Chapter Summary

Chapter 14 Summary: Sparse Reconstruction for RF Imaging

Key Points

• 1.
  LASSO and Basis Pursuit

  The LASSO $\min \frac{1}{2}\|\mathbf{A}\mathbf{c} - \mathbf{y}\|^2 + \lambda\|\mathbf{c}\|_1$ is the workhorse of sparse RF imaging. FISTA solves it in $O(1/t^2)$; ADMM handles composite penalties. The discrepancy principle is the recommended $\lambda$ selection strategy for radar (known noise level). Debiasing on the detected support removes the $\ell_1$ shrinkage bias.

• 2.
  Group Sparsity and MMV

  The group LASSO ($\ell_{2,1}$ penalty) exploits block structure; MMV handles multi-snapshot imaging with common support. Block soft-thresholding is the proximal operator. Group sparsity requires fewer measurements than element-wise sparsity. The Pesavento compact formulation provides sharper support detection via iterative reweighting.

• 3.
  Total Variation Reconstruction

  TV regularization preserves edges in piecewise-constant scenes. ADMM with FFT-based linear system solve provides efficient TV reconstruction. The $\ell_1$ + TV combination handles mixed scenes. TGV extends TV to piecewise-smooth scenes, avoiding staircase artifacts.

• 4.
  Greedy Algorithms

  OMP, CoSaMP, and IHT provide fast alternatives to convex relaxation. Best for real-time imaging with small known sparsity. Convex methods (FISTA, ADMM) are more robust for compressible or structured scenes.

• 5.
  Super-Resolution via Atomic Norm

  Basis mismatch from grid discretization is a fundamental limitation. Atomic norm minimization (SDP) provides exact gridless recovery in 1D via the Vandermonde decomposition. SPARROW offers a practical workflow: coarse-grid LASSO followed by continuous position refinement. The connection to MUSIC and subspace methods unifies classical spectral estimation with modern sparse recovery.