2.2. The Optimization-Based Radar Image Reconstruction Algorithm

Existing radar-based image reconstruction techniques perform one-step image reconstruction, as in DAS and DMAS, where the image is produced in one mathematical step. While these techniques allow for rapid reconstruction, they do not allow nonlinear components to be incorporated into the signal model. The implicit forward signal model assumed in the DAS beamformer is given by

where Sfwd(a,f,σ) is the forward model S-parameter obtained at measurement position a, frequency f, due to reflectivity profile σ. The DAS beamformer aims to reconstruct this reflectivity profile, σ(r), as the image I(r).

This model assumes plane-wave propagation from the antenna to each scattering point and from each scattering point to the antenna. The model assumes no signal losses due to distance-based attenuation, partial transmission at interfaces, and losses due to lossy propagation media. The model assumes an isotropic, point-source antenna with a uniform gain as a function of frequency and also assumes a uniform propagation speed throughout the imaging domain. These assumptions are all violated in a BMS system, and as a result, the performance of the DAS beamformer and its derivatives are constrained.

However, this radar forward model can be also used to produce an optimization problem,

where σimg is the reflectivity profile displayed in the reconstructed image and Sexpt(a,f) is the experimentally measured S-parameter at measurement position a and frequency f, and where the sum-of-squares,

was used as the loss function, l(σ).

To solve this optimization problem, we employed the gradient descent optimizer to find σimg,

where the conjugate of the gradient (with respect to the reflectivity profile, ∇σ) of the loss function is used due to Sfwd,Sexpt∈C [27].

Let us explore the relationship between the DAS reconstruction algorithm and this optimization problem formulation. Assume that the measured S11 is at only one position, a0, and one frequency f0, to examine the exact solution for σ(r), assuming the implicit forward model in Equation (4), by writing:

where Sfwd(f0,a0,σ(r)) is given as in Equation (4), so that

The reflectivity profile that solves this equation, σD(r), is

where V is the volume of the integration domain.

To prove this, substitute Equation (10) into Equation (9), to obtain

and the factor of V in the numerator of the right-hand side cancels with that in the denominator, and so the left-hand side and right-hand sides are identical, demonstrating that Equation (10) is a solution to the cost function l(σ) assuming the forward model to be implicit to DAS when one measurement position a0 and one frequency f0 are used,

The solution σD(r) in Equation (10) is also that given by the DAS beamformer, within a normalization factor of 1/V. For a single-position, single-frequency measurement, DAS therefore completely reconstructs the reflectivity profile σ(r) so as to minimize the loss function l(σ) (in this case, l(σ)=0). However, no current BMS system uses only a single-position, single-frequency measurement to reconstruct an image (due primarily to the non-unique time delays—i.e., a given time delay may correspond to many regions in the spatial domain). If we consider a BMS system that uses a single-position, dual-frequency measurement (f0 and f1), we observe that the reflectivity profile produced by the DAS beamformer, σD(r) as described in Equation (10), no longer reproduces the experimental measured S-parameter, Sexpt, for either frequency. To illustrate this, consider the reflectivity profile produced by DAS using the information from the dual-frequency measurement,

If this relationship is used in Equation (9), we obtain,

for Sexpt(f0,a0). Neglecting the prefactor 1/V and simplifying the integrand, this becomes

This demonstrates that the experimental S-parameters are not perfectly reconstructed when the DAS beamformer is used with multiple measurements. Therefore, the gradient descent algorithm may be able to provide an improvement with respect to the loss function l(σ). This mathematical discussion motivated the development of the optimization-based reconstruction approach, as the ORR may improve upon the DAS reconstruction, with respect to the loss function l(σ).

The initial reflectivity profile estimate for all reconstructions produced in this work was a uniform map of zeroes, ∀r,σ(0)(r)=0. The reconstruction procedure was stopped when the relative change in the loss function was less than 0.1%, i.e.,