achieved at a high precision and efficiency. It should be noted that the unknown DEM will cause the degradation of the focusing and positioning performance of the method in this paper so that we will study in the future work.

The data used in this article are obtained from a private data source.

The authors declare that they have no conflicts of interest.

This research was funded by National Natural Science Foundation of China (Grant no. 61303031) and the Fundamental Research Funds for the Central Universities (Grant no. JB181005).