Bayesian Deconvolution for Angular Super-Resolution in Forward

Sensors 2015, 15, 6924-6946; doi:10.3390/s150306924

OPEN ACCESS

sensors

ISSN 1424-8220 www.mdpi.com/journal/sensors Article

Bayesian Deconvolution for Angular Super-Resolution in Forward-Looking Scanning Radar Yuebo Zha *, Yulin Huang, Zhichao Sun, Yue Wang and Jianyu Yang School of Electronic Engineering, University of Electronic Science and Technology of China, 2006 Xiyuan Road, Gaoxin Western District, Chengdu 611731, China; E-Mails: [email protected] (Y.H.); [email protected] (Z.S.); [email protected] (Y.W.); [email protected] (J.Y.) * Author to whom correspondence should be addressed; E-Mail: [email protected]; Tel.: +86-28-6183-1369. Academic Editor: James Churnside Received: 16 January 2015 / Accepted: 17 March 2015 / Published: 23 March 2015

Abstract: Scanning radar is of notable importance for ground surveillance, terrain mapping and disaster rescue. However, the angular resolution of a scanning radar image is poor compared to the achievable range resolution. This paper presents a deconvolution algorithm for angular super-resolution in scanning radar based on Bayesian theory, which states that the angular super-resolution can be realized by solving the corresponding deconvolution problem with the maximum a posteriori (MAP) criterion. The algorithm considers that the noise is composed of two mutually independent parts, i.e., a Gaussian signal-independent component and a Poisson signal-dependent component. In addition, the Laplace distribution is used to represent the prior information about the targets under the assumption that the radar image of interest can be represented by the dominant scatters in the scene. Experimental results demonstrate that the proposed deconvolution algorithm has higher precision for angular super-resolution compared with the conventional algorithms, such as the Tikhonov regularization algorithm, the Wiener filter and the Richardson–Lucy algorithm. Keywords: deconvolution; Bayesian; radar imaging; super-resolution; convex optimization

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1. Introduction Due to the superiorities of microwave remote sensing over optical remote sensing, such as all-weather and all-timing imaging and long operating distance, microwave remote sensing has been widely used in many civilian and military fields [1–5]. The most popular microwave remote sensing system is radar, which achieves two-dimensional high resolution imaging via signal processing. The high range resolution can be obtained by transmitting wide-bandwidth waveforms and using the maximum autocorrelation or global optimization method [6]. As for the azimuth direction, the angular resolution can be obtained by Doppler beam sharpening or the synthetic aperture radar technique. However, these techniques are not suitable for forward-looking imaging in airborne radar and fixed ground-based radar, because the direction of the Doppler resolution and the range resolution are the same in the forward-looking area of the radar, which leads to the failure to use the frequency information [7]. One option to achieve high azimuth resolution in the forward-looking area of the radar platform is to separate the transmitting station and receiving station. This forms the bistatic SAR, which has many advantages in comparison with its monostatic counterpart [8]. However, there are several problems associated with bistatic SAR, including interferences from the geometry configuration and the complexity of the hardware. Scanning radar imaging belongs to real aperture imaging, working as a noncoherent sensor. The high range resolution of scanning radar can be obtained using pulse compression. The upper limit of the angular resolution is determined by the effective wavelength and the size of the antenna, which is restricted by the radar system. Improvement in the angular resolution of the scanning radar image can be accomplished by increasing the physical size of the antenna. Increasing the size of the radar antenna is costly, because the forward-looking scanning radar is often mounted on an airborne platform, where there may be insufficient space to accommodate a large-sized antenna [9]. Hence, the angular resolution of the forward-looking scanning radar is usually poor when compared with the achievable range resolution. Under the Born hypothesis [10], the echo in the azimuth can be modeled as the convolution of the transmitted signal with the reflectivity of the observed scene. Therefore, the deconvolution method can enhance the angular resolution of the scanning radar image in theory [11,12]. The data recorded at the output of the radar system are a low-pass-filtered version of the original scene due to the finite size of the antenna. The portions lost by the antenna are the high frequency spectral components [13]. Convolution in the Fourier domains corresponds to multiplication, while deconvolution is Fourier division [14]. The challenge is that the multipliers are often small for high frequencies, and division is unstable due to noise presented in the reflectivity data. This phenomenon also occurs in communication systems. Deconvolution aims at removing the system noise and other non-idealities of the image processing and is inherently an ill-posed problem [15]. Basically, there are two main categories of deconvolution techniques: set-theoretic estimation methods [16–18] and Bayesian approaches. One of the popular attempts at employing a set-theoretic framework for deconvolution problem is the regularization method. Recently, the application of regularization deconvolution can be found in [19–23] for microwave imaging. One of the popular regularization methods is Tikhonov regularization [24]. The main motivation behind the regularization methods lies in replacing the original deconvolution problem with a nearby well-conditioned problem. A scheme that uses the singular value decomposition (SVD) has

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been reported in [25]. However, this method in the space domain is sensitive to the noise and loses performance in terms of super-resolution. Recently, the authors of [3] presented a truncated singular value decomposition (TSVD) algorithm to enhance the spatial resolution of radiometer data. The main idea of TSVD is to choose a truncation parameter so that all of the noise-dominated SVD coefficients are discarded, which makes application of TSVD for angular super-resolution in scanning radar challenging. Another direction for the deconvolution problem that is receiving greater popularity is the use of Bayesian theory. Following the pioneering work [26,27], the Bayesian deconvolution algorithms have been studied extensively in radar imaging. In [28], the authors applied the Bayesian deconvolution algorithm for ISARimaging, resulting in better resolution, even under strong noise. It is also studied for ISAR imaging with sparse aperture in [29]. A Bayesian inversion approach to estimate Titan’s lake features has also been presented in [30]. An important advantage of Bayesian deconvolution algorithms over the regularization deconvolution algorithms is, as proposedlater, Bayesian deconvolution algorithms can use the statistic characteristic of prior information about the solution, which is often ignored by the regularization methods. On the other hand, the Bayesian methods are model-based and can handle observation noise and missing data [31]. This paper presents a deconvolution algorithm for angular super-resolution in scanning radar using the Bayesian theory. To use the Bayesian theory for solving the deconvolution problem, we first convert the deconvolution problem into an equivalent MAP estimation task that not only incorporates the prior information of the targets in the spirit of [28,29], but also considers the mixture noise in the echo data. In the proposed algorithm, we assume that the noise is composed of two mutually independent parts, a Gaussian signal-independent component and a Poisson signal-dependent component [32–34]. The application of the Poisson distribution can be found in [35] for optical imagery, in [36] for position emission tomography and in [37] for single positron emission computer tomography. However, no work about the Poisson distribution has been studied for angular super-resolution in scanning radar imaging. The Poisson component models the signal-dependent part of the errors, which is essentially due to the working mode of the scanning radar, while the Gaussian component represents the signal-independent parts of the errors, such as the thermal and electric noise. This forms the key feature of our approach to model noise in the proposed algorithm. On the other hand, the most crucial point of applying Bayesian methods is the choice of the form of the prior distribution. Inspired by Bayesian compressed sensing [38], the Laplace distribution is applied to represent the statistical characteristics of prior information about the targets. This leads to the objective function of the MAP estimation problem consisting of the so-called data term, which is nonlinear, plus a convex non-differentiable regularizer (the negative log-prior). We solve the MAP estimation problem in the framework of convex optimization by using an approximation method. Experimental results with synthetic data indicate that the proposed deconvolution algorithm for angular super-resolution has higher precision compared with conventional deconvolution algorithms (Richardson–Lucy deconvolution algorithm, Tikhonov regularization algorithm and Wiener filter). The rest of the paper is organized as follows. Section 2 presents the signal model of scanning radar and mathematically formulates the angular super-resolution problem as an equivalent deconvolution problem. Section 3 presents the Bayesian inversion approach for solving the deconvolution problem, including the likelihood function and prior information about the targets. In Section 4,

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experimental results with synthetic data indicate that the proposed deconvolution algorithm for angular super-resolution in scanning radar has higher precision compared to the conventional deconvolution algorithms. Finally, conclusions are drawn in Section 5. 2. Angular Super-Resolution Model In this section, we mainly describe the angular super-resolution model with emphasis on the echo formulation and then derive the mathematical formulation of angular super-resolution in forward-looking scanning radar. The construction of forward model consists of two steps: (1) formulating the signal model of forward-looking scanning radar; and (2) converting the problem of angular super-resolution into an equivalent deconvolution task. Suppose that scanning radar works in forward-looking mode. The forward model of forward-looking scanning radar is illustrated at the top of Figure 1. The radar platform is moving along axis X corresponding to the range direction at altitude H with constant velocity V ; the antenna beam scans the scene along the axis X corresponding to the azimuth direction with constant angular velocity ω and then receives the echo data from the observed scene. Firstly, conventional range compression and range cell migration are applied to the echo data with the current approaches [39]. Z

Amplitude

Forward model

V

Z



0



0

Y

X



0

Y ratio e mig Rang

Noise

(a)

X



(b)

r jecto n tra

yt

(c) Motion compensation Amplitude

Z Bayesian theory Maximum A Posteriori Estimation Task

0

0

t

Y

X

(e)

(f)



Deconvolution problem

Convex optimization

(d)

Figure 1. Diagram of forward-looking scanning radar used for angular super-resolution: (a) geometry model for scanning radar imaging; (b) noise; (c) data acquisition for scanning radar; (d) echo data after range compression and range cell migration; (e) deconvolution method; (f) angular super-resolution result. Under the Born hypothesis [19,40], the data after range compression and range cell migration gR (τ, t) can be modeled as the convolution of the antenna beam h (τ, t) with the reflectivity coefficients of the observed scene f (t), which is: Z+∞ gR (τ, t) = h (τ, t − t0 ) f (t0 )dt0 + n (τ, t) −∞

(1)

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where τ is the fast time, t is the slow time and n is the noise. In the case of the observation of a certain scene F , the reflectivity coefficients of the scatters in the observed scene can be represented as a 2D matrix:   f (x1 , y1 ) , f (x1 , y2 ) . . . f (x1 , yA )    f (x , y ) , f (x , y ) . . . f (x , y )  2 1 2 2 2 A   F =  .. .. .. ...   . . .   f (xR , y1 ) , f (xR , y2 ) . . . f (xR , yA ) where f (xp , yq ), (p = 1, 2, · · · , R, q = 1, 2, · · · , A) are the discrete equivalent backscattering coefficients at the p-th position of the range along the axis X at the q-th position along the azimuth of the observation scene, R is the number of discretization cells of the observed scene along the X axis and A is the number of discretization cells of the observed scene along the Y axis. Each row of matrix F represents the backscatter coefficients index set in each range cell. To denote the reflectivity coefficient of the observed scene, the 2D reflectivity coefficient matrix F should be reshaped to a column vector by stacking the rows of F : f = [f (x1 , y1 ) , f (x1 , y2 ) · · · f (x1 , yA ) , f (x2 , y1 ) · · · f (x2 , yA ) , · · · , f (xR , y1 ) · · · f (xR , yA )]T

(2)

where f is a RA × 1 target vector and RA is the total number of targets after the discretization of the scene. The 2D discrete time radar signal can be denoted as: ! RA p ,yq ) X τr − 2R(ta ,x 4πf c c × exp −j R (ta , xp , yq ) ge (τr , ta ) = f (i) × rect T c r (3) i=1 r = 1, 2, · · · R;

a = 1, 2, · · · A

where τr represents the r-th sample in fast time, ta is the a-th sample in slow time, f (i) is the i-th element in Equation (2), rect(·) is a rectangular function, Tr denotes the pulse duration, fc is the carrier frequency, c is the speed of light, R is the number of range samples and A is the number of azimuth samples. The R (ta , xp , yq ) is the range history between the antenna and a point target at (xp , yq ) when the azimuth time is ta . The equation for R (ta , xp , yq ) is shown in Equation (4) q R( ta , xp , yq ) = R02 + (V ta )2 − 2R0 V ta cos θ cos ϕ (4)

in which R0 is the range when the antenna beam center is across the target at (xp , yq ), V is the platform velocity, θ denotes the angle between the direction of the antenna and flight direction and ϕ represents the incident angle of the beam, which is usually smaller than 10◦ . The geometry relationship is illustrated in Figure 2. At the time when the antenna beam center crosses the target at (xp , yq ), cubic and higher order terms of the Taylor expansion for Equation (4) can be ignored in the azimuth phase history of the targets [39]. In this case, Equation (4) can be simplified to: R (ta , xp , yq ) = R0 − (V cos θ cos ϕ) ta +

V 2 sin2 θsin2 ϕ 2 ta + o (ta ) 2R0

(5)

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Z Flight direction

V 

H 

O

X x

p

, yq 

Y



Figure 2. The geometry relationship of scanning radar. Since the product of the velocity V and azimuth time ta is much smaller than R0 , Equation (5) can be rewritten as: R (ta , xp , yq ) ≈ R0 − (V cos θ cos ϕ) ta

(6)

which leads to the range migration trajectory presented as a straight line in Figure 1c. Substituting Equation (6) into Equation (3) and applying the range compression to the data ge (τr , ta ) yields: gc (τr , ta ) =

RA X i=1

2R (ta , xp , yq ) 4πfc f (i) ×δ τr − R (ta , xp , yq ) × A (ta ) × exp −j c c

(7)

where δ (·) is the impulse function and A (ta ) is the antenna pattern. After range cell migration correction to the data gc (τr , ta ), we have: g (τr , ta ) =

RA P

f (i) H (τr , ta , xp , yq ) + n (τr , ta )

(8)

i=1

where:

4πfc 2R0 H (τr , ta , xp , yq ) = δ B τr − × A (ta ) × exp −j R (ta , xp , yq ) c c

(9)

and B is the signal bandwidth; the corresponding data g (τr , ta ) are shown in Figure 1d, and n (τr , ta ) denotes noise. In Figure 1d, the amplitude of the received signal at the receiving output is proportional to the antenna pattern. If the two targets are close enough, the response of the two targets are proportional to two replicas of the antenna pattern, overlapped and added to get a composite response. Clearly, the space limitation at which targets are resolved is determined by the beam width of the antenna pattern. The resulting low-resolution signal is shown in Figure 1d. This phenomenon brings great difficulty in realizing the angular super-resolution imaging.

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When we take the noise into account, Equation (8) can be written in matrix form as: (10)

g =H⊗f +n

where g and n are of dimension RA × 1, H is a RA × RA matrix and ⊗ indicates the convolution operator. In Equation (10): (11)

g = [g (τ1 , t1 ) , · · · , g (τ1 , tA ) , · · · g (τR , t1 ) , · · · , g (τR , tA )]T

It can be shown in [1] that the convolution relationship holds between the expected value of |g (τr , ta )|2 and |f (xp , yq )|2 with the two-way antenna power pattern |A (ta )|2 . That is: E |g (τr , ta )|2 = |A (ta )|2 ⊗ E |f (xp , yq )|2 (12)

where E{·} denotes the expected value operator. Therefore, the as follows:  H1 HR HR−1 · · ·   H H1 HR · · ·  2  H= H2 H1 · · ·  H3  .  ..  HR HR−1 HR−2 · · ·

matrix H in Equation (10) is written

H2 H3 .. .

H1

         

(13)

Each element of matrix H is the module operation result of H (τr , ta , xp , yq ). Therefore, the matrix H can be written as:        H=    

H (τ1 , t1 , x1 , y1 ) .. .

···

H (τ1 , t1 , x1 , yA ) .. .

···

H (τ1 , tA , x1 , y1 ) .. .

H (τ2 , t1 , x1 , y1 ) .. .

···

H (τ2 , t1 , x1 , yA ) .. .

··· .. .

H (τ1 , tA , x1 , yA ) .. .

··· .. .

H (τ2 , tA , x1 , y1 ) .. .

... .. .

H (τ2 , tA , x1 , yA ) .. .

H (τR , t1 , xR , y1 ) .. .

···

H (τR , t1 , xR , yA ) .. .

···

H (τ1 , t1 , x1 , y1 ) .. .

···

H (τ1 , t1 , x1 , yA ) .. .

H (τR , tA , xR , y1 )

···

H (τR , tA , xR , yA )

···

H (τ1 , tA , x1 , y1 )

···

H (τ1 , tA , x1 , yA )

···

···

         

(14)

RA×RA

At this point, the goal of angular super-resolution imaging in forward-looking scanning radar is to infer, as accurately as possible, f from the samples g. This task is called the deconvolution problem in this paper. A naive method would be simple division of data g by transferring the function in the Fourier domain. Using the Fourier transform, Equation (10) can be written as: G (w) = H (w) F (w) + N (w)

(15)

where G(w), H(w), F(w) and N(w) are the Fourier transforms of g, H, f and n, respectively. The conventional deconvolution methods are that to find a linear operator T(w), such that: ˆ (w) = G (w) = F (w) + N (w) F T (w) H (w)

(16)

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where—denotes the matrix division operator. The challenge is that convolution in the Fourier corresponds to multiplication, and deconvolution is Fourier division. For the radar system, the multipliers 1 are often small for high frequencies, and the inverse filter T(w) is large for T (w) very small. This results in a large noise amplification and, thus, a poor angular resolution. To address this challenge, we propose a Bayesian deconvolution method for angular super-resolution imaging in scanning radar in the next section. 3. Bayesian Inversion Approach for Angular Super-Resolution The data g recorded at the output of the radar system are a low-pass-filtered version of the original data f . Thus, to recover the original scene from the recorded data g by the deconvolution approach, we firstly need to compensate for the loss of high-frequency information beyond the passband range. For this purpose, the obtained angular super-resolution model Equation (10) can be recast in the Bayesian framework. In this framework, we can handle prior information about the original scene and the likelihood function between the observation and the original scene. Starting from Equation (10), we first transform the deconvolution problem into an equivalent MAP estimation task, and the corresponding problem can be equivalently transformed into an unconstrained optimization problem by adopting the negative logarithm of the posterior probability p (f |g). By Equation (10), the MAP estimation task consists of finding a solution that satisfies the following criterion: fM AP = arg max p (f |g) = arg max p (g|f ) p (f ) f

f

= arg min {− ln p (g|f ) − ln p (f )}

(17)

f

where p (g|f ) represents the likelihood function pertaining to the angular super-resolution model Equation (10), which models all of the information coming from the data and their uncertainty, and p (f ) is the prior probability density function (pdf) that models the information coming from the other source [41]. The prior term p(f ) in Equation (17) is a function of f , which does not vary with the observation model. In Equation (17), the − ln p (g|f ) measures the violation of the relation between f and its observation g, and − ln p (f ) corresponds to the prior information about the f , which does not vary with the measurement in the framework of Bayesian theory. It is noted that Equation (17) looks similar to the regularization method, because they are philosophically similar. The main idea behind these methods is to find a solution to the deconvolution problem [42]. However, the implementation of Equation (17) requires the knowledge of a likelihood pdf p (g|f ) and the prior pdf p (f ). The details on choosing these functions are shown as follows. 3.1. Likelihood The choosing of the likelihood function mainly depends on the application. In radar signal processing [19,43], the likelihood function between the observation data and the original scene is approximated as a zero mean Gaussian distribution. However, there are problems with this assumption in angular super-resolution. The first is that such a continuous time process cannot exist, since it would

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have infinite power. For details about this reason, we refer the interested reader to [44]. In addition, the treatment of the likelihood function by the Gaussian distribution involves interesting ideas in the field of mathematical statistics [44]. In this paper, we assume that the noise is composed of two mutually independent parts, a Poisson signal-dependent component np and a Gaussian signal-independent component ng . The Poisson component np models the signal-dependent part of the errors, which is essentially due to the working mode of the scanning radar, while the Gaussian component ng represents the signal-independent parts of the errors, such as the thermal and electric noise. In applications of angular super-resolution, where a high number of data are collected, the echo data are inherently affected by signal-dependent noise. Notice that the Poisson distribution is not additive, and its strength is dependent on the point scatterer intensity [45]. For these reasons, the Poisson distribution is used to represent the likelihood function between the observation data and the observed scene in angular super-resolution, i.e., p [g(k)|(Hf )(k)] =

[(Hf )(k)]g(k) exp [−(Hf )(k)] (g(k))!

(18)

where (·) (k) represents the k-th element of the vector given by the expression inside the brackets. Assuming that the values of g(k) are independent and identically distributed, the likelihood function of the data g is also Poisson, i.e., p (g|Hf ) =

RA Y [(Hf )(k)]g(k) exp [−(Hf )(k)]

k=1

(g(k))!

(19)

3.2. Prior Law of the Targets Since the echo formulation includes a convolution operation, some prior information about the statistical characteristics of the true scene must be introduced to regularize the solution of the deconvolution problem. In Bayesian theory, the prior information about the true scene is modeled as random variables with an assigned probability distribution, which represents a function of the the original scene and does not vary with the measurement. The scanning radar image demonstrates the distribution and amplitude information of the point scatterers. Therefore, the dominant point scatterers can capture most of the information about the scene, and the weak scattering centers can be regarded as noise in the radar image. Note that the Laplace distribution has heavy tails, which means that the probability of strong scatters is large. Therefore, the statistical characteristics of strong scatters in the radar image can be modeled by an independent identical Laplace distribution with the same deviation [28,29,46,47]. As mentioned above, we utilize the Laplace distribution to represent the statistic characteristic of the dominant scatters in the scene. The law for f is also Laplace, so that the joint probability of f can be written as: ! √ RA Y 1 2 |f (i)| √ exp − p (f ) = (20) σ 2σ i=1 where σ denotes the deviation.

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Substituting Equations (19) and (20) into Equation (17), the final expression for the MAP estimation task becomes: ) ( RA !) ( RA √ Y 1 Y [(Hf )(k)]g(k) 2 |f (i)| √ exp − exp [−(Hf )(k)] × (21) fM AP = arg max (g(k)) ! σ 2σ f i=1 k=1 Maximizing Equation (21) is equivalent to minimizing: E (f ) = − ln p (g|Hf ) − ln p (f )

√ 2 = kf k1 {(Hf )(k) − g(k) ln [(Hf )(k)]} + σ k=1 RA X

with respect to f . The term kf k1 =

RA P

i=1

(22)

|f (i)| is the l1 -norm of f Note that we drop the ln [(g(k))!]

term in the summation for the simplicity of notation, since this term has no effect on the corresponding minimization problem. From the point of the convex optimization, problem Equation (22) is equivalent to the MAP task Equation (21); solving Equation (22) for f yields the desired solution in Equation (17). 3.3. Solution of the Unconstrained Problem

The non-differentiability of the kf k1 around the origin makes it impossible to solve Equation (22) by means of gradient-based optimization techniques. Fortunately, this problem can be solved by introducing 1 a small positive parameter ε in the l1 -norm (see [43,48]). Specifically, |f (i)| ≈ |f (i)|2 + ε 2 , where ε > 0 is a small constant. The role of ε is to ensure that the approximation is as rigid as possible; therefore, the parameter ε should be set small. In this paper, we set the approximate parameter ε to be 10−8 . This modified l1 -norm is a differentiable function, which enables us to solve the unconstrained problem Equation (22) within the framework of convex optimization by using the gradient-based approach. The first step in solving the unconstrained problem Equation (22) consists of replacing the l1 -norm by its differentiable approximation. After the smoothed approximation, the kf k1 has the following form: kf k1 ≈

RA h X i=1

1 i |f (i)|2 + ε 2

(23)

Substituting Equation (23) into Equation (22), we express Equation (22) as follows: E (f ) ≈

RA X k=1

{(Hf ) (k) − g (k) ln [(Hf ) (k)]} +

√

RA 1 2X |f (i)|2 + ε 2 σ i=1

(24)

Since the objective function in the right-hand of Equation (24) is convex with respect to f , searching for a minimum is equivalent to searching for a zero of the gradient of Equation (24). This leads to: g √2 T T H IRA − H + Λ (f ) f = 0 (25) Hf σ where the superscript the transpose of a matrix, IRA stands for RA × 1 vector of ones n T represents 1o − and Λ (f ) = diag |f (i)|2 +ε 2 is a diagonal matrix whose i-th diagonal element is given by the

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expression inside the brackets. We further assume that HT IRA = IRA , where IRA stands for the column vector consisting of RA ones. fM AP solves the problem Equation (25) if and only if the following equivalent statements hold: g T − λΛ (fM AP ) fM AP = IRA (26) H Hf M AP √

where λ = σ2 is known as the regularization parameter, which controls the weight of the data term and the prior information of the observed scene. The method for selecting this parameter in Equation (26) is presented in the next section. Solving Equation (26) above naturally calls for the fixed point iterative scheme; we can derive the following iteration form, g T − λΛ (fm ) fm (27) fm+1 = fm H Hfm

where fm and fm+1 represent the estimations of the true reflectivity coefficients of theas observed scene f corner of the angular super-resolution results. They are defined Y (λ). The in the m-th andfollows: (m + 1)-th iterations, respectively. Assuming that at convergence, the ratio identified of fm+1 /fm and the c is IRA . The greatest problem with multiplicative form Equation (27) is that the diagonal out elements as theof weight co Λ (fm ) correspond to the approximation processes Specially, L-curve ∥f ∥ in the iteration. ∥fM AP − f ∥

2 2 , ReErr = , the regularization p ∥fM AP − f ∥2 ∥f ∥2 4. Simulation and Experimental Results function ∥g − f ∥2 ISNR = 20log10 , ∥fMresults f ∥illustrate AP −to In this section, we present experimental the angular super-resolution performance 2 2ρ(f · (2µf · µfscanning ) of the proposed deconvolution algorithm in) forward-looking radar. We compare the proposed ,f ). SSIM = ( 2 M AP 2 ) ( 2M AP C (λ) = 2 deconvolution method with Wiener the+Richardson–Lucy (R-L) algorithm and the µfM APfilter + µfmethod, σfM AP σf

SNR = 20log10

Tikhonov regularization method.

4.1. Simulation where fM AP , f , and g corresponds to the obtained angular

Normalized Amplitude

where X (λ) = super-resolution image, the original image, and the observed log (∥D |fλ |∥2 ) are For experiments on synthetic data, weThe apply our method a synthetic composed and of sixthe point image, respectively. terms µ, σ, to and ρ(f ,fM APscene prior inform ) , are the targets with different reflectivity magnitudes. The synthetic scene is shown in Figure 3. The targets are mean, standard deviation of the vectors, the correlation coef- represents the diffe of unequal amplitude, means different ficientwhich corresponds to the scattering vector f coefficients. and fM AP .Some The related SSIMradar is a parameters In theare simulation ◦ set as follows: the pulse repetition frequency is 4000 and the antenna scanning speed ismethod. 30 /s. TheThis own quantitative measure between theHz,super-resolution result and ◦ bandwidth of the signal isThe 2 MHz, andofthe 3-dB width of the real is about . Theto treat p thetransmitted original scene. value SSIM is between −1 beam and 1, are 3able number of azimuth is 2666. andsamples 1 means full identical with the original scene. and robustness no m The L-curve meth 1.2 there are two para 1 proposed deconvolu 0.8 0.6 performance of the 0.4 of angular super-re 0.2 choice of the regula 0 −10 −8 −6 −4 −2 0 2 4 6 8 10 this paper. To deter Targets Distribution (Degree) firstly use the iterat Figure of 3. targets Location in scene the simulated scene. it for all the differe Fig. 3. Location in of thetargets simulate λ. Then, choose the In order to show the performance of the proposed deconvo- the obtained λ unde lution algorithm for angular super-resolution under different following simulatio

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To provide a quantitative evaluation for the following super-resolution simulations, relative error (ReErr), improved signal-to-noise ratio (ISNR), structure similarity (SSIM) [49], and the signal-to-noise ratio (SNR) are used to measure the quality of the angular super-resolution results. They are defined as follows: kfM AP − f k2 kf k2 , ReErr = kfM AP − f k2 kf k2 kg − f k2 ISNR = 20log10 kfM AP − f k2 2ρ(f · µf ) ,f ) · (2µf SSIM = 2 M AP 2 2M AP µfM AP + µf σfM AP + σf2 SNR = 20log10

where fM AP , f and g correspond to the obtained angular super-resolution image, the original image and the observed image, respectively. The terms µ, σ, and ρ(f ,fM AP ) are the mean and standard deviation of the vectors, and the correlation coefficient corresponds to the vector f and fM AP . The SSIM is a quantitative measure between the super-resolution result and the original scene. The value of SSIM is between −1 and one, and one means fully identical to the original scene. In order to show the performance of the proposed deconvolution algorithm for angular super-resolution under different noise levels, Gaussian noise models the signal-independent part of the errors, such as thermal and electric noise, which is stronger than the signal-dependent noise in scanning radar imaging. Using the approximation N (σ2 , σ2 ) = P oiss(σ2 ), the Poisson noise can be approximated by Gaussian noise. Then, the Gaussian noise with different levels is added to model the errors. 4.1.1. Parameter Values In most of the Bayesian deconvolution algorithms, the regularization parameter has to be chosen, so that it gives the best visual results. Both the data fidelity and the prior are presented in Equation (22), and their size depends on the regularization parameter λ. Small values of the regularization parameter tend to amplify the noise, while a large regularization parameter over smooths the radar image. Most of the referenced methods need to manually choose the regularizing parameter λ to control the weight of the prior, so that the result of super-resolution gives the best visual quality. However, this approach is time consuming and relies too much on subjective factors. To overcome this problem, a number of works on the selection of regularization parameter, such as the L-curve method [50], generalized cross-validation [51] and the unbiased predictive risk estimator method [52], have been reported. In order to determine the parameter λ in Equation (27), we choose the regularization parameter λ by means of the L-curve method, which plots the squared norm of the prior solution X(λ) against the squared norm of the corresponding residual solution Y (λ). The corner of the regularization parameter curve is identified, and the corresponding parameter value λ is picked out as the weight coefficient in the deconvolution process Equation (27). Specifically, L-curve method is proposed in [50], which selects the regularization parameter λ that maximizes the curvature function: X 00 (λ) Y 0 (λ) − X 0 (λ) Y 00 (λ) C (λ) = 3 X 0 (λ)2 + Y 0 (λ)2 2

(28)

e original scene.

and robustness no matter which deconvolution method is used. The L-curve method is useful for direct solvers. However, there are two parameters in the solution obtained using the Sensors 2015, 15 proposed deconvolution algorithm. This paper focuses on the 6936 performance of the proposed deconvolution algorithm in terms of angular super-resolution in scanning radar. The automatic where X (λ) = logchoice (kHf of gkregularization = log (kD |fλbeyond |k2 ) arethethe log-transform of the data λ− the λ is scope of 2 ) and Y (λ) parameter 2 4 6 8 and 10 the priorthis fidelity information, superscript (0) represents the differentiation with paper. Torespectively, determine a and goodthepair of parameters in (26),we on (Degree) firstly use the iterative solver with a lot of iterations, and re-use respect to λ. it for the different the regularization parameterThis is owed to the e scene In the simulations, theallparameter λ is choices selectedofusing the L-curve method. λ. Then, choose the good solution using L-curve. Fig.4 shows advantage that the L-curve method is able to treat perturbations consisting of corrected noise and ce of the proposed deconvo- the obtained λ under different noise levels, which is used in the no matterfollowing which deconvolution method is used. L-curve method simulations. The setting of theThe iteration number of is useful for direct r-resolutionrobustness under different solvers. However, there two parameters theway solution obtained using the proposed deconvolution R-L are algorithm is equal tointhe for the proposed algorithm. dels the signal-independent l and electric noise, which 2) Angular super-resolution Fig.5-Fig.7 show thealgorithm in terms of algorithm. This paper focuses on the performance ofresults: the proposed deconvolution ent noise in scanning radar angular super-resolution results under choice different levels. angular super-resolution in scanning radar. ∥g∥ The automatic of noise the regularization parameter λ is 2 n N (σ 2 , σ 2 ) = P oiss(σ 2 ), The BSNR = 20log10 ∥n∥ is used to measure the quality of beyond the scope of this paper. To determine a2 good pair of parameters in Equation (27), we firstly use imated by Gaussian noise. these echo data with different noise levels. The regularization theadded iterative solver with a lot of iterations and re-use it for all of the different choices of the regularization rent levels is to model parameter λ is computed by using Eq.(27), and Fig.4 shows parameter λ. Then, we theand good using thecurvature L-curve. as Figure 4 shows thechoose L-curve thesolution corresponding a function ofthe obtained λ under of the Bayesian deconvoludifferent noise levels,λ.which is used in following setting of the, iteration number of The number ofthe iterations in simulations. Fig.4 are 75,The 110, and 150 parameter the hasR-L to bealgorithm chosen isrespectively. equal to the way for the proposed algorithm. sults. Both the data fidelity ), and their size depends on Noise level:30dB mall values of regularization 10 Corner at λ=0.1406 e, while large regularization 10 Noise level: 20dB mage. Most of the referenced Corner at λ=0.4504 the regularizing parameter 10 Noise level:15dB prior so that the result of Corner at λ=0.8296 10 sual quality. However, this 7 elies too much on subjective 6 10 , a number of works on the 5 ter such as L-curve method 10 4 0.2 0.4 0.6 0.8 43], unbiased predictive risk residual norm || H f − g || 10 reported. 0 0.5 1 1.5 2 2.5 3 3.5 meter λ in (26) , we choose residual norm || H fλ− g ||2 means of L-curve method, the prior solution X(λ)4.a-The Fig. Figure L-curve atL-curve the different noise levels; the and corner corresponds to the point with 4. The at the different noise levels, the corner corresponds to the point with the maximum curvature. responding residual solution the maximum curvature. 6

4

3

2

1

0

solution norm || fλ ||2

solution norm || fλ ||2

5

λ

2

4.1.2. Angular Super-Resolution Results Figures 6–8 show the angular super-resolution results under different noise levels. The kgk BSNR = 20log10 knk2 is used to measure the quality of these echo data with different noise levels. The 2 regularization parameter λ is computed by using Equations (28), and Figure 4 shows the L-curve and the corresponding curvature as a function of λ. The number of iterations in Figure 4 are 75, 110 and 150, respectively. The angular super-resolution results by those methods under various noise levels are shown in Figures 5–7. The visual quality of super-resolution results using the proposed deconvolution algorithm are quite competitive with those using the Tikhonov regularization method, Wiener filter method and R-L algorithm. It is noted that the angular super-resolution results by using the proposed deconvolution algorithm exhibit a close match with the original scene.

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It can be seen from Figures 5–7 that the spikes of the targets in the results of the Tikhonov regularization and Wiener filter are more connected compared with the results of the R-L algorithm and the proposed method. The reason for this is that the Tikhonov regularization method is based on the noise power of the observed scene, which makes the profile of targets overly smooth. The Wiener filter obtains an optimal result in the sense of minimizing the mean square error between the obtained result and the true scene using the correlation information between the signal and noise. Therefore, the angular super-resolution performance of the Wiener filter is degraded when the received signal includes complicated targets and high noise. 8 8

10

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Figure 5. (a)The echo data added by Gaussian noise with 30 dB; (b) Angular super-resolution result of the Tikhonov regularization; (c) Angular super-resolution result (c) (c) (d) (d) (e) (e) (c) (c) (c) (d) (d) (d) result of the R-L algorithm (e) (e) (e) of the Wiener filter; (d) Angular super-resolution with 75 iters; Fig. 5. Fig. The5.first The row first is row the echo is theadded echo by added Gaussian by Gaussian noise with noise30dB; with 30dB; the second the second row is row the angular is the angular super-resolution super-resolution results results from the from leftthe to right: to (b) right: Angular (b) Angular (e) Angular super-resolution result of the proposed method with 75 and =left 0.1406. Fig. 5. Fig. The 5. Fig. first The 5. row first Theisrow first the isthe row echo theof isecho added the Tikhonov echo added byregularization. Gaussian added by Gaussian by noise Gaussian noise with noise with 30dB; with 30dB; the 30dB; second the second the row second isrow theisrow angular the isangular the super-resolution angular super-resolution super-resolution results results from results from the left from theλ toleft the right: toleft right: (b) toof Angular right: (b) (b)the Angular super-resolution super-resolution result of result Tikhonov the regularization. (c) Angular (c) Angular super-resolution super-resolution result of result the of Wiener the Wiener filter. (d) filter. Angular (d)iters Angular super-resolution super-resolution result result the Angular of R-L R-L Normalized Amplitude Normalized Amplitude



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−8 −10 −6 −8 −4 −6 −2 −4 0 −2 2 0 4 2 6 4 8 −10 −6 −8 −4 −6 −2 −4 0 −2 2 0 4 2 6 4 8 6 10 8 10 −10 1.2 1.2 0 0 0 0 (Degree) (Degree) (Degree) −10 −6 −8 −10 −4 −6 −8 Targets −2 −4 −6 distribution −2 0Targets −4 2−2 0distribution 420 (Degree) 642 864 −6 −8 −10 −4 −6 −8 Targets −2 −4 −6 distribution −2 0Targets −4 2−2 0distribution 420 1.2 642 864 1086 108 −1010 −8 1 1 Targets distribution Targets distribution Targets (Degree) distribution (Degree)(Degree) Targets distribution Targets distribution Targets (Degree) distribution (Degree) 1 (Degree) 1.2 0.8 0.8 1.2 0.8 1 0.6 0.6 1 0.6 0.8 0.4 0.4 0.8 0.4 0.6 0.2 0.2 0.6 0.2 0.4 0 0 0.4 0 −100.2 −8−10 6 10 8 10 −6 −8 −4 −6 −2 −4 0 −2 2 0 4 2 6 4 8 6 10 8 10 10 −10 −8 −6 −4 −2 0 2 4 6 8 10 Targets distribution Targets distribution (Degree) (Degree) 0.2 Targets distribution (Degree) 0 8 10 −10 −8 −6 −4 −2 0 2 4 6 8 10 0 Targets distribution (Degree) 6 8 10 −10 −8 −6 −4 −2 0 2 4 6 8 10 Targets distribution (Degree) Normalized Amplitude




−10 −8 −10 −6 −8 −4 −6 −2 −4 0 −2 2 0 4 2 6 4 8 6 10 8 10 −10 −8 0 0 0 0 0 1.2 distribution 1.2 distribution −10 −8 −10 −6 −8 −10 −4 −6 −8 Targets −2 −4 −6 −2 0Targets −4 2−2 0(Degree) 420 (Degree) 642 864 1086 108 −10 10 −8 −10 1.2 1 1(Degree) Targets distribution Targets distribution Targets distribution (Degree)(Degree) 1 1.2 0.8 0.8 0.8 1 1.2 0.6 0.6 0.6 0.8 1 0.4 0.4 0.4 0.6 0.8 0.2 0.2 0.2 0.4 0.6 0 0 0 −100.2 −8 0.4 −10 −6 −8 −4 −6 −2 −4 0 −2 2 0 4 2 6 4 8 −10 −8 −6 −4 −2 0 2 4 6 8 Tareget distribution Tareget distribution (Degree) (Degree) Tareget distribution (Degree) 0 0.2 −10 −8 −6 −4 −2 0 2 4 6 0 Tareget distribution (Degree) −10 −8 −6 −4 −2 0 2 4 Tareget distribution (Degree)








(a) (a) (b) (b) (c) (d) (d) (e) (e) (a) (a) (a) (b) (b) (b) (e) (d) (c) (d) (e) 1.2 1.2 1.2 1.2 (c) (d) (e) 1.2 1.2 1.2 1.2 1 with 1.2 20dB; 1.2 1with Fig.1.26.1Fig. The 6.11.2 first Therow firstisrow the is echo the added echo added by Gaussian by Gaussian noise noise 20dB; the second the second row isrow the is angular the angular super-resolution super-resolution results from the from leftthe toleft right: to right: (b) Angular (b) Angular 1.2 1 1.2 1.2 1results Fig. 6. first the second row is the angular super-resolution results from the left to right: (b) Angular 10.8 The 1 0.8 1 row is the echo added by Gaussian noise 1with 120dB; 0.8 0.8 1 10.8 (d)1 Angular 1 Angular 0.8(d) super-resolution super-resolution result result ofisthe ofTikhonov the Tikhonov regularization. Angular (c) Angular super-resolution super-resolution result of the ofWiener thefilter. Wiener filter. super-resolution super-resolution result result of (b) the ofR-L the R-L Fig. 6. The first row the echo added by regularization. Gaussian(c) noise with 20dB; the second rowofisresult the angular super-resolution results from the left to right: super-resolution super-resolution result the Wiener (d) filter. Angular super-resolution result oftothe R-LAngular 0.8 0.6 0.8 0.60.8 result of the Tikhonov regularization. (c)0.8Angular 0.6 0.8 0.60.8 0.80.6 0.8 0.60.8 Fig. 6. The first row is the echo added by Gaussian noise with 20dB; the second row is the angular super-resolution results from the left right: (b) Angular algorithm algorithm with 110 withresult iters. 110(e)(e) iters. Angular (e) Angular super-resolution super-resolution result of the of proposed the proposed method method with 110 with 110 and iters λand = 0.4504. λ = 0.4504. super-resolution of the Tikhonov regularization. Angular super-resolution result of iters the Wiener filter. Angular super-resolution result of the R-L 0.6 0.60.4(c) 0.4 0.6with 0.40.6110 iters. 0.40.6 algorithm Angular super-resolution result of0.6result the proposed method with 110 iters and λ Wiener = 0.4504. 0.6 0.4 0.6(d) 0.40.6 super-resolution result of the Tikhonov regularization. (c) Angular super-resolution result of the filter. (d) Angular super-resolution result of the R-L 0.4 0.40.2result 0.4 0.2 0.4 the proposed method with 110 iters and0.4 0.2 0.4 0.20.4 0.20.4 algorithm with 110 iters. (e) Angular super-resolution of λ0.2=0.40.4504. result and λ =00.2 0.4504. 0.2 algorithm 0.2 0 0.2 0 0.2 00.2 with 110 iters. (e) Angular super-resolution 0 0.2 00.2 of the proposed method with 110 iters 0.2 (c) (c)

6 10 8

1086

10

108

10

(e) (e) (e) (e) (e)

(e)

Fig. 6. Fig. The6.first The row first is row the echo is theadded echo by added Gaussian by Gaussian noise with noise20dB; with 20dB; the second the second row is row the angular is the angular super-resolution super-resolution results results from the from leftthe to right: left to (b) right: Angular (b) Angular Fig. 6. Fig.super-resolution The 6. Fig.first The 6. row first Theisrow first the isthe row echo theof isecho added the Tikhonov echo added byregularization. Gaussian added by Gaussian by noise Gaussian noise with noise with 20dB; with 20dB; the 20dB; second the second the row second isrow theisrow angular the isangular the super-resolution angular super-resolution super-resolution results from results from thesuper-resolution left from thetoleft the right: toleft right: (b) toof Angular right: (b) (b)the Angular super-resolution result of result Tikhonov the regularization. (c) Angular (c) Angular super-resolution super-resolution result of result the of Wiener the Wiener filter. (d) filter. Angular (d)results Angular super-resolution result result the Angular of R-L R-L super-resolution super-resolution super-resolution result of110 the result of Tikhonov the of(e) Tikhonov theAngular Tikhonov regularization. regularization. regularization. (c)result Angular (c) of Angular (c) super-resolution Angular super-resolution super-resolution result result of 110 the result ofWiener the ofWiener the filter. Wiener filter. (d) filter. (d)0.4504. Angular (d)super-resolution Angular super-resolution super-resolution resultresult of the result ofR-L theofR-L the R-L algorithm algorithm with result 110 with iters. (e) iters. Angular super-resolution super-resolution result the of proposed the proposed method method with with iters 110 and iters λ = and 0.4504. λAngular = algorithm algorithm algorithm with with 110 iters. with 110 iters. 110 (e) Angular iters. (e) Angular (e)super-resolution Angular super-resolution super-resolution resultresult of the result ofproposed theofproposed the method proposed method with method with 110 iters with 110 iters and 110 λiters and = 0.4504. λand = 0.4504. λ = 0.4504. (a)(a) (a) (b) (b) (b) (a) (b) 1.2 1.2 1.2 1.2 1.2 1.2 (a) (b) 1.2 1.2 1 1.2 1.2 1.2 11.2 1.2 1 1.2 1.2 1.2 11.2 Normalized Amplitude Normalized Amplitude






Figure 6. (a) The echo added by Gaussian noise with 20 dB; (b) Angular super-resolution result of the Tikhonov regularization; (c) Angular super-resolution result of the Wiener filter; (d) Angular super-resolution result of the Richardson–Lucy (R-L) algorithm with 110 iters; (e) Angular super-resolution result of the proposed method with 110 iters and λ = 0.4504. Normalized Amplitude Normalized Amplitude





Normalized Normalized AmplitudeAmplitude






1.2 1.2 1 1 1 1.2 1 10.8 1 0.8 1 1 1 10.8 1 0.8 1 1 1 1 1.2 0.8 1.2 0.80.8 0.8 0.8 0.80.6 0.8 0.60.8 1 1.2 0.80.6 0.8 0.60.8 0.8 0.8 0.8 0.8 1 1.2 0.6 0.6 1 1.2 0.6 0.6 0.6 0.60.4 0.6 0.40.6 0.6 0.6 0.8 1 0.60.4 0.6 0.40.6 0.6 0.6 0.8 1 0.8 1 0.4 0.40.4 0.4 0.4 0.4 0.4 0.4 0.4 0.40.2 0.4 0.20.4 0.40.2 0.4 0.20.4 0.6 0.8 0.6 0.8 0.6 0.8 0.2 0.20.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0 0.2 0.2 0 0.2 0.4 0.6 00.2 00.2 0.4 0.4 0.66 10 8 0.6 −10 −8 −10 −6 −8 −4 −6 −2 −4 0 −2 2 0 4 2 6 4 8 10 0 0 −10 −8 −10 −6 −8 −4 0−6 −2 10 0 −4 0 0−2 2 0 4 2 6 4 8 6 10 8 0 0 0 0 0 6 84 0 8 10 0 8 −100.2 −8−10−6−6 −8−4−4 −60−2 −2 −4 00 0 −2 20 2 0 4 4Tareget 2 6 6 distribution 48Tareget 8 610 distribution 10 8 10 −8−6 −10 distribution −6−4 −8 −4−2 −6 −2 −4 02 −2 24 0 46 2 68 4 8106 10 8 10 −10 −8 −80.4 −10 −6 −8 −4 −6 −2 −4 0 −2 6 10 10 −10 −80.4 0.2Targets −10 −10 −8 0 −10 2 2 40 4 62−10 (Degree) −10 distribution −8 −10 −6 −8 −10 −4 −6 −8 (Degree) −2 −4 −6 −2 0−4 2−2 0(Degree) 420 (Degree) 642 0.2 864 −6 1086 −4108 −2 10 0 −8 −10 −6 −8 −10 −4 −6 −8 Targets −2 −4 −6 distribution −2 0−4 0.4 2−2 0(Degree) 420 Targets 642 distribution 8Targets 64 108distribution 6(Degree) 108 (Degree) 10 Targets Targets distribution (Degree) Targets distribution Targets distribution (Degree) (Degree) Targets distribution (Degree) Targets distribution (Degree) Targets distribution (Degree) 0 0.2 Tareget distribution Tareget distribution Tareget (Degree) distribution (Degree)(Degree) 0 distribution 0 0.2 Targets distribution Targets Targets (Degree) distribution (Degree)(Degree) 0.2 −10 −8 −6 −4 −2 0 2 4 6 8 10 −10 −8 −6 −4 −2 0 2 4 6 8 10 −10 −8 −6 −4 −2 0 2 4 6 8 10 0 0 0 Targets distribution (Degree) Targets distribution (Degree) Targets distribution (Degree) −10 −8 −6 −4 −2 0 2 4 6 8 10 −10 −8 −6 −4 −2 0 2 4 6 8 10 −10 −8 −6 −4 −2 0 2 4 6 8 10 Targets distribution (Degree) Targets distribution (Degree) Targets distribution (Degree)

(a) (a) (b) (b) (c) (d)(d) (d) (a) (a) (a) (b) (b) (b) (e) (e) (e) (c) (d) (e) 1.2 1.2 1.2 1.2 (c) (d) (e) 1.2 11.2 Fig.1.27.7.1Fig. The 7.first first The row first row theecho is echo the added echobyadded byGaussian Gaussian by Gaussian noise noise 15dB; 15dB; the second therow second row isrow the is angular thesuper-resolution angular super-resolution super-resolution results from from left toleft right: to Angular right: (b) Angular (b) Angular Fig. The row isisthe added noise second is the angular the the left tothe right: (b) 1.2with 1.215dB; 1.2 the 1 with 1with 1.2 1 1.2results 1.2 from 1results 10.8 1 0.8 1 1Angular 1 0.8 1 super-resolution 0.8Angular 10.8 filter. 1 Angular 1 super-resolution 0.8(d) super-resolution result regularization. (c)(c) super-resolution result of result thethe Wiener (d) Angular result of the R-Lof super-resolution super-resolution resultofresult ofthe theTikhonov ofTikhonov the Tikhonov regularization. regularization. (c) Angular super-resolution result of ofWiener thefilter. Wiener filter. (d) Angular super-resolution super-resolution result result of the R-L the R-L Fig. The 0.8 0.8 0.6 0.8 first row is the echo added by Gaussian noise with 15dB; the second row is the angular super-resolution results from the left to right: (b) Angular 0.6 7. 0.80.6 0.8 0.60.8 0.80.6 0.8 0.60.8 algorithm with 150 iters. (e) Angular super-resolution result of the proposed method with 150 iters and λ = 0.8296. algorithm algorithm with 150 with iters. 150 (e) iters. Angular (e) Angular super-resolution super-resolution result result of the of proposed the proposed method method with 150 with iters 150 and iters λ and = 0.8296. λ = 0.8296. 0.6 0.6 7. Fig. rowofisthe theTikhonov echo added by Gaussian noise 15dB; the secondresult row is results from the leftresult to right: (b)R-L Angular 0.4 0.40.6The first super-resolution result regularization. super-resolution ofthe theangular Wiener super-resolution filter. Angular super-resolution of the 0.60.4(c) 0.6Angular 0.6 0.4with 0.6 0.4 0.6(d) 0.40.6 0.4 0.4 0.20.4 0.2 0.40.2result 0.4 0.4 the proposed 0.4 0.4 0.20.4 algorithm with 150result iters. of (e)the Angular super-resolution method with 150ofiters λ0.2=filter. 0.8296. super-resolution Tikhonov regularization. (c)0.2of Angular super-resolution result the and Wiener (d) Angular super-resolution result of the R-L 0.2 0 0.2 00.2 0.2 0 0.2 00.2 0.2 0 0.2 0.2 0 algorithm 150 iters. and =−100.8296. −10 −8 −10 −6 with −8 −4 −6 −2 −4 0 −2 (e) 2 0 Angular 4 2 6 4 super-resolution 8 6 10 8 10 −10 result −10 λ −8 −6 −8 −4 −6 −2 −4 0 −2 2 0 4 2 6 4 8 6 10 8 10 −8 −10−6of −8 the −4 −6proposed −2 −4 0 −2 2method 0 4 2 6with 4 8 150 6 10 iters 8 10 1086

108

0 distribution −6 −8 −10 −4 −6 −8 Targets −2 −4 −6 distribution −2 0Targets −4 2−2 0(Degree) 420 (Degree) 642 864 Targets distribution Targets distribution Targets (Degree) distribution (Degree)(Degree)

1086

108

0 −10 10


0 −8 −10


0 10 −10


0 (Degree) −6 −8 −10 −4 −6 −8 Targets −2 −4 −6 distribution −2 0Targets −4 2−2 0distribution 420 (Degree) 642 864 Targets distribution Targets distribution Targets (Degree) distribution (Degree)(Degree)


0 −8 −10



1.2


0 −10



(c)(c)

1.2

0 −8 −10

0 (Degree) −6 −8 −10 −4 −6 −8 Targets −2 −4 −6 distribution −2 0Targets −4 2−2 0distribution 420 (Degree) 642 864 Targets distribution Targets distribution Targets (Degree) distribution (Degree)(Degree)

1086

108

10

10 8

(c) (c) (c) (c) (c)

Normalize

Normal Normalize

Normalize

(d) (d) (d) (d) (d)

0.6 0.4 0.4 0.2

Normal Normalized

10 8 6

0.6 0.4 0.6 0.4 0.2 0.4 0.2 0.2 0 0 −8 −10 10 6 10 8 10 −10 0.2−6 −8 −4 −6 −2 −4 0 −2 2 0 4 2 6 4 8 6 10 8 0 0 Targets distribution (Degree) −10 −10 −8 −6 −6 −4Targets −4 −2distribution −20 02 24 (Degree) 46 68 10 8 10 10 8 10 0−8 −10 −8 Targets −6 Targets −4 distribution −2(Degree) 0(Degree) 2 4 6 8 10 6 8 10 distribution Targets distribution (Degree)

Norma Normalized

6 10 8

0.4 0.4 0.6 0.4 0.2 0.4 0.2 0.4 0.20 0.2 0 10 −10 0.2−6 −8 −4 −6 −2 −4 0 −2 2 0 4 2 6 4 8 −8 −10 0 0 Targets distribution (Degree) −10 −10 −8 −8 −6 −6 −4 Targets −4 −2 distribution −2 0 02 24 (Degree) 46 68 0 10 −10 −8Targets −6Targets −4 distribution −2 0 2 4 distribution (Degree) (Degree) Targets distribution (Degree)

Normalize Norma

Norma Normalized

Normaliz

Normaliz Norma

0.4 0.4 0.6 0.40.2 0.4 0.2 0.4 0.2 0 0.2 0 0.2−6 −8 −4 −6 −2 −4 0 −2 2 0 4 2 6 4 8 −10 −8 −10 0 0 (Degree) −10 −10 −8 −8 −60 −6 −4 Targets −4 −2 distribution −2Targets 0 02distribution 24 (Degree) 46 68 −10 −8 −4 −2 (Degree) 0 2 4 Targets−6 Targets distribution distribution (Degree) Targets distribution (Degree)

(e) (e) (e) (e) (e)

Fig. 6. Fig. The6.first The row first is row the echo is theadded echo by added Gaussian by Gaussian noise with noise20dB; with 20dB; the second the second row is row the angular is the angular super-resolution super-resolution results results from the from leftthe to right: left to (b) right: Angular (b) Angular Sensors 2015, 15 6938

TABLE I

1.2 1 1.2 0.8 1 0.6 0.8 0.4 0.6

1.2



1 1.2 0.8 10.8 1 1 0.6 0.80.6 0.8 0.8 0.4 0.60.4 0.6 0.6 0.2 0.40.2 0.4 0.4 0 0.2 0 0.2 0.2−6 −8 −4 −6 −2 −4 0 −2 2 0 4 2 6 4 8 −10 −8 −10 Tareget distribution Tareget distribution (Degree) (Degree) 0 0 0 −10 −10 −8 −8 −6 −4 −2 0 0 2 240 462 684 −10 −6 −8 −4 −6 −2−4 −2 TaregetTareget distribution distribution (Degree) (Degree)(Degree) Tareget distribution


1.2


1.2

1.2 1 1.2



Fig. Fig. 6. super-resolution 6. The first row isfirst the isthe echo theof added added by Gaussian by Gaussian noise noise with with 20dB; 20dB; the20dB; second the second row is the is angular the super-resolution super-resolution results from from the left the the to leftresult right: to right: (b) Angular (b)of Angular super-resolution result of result Tikhonov the regularization. regularization. (c) Angular (c) Angular super-resolution super-resolution result of result the of Wiener the Wiener filter. (d) filter. Angular (d) results Angular super-resolution super-resolution result the R-L the R-L Fig.The 6. first Therow row isecho the Tikhonov echo added by Gaussian noise with therow second row is angular the angular super-resolution results from left toof right: (b) Angular super-resolution super-resolution result of the of Tikhonov the Tikhonov regularization. regularization. (c) Angular (c) of Angular super-resolution super-resolution result of110 the ofiters Wiener the Wiener filter. filter. (d) (d) Angular super-resolution super-resolution result result ofresult the of R-L the R-LR-L algorithm algorithm with result 110 with iters. 110 (e) iters. Angular super-resolution super-resolution result result theAngular of proposed the proposed methodresult method with with 110 iters λ = and 0.4504. λfilter. =Angular 0.4504. super-resolution result of(e) theAngular Tikhonov regularization. (c) super-resolution result of and the Wiener (d) Angular super-resolution of the algorithm algorithm with with 110 110 iters. iters. (e) Angular (e) Angular super-resolution super-resolution result result of the of proposed the proposed method method with with 110 110 iters iters and and λ = λ 0.4504. = 0.4504. algorithm with 110 iters. (e) Angular super-resolution result of the proposed method with 110 iters and λ = 0.4504. 1 1.2 1 0.8 1 0.8 0.6 0.8 0.6 0.4 0.6 0.4 0.20.4

1.2

Relative Error

10 86

108

10


1.2

1

6 10

10 8 6

1.2

1.2

1 1.2 0.8 1 0.6 0.8 0.4 0.6 0.2 0.4 0 −10 10 0.2

1.2

8

10 8


108

0 0.2 0.2 00.2 −10 −8 −10 −6 −8 −4 −6 −2 −4 0 −2 2 0 4 2 6 4 8 6 10 8 10 Targets distribution Targets distribution (Degree) (Degree) 0 0 0 −10 −10 −8 −8 −6 −6 −4 −4 −2 −20 8 −10 −8 −6 −4 −202 024 246 468 610 810 10 TargetsTargets distribution distribution (Degree) (Degree) Targets distribution (Degree)

1 1.2 0.8 1 0.6 0.8 0.8 0.4 0.6 0.6 0.2 0.4 0.4 0 0.2 −8 −10 10 0.2−6 −8 −4 −6 −2 −4 0 −2 2 0 4 2 6 4 8 6 10 8 Targets distribution Targets distribution (Degree) (Degree) 0 0 0 −8 −10 −8 −6 −6 −4 −6 −4 −2 −4 −20 −202 8 −8 0 24 2 46 4 68 6 10 810 10 10−10 −10 TargetsTargets distribution distribution (Degree) (Degree) Targets distribution (Degree)


10 86

10

1 1.2 0.8 1 0.6 0.8 0.8 0.4 0.6 0.6 0.2 0.4 0.4 0 0.2 −8 −10 0.2−6 −8 −4 −6 −2 −4 0 −2 2 0 4 2 6 4 8 Targets distribution Targets (Degree) distribution (Degree) 0 0 0 10 −10 −10 −8 −8 −6 −6 −4 −4 −2 −2 0 02 20 4 426 6 −10 −8 −6 −4 −2 48 TargetsTargets distribution distribution (Degree) (Degree) Targets distribution (Degree)



6 10 8

1.2

1 1.2 0.8 1 0.6 0.8 0.4 0.6 0.2 0.4 0 10 0.2−10

6 10 8


1.2

1 1.2 1 1.2 1.2 0.8 0.8 1 1 1 0.6 0.6 0.8 0.8 0.8 0.4 0.4 0.6 0.6 0.6 0.2 0.2 0.4 0.4 0.4 0 0 0.2 −100.2 −8 −10 0.2−6 −8 −4 −6 −2 −4 0 −2 2 0 4 2 6 4 8 Targets distribution Targets distribution (Degree) (Degree) 0 0 0 −10 −10 −8 −8 −6 −4−8 −4 −2−6 −20−4 02 240 462 684 −10 −6 −2 TargetsTargets distribution distribution (Degree) (Degree) (Degree) Targets distribution


1.2



the spikes of the targets lo angular super-resolution res METHODS (a) (a) (b) (b) spikes of the targets looks m (a) (b) (a) (a) (a) (b) (b) (b) of the proposed method co Image Method SNR(dB) ISNR(dB) SSIM be appreciated in Fig.7, as Tikhonov 2.93 −0.43 0.61 Fig.5(a) Wiener 3.93 1.43 0.71 separated, and the noise am BSNR=14.91dB R-L 7.71 5.20 0.89 presented in the result by u Proposed 8.45 5.94 0.91 The reason is that the R(c) (c) Tikhonov (d) (d) 2.45 0.02 0.54 (e) (e) (c) (c) (c) (d) (d) (d) (e) (e) (e) likelihood criterion, which (c) (d)3.17 (e) Wiener 0.69 0.64 Fig. 7. Fig. The7.first The row first is row the echo is theadded echo Fig.6(a) by added Gaussian by Gaussian noise with noise15dB; with 15dB; the second the second row is row the angular is the angular super-resolution super-resolution results results from the from leftthe to right: left to (b) right: Angular (b) Angular method with no penalty fu super-resolution result of result Tikhonov the Tikhonov regularization. regularization. (c) Angular (c) Angular super-resolution super-resolution result of result the of Wiener the Wiener filter. (d) filter. Angular (d) results Angular super-resolution super-resolution result the R-L the R-L Fig.The 7. first Therow row isecho the echo added by Gaussian noise with therow second row is angular the angular super-resolution results from left toof right: (b) Angular BSNR=9.94dB R-L 6.28 3.74 0.83 Fig. Fig. 7. super-resolution 7. The first row isfirst the isthe echo theof added added by Gaussian by Gaussian noise noise with with 15dB; 15dB; the15dB; second the second row is the is angular the super-resolution super-resolution results from from the left the the to leftresult right: to right: (b) Angular (b)of Angular algorithm algorithm with 150 with iters. 150 (e) iters. Angular (e) Angular super-resolution super-resolution result of result the of proposed the proposed method method with 150 with iters 150 and iters λ = and 0.8296. λ = 0.8296. super-resolution result the Tikhonov regularization. (c) Angular super-resolution of Wiener the filter. Wiener (d) Angular super-resolution of the R-L maximum likelihood criteri super-resolution super-resolution result result of the of Tikhonov theofTikhonov regularization. regularization. (c) Angular (c) Angular super-resolution super-resolution result result ofresult the of Wiener the filter. (d)filter. Angular (d) Angular super-resolution super-resolution result result ofresult the of R-L the R-L 7.15 4.66 0.88 algorithm with 150 iters. (e)echo Angular super-resolution result the proposed method 150 iters =Angular 0.8296. algorithm algorithm withwith 1507. 150 iters. iters. (e) Angular (e) Angular super-resolution super-resolution result result ofProposed the of proposed theofproposed method method with with 150with 150 iters iters and and λ =and λ0.8296. =λ0.8296. Figure (a) The added by Gaussian noise with 15 dB; (b) super-resolution ment between the measurem Tikhonov 2.30 0.05 0.50 result of the Tikhonov regularization; (c) Angular super-resolution result of the Wiener high noise estimates, partic Fig.7(a) Wiener 2.83 0.47 0.58 ThisAngular conclusion is also su R-L of the 5.60 3.19 with0.80 filter; (d) AngularBSNR=7.69dB super-resolution result R-L algorithm 150 iters; (e) Proposed 6.71 4.36 0.84 evaluation parameters of t super-resolution result of the proposed method with 150 iters and λ = 0.8296. than the evaluation paramete methods. 0.95 Table I shows the com 0.9 regularization method, Wie 0.85 proposed deconvolution me 0.8 SNRs, ISNRs, and SSIMs. R 0.75 the proposed deconvolution 0.7 Tikhonov SSIM values while keepin 0.65 Wiener R−L the four methods, both Tik 0.6 Proposed Wiener filter use the inverse 0.55 R-L algorithm may be an 0.5 for super-resolution imagin 0.45 a small number of iteratio 0.4 This phenomenon is a gen 0.35 5 10 15 20 25 30 Noise Level (dB) leading to the larger error o resolution result. These resu FigureFig. 8. Relative error comparison under various noise levels. 8. Relative errorperformance performance comparison under various noise levels. deconvolution algorithm fo competitive over the conven results by thosesuper-resolution methods under resultInby In Figure 5, we canThe seeangular that thesuper-resolution visual quality of the angular using the addition, another noti Fig.6,method. and Fig.7. algorithm com R-L algorithm looksvarious similarnoise to thelevels resultarebyshown using intheFig.5, proposed In The Figuredeconvolution 6, the proposed visual quality ofresult, super-resolution results the proposed tion algorithms approach gives the super-resolution where the spikes of using the targets look fairly separate, whereasis that the deconvolution algorithm are quite competitive with those using suppresses the spurious p in the angular super-resolution result, by using the R-L algorithm, the method, spikes ofand the targets looks more the Tikhonov regularization method, Wiener filter resolution results under di connected. The improvement of the proposed method to R-L algorithm can also R-L algorithm. It is noted that compared the angular super-resolution to be a appreciated better angular superresults by using proposed deconvolution exhibitby the in Figure 7, as the spikes of the targetsthe have been separated, and thealgorithm noise amplified algorithm of R-L precision. Fig.8 shows close match to the original scene. the algorithms in the diffe is not presented in the result by using the proposed method. It can be seen from Fig.5Fig.7 that the spikes of the targets from the Fig.8 that the Tik The reason is that the R-L algorithm is based on the maximum likelihood criterion, which is identical in the results of the Tikhonov regularization and Wiener filter Wiener filter have higher r to the deterministic method with no penalty function or prior information. The maximum likelihood are more connected compared with the results of the R-L levels, while the proposed criterion aims at maximizing agreement between the measurement and theisobject, high algorithmthe and the proposed method. The reason for this that which better yields than the conventiona noise estimates, particularly whenregularization the noise levelmethod is low.isThis conclusion is also of supported by Table 1, the Tikhonov based on noise power The reason is that the prop the observed scene, which makes the profile of targets overly explored the prior informa smooth. The Wiener filter obtains an optimal result in the sense to a more stable solution of minimizing the mean square error between the obtained problem and the evaluation result and the true scene using the correlation information other comparative methods. 0.2 0.4

S UMMARY OF ANGULAR SUPER - RESOLUTION RESULTS FOR DIFFERENT

1.2 1

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in which the evaluation parameters of the proposed algorithm are better than the evaluation parameters associated with the comparative methods. Table 1. Summary of angular super-resolution results for different methods. improved signal-to-noise ratio; SSIM, structure similarity. Image

Method

SNR (dB)

ISNR (dB)

SSIM

Tikhonov

2.93

0.61

Figure 5a

Wiener

−0.43

3.93

1.43

0.71

BSNR = 14.91 dB

R-L

7.71

5.20

0.89

Proposed

8.45

5.94

0.91

Tikhonov

2.45

0.02

0.54

Figure 6a

Wiener

3.17

0.69

0.64

BSNR = 9.94 dB

R-L

6.28

3.74

0.83

Proposed

7.15

4.66

0.88

Tikhonov

2.30

0.05

0.50

Figure 7a

Wiener

2.83

0.47

0.58

BSNR = 7.69 dB

R-L

5.60

3.19

0.80

Proposed

6.71

4.36

0.84

ISNR,

Table 1 shows the comparisons between the Tikhonov regularization method, Wiener filter, R-L algorithm and the proposed deconvolution method on super-resolution results in SNRs, ISNRs and SSIMs. Referring to Table 1, it is shown that the proposed deconvolution algorithm produces the highest SSIM values while keeping ISNR at a high level. Among the four methods, both the Tikhonov regularization method and the Wiener filter use the inverse filtering, which gives poor results. The R-L algorithm may be an attractive deconvolution approach for super-resolution imaging, but the noise is amplified after a small number of iterations, and the false targets emerge. This phenomenon is a generic problem for the R-L algorithm, leading to the larger error of the corresponding angular super-resolution result. These results demonstrate that the proposed deconvolution algorithm for angular super-resolution is quite competitive over the conventional deconvolution algorithms. In addition, another noticeable advantage of the proposed deconvolution algorithm compared to conventional deconvolution algorithms is that the proposed deconvolution algorithm suppresses the spurious peak appearance in the angular-resolution results under different noise levels, which leads to a better angular super-resolution performance in terms of precision. Figure 8 shows the relative error performance of the algorithms in the different noise levels. It can be seen from Figure 8 that the Tikhonov regularization method and Wiener filter have higher relative errors under various SNR levels, while the proposed deconvolution algorithm is much better than the conventional algorithms in terms of precision. The reason is that the proposed deconvolution algorithm has explored the prior information about the

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targets. This leads to a more stable solution to the associated deconvolution problem, and the evaluation parameters are also better than other comparative methods. 4.2. Experiment In order to show the validity of the proposed method, we present the real data results. Figure 9 shows the tested scene in which three buildings and their distribution are shown. Employing the scanning radar system in Figure 10, the real data are acquired. The scanning radar system parameters for the experimental result are shown in Table 2. The resolution of the range dimension was calculated to be 1 m. The distance of each building is 45 m. The range from the scene center to the radar system is about 594 m, and the angular resolution in the scene center is 95 m. The echo from three buildings is much stronger than that from the other areas of the observed scene, so that the scene can be consider as consisting of three main targets. to bescattering 1m. Thepoint distance of each building is 45m. Fig.9 shows The range compression is applied the their recorded data. The imaging result the and the corresponding the three buildingsto and distribution. The range from scene center to the radar system is about 594m and the angular profile of the target area are shown in Figure 11a,b, respectively. From Figure 11b, we can see that the resolution in the scene is the 95m. The echo from three echo of adjacent buildings is overlapping and center covering building features. buildings is much stronger than that from the other areas of the observed scene2.so that the scene can be consider consisting Table Experimental parameters for real as data. of three main scattering point targets. Parameters

TABLE II

Value

E XPERIMENTAL PARAMETERS FOR

Carrier frequency

REAL DATA

10

Band width Parameters

75 Value Carrier frequency 10 Pulse duration 2 Band width 75 Sampling frequency 200 Pulse duration 2 Pulse repetition frequency 200 Antenna scanning velocity 70 Antenna scanning velocity 70 Antenna scanningarea area −45 45 Antenna scanning −45 ∼± +45 Main-lobe beam width 5

Main-lobe beam width

Units

5

GHz

MHz Units GHz µs MHz MHz µs MHz ◦ /s ◦/s ◦◦ ◦

◦

Height 45m

45m

Fig. 10. The scanning radar syst to collect scattering data. Azimuth

0 59

4m

Radar

Range

Fig. 9.

Figure The tested scene.

9. The tested scene.

The range compression is imaging result and the cor are shown in Fig.11(a) and Fig.11(b), we can see that overlap and cover the build As compared with other and the regularization para are hand-tuned alternative

width 75 as consisting MHz observed scene Band so that the scene can be consider Pulse duration 2 µs of three main scattering point targets. Pulse repetition frequency 200 TABLE II Antenna scanning velocity 70 E XPERIMENTAL PARAMETERS FOR REAL DATA Antenna scanning area −45 ∼ +45 Sensors 2015,beam 15 width Main-lobe 5

Parameters Value Carrier frequency 10 Band width 75 duration 2 is 45m. Fig.9 Pulse shows Height Pulse repetition frequency 200 45m 45m . The range from the Antenna scanning velocity 70 594m and the angular Antenna scanning area −45 ∼ +45 The echo fromMain-lobe three beam width 5

Units GHz MHz µs MHz ◦/s

MHz ◦/s ◦

6941

◦

10

0

(a)

Fig. 10. The s to collect scatter

Azimuth

The range imaging resu Radar ◦ are shown in ◦ the other areas of the Fig.11(b), we Range consider as consisting (b) overlap and c As compar Height Fig. 9. The tested scene. Fig. 10. The scanning radar system[(a)front side and (b)back side] utilized 45m 45m and the regu to collect scattering data. Azimuth REAL DATA 0 are hand-tun resolution re e Units 59 The range compression is applied to the recorded data. The dec 4m proposed GHz (b) (a) imaging result and area (b)the corresponding profile of targets Radar (a) MHz parameter λ = Fig. 10. The radar side and (b)back side] utilized arescanning shown insystem[(a)front Fig.11(a) and Fig.11(b), respectively. FromInthe µs Fig.12, w to collect scattering data. MHz Fig.11(b), canback see that echo oftoadjacent is of the Figure0 10. The scanningAzimuth radar Range system ((a) front side andwe(b) side)theutilized collect buildings mance ◦/s overlap and cover the building features. ◦ angular super +45 scattering data. comparedis with other methods, maximum iterations ◦Fig. 9. 59 tested scene. The The range As compression applied to the recordedthe data. The 4m The right col and the parameter the proposed algorithm imaging result and regularization the corresponding profile ofoftargets area Radar scene corresp hand-tuned so that theFrom corresponding superare shownare in Fig.11(a) and alternative Fig.11(b), respectively. the the angular s (a) best buildings visual quality. For the Fig.11(b), resolution we can see result that thepresents echo of the adjacent is Range larization me overlap and cover thedeconvolution building features. proposed algorithm, we used the regularization 1 results, while As compared with λ other methods, the iteration maximumnumber iterations parameter = 1.47 and the is 100. Fig. 9. The tested scene. 0.8 resolution res and the regularization of the algorithm In Fig.12, parameter we investigate theproposed angular super-resolution perfor0.6 algorithm, re are hand-tuned alternative so that the corresponding supermance of the approaches. The first column of Fig.12 gives the 0.4 (b) amplitude of resolution result presents the best visual quality. For the 0.2 angular super-resolution imaging results of different methods. proposed deconvolution algorithm, we used the regularization different. Fig. 10. The scanning radar system[(a)front side and (b)back side] The right column of Fig.12 presents the profile of the are targets 0 utilized parameter λ = 1.47 and 40 the iteration number is 100. 20 60 80 100 to collect scattering data. middle scene corresponding thesuper-resolution left column. The first row presents buildi Azimuthto (samples) Azimuth In Fig.12, we investigate the angular perforthe approaches. angular super-resolution using thetheTikhonovbetween regu- the (a) mance of the The (b) first columnresults of Fig.12 gives (a) (b) This leads to larization method. The second row contains the Wiener filter The range compression is applied to the recorded The angulardata. super-resolution imaging results of different methods. the other two Fig. 11. The received echo signal of the tested scene after range compression 1 while the presents third andthebottom imaging result and the corresponding profile The of targets area of right results, column Fig.12 profilerows of thecontain targets angular superand corresponding profile of the target area. 11. (a) The received echo signal of the tested scene after range compression; (b) The also fits their 0.8 Figure are shown in Fig.11(a) and Fig.11(b), respectively. From the resolution using The R-Lfirst algorithm and the proposed scene corresponding toresults the leftby column. row presents 0.6 Fig.11(b), we can see that the echo of adjacent buildings is the angular super-resolution results using the Tikhonov regualgorithm, respectively. It can be obviously observed that the profile of the target ares. (a) Range 0.4 overlap and cover the building features. larization amplitude method. Theofsecond row contains thethe Wiener the targets profile in rightfilter column of Fig.12 0.2 As compared othermethods, methods, thethe maximum 1 results,iterations while the third andThe bottom superareiterations different. reason forcontain this isangular that the distance from As compared withwith other maximum and therows regularization parameter of the 0 0.8 and20the regularization parameter of the proposed algorithm resolution results building by using to R-L algorithm and the proposed 40 60 80 100 middle the radar is about 594m, while the distance Azimuth (samples) 0.6 the proposed algorithm are hand-tuned so that the Itcorresponding super-resolution result are hand-tuned alternative so that the alternatives, corresponding algorithm,superrespectively. can be observed that theis about between the radar andobviously the left/right building 597m. 0.4 resolution result presents the best visual quality. For the amplitude of the targets profile in the right column of Fig.12 (b) presents the best visual quality. For the proposed deconvolution algorithm, we used the regularization This leads to the profile of the middle building is higher than 0.2 proposed deconvolution algorithm, we used the areregularization different. The reason for this is that the distance from the 0 the other two profiles in the right column of Fig.12, which Fig. 11. The received signal of the and tested scene after number range compression parameter λ echo = the iteration is 100. λ= 1.47 the iteration number is 100. 20 parameter 40 1.47, 60 and 80 100 middle building to the radar is about 594m, while the distance Azimuth (samples) and corresponding profile of thewe target area. the angular super-resolution perforalsoradar fits and theirthe physical In Fig.12, investigate between the left/righttruth. building is about 597m.

0

4m

Normalized amplitude



59

In Figure 12, (b) we investigate the angular super-resolution performance of the approaches. The first mance of the approaches. The first column of Fig.12 givestothe This leads the profile of the middle building is higher than column of Figure 12 gives the angular super-resolution imaging results of different methods. The right angular super-resolution imaging results of different methods. Fig. 11. The received echo signal of the tested scene after range compression the other two profiles in the right column of Fig.12, which The right column of fits the their targets and corresponding profile of the target area. of Fig.12 presents the profile also physical truth. column of Figure 12 presents the profile of the target scene corresponding to the left column. The first scene corresponding to the left column. The first row presents the angular super-resolution results usingresults the Tikhonov row presents the angular super-resolution usingreguthe Tikhonov regularization method. The second larization method. The second row contains the Wiener filter row contains the Wiener filter results, while the third and bottom rows contain angular super-resolution results, while the third and bottom rows contain angular superresults using R-Land algorithm and the proposed results byresolution using the R-Lby algorithm the proposed algorithm, respectively. It can be obviously observed algorithm, respectively. It can be obviously observed that the that the amplitudes of targets the target in the right column of Figure 12 are different. The reason for amplitude of the profileprofiles in the right column of Fig.12 arethe different. The reason for this is thatbuilding the distance the is about 594 m, while the distance between this is that distance from the middle to from the radar 100 middle building to the radar is about 594m, while the distance the radarbetween and thetheleft/right is about 597 m. This leads to the profile of the middle building being radar andbuilding the left/right building is about 597m. Thisthe leads to thetwo profile of the middle is higherof than higher than other profiles in the building right column Figure 12, which also fits their physical truth.

ne after range compression

the other two profiles in the right column of Fig.12, which also fits their physical truth.

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11 11 11 11

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(a)

(c)

(c)

(c)

(c) (c) (c) (c)

(c) (c)

(e)

(e) (e)

(e) (e)

(e) (e) (e) (e)

0.60 1 0.4 0.8 0.2 0.6 0 0.4 0.2 1 0 0.8

0.6 1 0.4 0.8 0.2 0.6 0 1 0.4 0.8 0.2 0.6 10 0.4 0.8 0.2 0.6 0 0.4 0.2 1 0 0.8 0.6 0.41

0.8 0.2 0.6 0 1 0.4 0.8 0.2 0.6 10 0.4 0.8 0.2 0.6 0 0.4 0.2 1 0 0.8

Normalized Normalized Normalized Normalized amplitude amplitude amplitude amplitude

0.8 0.2

11

1 0.8 1 0.6 0.8 0.6 0.8 0.4 0.6 0.4 0.6 0.2

0.4 0.2 0.4 0 20 0.2 0 0.2 0 0

11 20 60 4080 60 80 100 (samples) 80 Azimuth 20 (samples)40 Azimuth60

100 100

(samples) 80 20 40 Azimuth60 (b) 20 40 80 (b) 60 40 60 80 100 Azimuth (samples) Azimuth (samples) Azimuth (b) (samples)

100 100

40

(b)

20

(b) 20


(a)

0.6 1 0.4 0.8 0.2 0.6 0 1 0.4

1 1 0.8 1 0.8 1 0.6 0.8 0.6 0.8 0.4 0.6 0.6 20 0.4 0.2 0.4 0.2 0.4 0 20

20

0.2 0 0.2 0 0

20

40

60 80 Azimuth (samples)

40


40

40

0.4

0.2 20 0.4 0

0.2 0 0.2 0 20 0

20

40

0 0.6 1

100


100


80 80

100 100

80 80

100 100

80 80

100 100

80 80

100 100

(d) (d) (d)

100

(d)


100


100

(f) 40

60 80 20 (samples) 40 Azimuth

100

60 (samples) 40 Azimuth60 (f) Azimuth (samples) 40 100 20 60 4080 60 20 (samples)40 Azimuth (f) 60 Azimuth (samples) Azimuth (samples) 20

40

(f)


(f) (f) (f)

(f)

100

(f)

0.41

0.4 0.8

(b)

(d)

0.6

0.2 0.8

100

20 40 60 (d) Azimuth60 (samples) 20 60 40 80 40 100 Azimuth (samples) Azimuth (samples) 20 40 (d) 60 20 40 Azimuth 60 (samples) Azimuth (d) (d) (samples)

1 1 0.8

1 0.8 1 0.6 20 0.8 0.6 0.8 0.4 0.6 0.4 0.6 0.2

(b) (b)

(b)

40

20


(a)

(a) (a) (a) (a)

1 0.8


(a)

(a)

NormalizedNormalized amplitudeNormalized amplitude Normalized NormalizedNormalized amplitude amplitudeamplitude Normalized amplitude NormalizedNormalized amplitude Normalized amplitude Normalized NormalizedNormalized amplitude amplitude amplitude Normalized amplitude amplitude amplitude Normalized amplitudeamplitude Normalized

11 1 1 0.8

1 1 20 0.8 1 0.8 1 0.6 0.8 0.6 0.8 0.4 20 0.6 0.4 0.6 0.2

40


100

0.2 0.6 (g) (h) 1 0.40 40 60 80 100 0.8 Fig. 12. The results of the angular super-resolution using different methods. 0.2 Left column from top to Azimuth bottom, angular super-resolution result obtained by (samples) 0.6 0.2 using Tikhonov regularization method refer to angular super-resolution result obtained by using0.4 0.4 0 0Wiener filter method, angular super-resolution result obtained 20 40 60 80 20 40 60bottom corresponds 80 100 (g)super-resolution result using the proposed0.4method, respectively; (h)column by using R-L algorithm, and angular Right from top to100 to the 0.2 0 Azimuth (samples) 0.2 (samples) 80 20 40 Azimuth60 100 0.2 profile of the left column in the target area.

0

(samples) 80 0 top to (h) Fig. 12. The results of the angular(g) super-resolution using different methods. Left column from bottom, angular super-resolution result obtained 20 40 Azimuth 60 100 by 0 20 60 60 80 20 40 filter 80 Azimuth 100 (h)super-resolution (samples) using Tikhonov regularization method refer (g) to angular super-resolution result obtained by using Wiener method,40angular result 100 obtained Azimuth (samples) Azimuth (samples) (g) (h) byFig. using super-resolution result different using themethods. proposedLeft method, column from top to bottomresult corresponds 12. R-L Thealgorithm, results of and the angular angular super-resolution using columnrespectively; from top toRight bottom, angular super-resolution obtainedtobythe The left column of Fig.12 shows the angular superright column of Fig.12. The spikes of buildings in the result of (g) (h) profile of12. the left column in the target area. (g) Fig.Tikhonov The results ofmethod the angular using result different methods. column from top to bottom, angular result super-resolution result obtained by (g) (h)method, using regularization refer tosuper-resolution angular super-resolution obtained by usingLeft Wiener filter angular super-resolution obtained (h) The results theItangular using differentthe methods. Left column from topfrom toseparated bottom, angular super-resolution result obtained by resolution results. can besuper-resolution notedtothat the proposed proposed method look fairly whereas thesuper-resolution spikes using Tikhonov regularization method refer angular super-resolution result obtained byRight using Wiener filter angular result obtained byFig. using12. R-Limaging algorithm, andofangular super-resolution result using the proposed method, respectively; column top method, to bottom corresponds to the Fig. 12. The results of the angular super-resolution using different methods. Left column from top to bottom, angular super-resolution result obtained by profile of the left column in the target area. using Tikhonov regularization method refer to angular super-resolution result obtained by using Wiener filter method, angular super-resolution result obtained Fig. 12. The results of the angular super-resolution using different methods. Left column from top to bottom, angular super-resolution result obtained by by using R-L algorithm, and angular super-resolution result using the proposed method, respectively; Right column from top to bottom corresponds to the method gives theresults best results in termssuper-resolution of visible quality. This is ofmethods. buildingsLeft in column the other three results look more super-resolution connected. Fig. 12. The of the angular using different from top to bottom, angular result obtained by using Tikhonov regularization method refer to super-resolution angular super-resolution result obtained by using Wienerbyfilter method, angular super-resolution result obtained by using algorithm, and angular result using the proposed method, respectively; Right column from topsuper-resolution to bottom corresponds to the using Tikhonov regularization method refer to angular super-resolution result obtained using Wiener filter method, angular result obtained profile of R-L the left column in the target area. the fact that the proposed method is able to suppress the noise Therefore, we believe that the proposed method for angular The left column of Fig.12 shows the angular superright column of Fig.12. The spikes of buildings in the result of using Tikhonov regularization method refer to angular super-resolution result obtained by using Wiener filter method, angular super-resolution result obtained byby using R-L algorithm, and angular super-resolution result usingresult the proposed method, respectively; Right column from top tocolumn bottom from corresponds tobottom the profile of the left column in the target area. using R-L algorithm, and angular super-resolution using the proposed method, respectively; Right top to corresponds to the by using R-L algorithm, super-resolution result using respectively; column from top bottom corresponds to the amplification incorporating the prior information the the super-resolution in scanning radarRight is useful inwhereas real applications. profile of theimaging leftby column in theand target area. resolution results. It angular can be noted that the about proposed theproposed proposedmethod, method look fairly separated thetospikes

(g)

(h)

Figure 12. The results ofarea. the angular super-resolution using different methods. (a) Angular profile of left column in the target The left column of the Fig.12 shows the angular results superright column of Fig.12. The spikes of buildings in the result of profile of the the leftbest column in in the target area. targets. We can see that angular super-resolution method gives the results terms of visible quality. This is of buildings in the other three results look more connected. resolution imaging results. It can be noted that the proposed the proposedright method look fairly separated thebuildings V.that Cof ONCLUSION obtained by using the Tiknonov method and Tikhonov The left column of Fig.12 shows the supercolumn Fig.12. Thewhereas spikes of the resultof of the fact that the proposed method isregularization able to suppress theangular noise Therefore, weregularization believe the proposed method for angular super-resolution result obtained using method; (b)spikes The inprofile method the best results terms ofparticularly visible quality. This is right ofsuperbuildings in the other results lookradar more connected. Thegives left column ofinthe Fig.12 shows the angular right column of Fig.12. The spikes ofreceived buildings in thethe result of The left column of Fig.12 the noted angular supercolumn ofsuper-resolution Fig.12. Thethree spikes of buildings in separated the result ofwhereas The angular in scanning has Wiener filter method areresults. still noisy, around the amplification by incorporating prior information about the proposed super-resolution in scanning radar islook useful in real applications. resolution imaging Itshows can be that the the proposed method fairly spikes the fact that the proposed method isbeable to the noise Therefore, we believe thatyears, theFig.12. proposed method for The left column of Fig.12 shows the angular superright of The spikes of buildings in the result of resolution imaging results. It angular can noted that the proposed the proposed method look fairly separated whereas theangular spikes resolution imaging results. It can besuppress noted the proposed the proposed method look fairly separated whereas the spikes much attention incolumn recent however, limited work about building boundaries. The improvement ofresult the proposed method The left column of Fig.12 shows thethat angular superright column of Fig.12. Thethree spikes of buildings inthe the result of Angular super-resolution obtained using Tikhonov regularization method in target targets. We can see that the super-resolution results method gives the best results in terms of visible quality. This is of buildings in the other results look more connected. amplification by incorporating prior information about the super-resolution in deconvolution scanning radar is useful infor real applications. resolution imaging results. Itregularization can beappreciated noted that the proposed the proposed method look fairly separated whereas the spikes spikes method gives the best results in the terms of visible quality. This is proposed of buildings inbuildings the other results look more connected. V. three C ONCLUSION the application of algorithm angular supercompared to the other methods can also be in the method gives the best results in terms of visible quality. This is of in the other three results look more connected. obtained by using the Tiknonov method and resolution imaging results. It can be noted that the the proposed method look fairly separated whereas the the fact that the proposed method is able to suppress the noise Therefore, we believe that the proposed method for angular targets. We can see that the angular super-resolution results themethod fact that the proposed isinable to able suppress the quality. noise Therefore, we believe that the proposed method for angular gives the best results terms of This is of buildings in the other three results look more connected. resolution in scanning radar has been Since the area; (c) Angular super-resolution result obtained using Wiener filter method; (d) The The super-resolution scanning radar received Wiener filter method are method still noisy, particularly around the the the fact that the proposed method is to suppress noise Therefore, believe thatreported. the proposed method for angular method the best results in the terms of visible visible quality. This isangular of buildings the other three results look moreapplications. connected. V. we Cin ONCLUSION amplification bythe incorporating prior information the super-resolution ininscanning radar ishasuseful in real obtained bygives using Tiknonov regularization method and amplification bythe incorporating the prior about the about in radar is useful realproposed applications. the fact that proposed method is able to suppress suppress thesuper-resolution noise Therefore, weyears, believe thatinlimited the method for angular angular much attention in scanning recentwe however, work about building boundaries. The improvement ofinformation the proposed method amplification by incorporating the prior information about the super-resolution in scanning radar is useful in real applications. the fact that the proposed method is able to the noise Therefore, believe that the proposed method for The angular super-resolution in scanning radar method has received in the target Wiener filter method are the stillsuper-resolution noisy, particularly around the targets. We can see that the also angular super-resolution results targets. We can see that angular super-resolution results profile of Angular result obtained Wiener filter the application ofusing deconvolution algorithm for angular super-real applications. compared to the other methods can be appreciated in the about amplification by incorporating the prior information the super-resolution in scanning radar is targets. can see thatTiknonov theregularization angular super-resolution results V. Climited ONCLUSION V. C ONCLUSION much attention in recent years, work aboutin amplification bythe incorporating the prior information about the super-resolution inhowever, scanning radar is useful useful in real applications. building boundaries. The improvement of the proposed method obtained byWe using Tiknonov method and obtained by using the regularization method and resolution in scanning radar has been reported. Since the V. C ONCLUSION targets. We can see that the angular super-resolution results the application of deconvolution algorithm for angular supercompared to the other methods can also be appreciated in the obtained by using the Tiknonov regularization method and targets. We can see that the angular super-resolution results The super-resolution in scanning radarinhas received Wiener filter method are still noisy, particularly around the area; (e) Angular super-resolution result obtained using algorithm; (f) The of angularR-L super-resolution scanning radarprofile has received Wiener filter method are still noisy, particularly around theangularThe V. C Creported. ONCLUSION obtained by Tiknonov regularization method and resolution in The scanning radar has been Sinceabout the radar has received V. ONCLUSION angular super-resolution inhowever, scanning Wiener filter method are still noisy, particularly around the much attention in recent years, limited work building boundaries. The the improvement of the proposed method obtained by using using the Tiknonov regularization method and much attention in however, recent years, limited work about building boundaries. The improvement of the proposed method The angular super-resolution super-resolution inhowever, scanning radar has received Wiener filter method areimprovement stillalso noisy, particularly around theusing the ofWiener deconvolution algorithm for angular compared tofilter the other methods can be appreciated in the Angular super-resolution result obtained byapplication filter method insuperthelimited target area; attention in recent years, work about building boundaries. The of be the proposed method angular in scanning radar has received Wiener method are still noisy, particularly around the theThe application of deconvolution algorithm for angular supercompared to the other methods can also appreciated in the much much attention in recent however, limited work about resolution scanning radarof has been years, reported. Since the building The improvement of the proposed the application deconvolution algorithm for angular supercomparedboundaries. to the other methods can also in the in much attention in recent years, however, limited work building boundaries. The improvement of be theappreciated proposed method method resolution method; in scanning (h) radarThe has profile been reported. Sinceabout the (g)compared Angular super-resolution result using in the Angular the application of deconvolution deconvolution algorithm forof angular supercompared to the the other methods methods can can also also be appreciated appreciated in theproposed resolution in scanning radar hasalgorithm been reported. Since the the application of for angular superto other be the resolution in scanning radar has been reported. Since the resolution in scanning radar has been reported. Since the super-resolution result obtained using proposed method in the target area. The left column of Figure 12 shows the angular super-resolution imaging results. It can be noted that the proposed method gives the best results in terms of visible quality. This is due to the fact that the proposed method is able to suppress the noise amplification by incorporating the prior information about the targets. We can see that the angular super-resolution results obtained by using the Tikhonov regularization method and Wiener filter method are still noisy, particularly around the

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building boundaries. The improvement of the proposed method compared to the other methods can also be appreciated in the right column of Figure 12. The spikes of buildings in the result of the proposed method look fairly separated, whereas the spikes of buildings in the other three results look more connected. Therefore, we believe that the proposed method for angular super-resolution in scanning radar is useful in real applications. 5. Conclusions The angular super-resolution in scanning radar has received much attention in recent years; however, limited work about the application of the deconvolution algorithm for angular super-resolution in scanning radar has been reported. Since the conventional linear deconvolution approaches result in large noise amplification and, thus, a poor angular resolution, this paper proposes a deconvolution algorithm for angular super-resolution in scanning radar. This algorithm can be interpreted as solving a deconvolution problem, corresponding to the original angular super-resolution problem under the MAP criterion. To use the MAP criterion for realizing the angular super-resolution, we first transform the original angular super-resolution problem into an equivalent maximum a posteriori estimation task, so that the prior information about the statistical characteristics of the original scene can be incorporated. In this paper, the Laplace distribution is used to represent the prior information about the targets. This makes the resulting MAP estimation task challenging due to the presence of a “non-smooth” function in the cost function, which calls for an effective and robust differentiable approximation. Therefore, a convex optimization method with differentiable approximation is employed to solve the associated MAP estimation task. In our experiments with synthetic data, the proposed deconvolution algorithm for angular super-resolution has higher precision and suppresses the noise amplification in the angular super-resolution results. Acknowledgments This work was supported by the National Natural Science Foundation of China (No. 61201272). Author Contributions Yuebo Zha: Field data acquisitions, algorithm developing, data processing, writing of the paper; Yulin Huang: Field data acquisitions, project managing and managing; Zhichao Sun; Field data acquisitions, data processing; Yue Wang: Data processing; Jianyu Yang:Project managing and managing. Conflicts of Interest The authors declare no conflict of interest. References 1. Richards, M.A. Iterative Noncoherent Angular Superresolution. In Proceedings of the 1988 IEEE National Radar Conference, Ann Arbor, MI, USA, 20–21 April 1988; pp. 100–105.

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2. Christianson, P. Method for Cross-Range Enhancement of Real-Beam Radar Imagery. U.S. Patent 8,063,817, 22 November 2011. 3. Lenti, F.; Nunziata, F.; Migliaccio, M.; Rodriguez, G. Two-Dimensional TSVD to Enhance the Spatial Resolution of Radiometer Data. IEEE Trans. Geosci. Remote Sens. 2014, 52, 2450–2458. 4. Schiavulli, D.; Nunziata, F.; Pugliano, G.; Migliaccio, M. Reconstruction of the Normalized Radar Cross Section Field From GNSS-R Delay-Doppler Map. IEEE J. Sel.Top. Appl. Earth Obs. Remote Sens. 2014, 7, 1573–1583. 5. Valencia, E.; Camps, A.; Rodriguez-Alvarez, N.; Park, H.; Ramos-Perez, I. Using GNSS-R imaging of the ocean surface for oil slick detection. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens. 2013, 6, 217–223. 6. Du, X.; Duan, C.; Hu, W. Sparse Representation Based Autofocusing Technique for ISAR Images. IEEE Trans. Geosci. Remote Sens. 2013, 51, 1826–1835. 7. Qiu, X.; Hu, D.; Ding, C. Some reflections on bistatic SAR of forward-looking configuration. IEEE Geosci. Remote Sens. Lett. 2008, 5, 735–739. 8. Wu, J.; Yang, J.; Huang, Y.; Yang, H.; Wang, H. Bistatic forward-looking SAR: Theory and challenges. In Proceedings of the 2009 IEEE Radar Conference, Pasadena, CA, USA, 4–8 May 2009; pp. 1–4. 9. Borden, S.C.; Brandt, T.E.; Newcomb, G.L.; Stevenson, J. Method and System for Real Aperture Radar Ground Mapping. U.S. Patent 4,978,960, 18 December 1990. 10. Bamler, R.; Hartl, P. Synthetic aperture radar interferometry. Inverse Probl. 1998, 14, doi:10.1088/0266-5611/14/4/001. 11. Migliaccio, M.; Gambardella, A. Microwave radiometer spatial resolution enhancement. IEEE Trans. Geosci. Remote Sens. 2005, 43, 1159–1169. 12. Gambardella, A.; Migliaccio, M. On the superresolution of microwave scanning radiometer measurements. IEEE Geosci. Remote Sens. Lett. 2008, 5, 796–800. 13. Sundareshan, M.K.; Bhattacharjee, S. Enhanced iterative processing algorithms for restoration and superresolution of tactical sensor imagery. Opt. Eng. 2004, 43, 199–208. 14. Getreuer, P. Total Variation Deconvolution using Split Bregman. Image Process. On Line 2012, 10, doi:10.5201/ipol.2012.g-tvdc. 15. Shah, S. Deconvolution Algorithms for Fluorescence and Electron Microscopy. Ph.D. Thesis, The University of Michigan, Ann Arbor, MI, USA, February 2006. 16. Combettes, P.L.; Trussell, H.J. The use of noise properties in set theoretic estimation. IEEE Trans. Signal Process. 1991, 39, 1630–1641. 17. Gerchberg, R. Super-resolution through error energy reduction. J. Mod. Opt. 1974, 21, 709–720. 18. Lent, A.; Tuy, H. An iterative method for the extrapolation of band-limited functions. J. Math. Anal. Appl. 1981, 83, 554–565. 19. Tello Alonso, M.; López-Dekker, P.; Mallorqui, J.J. A novel strategy for radar imaging based on compressive sensing. IEEE Trans. Geosci. Remote Sens. 2010, 48, 4285–4295. 20. Huang, Q.; Qu, L.; Wu, B.; Fang, G. UWB through-wall imaging based on compressive sensing. IEEE Trans. Geosci. Remote Sens. 2010, 48, 1408–1415.

Sensors 2015, 15

6945

21. Estatico, C.; Pastorino, M.; Randazzo, A. A Novel Microwave Imaging Approach Based on Regularization in Banach spaces. IEEE Trans. Antennas Propag. 2012, 60, 3373–3381. 22. Lenti, F.; Nunziata, F.; Migliaccio, M.; Estatico, C. On the spatial resolution enhancement of microwave radiometer data in Banach spaces. IEEE Trans. Geosci. Remote Sens. 2014, 52, 1834–1842. 23. Zhu, X.X.; Bamler, R. Tomographic SAR Inversion by L1 Norm Regularization—The Compressive Sensing Approach. IEEE Trans. Geosci. Remote Sens. 2010, 48, 3839–3846. 24. Tikhonov, A.N. Numerical Methods for the Solution of Ill-Posed Problems; Springer Science & Business Media: New York, NY, USA, 1995 25. Rothwell, E.; Sun, W. Time domain deconvolution of transient radar data. IEEE Trans. Antennas Propag. 1990, 38, 470–475. 26. Lucy, L. An iterative technique for the rectification of observed distributions. Astron. J. 1974, 79, 745, doi:10.1086/111605. 27. Richardson, W.H. Bayesian-based iterative method of image restoration. JOSA 1972, 62, 55–59. 28. Xu, G.; Xing, M.; Zhang, L.; Liu, Y.; Li, Y. Bayesian inverse synthetic aperture radar imaging. IEEE Geosci. Remote Sens. Lett. 2011, 8, 1150–1154. 29. Zhang, L.; Qiao, Z.J.; Xing, M.D.; Sheng, J.L.; Guo, R.; Bao, Z. High-resolution ISAR imaging by exploiting sparse apertures. IEEE Trans. Antennas Propag. 2012, 60, 997–1008. 30. Notarnicola, C.; Ventura, B.; Casarano, D.; Posa, F. Cassini radar data: Estimation of Titan’s lake features by means of a Bayesian inversion algorithm. IEEE Trans. Geosci. Remote Sens. 2009, 47, 1503–1511. 31. Yildirim, S.; Cemgil, A.; Aktar, M.; Ozakin, Y.; Ertuzun, A. A Bayesian deconvolution approach for receiver function analysis. IEEE Trans. Geosci. Remote Sens. 2010, 48, 4151–4163. 32. Luisier, F.; Blu, T.; Unser, M. Image denoising in mixed Poisson–Gaussian noise. IEEE Trans. Image Process. 2011, 20, 696–708. 33. Foi, A.; Trimeche, M.; Katkovnik, V.; Egiazarian, K. Practical Poissonian-Gaussian noise modeling and fitting for single-image raw-data. IEEE Trans. Image Process. 2008, 17, 1737–1754. 34. Zhang, B.; Fadili, M.; Starck, J.L.; Olivo-Marin, J.C. Multiscale variance-stabilizing transform for mixed-Poisson-Gaussian processes and its applications in bioimaging. In Proceedings of the IEEE International Conference on Image Processing, San Antonio, TX, USA, 16 September–19 October 2007; Volume 6, pp. 6–233. 35. Shaked, E.; Michailovich, O. Iterative shrinkage approach to restoration of optical imagery. IEEE Trans. Image Process. 2011, 20, 405–416. 36. Green, P.J. Bayesian reconstructions from emission tomography data using a modified EM algorithm. IEEE Trans. Med. Imag. 1990, 9, 84–93. 37. Mignotte, M.; Meunier, J.; Soucy, J.P.; Janicki, C. Comparison of deconvolution techniques using a distribution mixture parameter estimation: Application in single photon emission computed tomography imagery. J. Electron. Imag. 2002, 11, 11–24. 38. Ji, S.; Xue, Y.; Carin, L. Bayesian compressive sensing. IEEE Trans. Signal Process. 2008, 56, 2346–2356.

Sensors 2015, 15

6946

39. Li, W.; Yang, J.; Huang, Y. Keystone transform-based space-variant range migration correction for airborne forward-looking scanning radar. Electron. Lett. 2012, 48, 121–122. 40. Andersson, F.; Moses, R.; Natterer, F. Fast Fourier methods for synthetic aperture radar imaging. IEEE Trans. Aerosp. Electron. Syst. 2012, 48, 215–229. 41. Orieux, F.; Sepulveda, E.; Loriette, V.; Dubertret, B.; Olivo-Marin, J.C. Bayesian estimation for optimized structured illumination microscopy. IEEE Trans. Image Process. 2012, 21, 601–614. 42. Mohammad-Djafari, A. Bayesian Approach for Inverse Problems in Optical Coherent and Noncoherent Imaging. In Proceedings of Optical Science and Technology, SPIE’s 48th Annual Meeting, San Diego CA, USA, 3–8 August 2003; pp. 209–218. 43. Cetin, M. Feature-Enhanced Synthetic Aperture Radar Imaging. Ph.D. Thesis, University of Salford, Manchester, UK, February 2001. 44. Moran, B. Mathematics of radar. In Twentieth Century Harmonic Analysis—A Celebration; Springer Netherland: Heidelberg, Germany, 2001; pp. 295–328. 45. Giryes, R.; Elad, M. Sparsity Based Poisson Denoising with Dictionary Learning. IEEE Trans. Image Process. 2014, 23, 5057–5069. 46. Samadi, S.; Çetin, M.; Masnadi-Shirazi, M.A. Sparse representation-based synthetic aperture radar imaging. IET Radar Sonar Navig. 2011, 5, 182–193. 47. Xu, G.; Sheng, J.; Zhang, L.; Xing, M. Performance improvement in multi-ship imaging for ScanSAR based on sparse representation. Sci. China Inf. Sci. 2012, 55, 1860–1875. 48. Çetin, M.; Karl, W.C. Feature-enhanced synthetic aperture radar image formation based on nonquadratic regularization. IEEE Trans. Image Process. 2001, 10, 623–631. 49. Wang, Z.; Bovik, A.C. Mean squared error: love it or leave it? A new look at signal fidelity measures. IEEE Signal Process. Mag. 2009, 26, 98–117. 50. Hansen, P.C. Regularization Methods at Work: Solving Real Problems. In Discrete Inverse Problems: Insight and Algorithms; SIAM: Philadelphia, PA, USA, 2010; Volume 7. 51. Sourbron, S.; Luypaert, R.; van Schuerbeek, P.; Dujardin, M.; Stadnik, T. Choice of the regularization parameter for perfusion quantification with MRI. Phys. Med. Biol. 2004, 49, 3307. 52. Vogel, C.R. Computational Methods for Inverse Problems; SIAM: Philadalphia, PA, USA, 2002. c 2015 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article

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Bayesian Deconvolution for Angular Super-Resolution in Forward

Bayesian Deconvolution for Angular Super-Resolution in Forward

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