Bayesian Deconvolution for Angular Super-Resolution in Forward-Looking Scanning Radar

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.

Under the Born hypothesis [19,40], the data after range compression and range cell migration g R (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: Sensors 2015, 15

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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: where f (x p , y q ), (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 : 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: x p , y q ) r = 1, 2, · · · R; a = 1, 2, · · · A where τ r represents the r-th sample in fast time, t a is the a-th sample in slow time, f (i) is the i-th element in Equation (2), rect(·) is a rectangular function, T r denotes the pulse duration, f c 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 (t a , x p , y q ) is the range history between the antenna and a point target at (x p , y q ) when the azimuth time is t a . The equation for R (t a , x p , y q ) is shown in Equation (4) R( t a , x p , y q ) = R 2 0 + (V t a ) 2 − 2R 0 V t a cos θ cos ϕ (4) in which R 0 is the range when the antenna beam center is across the target at (x p , y q ), 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 (x p , y q ), 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: x y Figure 2. The geometry relationship of scanning radar.
Since the product of the velocity V and azimuth time t a is much smaller than R 0 , Equation (5) can be rewritten as: 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 g e (τ r , t a ) yields: where δ (·) is the impulse function and A (t a ) is the antenna pattern. After range cell migration correction to the data g c (τ r , t a ), we have: f (i) H (τ r , t a , x p , y q ) + n (τ r , t a ) where: and B is the signal bandwidth; the corresponding data g (τ r , t a ) are shown in Figure 1d, and n (τ r , t a ) 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.
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: 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: 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 are often small for high frequencies, and the inverse filter 1 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.

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: 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.

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 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 n p and a Gaussian signal-independent component n g . The Poisson component n p models the signal-dependent part of the errors, which is essentially due to the working mode of the scanning radar, while the Gaussian component n g 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., 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.,

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: where σ denotes the deviation.
Substituting Equations (19) and (20) into Equation (17), the final expression for the MAP estimation task becomes: Maximizing Equation (21) is equivalent to minimizing: with respect to f . The term 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).

Solution of the Unconstrained Problem
The non-differentiability of the f 1 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 a small positive parameter ε in the l 1 -norm (see [43,48]). Specifically, |f (i)| ≈ |f (i)| 2 + ε 1 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 l 1 -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 l 1 -norm by its differentiable approximation. After the smoothed approximation, the f 1 has the following form: Substituting Equation (23) into Equation (22), we express Equation (22) as follows: 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: where the superscript T represents the transpose of a matrix, I RA stands for RA × 1 vector of ones is a diagonal matrix whose i-th diagonal element is given by the expression inside the brackets. We further assume that H T I RA = I RA , where I RA stands for the column vector consisting of RA ones. f M AP solves the problem Equation (25) if and only if the following equivalent statements hold: where λ = √ 2 σ 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, where f m and f m+1 represent the estimations of the true reflectivity coefficients of the observed scene f in the m-th and (m + 1)-th iterations, respectively. Assuming that at convergence, the ratio of f m+1 /f m is I RA . The greatest problem with multiplicative form Equation (27) is that the diagonal elements of Λ (f m ) correspond to the approximation processes in the iteration.

Simulation and Experimental Results
In this section, we present experimental results to illustrate the angular super-resolution performance of the proposed deconvolution algorithm in forward-looking scanning radar. We compare the proposed deconvolution method with Wiener filter method, the Richardson-Lucy (R-L) algorithm and the Tikhonov regularization method.

Simulation
For experiments on synthetic data, we apply our method to a synthetic scene composed of six point targets with different reflectivity magnitudes. The synthetic scene is shown in Figure 3. The targets are of unequal amplitude, which means different scattering coefficients. Some related radar parameters are set as follows: the pulse repetition frequency is 4000 Hz, and the antenna scanning speed is 30 • /s. The bandwidth of the transmitted signal is 2 MHz, and the 3-dB width of the real beam is about 3 • . The number of azimuth samples is 2666.
of the angular super-resolution results. They are defined as follows: where f M AP , f , and g corresponds to the obtained angular super-resolution image, the original image, and the observed image, respectively. The terms µ, σ, and ρ (f ,f M AP ) , are the mean, standard deviation of the vectors, the correlation coefficient corresponds to the vector f and f M AP . The SSIM is a quantitative measure between the super-resolution result and the original scene. The value of SSIM is between −1 and 1, and 1 means full identical with the original scene. where X (λ) = log (∥D |f λ |∥ 2 ) are and the prior inform represents the diffe In the simulation method. This own are able to treat p and robustness no m The L-curve meth there are two para proposed deconvolu performance of the of angular super-re choice of the regula this paper. To deter firstly use the iterat it for all the differe λ. Then, choose the the obtained λ unde following simulatio 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: where f M 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 ,f M AP ) are the mean and standard deviation of the vectors, and the correlation coefficient corresponds to the vector f and f M 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 (σ 2 , σ 2 ) = P oiss(σ 2 ), the Poisson noise can be approximated by Gaussian noise. Then, the Gaussian noise with different levels is added to model the errors.

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: where X (λ) = log ( Hf λ − g 2 ) and Y (λ) = log ( D |f λ | 2 ) are the log-transform of the data fidelity and the prior information, respectively, and the superscript ( ) represents the differentiation with respect to λ.
In the simulations, the parameter λ is selected using the L-curve method. This is owed to the advantage that the L-curve method is able to treat perturbations consisting of corrected noise 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 proposed deconvolution algorithm. This paper focuses on the performance of the proposed deconvolution algorithm in terms of angular super-resolution in scanning radar. The automatic choice of the regularization parameter λ is beyond the scope of this paper. To determine a good pair of parameters in Equation (27), we firstly use the iterative solver with a lot of iterations and re-use it for all of the different choices of the regularization parameter λ. Then, we choose the good solution using the L-curve. Figure 4 shows the obtained λ under different noise levels, which is used in the following simulations. The setting of the iteration number of the R-L algorithm is equal to the way for the proposed algorithm. original scene. scene e of the proposed deconvo--resolution under different els the signal-independent and electric noise, which nt noise in scanning radar N (σ 2 , σ 2 ) = P oiss(σ 2 ), mated by Gaussian noise. ent levels is added to model f the Bayesian deconvoluarameter has to be chosen ults. Both the data fidelity , and their size depends on all values of regularization , while large regularization age. Most of the referenced the regularizing parameter rior so that the result of ual quality. However, this lies too much on subjective a number of works on the er such as L-curve method 3], unbiased predictive risk eported. eter λ in (26) , we choose means of L-curve method, the prior solution X(λ) aesponding residual solution 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 proposed deconvolution algorithm. This paper focuses on the performance of the proposed deconvolution algorithm in terms of angular super-resolution in scanning radar. The automatic choice of the regularization parameter λ is beyond the scope of this paper. To determine a good pair of parameters in (26),we firstly use the iterative solver with a lot of iterations, and re-use it for all the different choices of the regularization parameter λ. Then, choose the good solution using L-curve. Fig.4 shows the obtained λ under different noise levels, which is used in the following simulations. The setting of the iteration number of R-L algorithm is equal to the way for the proposed algorithm.
2) Angular super-resolution results: Fig.5-Fig.7 show the angular super-resolution results under different noise levels. The BSNR = 20log 10 ∥g∥ 2 ∥n∥ 2 is used to measure the quality of these echo data with different noise levels. The regularization parameter λ is computed by using Eq. (27), and Fig.4 shows the L-curve and the corresponding curvature as a function of λ. The number of iterations in Fig.4 are 75, 110, and 150 , respectively.   is used to measure the quality of these echo data with different noise levels. The 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.
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.                                  The angular super-resolution results by those methods under various noise levels are shown in Fig.5, Fig.6, and Fig.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 close match to the original scene.
It can be seen from Fig.5-Fig.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 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 the spikes of the targets lo angular super-resolution res spikes of the targets looks m of the proposed method co be appreciated in Fig.7, as separated, and the noise am presented in the result by u The reason is that the Rlikelihood criterion, which method with no penalty fu maximum likelihood criteri ment between the measurem high noise estimates, partic This conclusion is also su evaluation parameters of t than the evaluation paramete methods. Table I shows the com regularization method, Wie proposed deconvolution me SNRs, ISNRs, and SSIMs. R the proposed deconvolution SSIM values while keepin the four methods, both Tik Wiener filter use the inverse R-L algorithm may be an for super-resolution imagin a small number of iteratio This phenomenon is a gen leading to the larger error o resolution result. These resu deconvolution algorithm fo competitive over the conven In addition, another noti deconvolution algorithm com tion algorithms is that the suppresses the spurious p resolution results under di to a better angular superof precision. Fig.8 shows the algorithms in the diffe from the Fig.8 that the Tik Wiener filter have higher r levels, while the proposed better than the conventiona The reason is that the prop explored the prior informa to a more stable solution problem and the evaluation other comparative methods. In Figure 5, we can see that the visual quality of the angular super-resolution result by using the R-L algorithm looks similar to the result by using the proposed method. In Figure 6, the proposed approach gives the super-resolution result, where the spikes of the targets look fairly separate, whereas in the angular super-resolution result, by using the R-L algorithm, the spikes of the targets looks more connected. The improvement of the proposed method compared to R-L algorithm can also be appreciated in Figure 7, as the spikes of the targets have been separated, and the noise amplified by the R-L algorithm is not presented in the result by using the proposed method.
The reason is that the R-L algorithm is based on the maximum likelihood criterion, which is identical to the deterministic method with no penalty function or prior information. The maximum likelihood criterion aims at maximizing the agreement between the measurement and the object, which yields high noise estimates, particularly when the noise level is low. This conclusion is also supported by Table 1, in which the evaluation parameters of the proposed algorithm are better than the evaluation parameters associated with the comparative methods.  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 targets. This leads to a more stable solution to the associated deconvolution problem, and the evaluation parameters are also better than other comparative methods.

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 scattering point targets.
The range compression is applied to the recorded data. The imaging result and the corresponding profile of the target area are shown in Figure 11a,b, respectively. From Figure 11b, we can see that the echo of adjacent buildings is overlapping and covering the building features.

Main-lobe beam width 5
• to be 1m. The distance of each building is 45m. Fig.9 shows the three buildings and their distribution. The range from the scene center to the radar system is about 594m and the angular resolution in the scene center is 95m. 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 scattering point targets.  Fig. 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 Figure 9. The tested scene. 10 is 45m. Fig.9 shows The range from the 94m and the angular he echo from three the other areas of the onsider as consisting The range compression is applied to the recorded data. The imaging result and the corresponding profile of targets area are shown in Fig.11(a) and Fig.11(b), respectively. From the Fig.11(b), we can see that the echo of adjacent buildings is overlap and cover the building features.
As compared with other methods, the maximum iterations and the regularization parameter of the proposed algorithm are hand-tuned alternative so that the corresponding superresolution result presents the best visual quality. For the proposed deconvolution algorithm, we used the regularization parameter λ = 1.47 and the iteration number is 100.
In Fig.12, we investigate the angular super-resolution performance of the approaches. The first column of Fig.12 gives the angular super-resolution imaging results of different methods. The right column of Fig.12 presents the profile of the targets scene corresponding to the left column. The first row presents the angular super-resolution results using the Tikhonov regularization method. The second row contains the Wiener filter results, while the third and bottom rows contain angular superresolution results by using R-L algorithm and the proposed algorithm, respectively. It can be obviously observed that the amplitude of the targets profile in the right column of Fig.12 are different. The reason for this is that the distance from the middle building to the radar is about 594m, while the distance between the radar and the left/right building is about 597m. This leads to the profile of the middle building is higher than the other two profiles in the right column of Fig.12, which also fits their physical truth.
(a) observed scene so that the scene can be consider as consisting of three main scattering point targets.  The range compression is applied to the recorded data. The imaging result and the corresponding profile of targets area are shown in Fig.11(a) and Fig.11(b), respectively. From the Fig.11(b), we can see that the echo of adjacent buildings is overlap and cover the building features.
As compared with other methods, the maximum iterations and the regularization parameter of the proposed algorithm are hand-tuned alternative so that the corresponding superresolution result presents the best visual quality. For the proposed deconvolution algorithm, we used the regularization parameter λ = 1.47 and the iteration number is 100.
In Fig.12, we investigate the angular super-resolution performance of the approaches. The first column of Fig.12 gives the angular super-resolution imaging results of different methods. The right column of Fig.12 presents the profile of the targets scene corresponding to the left column. The first row presents the angular super-resolution results using the Tikhonov regularization method. The second row contains the Wiener filter results, while the third and bottom rows contain angular superresolution results by using R-L algorithm and the proposed algorithm, respectively. It can be obviously observed that the amplitude of the targets profile in the right column of Fig.12 are different. The reason for this is that the distance from the middle building to the radar is about 594m, while the distance between the radar and the left/right building is about 597m. This leads to the profile of the middle building is higher than the other two profiles in the right column of Fig.12, which also fits their physical truth. The range compression is applied to the recorded data. The imaging result and the corresponding profile of targets area are shown in Fig.11(a) and Fig.11(b), respectively. From the Fig.11(b), we can see that the echo of adjacent buildings is overlap and cover the building features.
As compared with other methods, the maximum iterations and the regularization parameter of the proposed algorithm are hand-tuned alternative so that the corresponding superresolution result presents the best visual quality. For the proposed deconvolution algorithm, we used the regularization parameter λ = 1.47 and the iteration number is 100.
In Fig.12, we investigate the angular super-resolution performance of the approaches. The first column of Fig.12 gives the angular super-resolution imaging results of different methods. The right column of Fig.12 presents the profile of the targets scene corresponding to the left column. The first row presents the angular super-resolution results using the Tikhonov regularization method. The second row contains the Wiener filter results, while the third and bottom rows contain angular superresolution results by using R-L algorithm and the proposed algorithm, respectively. It can be obviously observed that the amplitude of the targets profile in the right column of Fig.12 are different. The reason for this is that the distance from the middle building to the radar is about 594m, while the distance between the radar and the left/right building is about 597m. This leads to the profile of the middle building is higher than the other two profiles in the right column of Fig.12, which also fits their physical truth.  The range imaging resu are shown in Fig.11(b), we overlap and c As compar and the regu are hand-tun resolution re proposed dec parameter λ = In Fig.12, w mance of the angular super The right col scene corresp the angular s larization me results, while resolution res algorithm, re amplitude of are different. middle buildi between the This leads to the other two also fits their (b) Figure 11. (a) The received echo signal of the tested scene after range compression; (b) The profile of the target ares.
As compared with other methods, the maximum iterations and the regularization parameter of the proposed algorithm are hand-tuned alternatives, so that the corresponding super-resolution result presents the best visual quality. For the proposed deconvolution algorithm, we used the regularization parameter λ = 1.47, and the iteration number is 100.
In Figure 12, we investigate the angular super-resolution performance of the approaches. The first column of Figure 12 gives the angular super-resolution imaging results of different methods. The right column of Figure 12 presents the profile of the target scene corresponding to the left column. The first row presents the angular super-resolution results using the Tikhonov regularization method. The second row contains the Wiener filter results, while the third and bottom rows contain angular super-resolution results by using the R-L algorithm and the proposed algorithm, respectively. It can be obviously observed that the amplitudes of the target profiles in the right column of Figure 12 are different. The reason for this is that the distance from the middle building to the radar is about 594 m, while the distance between the radar and the left/right building is about 597 m. This leads to the profile of the middle building being higher than the other two profiles in the right column of Figure 12, which also fits their physical truth. The left column of Fig.12 shows the angular superresolution imaging results. It can be noted that the proposed method gives the best results in terms of visible quality. This is 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 Tiknonov regularization method and Wiener filter method are still noisy, particularly around the building boundaries. The improvement of the proposed method compared to the other methods can also be appreciated in the right column of Fig.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.

V. CONCLUSION
The angular super-resolution in scanning radar has received much attention in recent years, however, limited work about the application of deconvolution algorithm for angular superresolution in scanning radar has been reported. Since the The left column of Fig.12 shows the angular superresolution imaging results. It can be noted that the proposed method gives the best results in terms of visible quality. This is 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 Tiknonov regularization method and Wiener filter method are still noisy, particularly around the building boundaries. The improvement of the proposed method compared to the other methods can also be appreciated in the right column of Fig.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.

V. CONCLUSION
The angular super-resolution in scanning radar has received much attention in recent years, however, limited work about the application of deconvolution algorithm for angular superresolution in scanning radar has been reported. Since the The left column of Fig.12 shows the angular superresolution imaging results. It can be noted that the proposed method gives the best results in terms of visible quality. This is 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 Tiknonov regularization method and Wiener filter method are still noisy, particularly around the building boundaries. The improvement of the proposed method compared to the other methods can also be appreciated in the right column of Fig.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.

V. CONCLUSION
The angular super-resolution in scanning radar has received much attention in recent years, however, limited work about the application of deconvolution algorithm for angular superresolution in scanning radar has been reported. Since the (c) 11 (a) 20 40 The left column of Fig.12 shows the angular superresolution imaging results. It can be noted that the proposed method gives the best results in terms of visible quality. This is 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 Tiknonov regularization method and Wiener filter method are still noisy, particularly around the building boundaries. The improvement of the proposed method compared to the other methods can also be appreciated in the right column of Fig.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.

V. CONCLUSION
The angular super-resolution in scanning radar has received much attention in recent years, however, limited work about the application of deconvolution algorithm for angular superresolution in scanning radar has been reported. Since the (d) 11 (a) 20 40 The left column of Fig.12 shows the angular superresolution imaging results. It can be noted that the proposed method gives the best results in terms of visible quality. This is 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 Tiknonov regularization method and Wiener filter method are still noisy, particularly around the building boundaries. The improvement of the proposed method compared to the other methods can also be appreciated in the right column of Fig.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.

V. CONCLUSION
The angular super-resolution in scanning radar has received much attention in recent years, however, limited work about the application of deconvolution algorithm for angular superresolution in scanning radar has been reported. Since the (e) 11 (a) 20 40 The left column of Fig.12 shows the angular superresolution imaging results. It can be noted that the proposed method gives the best results in terms of visible quality. This is 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 Tiknonov regularization method and Wiener filter method are still noisy, particularly around the building boundaries. The improvement of the proposed method compared to the other methods can also be appreciated in the right column of Fig.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.

V. CONCLUSION
The angular super-resolution in scanning radar has received much attention in recent years, however, limited work about the application of deconvolution algorithm for angular superresolution in scanning radar has been reported. Since the The left column of Fig.12 shows the angular superresolution imaging results. It can be noted that the proposed method gives the best results in terms of visible quality. This is 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 Tiknonov regularization method and Wiener filter method are still noisy, particularly around the building boundaries. The improvement of the proposed method compared to the other methods can also be appreciated in the right column of Fig.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.

V. CONCLUSION
The angular super-resolution in scanning radar has received much attention in recent years, however, limited work about the application of deconvolution algorithm for angular superresolution in scanning radar has been reported. Since the (g) 11 (a) 20 40 The left column of Fig.12 shows the angular superresolution imaging results. It can be noted that the proposed method gives the best results in terms of visible quality. This is 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 Tiknonov regularization method and Wiener filter method are still noisy, particularly around the building boundaries. The improvement of the proposed method compared to the other methods can also be appreciated in the right column of Fig.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.

V. CONCLUSION
The angular super-resolution in scanning radar has received much attention in recent years, however, limited work about the application of deconvolution algorithm for angular superresolution in scanning radar has been reported. Since the (h) 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 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.

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.