Reconstruction and Compressed Sensing
5.1 INTRODUCTION
5.2 CS Theory . . . . 150 5.3 SR Algorithms . . . . 166 5.4 Sample Radar Applications . . . . 183 5.5 Summary . . . . 196 5.6 Further Reading . . . . 196 5.7 Acknowledgments . . . . 197 5.8 References . . . 197 5.9 Problems . . . . 207
5.1 INTRODUCTION
Sparse reconstruction and design through randomization have played significant roles in the history of radar signal processing. A recent series of theoretical and algorithmic results known as compressive or compressed sensing (CS) has ignited renewed interest in applying these ideas to radar problems. A flurry of research has explored the application of CS approaches as well as closely related sparse reconstruction (SR) techniques to a wide range of radar problems. This chapter will provide some historical context for CS, describe the existing theoretical results and current research directions, highlight several key algorithms that have emerged from these investigations, and offer a few examples of the application of these ideas to radar.
5.1.1 Organization
The chapter is organized into three sections. Section 5.2 develops the motivation and theoretical framework for SR and CS. We attempt to motivate these ideas from a radar perspective while also highlighting intuitions and connections with basic linear algebra and optimization theory. Section 5.3 explores the myriad of available algorithms for solving SR problems and the provable performance guarantees associated with these algorithms when used in conjunction with measurements which satisfy CS design criteria. Section 5.4 concludes with a selection of examples that illustrate how to apply these ideas to radar problems.
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148 C H A P T E R 5 Radar Applications of Sparse Reconstruction 5.1.2 Key Points
• Many interesting radar problems in imaging, detection, tracking, and identification can be formulated in a linear model framework that is amenable to SR and CS techniques. In fact, the linear model framework is not limited to point scatterer assumptions, free-space propagation, or weak scattering scenes.
• Radar signals often exhibit sparsity or compressibility in a known basis, such as the basis of point scatterers for high frequency synthetic aperture radar (SAR) data.
• Many radar problems are underdetermined and must be regularized with additional in- formation to obtain unique solutions. Sparsity in a known basis represents one appealing candidate but leads to intractable NP-hard optimization problems.
• A key notion in CS theory is to overcome the combinatorial complexity of seeking sparse solutions through convex relaxation with the1norm.
• SR by itself is not CS. CS involves combining SR algorithms with constraints on the measurement process, typically satisfied through randomization, which leads to provable performance guarantees.
• CS performance guarantees are predicated on conditions on the forward operator like the restricted isometry property or mutual coherence.
• A wide range of SR algorithms are available.
• Penalized least squares techniques directly solve a convex (or nonconvex) relaxation of the0optimization problem and offer excellent performance coupled with relatively high computational burden.
• Iterative thresholding methods offer simplified algorithms with performance guaran- tees, but in some cases these guarantees are weaker than those for penalized least squares.
• Greedy algorithms offer faster heuristic approaches to SR problems that provide limited performance guarantees.
• New trends in CS include approaches inspired by Bayesian and information theoretic approaches.
• Many CS guarantees are sufficient, conservative guarantees. SR algorithms may pro- duce desirable results for radar problems even when the conditions for these guarantees are not met. Phase transition plots offer a concise method for representing typical numerical algorithm performance on problems of interest.
• By incorporating structured sparsity information, CS performance can be further enhanced.
5.1.3 Notation
Variables used in this chapter include the following:
xtrue=true value of the unknown vector
A=the forward operator relating unknowns to measurements y=the vector of collected measurements
e=the additive disturbance signal
N =number of elements in the unknown vector M =number of measurements collected s=xtrue0, the sparsity of the unknown vector
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5.1 Introduction 149
δ=M/N , the undersampling ratio ρ=s/M, the normalized sparsity
σ =the energy bound on the additive disturbance signal e f0 =radar center frequency (Hz)
λ=wavelength (meters)
ω0 =2πf0, center frequency in (radians per second) c=speed of light (meters per second)
d=array inter element spacing (meters) f =c/λ, frequency (Hz)
k=2π/λ, wavenumber (radians per meter) J =number of spatial channels
K =number of slow-time pulses L=number of range gates in fast time T =slow-time sampling period (seconds).
5.1.4 Acronyms
Acronyms used in this chapter include the following:
AMP approximate message passing BP basis pursuit
BPDN basis pursuit denoising
CoSaMP compressive sampling matching pursuits CPI coherent processing interval
CS compressed sensing DFT discrete Fourier transform DLS data least squares
DWT discrete wavelet transform EM expectation maximization
FISTA fast iterative shrinkage-thresholding algorithm FOCUSS focal undetermined system solver
IHT iterative hard thresholding IRLS iterative reweighted least squares
ISTA iterative shrinkage-thresholding algorithm LaMP lattice matching pursuit
LASSO least absolute shrinkage and selection operator MAP maximum a posteriori
MIMO multiple input multiple output ML maximum likelihood
MM majorization minimization MMSE minimum mean squared error MP matching pursuits
MTI moving target indication NESTA Nesterov’s algorithm
OMP orthogonal matching pursuits PSF point spread function
RIC restricted isometry constant RIP restricted isometry property SAR synthetic aperture radar
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150 C H A P T E R 5 Radar Applications of Sparse Reconstruction SP subspace pursuit
SR sparse reconstruction
STAP space time adaptive processing SVD singular value decomposition TV total variation
ULA uniform linear array.