Principles of Modern Radar - Volume 2 1891121537

172 CHAPTER 5 Radar Applications of Sparse Reconstructionin (5.25) that the TV norm of the magnitude of the image rather than the complex-valuedreflectivity, is penalized. This choice is made to allow rapid phase variations. 24 Noticealso that the l p norm is used with 0 < p ≤ 2. As we have mentioned, selecting p < 1can improve performance but yields a nonconvex problem. Cetin and Karl replace the l pterms in the cost function with differentiable approximations, derive an approximation forthe Hessian of the cost function, and implement a quasi-Newton method.Kragh [77] developed a closely related algorithm (for the case with λ 2 = 0) alongwith additional convergence guarantees leveraging ideas from majorization minimization(MM). 25 For the case with no TV penalty, both algorithms 26 end up with an iteration ofthe formˆx k+1 = [ A H A + h( ˆx k ) ] −1A H y (5.26)where h(·) is a function based on the norm choice p. The matrix inverse can be implementedwith preconditioned conjugate gradients to obtain a fast algorithm. Notice that A H A oftenrepresents a convolution that can be calculated using fast Fourier transforms (FFTs). Amore detailed discussion of these algorithms and references to various extensions to radarproblems of interest, including nonisotropic scattering, can be found in [17].These algorithms do not begin to cover the plethora of existing solvers. Nonetheless,these examples have proven useful in radar applications. The next section will considerthresholding algorithms for SR. As we shall see, these algorithms trade generality forfaster computation while still providing solutions to QP λ .5.3.2 Thresholding AlgorithmsThe algorithms presented at the end of Section 5.3.1.2 both require inversion of a potentiallymassive matrix at each iteration. While fast methods for computing this inverse exist, weare now going to consider a class of algorithms that avoids this necessity. These algorithmswill be variations on the iterationˆx k+1 = η { ˆx k − μA H ( A ˆx k − y )} (5.27)where η{·} is a thresholding function. The motivation for considering an algorithm ofthis form becomes apparent if we examine the cost function for QP λ . First, for ease ofdiscussion let us define two functions that represent the two terms in this cost function.Adopting the notation used in [79]f (x) = ‖Ax − y‖ 2 2g(x) = λ ‖x‖ 124 Imagine, for example, the phase response of a flat plate to see intuitively why penalizing variation inthe phase would be problematic.25 MM relies on replacing the cost function of interest with a surrogate function that is strictly greater(hence majorization) but easier to minimize (hence minimization). The idea is to majorize the functionnear the current estimate, minimize the resulting approximation, and repeat. The perhaps more familiarexpectation maximization algorithm is actually a special case of MM, as detailed in an excellenttutorial [78].26 A particular step size must be selected to obtain this form for the algorithm in [75]. It is also worthemphasizing that this version can handle the more general problem with λ 2 ≠ 0. The resulting algorithmsimply uses a more complicated expression for the function h.