Principles of Modern Radar - Volume 2 1891121537

5.3 SR Algorithms 171takes special advantage of the structure of the Pareto frontier to obtain the desired solution.In particular, van den Berg and Friedlander develop a fast projected gradient technique forobtaining approximate solutions to LS τ . The goal is to approximately solve a sequenceof these LASSO problems so that τ 0 ,τ 1 ,...τ k approaches τ σ , which is the value for τwhich renders the problem equivalent to BP σ . While slower than solving the unconstrainedproblem, the fast approximate solutions to these intermediate problems allow the algorithmto solve the BP σ in a reasonable amount of time.Let us consider a step of the algorithm starting with τ k . First, we compute the correspondingsolution ˆx k τ . As discussed already, this provides both the value and an estimateof the slope of the Pareto curve asφ(τ k ) = ∥∥ A ˆx k τ − y∥ ∥ ∥ 2φ ′ A H r k ∥∥∥∥∞(τ k ) =−∥ ∥ r k∥ ∥2r k = A ˆx k τ − y (5.22)We will choose the next parameter value as τ k+1 = τ k +τ k . To compute τ k , the authorsof [63] apply Newton’s method. We can linearize the Pareto curve at τ k to obtainφ(τ) ≈ φ(τ k ) + φ ′ (τ k )τ k (5.23)We set this expression equal to σ and solve for the desired step to obtainτ k = σ − φ(τ k)φ ′ (τ k )(5.24)The authors of [63] provide an explicit expression for the duality gap, which provides abound on the current iteration error, and prove several results on guaranteed convergencedespite the approximate solution of the sub-problems. Further details can be found in [63],and a MATLAB implementation is readily available online. We should also mentionthat the SPGL1 algorithm can be used for solving more general problems, includingweighted norms, sums of norms, the nuclear norm for matrix-valued unknowns, and othercases [74].We will now discuss two closely related algorithms that were developed in the radarcommunity for SAR imaging for solving generalizations of QP λ . The algorithms can beused to solve the l 1 problem specifically, and hence inherit our RIP-based performanceguarantees, but they can also solve more general problems of potential interest to radarpractitioners. First, we will consider the algorithm developed in [75] which addresses themodified cost functionˆx = argminxλ 1 ‖x‖ p p + λ 2 ‖D|x|‖ p p + ‖Ax − y‖2 2 (5.25)where D is an approximation of the 2-D gradient of the magnitude image whose voxelvalues are encoded in |x|.This second term, for p = 1, is the total variation norm of the magnitude image. TheTV norm is the l 1 norm of the gradient. In essence, this norm penalizes rapid variationand tends to produce smooth images. As Cetin and Karl point out, this term can help toeliminate speckle and promote sharp edges in SAR imagery. Indeed, TV minimization hasseen broad application in the radar, CS, and image processing communities [76]. Notice