# A Sparse Bayesian Approach for Forward-Looking Superresolution Radar Imaging
## 3 Sparse Bayesian Algorithm for High Cross-Range Resolution
### Sparse Bayesian Superresolution Algorithm  
The maximum a posteriori (MAP) estimation can be considered as a Bayesian version of maximum likelihood (ML) estimation, with prior information of the target included. The basic form of the MAP estimator is:  
\[\hat{\mathbf{x}}=\arg\ \max_{\mathbf{x}}p\left(\mathbf{x}|\mathbf{g}\right)= \arg\ \max_{\mathbf{x}}\left[p\left(\mathbf{g}|\mathbf{x}\right)p\left(\mathbf{x} \right)\right] \tag{9}\]  
where \(p\left(\cdot\right)\) represents the probability density function, \(p\left(\mathbf{x}|\mathbf{g}\right)\) is the a posteriori probability density function, \(p\left(\mathbf{g}|\mathbf{x}\right)\) is the likelihood probability density function (which is determined by the statistical properties of noise), and \(p\left(\mathbf{x}\right)\) is the prior information of the target scattering coefficient. The MAP estimator is equivalent to the ML estimator when there is no prior information about the target distribution. Though many Bayesian approaches have been proposed to deal with the signal recovery problem of forward-looking superresolution imaging, most of them adopt Poisson statistics, which do not apply to the deconvolution of the motion platform. These approaches are valid for noncoherent radar, but such an imaging system has a small set of applications, and the uncorrectable assumption of the noise statistic causes the degradation of superresolution performance. In radar imaging, the statistics of noise often obey a Gaussian distribution. Meanwhile, in order to overcome the complex-valued signal recovery problem, we assume that the observation noise is statistically independent and obeysa circular complex Gaussian distribution. The likelihood probability density function (pdf) can be written as:  
\[p\left(\mathbf{x}|\mathbf{g}\right)=\frac{1}{\left(\pi\sigma^{2}\right)^{MK}} \exp\left[-\frac{1}{\sigma^{2}}\left\|\mathbf{g}-\mathbf{A}\mathbf{x}\right\| _{2}^{2}\right]. \tag{10}\]  
Another important task is determining how to correctly select prior information of target scatters. In [23], a sparsity promoting prior was given:  
\[p\left(\mathbf{x}\right)\propto\prod_{n=1}^{NK}\exp\left[-\frac{2}{q}\left( \left|x_{n}\right|^{q}-1\right)\right] \tag{11}\]  
where \(0<q\leq 1\). When \(q=1\), \(p\left(\mathbf{x}\right)\propto\exp\left(-2\left\|\mathbf{x}\right\|_{1}\right)\) becomes the Laplace distribution. When \(q\to 0\), the prior term becomes \(p\left(\mathbf{x}\right)\propto\prod\limits_{n=1}^{NK}\left(1\middle/\left|x_ {n}\right|^{2}\right)\). In general, the prior corresponding to a smaller one has a sharper peak at zero and provides sparser estimates in Bayesian inference. This distribution is used to express the prior information. Since the main applications of forward-looking imaging are target searching, location, and tracking, the target distribution corresponding to the whole forward-looking imaging region is independent and sparse. In addition, because noise is an important factor in the performance of deconvolution, the sparseness prior in signal processing has denoising capability [31]. The objective function can be written as:  
\[p\left(\mathbf{g}|\mathbf{x}\right)\cdot p\left(\mathbf{x}\right)=\frac{1}{ \left(\pi\sigma^{2}\right)^{MK}}\exp\left[-\frac{1}{\sigma^{2}}\left\|\mathbf{ g}-\mathbf{A}\mathbf{x}\right\|_{2}^{2}\right]\prod_{n=1}^{NK}\exp\left[-\frac{2}{q} \left(\left|x_{n}\right|^{q}-1\right)\right]. \tag{12}\]  
It is difficult to directly calculate the MAP solution of Equation (12). Because \(\ln\left(\mathbf{x}\right)\) is a monotonic function in terms of \(\mathbf{x}\), it is much easier to calculate the maximum value of \(ln\left(p\left(\mathbf{g}|\mathbf{x}\right)p\left(\mathbf{x}\right)\right)\) than \(p\left(\mathbf{g}|\mathbf{x}\right)p\left(\mathbf{x}\right)\).