SAR imaging from randomly sampled phase history

using Compressive Sensing

Amit Kumar Mishra

Department of Electrical Engineering

University of Cape Town

South Africa

Email: akmishra@ieee.org

Rohan Phogat and Shikhar Mann

Department of Electronics and Communication Engineering

Indian Institute of Technology

Guwahati, India 781039

Email: s.mann@iitg.ernet.in

Abstract—Reconstructing synthetic aperture Radar (SAR) im-ages from gapped phase history or k space data, is a majorproblem for SAR engineers. In this work we use the newlyproposed compressive sensing (CS) algorithms to form SARimages of randomly and sparsely sampled k space data. We alsoinvestigate the effect of adding phase noise of various degrees ofseverity in the sparse and random k space data. We show that CSbased algorithms can intelligibly reconstruct SAR images fromrandomly sparse phase history data and can tolerate a goodamount of phase noise corruption. Dantzig selector based CSalgorithm was found to perform better than the usual l1 normbased CS algorithm.

I. INTRODUCTION

Synthetic aperture radar (SAR) imaging is one of the

major use of a modern radar system. However, the image

generated by SAR is sensitive to many artifacts during the

data capturing phase. One of the major challenges in SAR

imaging is the lack of exact information about the position of

each data point. This arises mainly because of uncertainties

in the radar platform trajectory. The problem of SAR image

recovery from gapped phase history has been studied for a

while and many algorithms have been suggested [1]–[4]. Most

of these consider regularly gapped phase history [3], [5]. The

algorithms used in these work range from the application of

iterative Fourier transform to sophisticated optimization based

algorithms.

In the current work, we propose to use the recently proposed

compressive sensing (CS) algorithms [6], [7] to form the image

out of a subset of randomly gapped phase history. In one of our

previous works [8] we have used Dantzig filter based CS to

show that using CS not only is capable of forming SAR images

out of severely under-sampled phase history data, the resulting

image also contains less speckle. In the current work, we show

that using CS based approach we can also form intelligible

SAR images out of randomly gapped phase history. In a second

part of the work we show that the proposed approach also can

tolerate phase noise to a large extent. Hence the CS based

algorithm carries with it four advantages. First of all it does

not require the exact position of the data points in the k-space

or phase history. Secondly, it can frame good images out of

sparse data and thereby reducing the load on the sampling

system. Thirdly, it is reasonably tolerant to phase noise. Lastly,

using Dantzig based algorithm, CS imaging also gets rid of

much of the sidelobe and speckle in the SAR image.

Rest of the paper is organized as follows. Section II gives a

short review of compressive sensing using Dantzig filtering

algorithm. Section III describes the experimental setup. In

Section IV we present the results and the paper is concluded

in the last section.

II. COMPRESSIVE SENSING BACKGROUND

Compressive sensing contradicts conventional sampling

techniques which employ Shannon-Nyquist theorem by ac-

quiring and reconstructing subsampled signals that are sparse

in a certain domain. CS is governed by two basic principles,

viz. sparsity and incoherence. Most real-life signals have a

very concise representation when expressed in an appropriate

domain. A signal vector is said to be S-sparse in a domain if

at most S of its coefficients are non zero in that domain. The

signal is sensed in a domain that has high incoherency with

the representation basis so that we do not miss the significant

coefficients which will have a higher spread in the incoherent

sensing basis.

For an N dimensional scene vector ‘s’, we randomly take

any M of its measurements by applying φm, m = 1, 2...M to

it, where Φ = [φ1, φ2, ...φM ]T is the M ×N sensing matrix.

The sensing process can be represented by the matrix equation:

y = Φs, where s can be written in its expanded form as:

s = Σni=0

xiψi. The measurement process can be expressed by

the following expression.

y = Ax with A = Φψ (1)

In applying CS, the above equation forms an under-

determined system of linear equations which can be solved us-

ing various optimization techniques, given the apriori knowl-

edge about sparsity of the signal to be recovered. A popular

optimization technique is by using l1 norm, where l1 norm

implies sum of absolute values of x′n. It requires solving of

the following problem:

x = argmin{||x′||1 : x′ǫ ℜn, y = Ax′}, (2)

where kS incoherent measurements suffice to give efficient

results for S sparse signal. Here S can be of the order of

magnitude less than N and k is a constant.

IRS 2012, 19th International Radar Symposium, May 23-25, Warsaw, Poland

978-1-4577-1837-3/12/$26.00 ©2012 IEEE 221

For noisy data, matrix equation (1) is modified as:

y = Ax+ z, (3)

where z is stochastic or deterministic unknown error term. An

efficient approach for recovering signal from noisy data is that

by using Dantzig selector [9]. In this approach, we solve for

following program.

min||x||1 subjectto ||A∗(Ax− b)||∞ ≤ γ, (4)

where γ is a user defined parameter and A∗(Ax − b) is the

measurement of the correlation between the residual and each

of the columns of A. This approach requires that the residual

Ax− b of a candidate vector x to be almost orthogonal with

any of the columns of A.

III. EXPERIMENTAL SETUP

We have used two types of data to show the efficiency of

CS for gapped phase-history SAR. We use a simulated scene

(as shown in Figure 1 (d)). We also validate the process using

real phase history of a millitary target obtained from the GTRI

data base [10], [11]. The SAR image of the target is shown

in Figure 1 (c).

We will now reframe the problem of radar imaging into a

form which enables us to apply CS algorithms. Let F denote

the n × n 2D matrix representing the phase history data of

the scene detected by the radar. This data will have a total of

N = n× n points. The matrix F can be related to the scene

matrix S by the following equation:

F = AS, (5)

where A is n×n inverse Fourier matrix, which in vector form

can also be written as

f = As, (6)

where f is the signal vector and s is our scene vector, both

being N dimensional and A is a N × N matrix. For most

radar imagery, the scene vector is sparse as there are very few

scattering targets in the view, thereby allowing us to implement

CS technique to it. The sparse sampling can be obtained by

selecting fewer frequencies and pulse emission times for a

given aperture angle φm. It can be realized by taking only M

out of N points in the (kx, ky) domain using the grid generated

via random sampling along horizontal and vertical lines in

k space as shown in figure 1(a) or along randomly sampled

points in k space as shown in figure 1(b). On implementing

this sampling mechanism Eq.(6) becomes:

f = As (7)

which is similar to Eq.(1) and hence can be solved by using

optimization techniques discussed in section II.

Two sets of experiments were carried out. In the first set, CS

and Dantzig based CS were applied to randomly sparse phase

history to form SAR images. Two ways were examined to

make the phase history sparse. Firstly random lines (which

correspond to frequency and angle domains in the phase

history) were removed from the phase history. This type of

sparsification is shown in Figure 1 (a). Secondly points from

the phase history were eliminated at random. This type of

sparsification is shown in Figure 1 (b).

In the second set of experiments, phase noise was also added

to the gapped phase history and the effect of this was studied

on the CS made images.

IV. RESULTS AND DISCUSSIONS

In this section we present the results from the simulation

exercises. The sparsity of the phase history for all the exper-

iments was kept around 45% of the actual amount of phase

history.

A. Reconstruction of Real SAR image

In this subsection we discuss the simulations using the real

data of a T-72 tank, containing high levels of speckle noise

as shown in figure 1(c). Here we analyze the results using

aforementioned sampling mechanism. Due to noise, the given

image is not strictly sparse. As a result the reconstruction using

l1-norm minimization (figure 2(a) and 2(b)) did succeed in

exactly preserving the main targets but could not eliminate

the noise in the scene. However in the reconstruction using

the Dantzig selector (figure 2(c) and 2(d)), not only a good

reconstruction of the main target was observed, but also the

speckle noise present in the background is removed very

efficiently.

B. Reconstruction of Synthetic SAR images

In this subsection we discuss the simulations using the

synthesized SAR scene as shown in figure 1(d). Here due to

absence of speckle noise, the difference in reconstructions of

main targets using l1-norm minimization and Dantzig selector

along randomly sampled grid lines and points become more

conspicuous. The reconstruction using l1-norm minimization

(figure 3(a) and 3(b)) gave exact reconstruction of the main

targets but failed to suppress the side lobes. However in the

reconstruction using the Dantzig selector (figure 3(c) and 3(d)),

an exact reconstruction of the main target along with side lobe

suppression is achieved.

C. Effect of phase noise

In this subsection we show the effect of adding phase noise

to the gapped phase history, on the CS based SAR imaging

scheme. Figures 4 and 5 show the results of CS-based SAR

image formation from gapped phase history with phase noise.

Fugure 4 shows the results when we used l1 norm based

algorithm, while Figure 5 shows the results when we used

Dantzig selector based method. For both the cases we show the

results for cases with various degrees of phase noise. As can

be observed, CS based schemes can still retrieve intelligible

image with phase noise as large as π. Secondly, like cases

discussed in the last subsection, Dantzig selector based scheme

forms better images.

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(a) (b) (c) (d)

Fig. 1. (a) Sampling grid lines (b) Sampling grid points (c) Original SAR image of a tank (d) Original synthesized image of four scattering centers

(a) (b) (c) (d)

Fig. 2. SAR image of the tank reconstructed using l1-norm minimization from data sampled along Grid lines (a) and Grid points (b); and using Dantzigselector for data sampled along Grid lines (c) and Grid points (d).

(a) (b) (c) (d)

Fig. 3. SAR image of the simulated scene reconstructed using l1-norm minimization from data sampled along Grid lines (a) and Grid points (b); and usingDantzig selector for data sampled along Grid lines (c) and Grid points (d).

(a) (b) (c)

Fig. 4. SAR image of the tank reconstructed using l1-norm minimization from gapped phase history with maximum phase noise of π (a), π/2, and π/4.

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(a) (b) (c)

Fig. 5. SAR image of the tank reconstructed using Dantzig selector based minimization from gapped phase history with maximum phase noise of π (a),π/2, and π/4.

V. CONCLUSION

This work describes CS based approach to reconstructing

SAR image of man made targets from distorted and randomly

sampled k-space or phase history data. Results are shown us-

ing simulated as well as real data. We show that for man made

targets, CS based algorithms can reconstruct SAR images from

highly sparse k space data without any knowledge of the

position of the missing data points. Secondly, the algorithms

can provide decent results even when the sparse k space data

has been severely affected with phase noise.

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