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# Compressive sensing imaging for general synthetic aperture radar echo model based on Maxwell’s equations

- Bing Sun
^{1, 2}, - Yufeng Cao
^{2, 3}, - Jie Chen
^{1}Email author, - Chunsheng Li
^{1}and - Zhijun Qiao
^{2}

**2014**:153

https://doi.org/10.1186/1687-6180-2014-153

© Sun et al.; licensee Springer. 2014

**Received:**27 February 2014**Accepted:**3 October 2014**Published:**10 October 2014

## Abstract

A general echo model is derived for the synthetic aperture radar (SAR) imaging with high resolution based on the scalar form of Maxwell’s equations. After analyzing the relationship between the general echo model in frequency domain and the existing model in time domain, a compressive sensing (CS) matrix is constructed from random partial Fourier matrices for processing the range CS SAR imaging. Simulations validate the orthogonality of the proposed CS matrix and the feasibility of CS SAR imaging based on the general echo model.

## Keywords

- Compressive sensing
- General echo model
- Maxwell’s equations
- Synthetic aperture radar

## 1 Introduction

Synthetic aperture radar (SAR) has been in development for more than 60 years. Many operations, including strip-map, spotlight, scan, and multiple platform-borne SAR systems have become more and more popular in recent years. SAR systems have been used in many fields, such as soil moisture, forestry, wetland, and agriculture. Due to the higher resolution of SAR image required, the accuracies of the echo models and imaging algorithms need improvement. Because the echo model is a kind of output, it forms a theoretical basis for all SAR imaging algorithms. From an engineering point of view, the traditional echo model is the time-delayed signal of the transmitted signal. There are lots of approximations for echo model. Let us recall the echo signal model. The SAR echo is one of electromagnetic wave forms, and Maxwell’s equations are the basic and accurate tools for electromagnetic wave measurement. Mathematically, the SAR imaging procedure is an inverse problem of the electromagnetic wave. Many mathematical and practical researchers are interested in these types of inverse problems [1–5].

In recent studies of SAR and inverse synthetic aperture radar (ISAR), the data size trends larger and larger as the need for high resolution images becomes greater and greater. It costs too much and many times, the data cannot be downloaded in real time from some space-borne platforms, such as satellites. How to disperse the data size efficiently and decrease the data ratio is a real problem for engineers. The good news is that compressive sensing (CS) theory addresses at least some of these problems [6]. Random sampling theory, the CS sampling, and construction proposed by Donoho et al. in 2006 [7–9] and Baraniuk in 2007 [10] may also help in this endeavor. Many publications in the literature cast the CS into radar imaging [11–16], where they analyzed the sparse characteristics of SAR signals in different domains. Those are the fundamentals of reconstructing the scenes from the equivalent down-sampling data set. Another application of the CS SAR imaging is to construct the CS matrix and its optimization [17]. However, most of the past research works are focused on the classical echo model for SAR [18] and ISAR image [19].

In this paper, our goal is to analyze a general echo model and construct a new CS matrix for the SAR imaging. We use the partial differential Maxwell’s equations to derive the electronic field and then obtain the general echo model in frequency domain and time domain. Based on this general model, considering the sparse characteristic of the scene, we will construct the corresponding orthogonal CS matrix for the SAR imaging.

Our paper is organized as follows. Section 2 derives the general SAR echo model based on Maxwell’s equations. In Section 3, a new CS matrix is constructed for the CS imaging based on the above general echo model in frequency domain. In Section 4, simulations are given to validate the orthogonality of the CS matrix and the imaging performance. Some comparisons of two CS methods and the evaluation method with indices are also provided. Section 5 concludes the paper.

## 2 General SAR echo model

From the scalar form of Maxwell’s equations, the incident and scattering fields are analyzed first, and then the general SAR echo model [20] is derived according to the antenna theory.

### 2.1 Maxwell’s equations and field expressions

**x**is the three-dimensional position vector,

*c*(

**x**) is the local propagation speed of electromagnetic waves and

*c*(

**x**) =

*c*

_{0}in free space (usually,

*c*

_{0}is the speed of light),

*ε*

^{tot}(

*t*,

**x**) and

*j*(

*t*,

**x**) is the total scalar field and the current density on the antenna, respectively.

*c*(

**x**) satisfies ${c}^{-2}\left(\mathbf{x}\right)={c}_{0}^{-2}-V\left(\mathbf{x}\right)$, where

*V*(

**x**) stands for the target reflectivity function, which will be reconstructed from radar echoes.

*ε*

^{tot}(

*t*,

**x**) =

*ε*

^{in}(

*t*,

**x**) +

*ε*

^{sc}(

*t*,

**x**), where

*ε*

^{in}(

*t*,

**x**) and

*ε*

^{sc}(

*t*,

**x**) are the incident scalar field and the scattered scalar field, respectively. And

*ε*

^{in}(

*t*,

**x**) satisfies

where $g\left(t,\mathbf{x}\right)=\frac{\delta \left(t-\left|\mathbf{x}\right|/{c}_{0}\right)}{4\pi \left|\mathbf{x}\right|}$, called Green’s function [22], is the fundamental solution of the partial differential equation $\left({\nabla}^{2}-{c}_{0}^{-2}{\partial}_{t}^{2}\right)g\left(t,\mathbf{x}\right)=-\delta \left(t\right)\delta \left(\mathbf{x}\right)$.

*ω*instead of time domain

*t*.

*E*

^{sc}(

*ω*,

**x**) and

*E*

^{in}(

*ω*,

**x**) are the Fourier transforms of

*ε*

^{in}(

*t*,

**x**) and

*ε*

^{in}(

*t*,

**x**), respectively; $G\left(\omega ,\mathbf{x}\right)=\frac{{e}^{-\mathit{\text{ik}}\left|\mathbf{x}\right|}}{4\pi \left|\mathbf{x}\right|}$ is the frequency expression of Green’s function

*g*(

*t*,

**x**); $k=\frac{\omega}{c}$ is the wavenumber in range direction; and

*J*(

*ω*,

**x**) is the current source in the frequency domain. Then, the scattered field with a theoretical point antenna is

### 2.2 Mathematical signal model

#### 2.2.1 Radiation pattern for a SAR antenna

*a*,

*a*]×[-

*b*,

*b*] is analyzed and the current density

*I*is constant; then, the radiation scalar

*F*(

*k*,

**x**) can be expressed by

where $\text{sinc}\left(x\right)=\frac{\text{sin}\left(x\right)}{x}$, $\xea=\left({\xea}_{1},{\xea}_{2}\right)$ is corresponding to the antenna direction.

*p*(

*t*) be the transmitted signal; then, the current density on antenna

*j*(

*t*,

**x**) is proportional to

*p*(

*t*) and independent of position. So,

*J*(

*ω*,

**x**) is proportional to the spectrum of transmitted signal

*P*(

*ω*). Then,

where ${G}_{a}\left(k,\widehat{\mathbf{x}},\xea\right)$ is just an amplitude function independent of the transmitted signal, with a function of wavenumber *k*, which varies for a wideband signal even within a short pulse time duration. In fact, this corresponds to the frequency characteristics of antenna, especially under wide bandwidth.

#### 2.2.2 Mathematical model of received echo

*x*

_{0}, the incident field

*E*

^{in}(

*ω*,

**x**) and scattered field ${E}_{B}^{\text{sc}}\left(\omega ,\mathbf{x}\right)$ are

*ω*,

**y**) is the weight function of the antenna cell at the position

**y**. Substitute Equation 12 into Equation 13, we have

**y**-

**z**|

^{-1}≈|

**z**-

**x**

_{0}|

^{-1}are adopted to produce the following form

*W*(

*ω*,

**y**) = 1 yields

*p*(

*t*), ${g}_{a}\left(t,\hat{\mathbf{z}-{x}_{0}},\xea\right)$ is the inverse Fourier transform of ${G}_{a}^{2}\left(k,\hat{\mathbf{z}-{x}_{0}},\xea\right)$, and ⊗ is convolution on

*t*. If we neglect the antenna’s variety with frequency

*ω*or wavenumber

*k*, that’s ${G}_{a}\left(k,\widehat{\mathbf{x}},\xea\right)\approx {G}_{a}\left({k}_{0},\widehat{\mathbf{x}},\xea\right)$, where ${k}_{0}=\frac{{\omega}_{0}}{c}$ is the wavenumber corresponding to the carrier frequency, the above expression can be simplified by

Even if some approximation about antenna radiation has been made, we can also find the difference between the above model and the transitional echo model, which is as simple as the summing the delayed signal of the transmitted signal. That is, the general echo is not the direct delay of the transmitted signal but the second order differential function of the transmitted signal. However, because Doppler phase is much more important in SAR imaging than the complex amplitude, the amplitude modulation is not serious for traditional SAR systems. In [20], the relation and difference between the general echo model and the classical model were minutely analyzed. However, for some high resolution applications, the differences may not be ignored. In this paper, to avoid error as much as possible, we try to find a CS matrix for the general echo model in frequency domain directly.

## 3 Compressive sensing imaging for SAR

### 3.1 Basics of compressive sensing

*Ψ*∈

**C**

^{N × N}(in fact, the matrix was defined on

**R**

^{N × N}originally and could be expanded to the complex matrix) is the orthogonal basis matrix and

**s**∈

**R**

^{ N }is the coefficients vector, the signal

**x**∈

**R**

^{ N }can be expressed by

**s**can be calculated by

**s**=

*Ψ*

^{-1}

**x**mathematically. If there are only

*K*(≪

*N*) non-zero values (or small absolution) in

**s**, the signal

**x**is sparse in the corresponding domain and can be reconstructed by a few random samples with very high probability. Suppose the linear observing process is

*Φ*∈

**R**

^{M × N}, where

*M*<

*N*, the observation data

**y**∈

**R**

^{ M }is

*Θ*=

*Φ*

*Ψ*∈

**C**

^{M × N}. Based on CS theory, the reconstruction of the sparse coefficient

**s**can be resolved by the following optimization problem.

Because *l*_{0} normalization optimization problem is difficult to resolve, *l*_{0} normalization is replaced by *l*_{1} normalization for the actual solution. Then, the signal **x** is to be estimated by $\widehat{\mathbf{x}}=\Psi \widehat{\mathbf{s}}$.

### 3.2 Compressive sensing matrix for SAR imaging

where $\text{chirp}\left(i\right)=\text{rect}\left(\frac{2i-{N}_{\tau}}{2{N}_{\tau}}\right)\text{exp}\left\{\mathrm{j\pi b}{\left(\frac{i-0.5\ast {N}_{\tau}}{\mathit{\text{Fs}}}\right)}^{2}\right\}$ is the transmitted linear frequency modulation (LFM) chirp signal, *b* is the frequency modulation ratio of the chirp signal, *f*_{
s
} is the sampling frequency, and *τ* and *N*_{
τ
} = *τ* *f*_{
s
} are the pulse width and point numbers of the chirp signal, respectively.

This CS matrix is based on the idea that the echo signal is the delay of the transmitted signal, and its orthogonality also was validated. However, according to Section 2 of this paper, we have found that the general echo model is the expansion of the simple delay model of the transmitted signal, and it is more accurate for some high resolution applications. Our goal is to find a CS matrix corresponding to the general echo model directly.

*m*represents the frequency number for fast time of echo,

*n*represents the target index,

*N*

_{ p }is the number of the discrete targets,

*R*

_{ n }is the range between the

*n*th target point located at

**z**

_{ n }and the center of the antenna located at

**x**

_{0},

*ω*

_{ m }= 2

*π*

*f*

_{ s }(

*m*/

*N*-1/2) is the discrete frequency,

*m*= 0,1,…,

*N*-1, and

*N*is the number of Fourier transform. Suppose the scatter coefficients vector $\mathbf{\Sigma}={\left[{\sigma}_{1},{\sigma}_{2},\dots ,{\sigma}_{{N}_{p}}\right]}^{T}$, the (

*m*,

*n*)-th element of the CS matrix

*Ψ*can be constructed as

*D*

_{ rec }(

*x*

_{0}) =

*Ψ*

**Σ**. Because we use the general echo model in frequency domain, the CS matrix

*Ψ*is also the function of the general transmitted signal in frequency domain

*P*(

*ω*

_{ m }). For instance, if the transmitted signal is a chirp signal, according to the principle of stationary phase, we can get $P\left({\omega}_{m}\right)=A{e}^{-i\left(\frac{{\omega}_{n}^{2}}{4\mathrm{\pi b}}+\frac{\pi}{4}\right)}$, where

*A*is a complex constant, and then the CS matrix

*Ψ*becomes

It is not hard to find that $\left|\text{Cor}\left({\Psi}_{{m}_{1}n},{\Psi}_{{m}_{1}n}\right)\right|\gg \left|\text{Cor}\right({\Psi}_{{m}_{1}n},{\Psi}_{{m}_{2}n}\left)\right|,({m}_{1}\ne {m}_{2})$ because of the cophasal stacking effect. And after normalization for each row, the correlation matrix is approximately an identical matrix. The orthogonality will be validated in the next section. On the other hand, besides Gaussian and Bernoulli matrices, another very important class of structured random matrices is the random partial Fourier matrix, which is also the object of study in the very first paper on CS [9]. In fact, a random partial Fourier matrix relates the time domain signal and the sparse spectrum items; also, it is the first time to construct an orthonormal basis in **C**^{N × N} rather than **R**^{N × N}. It has been proved that the Fourier matrix satisfied the restricted isometry property (RIP) and can be applied to CS reconstruction. The proposed matrix defined by Equations 24 or 25 can be thought as one form of the Fourier matrix and can be applied for CS imaging for the general SAR echo model. Randomly selected M rows from *Ψ* will generate a random partial Fourier matrix *Θ*; the range signal reconstruction from the SAR echoes can be accomplished in range frequency domain instead of the traditional match filtering.

In this paper, the orthogonal match pursuit (OMP) algorithm [25] is unitized for the reconstruction of the sparse signal, which corresponds to the discrete scattering coefficients in SAR imaging. The orthogonality of the CS matrix makes sure of the maximum probability of reconstruction quickly.

## 4 Simulations

In this section, the properties of the CS matrix *Ψ* are analyzed first, then the CS imaging for a point scene is simulated with both the previous time-domain CS matrix like Equation 22 and the proposed matrix corresponding to the general echo model defined by Equation 24. The comparison of the range reconstruction results after CS and the final imaging results are given.

### 4.1 Simulation parameters

**Main simulation parameters**

Parameter | Value |
---|---|

Height of antenna | 5000 m |

Velocity of antenna | 200 m/s |

Look angle | 45° |

Frequency of carrier | 1 GHz |

Bandwidth of chirp signal | 400 MHz |

Sampling frequency | 500 MHz |

Pulse width of chirp signal | 1 µs |

Pulse repetition frequency | 500 Hz |

Number of range cells | 1,024 |

Number of down-sampled range cells | 256 |

Synthetic aperture time | 5 s |

### 4.2 Results and discussion

#### 4.2.1 Compressive sensing matrix properties

*Ψ*by numerical simulation.

According to Figure 1, the diagonal elements of the correlation matrix trend to 1, and others are very small. This correlation matrix is approximate to the identical matrix, so the CS matrix *Ψ* can be thought as orthogonal matrix.

#### 4.2.2 Imaging results and evaluation

Before analyzing the simulation results, we discuss the evaluation method first. Fourier interpolation is often applied into the evaluation of the traditional range compressed result or SAR image. However, in the previous CS reconstruction, it is acceptable that the range reconstruction result has no sidelobe [18, 26, 27], and Fourier interpolation is not suitable to evaluate the result. This phenomenon might be explained that when *K* is 1 or a very small number, after reconstruction of compressive sensing, there is at most only 1 non-zero value in each range profile, like a delta function, and any interpolation does not fit during evaluation. According to our simulation during range CS reconstruction, we still set different sparse coefficients *K* for a single target, and a bigger *K* will expose the more sidelobes.

*K*without interpolation. The bigger

*K*is, the more obvious the sidelobes are. For example, when

*K*= 1, there is only 1 non-zero point, which likes a delta function and when

*K*= 15, there are at least two pairs of sidelobes nearby the peak. Also Fourier interpolation can be carried out for this construction with some sidelobes. Meanwhile, we set

*K*= 11 when we apply the previous CS matrix to the same echoes.To quantitatively analyze the effect of the CS imaging, Figures 3 and 4 show the corresponding results of range reconstruction and final imaging with two kinds of CS matrices.According to the above results, it is easy to find that there are a few sidelobes in both range reconstructions in Figure 3b and Figure 4b. The range reconstruction results based on our proposed method shown in Figure 3 are better than those based on the previous matrix shown in Figure 4. Also, it is easy to find the symmetry of the range result after CS, and the final result by our matrix is better.

**Point evaluation indices of two CS methods**

Index | Slant range | Azimuth | ||
---|---|---|---|---|

Resolution (m) | PSLR (dB) | Resolution (m) | PSLR (dB) | |

Results based on the previous matrix | 0.2860 | -12.60 | 1.0094 | -12.68 |

Results based on our proposed matrix | 0.3277 | -13.39 | 0.9867 | -13.79 |

Theoretical values | 0.3322 | -13.26 | 0.9397 | -13.26 |

## 5 Conclusions

In this paper, the general echo model is derived from Maxwell’s equations. The general echo expressions in both frequency domain and time domain are given after generating the scatter field. The general echo model is the expansion of the classical echo model. Based on the general echo model in the frequency domain, a new CS matrix like a random partial Fourier matrix is constructed to apply for the CS imaging. Simulation results validate the orthogonality of the proposed CS matrix and the indices of the CS imaging by our model approach the theoretical values better. Also a bigger sparse number *K* will expose the sidelobes of the reconstruction, and Fourier interpolation can be applied into evaluating the imaging results.

## Declarations

### Acknowledgements

This work is supported by the National Natural Science Fund of China (Grant Nos. 61301187 and 61328103), and the Fundamental Research Funds for the Central Universities. ZQ thanks the U.S. Department of Education GAANN project (P200A120256) for supporting the UTPA mathematics graduate program. The authors also thank Mr. John Montalbo for his communication and the anonymous reviewers for their valuable suggestions.

## Authors’ Affiliations

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