Principles of Modern Radar - Volume 2 1891121537

190 CHAPTER 5 Radar Applications of Sparse ReconstructionRange (m)Transmitters, Receivers andScatterers Positions80TxRx60Target40200−20−40−60−80−100 −50 0Range (m)50 100(a)Range (m)3020100−10−20−30 −30 −20 −10 0 10 20 30Range (m)FIGURE 5-12 (a) Sensor geometry and target and (b) a horizontal cross section of theTikhonov-regularized image.A common way to obtain an image x when A is ill-conditioned is to penalize largevalues of ‖x‖ 2 by solvingmin ‖Ax − y‖ 2 + α‖x‖ 2 (5.48)x∈C NThe tuning parameter α in the Tikhonov formulation (5.48) can be chosen by a varietyof methods such as L-curve [69,176], generalized cross-validation [177], or Morozov’sdiscrepancy principle [178]. Unfortunately, (5.48) fails to obtain a reasonable image (seeFigure 5-12) because in this scenario, l 2 penalization provides an insufficient amount ofa priori information to effectively regularize the problem without destroying the qualityof the solution. Fortunately for the tunnel-detection problem, our a priori knowledge alsoincludes sparsity in both the image itself and its spatial gradient as well as physicallymotivated upper and lower bounds on the reflectivity.We consider two different objective functions that seek to incorporate this knowledge.The first ismin ‖Ax − y‖ 2 + α‖x‖ 2 + β‖x‖ 1 (5.49)x∈C Ns.t. τ min ≤|x n |≤τ max , n = 1,...,N (5.50)To solve the problem (5.49)–(5.50) efficiently, we iteratively project solutions of (5.49)onto the feasible set (5.50). Decomposing the problem in this manner allows us to choosefrom several large-scale solvers (for example, LARS [179] was developed specificallyfor (5.49)). We choose to use the FISTA algorithm at each step by rewriting (5.49) and(5.50) as∥[ ] [ ]∥ ∥∥∥ Amin √ y ∥∥∥2x − + β‖x‖x∈C N α I 01 (5.51)which is the form of QP λ .s.t. τ min ≤|x n |≤τ max , n = 1,...,N (5.52)(b)