1. Introduction
As a microwave remote sensing imaging radar technique, synthetic-aperture radar (SAR) has the characteristics of high azimuth resolution and all-time, all-weather, and long-distance observation ability [
1]. However, SAR is susceptible to interference from various factors in practice, which results in missing sampling data and a decreased signal-to-noise ratio (SNR), which lead to degraded imaging quality. Generally, signal sampling must satisfy the Nyquist sampling theorem. The amount of information contained in SAR echo data increases with the amount of echo data obtained via traditional sampling methods, which imposes higher requirements for storage and calculation. Compressive sensing (CS) theory was proposed to cope with this problem, i.e., when the signal is sparse, sparse signal reconstruction can be achieved with less sampling [
2]. CS theory has been combined with radar imaging, and theoretical and simulation analyses have verified the feasibility of combining CS theory with SAR imaging [
3]. Research has shown that when the imaging scene is sparse, a small number of radar echo data are needed to obtain high-resolution and high-quality SAR images. Sparse SAR imaging technology can effectively reduce the data acquisition and storage requirements of radar receivers. Hence, the correlation imaging method has received extensive attention [
4,
5,
6,
7,
8].
Traditional sparse SAR imaging requires the construction of an observation matrix, which is closely related to radar system parameters and imaging geometry. The properties of the observation matrix directly affect the signal reconstruction performance. Errors in the observation matrix will lead to phase errors in the SAR data, which can cause the defocusing of the reconstructed image. There are some methods to improve the quality of sparse SAR imaging. A sparse-based SAR joint imaging and phase error correction method has been proposed, which includes image generation and model error correction at each iteration to generate high-quality SAR images [
9]. A sparse radar imaging method based on L
1/2 regularization theory has been proposed, which uses L
1/2 regularization instead of the L
1 norm and requires no observation matrix; by constructing a radar echo simulation operator to achieve imaging, the complexity of the algorithm is reduced greatly [
10]. A fast compressed sensing SAR imaging method has been proposed that derives an approximate SAR observation model from the inverse of the focusing process, which combines CS and a traditional matched filter into a sparse regularization framework. Then, a fast iterative threshold algorithm is used to solve the sparse image reconstruction problem, which greatly reduces time and storage costs [
11]. A sparse autofocus method based on approximate observation and a minimum entropy constraint has been proposed, which introduces a minimum entropy constraint to estimate the error phase and obtains the initial phase via the phase gradient autofocus (PGA) algorithm to reduce the number of iterations [
12]. In addition, a sparse SAR autofocus model combining phase error estimation and the L
1 norm regularization problem has been proposed. The algorithm can estimate the phase error and obtain high-quality sparse images [
13]. A sub-aperture motion state estimation method for airborne SAR platform segment based on parametric sparse representation has been proposed. The SAR echo is expressed as a joint sparse signal by the parametric dictionary matrix, and the SAR motion state estimation is transformed into the dynamic representation of the joint sparse signal, which has a good imaging effect in complex motion environments [
14]. An efficient SAR imaging algorithm based on compressed sensing and autofocus fusion has been proposed. The algorithm combines the Tikhonov regularization method and uses an efficient iterative method to effectively solve the optimization problem of spatial phase error variation based on the azimuth domain and low sampling rate [
15]. A sparse imaging method based on fusion motion error estimation and compensation has been proposed. This method combines the range-Doppler algorithm with motion error correction to invert the observation model to obtain more accurate estimation results [
16].
In the case of SAR with sparse sampling, when the sampling rate of echo data is low, sparse SAR imaging based on L
1 regularization has a slow iteration speed and is prone to reconstruction failure [
17]. We constructed a new sparse reconstruction model by combining the approximate observation operator [
18,
19,
20] and a new algorithm that can quickly solve the L
1/2 regularization problem [
21]. The model replaces the observation matrix with an approximate observation operator, which reduces memory usage and accelerates reconstruction. Point target simulation experiments were performed under the conditions of full sampling and undersampling, the results of which show that the proposed sparse reconstruction model shows a certain improvement in image quality compared with the traditional
ω-
k imaging and L
1 threshold iteration algorithms. Moreover, the effectiveness of the proposed sparse reconstruction model was demonstrated by processing spaceborne SAR-measured data.
2. A Sparse Reconstruction Model Based on Approximate Observation
This article mainly studies strip SAR imaging with a side-looking array on airborne or spaceborne data. The range resolution of SAR is achieved by transmitting linear frequency-modulated (LFM) signals, while the azimuth resolution is achieved using the synthetic-aperture principle and coherent-stacking imaging method.
The process of obtaining the scene scattering coefficient using a compressed sensing SAR imaging system can be expressed by a linear time-invariant system, shown as:
where
y ∈
CM×1 is the echo data vector,
x ∈
CN×1 is the scene scattering coefficient,
n ∈
CM×1 is a Gaussian white noise vector,
Θ =
HΨ ∈
CM×N is the observation matrix,
H ∈
CM×M is the sparse microwave imaging undersampling matrix, and
Ψ ∈
CM×N is the measurement matrix.
M is the number of sampling points of echo data, which is determined using the radar resolution theory and Nyquist sampling theorem.
N is the number of sampling points in the scene, which is determined by the size of the observation scene.
Formula (1) can be solved via L
1 norm regularization, expressed as:
where
λ > 0 is the regularization parameter, and ‖·‖
2 and ‖·‖
1 represent the L
2 norm and L
1 norm, respectively.
When the SAR echo data have a phase error, Formula (2) is rewritten as [
9]:
where
Φ is the phase error matrix, i.e.:
When using a sparse reconstruction algorithm for SAR imaging, it is necessary to change the echo data from a two-dimensional matrix to a one-dimensional vector. If the echo data have more sampling values, or the spatial scale of the imaging scene is large, the data consume a large amount of memory, and imaging requires much time. Hence, imaging based on sparse reconstruction has high computational complexity and poor real-time performance, which is not suitable for the real-time imaging of large scenes. Given the above problems, a sparse imaging method based on approximate observation is proposed. The approximate observation operator was used instead of matrix multiplication to construct a new sparse autofocus model, which greatly reduced the dimensions of the observation matrix and improved the reconstruction efficiency and imaging quality.
In this paper, we took the wavenumber domain (
ω-
k) imaging algorithm as an example to construct the approximate observation operator [
22]. This algorithm uses the reference function of the azimuth–range two-dimensional frequency domain to realize two-dimensional pulse compression and uniform range migration correction, and uses Stolt interpolation to realize the focusing of other targets.
The imaging operator based on the
ω-
k algorithm is expressed as:
where
Y represents the original echo data of two-dimensional SAR,
Fa and
Fr are the respective azimuth and range Fourier transforms,
Fa−1 and
Fr−1 are the respective azimuth and range Fourier inverse transforms,
θref is the uniform range migration correction matrix, and
S(·) is the Stolt interpolation operator.
The expression of
θref is:
where
ft and
fτ are the respective azimuth and range frequency,
Rref is the reference slant range,
fc is the signal carrier frequency,
vrc is the radar effective speed,
Kr is the range modulation frequency, and
c is the speed of light.
The expression of
S(·) is [
23]:
The approximate observation operator is obtained by inverting the imaging operator, shown as:
where
X is the two-dimensional focused SAR image of the observed scene,
S−1(·) is the inverse mapping of
S(·), and (·)* is the conjugate of a matrix.
By introducing the approximate observation operator into formula (3), a sparse autofocus model based on approximate observation can be obtained as:
where
L is the undersampling matrix of the echo.
To introduce the traditional matched filtering algorithm into the above model, the imaging operator
Iω-k(·) is decomposed into the phase correction operator
Ω(·) and azimuth Fourier transform operator
Fa(·), which are expressed as [
12]:
The
Γ(·) is the imaging operator, which can be expressed as:
and we rewrite Formula (9) as:
where
Ω−1(·) denotes the inverse-phase correction operator.
4. Sparse Reconstruction Algorithm Based on Approximate Observation and L1/2 Threshold Iteration
We constructed a new sparse reconstruction model by combining the threshold iteration algorithm, which can quickly solve the L
1/2 regularization problem, with the approximate observation operator based on the
ω-
k algorithm. The model is solved via a two-step iteration, including the reconstruction of
x and
Φ. The pseudocode of the solution process is given as follows Algorithm 1.
Algorithm 1 Sparse Reconstruction Algorithm Based on Approximate Observation and L1/2 Threshold Iteration |
Initialization: i = 0, Imax is the maximum number of iterations, R0 = 0, t0 = 1, Φ0 = PGA(Γ(L⊙y)). |
Iterative process: for i = 1: Imax. |
|
|
|
|
|
The algorithm stops when ‖xi+1-xi‖2/‖xi‖2 < ε. |
In Algorithm 1, the appropriate
Φ0 can reduce the number of iterations, so the phase gradient autofocus (PGA) algorithm [
27] is used to provide
Φ0.
μ usually takes a value of 1 to ensure convergence. The parameter
λ is related to the sparsity
K of the imaging scene, which is expressed as:
where |
Ri|
K+1 is the amplitude value of (
K + 1) in the order of magnitude of all elements in the matrix
Ri.
Therefore, after obtaining the sparsely sampled data and the imaging operator based on the matched filtering algorithm, sparse SAR imaging can be realized using the above algorithm.