Principles of Modern Radar - Volume 2 1891121537

192 CHAPTER 5 Radar Applications of Sparse Reconstructionthe Lippman-Schwinger integral equation for the scalar wave equation [180]. Here weillustrate one approach for addressing the multiple-scattering issue that is based on theLSM [181]. The exposition given here, which sketches the results in [182,183], considersthe 2-D acoustic-scattering case. We note that this 2-D acoustic formulation can easily beextended to the 3-D electromagnetic case by using the analogous far-field relation usedin [184,185].The LSM solves a linear equation satisfied by the far-field data at a single frequencyω = kc. Denote the far-field data for wavenumber k by E ∞ (̂γ(θ t ), ̂γ(θ r ), k), wherêγ(θ t ) = [cos θ t , sin θ t ] and ̂γ(θ r ) = [cos θ r , sin θ r ] are the directions of the transmit andreceive antennas, respectively, and assume we are imaging a target whose support is Ɣ.Then for a given incident plane wave from direction ̂γ(θ t ) and wavenumber k, the receiveddata satisfy [186]∫ 2π0E ∞ (̂γ(θ t ), ̂γ(θ r ), k)g p (θ r , k)dθ r = e j (π/4−k̂γ(θ t )· p)√8πk(5.57)when p ∈ R 2 is within the scene. In (5.57), the unknown variable is the Herglotz kernelfunction, g p . 39 Given a discrete number of antennas (assume all bistatic geometries areavailable), we arrive at the linear system F (k) g (k)p ≈ v (k) ( p), where the elements of F (k)and v (k) satisfyF (k)m,n = E ∞(̂γ(θ m ), ̂γ(θ n ), k)v (k)m ( p) = e j (π/4−k̂γ(θ m)· p)√8πkThe image is recovered by plotting the indicator function ‖g (k)p ‖2 2 , which takes on largevalues for pixel location p outside 40 Ɣ and is small inside Ɣ. Typically, each p-dependentsystem of equations is solved in parallel by forming the block-diagonal system⎡⎢⎣F (k) ⎤0 ... 00 F (k) ... 0⎥.. ⎦0 ... 0 F (k)} {{ }A⎡⎢⎣g (k) ⎤p 1g (k)p 2⎥. ⎦g (k)p P} {{ }x⎡v (k) ( p 1 )v (k) ( p 2 )⎤= ⎢ ⎥⎣ . ⎦v (k) ( p P )} {{ }y(5.58)which we write as Ax = y for brevity, where the number of rows of A is M =(Number of pixels) × (Number of transmitters) = PT and the number of columns isN = (Number of pixels)× (Number of receivers) = PR. Note that the far-field data composesthe A matrix, while y is noiseless. The resulting type of least squares problem isreferred to as a data least squares (DLS) problem [190]. Depending on the characteristicsof the noise, specialized DLS techniques could be used. However, for convenience weignore this subtlety and continue with the more common least-squares framework.Because the validity of (5.58) depends on how well the receiving antennas populatethe viewing sphere, it is clear that this method is very data-hungry and may require39 The function g p is known as the Herglotz kernel function because is the kernel of the Herglotz waveoperator (Hg)( p) = ∫ S eik̂γ · p g(̂γ)d̂γ .40 The reader may notice that this method is reminiscent of other imaging techniques such as the MUSICalgorithm [187]. The techniques are in fact related, and their similarities are discussed in [188,189].