ML-based Approaches for Joint SAR Imaging and Phase Error Correction

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International Journal of Remote Sensing Applications (IJRSA) Volume 6, 2016 doi: 10.14355/ijrsa.2016.06.002

ML-based Approaches for Joint SAR Imaging and Phase Error Correction Habti Abeida Department of Electrical Engineering, University of Taif, Al-Haweiah, 21974, Saudi Arabia abeida3@yahoo.fr Abstract This paper addresses a series of iterative sparse recovery approaches with application to the synthetic aperture radar (SAR) imaging which suffers from motion-induced model errors. These types of errors result in phase errors in SAR data, which cause defocusing of the reconstructed images. The proposed phase-error correction approaches combine the maximum a posterior (MAP) algorithm and the iterative sparse maximum likelihood-based (SMLA) approaches (referred to as the PE-MAP-SMLA approaches) to solve a joint optimization problem to achieve phase errors estimation and SAR image formation simultaneously. A new PESLIM approach is also proposed that extends the idea of the classical sparse and learning via iterative minimization (SLIM) approach. A closed-form expression for the recursive estimate of the phase errors parameters is derived. A general form of each of these iterative approaches consists of three steps, the first of which is for image formation, the second is for phase errors estimation and the last is for nuisance parameters estimation. The proposed approaches can correct the phase errors accurately, and the reconstruction quality of the SAR images can be improved significantly. Finally, simulation results of 1-D spectral estimation and 2-D SAR imaging examples are generated to show the effectiveness of the proposed approaches.

Keywords Initials in Capitals; Separate with Semicolons Synthetic Aperture Radar (SAR); Phase Errors; SAR Imaging; Sparse Signal Recovery; Maximum Likelihood (ML); Maximum A Posteriori (MAP); SLIM Approach; Iterative Sparse Maximum Likelihood-Based (SMLA) Approaches

Introduction Synthetic aperture radar (SAR) has been widely used in a variety of applications such as geosciences, remote sensing and defense [1]. SAR is an active sensor and has all-weather and day/night imaging capabilities. There are four distinct modes in which a SAR imaging system can operate: scan, stripmap, spotlight, and inverse SAR (ISAR). SAR images usually suffer from one-dimensional (1-D) phase errors due to unknown system platform motion, target motion, system phase instabilities, and propagation through atmospheric turbulence (e.g., [2, 3]). These types of errors cause defocusing of the sparse reconstructed images. If the phase error is too large, the sparse reconstruction might even fail. The post-processing techniques applied to estimate and compensate the unknown phase errors are often called autofocus techniques.

Conventional techniques, including phase gradient autofocus (PGA) [3, 4], multichannel autofocus (MCA) method [8], and minimum entropy method (MEM) [5], [6], [7], have been developed to compensate for the remaining phase errors.

However, these methods commonly postprocess conventionally reconstructed defocused images (i.e., polar-format algorithm [15]) to estimate phase errors, they require the fully-sampled data and cannot provide the high-resolution target image and estimate phase error simultaneously. Recently, phase error correction has been also considered by the sparse recovery techniques including ď Ź p (0 < p ď&#x201A;Ł 1) regularization-based SAR techniques (e.g., [9-14]) which have been shown to offer certain improvements over the conventional techniques. Although these techniques vary in formulations, the main idea is to obtain sparse scattering coefficients and phase errors. However, these methods are gradient descent based methods that suffer from either slow convergence or numerical instability, especially for high dimensional applications. Thus, a lot of iterations are often necessary to yield a satisfactory result.

On the other hand, with absence of phase errors, the high-resolution spectral estimation approaches including iterative adaptive approach (IAA) [16, 17], the sparse and learning via iterative minimization (SLIM) algorithm [18] and the recent iterative sparse maximum likelihood-based approaches (SMLAs) [19, 20], have been shown recently to offer excellent performance with high-resolution and low side lobe levels for both complete and incomplete SAR data

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sets. These approaches give closed-form expressions for the recursive estimates of SAR image formation and some nuisance parameters. To alleviate the computational overhead, efficient implementations of these approaches have also been proposed in [17, 21]. However, all these methods could not be directly applied to the SAR image formation in the presence of phase errors. This paper presents a series of iterative sparse PE-MAP.SMLA approaches by combining the maximum a posterior (MAP) algorithm and the iterative sparse maximum likelihood-based (SMLA) approaches to solve a joint optimization problem to achieve SAR image formation and phase errors estimation simultaneously. The SMLAs algorithms aim to provide an estimate of the sparse signal powers and noise power, while the MAP approach provides a joint estimate of SAR image formation and phase errors parameters.

The rest of this paper is organized as follows. In Section 2, we present briefly the complex-valued SAR signal model corrupted by noise errors. In Section 3, we introduce an auxiliary cost function from which the PE-MAP-SMLA approaches are derived in detail. Section 4 presents the PE-SLIM approach. Section 5 illustrates the performance of the proposed approaches using experimental 1-D spectral estimation and 2-D SAR imaging examples. Finally, conclusions are given in Section 6.

The following notations are used throughout the paper. Matrices and vectors are represented by bold upper and lower case characters, respectively. Vectors are by default in column orientation, while superscript T , H and  stand for transpose, conjugate transpose and complex conjugate, respectively. E() , Tr (), det () and Diag () are the expectation, trace, determinant and diagonal operators, respectively. I is the identity matrix, and em denotes the mth column of the identity matrix I. y(k ) denotes the kth element of the vector y and z is the phase angle of a complex vector z.

Proplem Formulation The phase history data used in the presence of phase errors to form complex-valued SAR images can in general be expressed as (e.g., [1, 9]) y = ΦAx  e,

(1)

where y  CM is a vector of measurements after appropriate pre-processing, e is a complex-valued Gaussian noise def

K vector, x  C is the underlying scene reflectivity (the pixels in a scene form a vector), and A = [a1 ,, aK ] CM  K is def

 φ

a rank-deficient (M << K ) “sensing matrix”. The phase error matrix is denoted by Φ = Diag e

i

where

def

φ = [1,, M ]T , which is a diagonal matrix representing time variant errors. We assume that the components of the noise vector e are independent, zero-mean complex Gaussian random variables with unknown variance  . We also def

def

assume that e and x are independent, and P = Diag (p), where p = [ p1,, pK ]T is the signal power vector. The covariance matrix of y that contains information about the unknown parameters p, φ and  is given by def

R = ΦAPA H Φ H  I. To form SAR images, we need to estimate the sparse vector

x

from

y

in the presence of the nuisance parameter

def

vector θ n = (pT , φT ,  )T . Given Φ, the most straightforward method is to perform the inverse Fourier transform, which gives the conventional delay-and-sum (DAS) or matched filtering approach [28]. However, the quality of the DAS image is usually unsatisfactory due to the low resolution and high sidelobe problems. This paper presents a series of iterative sparse PE-MAP.SMLAs approaches to jointly estimate the sparse scattering image coefficient x and the vector of nuisance parameters θ n from the phase error noisy SAR data y.

PE-MAP-SMLA Approaches Note that the existing SMLA-MAP algorithms [19] assume that data contain no phase errors and have been shown to offer excellent performance with high-resolution and low side lobe levels for SAR image data sets. This section presents the PE-MAP.SMLAs approaches to solve a joint optimization problem to achieve phase errors estimation and image formation simultaneously by minimizing the following auxiliary cost function:

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 y (x, φ, p, ) =

1



|| y  ΦAx ||2 x H P 1x  log(det (R)).

(2)

A cyclic optimization technique is applied to reach the optimal value of x, φ, p and  where each iteration in the algorithm consists of essentially three steps: optimizing x with fixed φ, p and  ; optimizing φ with fixed x, p and

 ; optimizing p and  with fixed x and φ . The update formulas of PE-MAP.SMLA are as follows: (i ) Step 1: Suppose that you have xˆ (i ) , φˆ (i ) , pˆ (i ) and ˆ in the ith iteration. Minimizing (2) w.r.t. x is equivalent to

minimizing the following MAP-based cost function w.r.t. x with fixed φ, p and  :

xˆ = arg min  MAP (x) = x

1



|| y  ΦAx ||2 x H P 1x.

(3)

The solution to (3) can be obtained explicitly as 1) xˆ (Mi AP (φ, p, ) = P(i ) A H R 1(i )Φ(i ) H y. def

Where Φ(i ) = Diag (ˆ1(i ) ,

def

def

, ˆK(i ) ) and R(i ) = AP(i ) A H   (i ) I with P(i ) = Diag ( pˆ (i ) ,

(4) (i ) , pˆ M ).

The next step is to minimize (2) w.r.t. p, φ and  with fixed x . First, inserting the solution xˆ MAP (φ, p, ) into (2) and using the following key equality 1



H 1 ˆ MAP = y H ΦR 1Φ H y, || y  ΦAxˆ MAP ||2 xˆ M AP P x

(5)

the cost function (2) becomes the stochastic Maximum Likelihood cost function  SML (p, φ,  ) = y H ΦR 1ΦH y  log(det (R)).

(6)

Step 2: First, note that the second term in (6) det (R)  det(APAH  I) does not depend on φ. Hence, minimizing (6) w.r.t φ is equivalent to solve the following constant modulus quadratic program (CMQP) problem βˆ  Arg min y H ΦR 1Φ H y  β H YR 1Y H β s.t | i | 1 β

def

def

where Y = Diag( y ) and β = e

iφ

(6a)

.

Note that numerical approaches for solving CMQP problem were presented in [25] and [26]. This paper adopts a cyclic optimization approach to solve the CMQP problem. Assuming that the kth phase k is an unknown variable at the (i+1)th iteration and the phases  j , j  k are fixed. Using

Φ  eik ek ekT  Φk ,

(6b)

def

where Φk = Diag (ei1 ,...,eik 1 ,0, eik 1 ,...,eiM ) is a phase diagonal matrix with the kth diagonal element equal to zero.

Substituting (6b) into (6a), yields a cost function depends only on k as (after dropping the constant term)

y H ΦR 1Φ H y  e i y H ek ekT R 1Φ kH y  e i y H Φ k R 1ek ekT y. k

k

(6c)

Hence, the ˆk(i 1) , k  1,...,M , that minimize (6c) can be obtained recursively as

ˆk(i 1) = (y* (k )z (ki ) (k )), k = 1,, M

(6d)

where z (ki ) = R 1(i )Φ(ki ) H y . This yield ( i 1)

ˆ (i 1) = Diag(eiˆ1 Φ

ˆ ( i 1)

,, eiM ).

(7)

Step 3: As shown in [19, 20], the value of p and ˆ minimizing (6) can be obtained by the family of SMLAs consisting of four separate algorithms, termed SMLA-  , for  = 0,,3 . At the (i+1)th iteration, the signal and noise power

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estimates are computed based on the previous iterations.

SMLA-0 Approach: The estimates of { pk }kK=1 and  are updated at the (i+1)th iteration as: pˆ k(i 1) = pˆ k2(i ) (| akH R1(i ) ΦH (i ) y |2 ), k = 1,

ˆ (i 1) =

(8)

,K

y H Φ(i ) R 2(i ) Φ H (i ) y .

Tr (R 2(i ) )

(9)

SMLA-1 Approach: The estimates of { pk }kK=1 and  are updated at the (i+1)th iteration as:

pˆ k(i 1) =

| akH R 1(i ) Φ H (i ) y |2 (akH R 1(i ) ak )2

ˆ (i 1) =

, k = 1,

(10)

, K,

y H Φ(i ) R 2(i ) Φ H (i ) y .

Tr (R 2(i ) )

(11)

SMLA-2 Approach: The estimates of { pk }kK=1 and  are updated at the (i+1)th iteration as:

pˆ k(i 1) = pˆ k(i )

| akH R 1(i ) Φ H (i ) y |2 akH R 1(i ) ak

ˆ (i 1) =

, k = 1,

(12)

, K,

y H Φ(i ) R 2(i ) Φ H (i ) y .

Tr (R 2(i ) )

SMLA-3 Approach: The estimates of { pk }kK=1 and  are updated at the (i+1)th iteration as:

pˆ k(i 1) =

| akH R11(i ) Φ H (i ) y |2 (akH R 1(i ) ak )2

ˆ (i 1) = where R1(i ) = AP1(i ) A H   (i ) I and P1(i ) = Diag ( pˆ1(i ) ,

, k = 1,

(13)

, K,

y H Φ(i ) R 2(i ) Φ H (i ) y .

Tr (R 2(i ) )

, pˆ K(i ) ), with pˆ k(i ) =

1 , k = 1, akH R 1(i ) ak

, K.

Hence once p SMLA and ˆ SMLA are obtained (at convergence of a SMLA approach), and using (4) and (7), the following result is proved.

Result 1 The final MAP estimate of the unknown complex-valued SAR image formation x is given by 1 ˆ H y, xˆ MAP  xˆ MAP (pˆ SMLA , ˆSMLA )  PSMLA A H RSMLA Φ SMLA

(14)

where R SMLA = APSMLAA H  ˆ SMLAI M , with PSMLA = Diag (pˆ SMLA ) .

Note that the direct implementation of these approaches can be computationally intensive and require a large auxiliary memory space for computation. However, these issues can be alleviated by following the fast implementation steps of the SMLAs algorithms developed in [21], which exploit the structural properties of the covariance matrices and the pertinent trigonometric polynomials involved in each of the SMLA algorithms.

The implementation steps of these approaches are summarized in Table 1.

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International Journal of Remote Sensing Applications (IJRSA) Volume 6, 2016 TABLE 1. THE PE-MAP.SMAL APPROACHES

The PE-MAP.SMAL approaches Initialization: Φ(0) = I, p(0) = DFT (y) and ˆ (0) =|| y ||2 /M . repeat  Update R(i)=AP(i) AH ˆ (i)I or R1(i) = AP1(i) AH ˆ (i) I .



Update

pˆ k(i 1) using SMLAs identities (8), (10), (13), or (12).

ˆ (i 1) using (11).  Update ˆ(i 1) using (7). 

Update



Final estimate of SAR image formation x is given by (14).

Remark 1: It is important to note that the PE-SMLA approaches can be viewed as approximate solutions of the stochastic ML criterion by minimizing the cost function  SML (.) . Note also that the estimates of { pk }kK=1 can be a good tool to determine whether {| xk |}kK=1 are large or small, which is mainly required for signal detection applications.

However, the powers { pk }kK=1 do not contain any information on the phases of the complex-valued signals {xk }kK=1 needed by some practical sparse signal recovery applications. This difficulty is overcome by result 1 which gives an estimate of the complex-valued signal by combining a SMLAs approaches and the MAP approach, resulting in the PE-MAP-SMLA approaches.

PE- SLIM Algorithm In the absence of phase errors, the SLIM algorithm [18, 19] has been shown to offer excellent performance with highresolution and low side lobe levels for both complete and incomplete data sets [18, 19, 21, 22, 23, 24]. This section presents a new PE-SLIM approach that extends the main iteration steps of the classical SLIM approach. This approach solves a joint optimization problem in an analytical manner to achieve phase errors estimation and image formation simultaneously.

In the presence of phase errors, PE-SLIM approach can be derived based on the following hierarchical Bayesian model: y | x, φ,  ~Nc (ΦAx,  I), fe ( )  1 def K

f q ( x) =  e

2 q  (| x | 1) q k

(15) (16)

,

k =1

where f q (x) is a sparsity-promoting prior for 0 < q  1 . Then, the PE-SLIM estimates of x, φ and



are obtained by

minimizing the following MAP criterion: {xˆ , φˆ , ˆ } = arg min q (x, φ,  ) x , ,

q (x, φ,  )   log( ( y | x, φ,  ) f q ( x) f e ( )).

(17)

Using (15) and (16), we have q (x, φ,

def

 ) = LDML (x, φ,  )  LRqeg (x),

where LDML (x,  ) is the negative log-likelihood function for the deterministic ML approach: def

LDML (x, φ,  ) = M log( ) 

1



|| y  Ax ||2 ,

and the regularized function LRqeg (x) is defined as the negative of the function f q (x) given by def

LRq eg (x) =  log( f q (x)) =

2 K q  (| xk | 1). q k =1

ˆ (0) and ˆ (0) are given. A cyclic optimization technique is also applied to Assume that the initial estimates xˆ (0) , φ

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reach the optimal value of x, φ and  where each iteration in the algorithm consists of essentially three steps: optimizing x with fixed φ and  , optimizing φ with fixed x and  , and finally optimizing  with fixed x and

φ. The update formulas of PE-SLIM are as follows: ˆ (i ) and ˆ (i ) in the ith iteration. By replacing A by ΦA in [18, Eq. (25)], the Step 1: Suppose that you have xˆ (i ) , φ estimate of x at the (i+1) iteration can be updated as ˆ (i ) A H Φ(i ) H R 1(i ) y, xˆ (i 1) = P def

def

where R(i ) = Φ(i ) APˆ (i ) A H Φ(i ) H   (i ) I with Pˆ (i ) = (1/ p1(i ) ,

(18)

,1/ pK(i ) ) , and pk(i ) =| xˆk(i ) |q 2 .

ˆ (i ) and ˆ (i ) are available.

Step 2: This step gives an estimate of φ at the (i+1)th iteration by assuming that xˆ (i ) , φ After inserting (18) into (17), we obtain the same cost function (after neglecting constant term) given by (6a) def

q (φ) =

K

1

|| y  ΦAxˆ (i ) ||2 

 (i )

~p

(i ) k

| xˆk |2

k =1

=

1



(i )

H || y  ΦAxˆ (i ) ||2  xˆ (i ) Pˆ 1(i ) xˆ (i )

 y H ΦR 1(i )Φ H y,

and hence k k 1 that minimize q (φ) are given by (6d) where the matrix Φ can be updated at the (i+1)th iteration M

ˆ (i 1) = Diag (eiˆ as Φ

( i 1) 1

,

, eiˆ

( i 1) M

).

Step 3: By replacing A by ΦA in [18, Eq. (26)] the estimate of  at the (i+1)iteration can be updated as

ˆ (i 1) =

1 || y  Φ(i 1) Axˆ (i 1) ||2 .

M

(20)

Therefore, by substituting xˆ (i 1) into (20), we obtain y  Φ(i 1) Axˆ (i 1) = Ω(i 1) y,

where Ω(i 1)  I  (Φ(i ) AP(i ) A H Φ(i ) H )R1(i ) =  (i ) R1(i) .

Then, the estimate of the noise power can be updated in each iteration using the following equation instead of (20):

ˆ (i 1) = ˆ 2(i )

y H R 2(i ) y .

M

(21)

Numerical and Experimental Examples This section evaluates the performances of the proposed PE-MAP-SMLAs and PE-SLIM approaches using onedimensional spectral estimation and two-dimensional SAR imaging examples.

1-D Spectral Estimation Let us consider a mixture of four sinusoidal signals observed in additive zero-mean complex circular Gaussian noise.

The four sinusoids are located at frequencies 0.05, 0.065, 0.27, and 0.28 and have the same unit amplitude except that the fourth sinusoid has amplitude of 0.5. The noise variance  , the data length M, and the number of the equally spaced frequency grid points K1 are set equal to 0.01, 100 and 1000, respectively. The number of iterations1 is fixed at 15 for all iterative algorithms. Fig.1 shows the obtained performance of the proposed approaches where superimposed spectra over 100 independent experiments are shown for each approach variant. Then a 1-D 1

In the examined examples, no significant further improvement was achieved after the specified number of iterations.

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crossrange-varying random phase error, i.e., uniformly distributed in   / 2,  / 2 has been added to the data.

Fig.1 compares the performance of the proposed algorithms PE-SLIM-0 and PE- SMLAs algorithms with the conventional SLIM and SMLAs algorithms without compensation of the phase errors and with the conventional SMLAs and SLIM-0 algorithms from the data without phase errors. This figure shows also the performance of the combined PE- SMLAs algorithms and SLIM-0 (or MAP algorithm) algorithm.

Firstly, it can be seen from Fig. 1 (a), (e) and (e) that the proposed approaches PE-SLIM-0, PE-SLMA-0 and PE-SLMA3 can estimate the phase errors accurately. Fig. 1 (c), (g), (k), (n) and (s) show the results of the conventional SMLAs and SLIM-0 approaches without compensation of the phase errors. These approaches suffer from the low resolution and high sidelobe level problems due to the phase errors. In addition, SMLA-3 fails to detect the fourth frequencies.

In comparison to the conventional SMLAs and SLIM-0 approaches (Fig. 1 (d), (h), (l), (p) and (t)) from the data without phase errors, it can be seen that the proposed approaches estimate the fourth frequencies accurately as shown in Fig. 1 (b), (f), (j), (m) and (r). More precisely, PE-SLIM-0, PE-SMLA-0 and PE-SMLA-3 provide sparse spectral estimates then PE-SMLA-1 and PE-SMLA-2 which provide dense estimation results. Fig. 1 (q) and (u) show the performance of the PE-SMLA-1 and PE-SMLA-2 approaches followed by a single SLIM-0 (referred to as a MAP step) as shown in result 1 and the resulting approaches are PE-MAP-SMLA-1 and PE-MAP-SMLA-2, respectively. It can be seen from these figures the effectiveness of the MAP step by attenuating the sidelobe levels.

2-D SAR Images Of The Slicy Data In this section, we present the results of the proposed PE-MAP- SMLA algorithms in comparison to the uncompensated SMLA algorithms [19, 20]. The simulation results have been performed on two-dimensional phasehistory data of the Slicy target generated at azimuth angle 0o using XPATCH [27], a high frequency electromagnetic scattering prediction code for complex 3-D objects. A photo of the Slicy target at 30  rotation is shown in Fig.1. For all obtained images, a 40  40 center block of the phase-history data is used with K1  K 2  400 uniformly spaced 2-D frequency points. Then a 1-D crossrange-varying random phase error, i.e., uniformly distributed in   / 2,  / 2 , has been added to the data. Unless otherwise specified, the number of iterations is fixed at 20 for all iterative algorithms.

Fig. 2 shows the estimated phase errors (using Eq. (7)) and the true phase errors. It can be seen also that the proposed algorithms provide accurate estimate of the phase errors in the case of 2-D SAR data.

Fig. 3 shows the SAR images obtained by the proposed PE-MAP.SMLAs algorithms (left subfigures), the conventional imaging SMLA-MAPs algorithms without compensation of the phase error (middle subfigures) and the conventional imaging SMLA-MAPs algorithms from the data without phase error (right subfigures).

14

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

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(i)

(j)

(k)

(l)

(m)

(n)

(p)

(q)

(r)

(s)

(t)

(u)

FIG.1 ESTIMATED (DASH-DOT CURVES) AND TRUE (SOLID CURVES) PHASE ERRORS VIA THE PROPOSED ALGORITHMS: (a) PE-SLIM-0, (e) PE-SMLA-0 ALGORITHM, (i) PE-SMLA-3 ALGORITHM. SPECTRAL SIMULATION RESULTS OBTAINED VIA DIFFERENT PROPOSED AND CONVENTIONAL ALGORITHMS: (b) PROPOSED PE-SLIM-0 ALGORITHM, (c) UNCOMPENSATED CONVENTIONAL SLIM-0 ALGORITHM, (d) CONVENTIONAL SMLA-0-MAP ALGORITHM WITHOUT PHASE ERROR, (f) PROPOSED PE-SMLA-0 ALGORITHM, (g) UNCOMPENSATED CONVENTIONAL SMLA-0 ALGORITHM, (h) CONVENTIONAL SMLA-0 ALGORITHM WITHOUT PHASE ERROR, (j) PROPOSED PE-SMLA-3 ALGORITHM, (k) UNCOMPENSATED CONVENTIONAL SMLA-3 ALGORITHM, (l) CONVENTIONAL SMLA-3 ALGORITHM WITHOUT PHASE ERROR, (m) PROPOSED PE-SMLA-1 ALGORITHM, (n) UNCOMPENSATED CONVENTIONAL SMLA-1 ALGORITHM, (p) CONVENTIONAL SMLA-1 ALGORITHM WITHOUT PHASE ERROR, (q) PROPOSED PE-MAP-SMLA-1 ALGORITHM, (r) PROPOSED PE-SMLA-2 ALGORITHM, (s) UNCOMPENSATED CONVENTIONAL SMLA-2 ALGORITHM, (t) CONVENTIONAL SMLA-2 ALGORITHM WITHOUT PHASE ERROR, (u) PROPOSED PE-MAP-SMLA-2 ALGORITHM. (a)

(b)

(c)

(d)

FIG.2 ESTIMATED (DASH-DOT CURVES) AND TRUE (SOLID CURVES) PHASE ERRORS VIA THE PROPOSED ALGORITHMS: (a) PEMAP-SMLA-0, (b) PE-MAP-SMLA-1 ALGORITHM, (c) PE-MAP-SMLA-2 ALGORITHM AND (d) PE-MAP-SMLA-3 ALGORITHM.

In Fig. 3 (b), (e), (h) and (k), it is observed that the conventional uncompensated images suffer from degradation due to phase errors. In contrast, the figures in Fig. 3 (a), (d), (g) and (j) demonstrate the effectiveness of the proposed PE-MAP-SMLAs algorithms in correcting the phase errors and provide images with high resolution and suppressed sidelobes. In comparison to the reference images obtained by the conventional SMLA-MAPs

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algorithms from the data without phase error (right subfigures), PE-MAP-SMLA-3 is sparser than the other proposed PE-MAP-SMLAs algorithms and can obtain the most focused and concentrated image. (a)

(b)

(c)

(d)

(e)

(f)

(h)

(i)

(k)

(l)

(g)

(j)

FIG.3 2-D SAR IMAGES ARE OBTAINED VIA DIFFERENT ALGORITHMS: (a) PROPOSED PE-MAP-SMLA-0 ALGORITHM , (b) UNCOMPENSATED CONVENTIONAL SMLA-0-MAP ALGORITHM, (c) CONVENTIONAL SMLA-0-MAP ALGORITHM WITHOUT PHASE ERROR, (d)PROPOSED PE-MAP-SMLA-1 ALGORITHM, (e) UNCOMPENSATED CONVENTIONAL SMLA-1-MAP ALGORITHM, (f) CONVENTIONAL SMLA-1-MAP ALGORITHM WITHOUT PHASE ERROR, (g) PROPOSED PE-MAP-SMLA-3 ALGORITHM, (h) UNCOMPENSATED CONVENTIONAL SMLA-3-MAP ALGORITHM, (i) CONVENTIONAL SMLA-3-MAP ALGORITHM WITHOUT PHASE ERROR, (j) PROPOSED PE-MAP-SMLA-2 ALGORITHM, (k) UNCOMPENSATED CONVENTIONAL SMLA-2-MAP ALGORITHM, (l) CONVENTIONAL SMLA-2-MAP ALGORITHM WITHOUT PHASE ERROR.

Conclusions We have presented a series of iterative sparse recovery PE-MAP-SMLAs approaches to combine SMLA variants with the MAP approach for joint SAR imaging and phase error correction. A novel PE-SLIM approach is also presented as an extension of the well-known SLIM approach for phase error correction. Each approach is cycles

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through iterative steps of image reconstruction and phase error estimation and nuisance parameters estimation. In the step of image reconstruction, a MAP reconstruction algorithm is used. A closed-form expression is derived in the phase error estimation step. In the nuisance parameters (i.e., noise and signal power parameters) estimation step, the conventional SMLA variants are used. The proposed approaches can correct the phase errors accurately and provide enhanced image resolution and suppressed sidelobe levels. Among all the competing approaches, the simulation results of 1-D spectral estimation and 2-D SAR imaging using Slicy data show that the proposed PEMAP-SMLA-3 approach offers superior performance in terms of spectral estimation, SAR imaging quality and robustness against the presence of phase errors.

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Habti Abeida was born in Settat, Morocco, on October 20, 1977. He received the B.Sc. degree in engineering applied mathematics from Hassan II University, Casablanca, Morocco, and Descartes University, Paris, France, in 2000 and 2001, respectively, and the M.S. degree in statistics and signal processing from Pierre and Marie Curie University, Paris, France, in 2002. He received the Ph.D. degree in signal processing and applied mathematics from Pierre and Marie Curie University, Paris, France, in collaboration with Telecom SudParis, France, in 2006. He was a Lecturer at the Paris-Dauphine and at the Vincennes-Saint-Denis Paris 8 University, in 2006 and 2007, respectively. He was a Postdoctoral Research Associate at the Superior school of electricity, Supelec, Paris, France, at the GIPAS-lab, Grenoble, France, at the Electrical Engineering Department, KFUPM, Saudi Arabia, and at the spectral analysis laboratory, University of Florida, United States, in 2008, 2009, 2010, and 2011, respectively. He is currently working as an assistant professor at the Electrical Engineering Department, Taif University, Saudi Arabia. His research interests are in the areas of statistical signal processing with application to communications and antenna array.

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