We have recently implemented a microwave imaging algorithm which incorporated scalar 3D wave propagation while reconstructing a 2D dielectric property profile. This is a preliminary step in reaching a full 3D image reconstruction approach but allows us to investigate important issues associated with speed of reconstruction and problem size. Key contents developed during our 2D system evaluations have also proved to be translatable to the 3D approach and have accelerated the overall implementation. Abstract

Apparatus DAQ System Data Acquisition Data Pre-Processing Noise Filtering Reconstruction Program Forward SolverJacobian Builder Reconstruction Algorithm Vector/Scalar Time /Freq Domain 2D/3D 2D Hybrid Elem. Solver 2D FEM Solver 3D FEM Solver 3D Vector FEM Solver 3D TD/vector Solver …… Influence Coefficient Method Sensitivity Equation Method Adjoint Method Approximated Adjoint Method 2D/3D reconstructor Tikhonov /LM regularization SVD/COD 2D Newton Method w/ LM 2.5D Newton Method w/LM/SVD/COD Full 3D reconstructor SA/GA Reconstructor Global Optimization methods …… Property Distribution Structure of Microwave Imaging Current Status Goal: A Fast, Accurate, Global Convergence, High Efficiency Iterative Approach Initial Property Estimation

Forward & Reconstruction Model 3D Scalar Helmholtz Equation Previously the most time-consuming part in the reconstruction. Significant improvement has been achieved by implementing the Adjoint Method.

Full 3D Forward Solution Scattering Pattern of an Inhomogeneous Object due to a Monopole Source Homogeneous Solution (Amplitude) at Mesh Perimeter

Dual-mesh scheme: Performing the Electric Field Forward Solution and Property Reconstruction on Separate Meshes, and Defining a Set of Mapping Rules Between Two Meshes. Forward Solver -> Fine Mesh Reconstruction -> Reconstruction Mesh The implementation of dualmesh makes it possible to choose any computational model for forward solver, and any reconstruction method as reconstructor. They can be 2D or 3D methods, and their discretization elements also can be determined arbitrarily, i.e. linear or higher order elements. Dual-Mesh Scheme FE 2D/2D dualmesh FD 2D/2D dualmesh Dualmesh Bi-Direction Mapping The 3D Forward/2D Reconstruction dualmesh case allows for 3D propagation of the signals while restricting the reconstruction problem to a manageable number of parameters. This is a natural intermediate step for a full 3D forward/3D reconstruction algorithm.

Source=1, multiple receivers Source=2, multiple receivers Source=ns, multiple receivers Source ID Receiver ID Parameter node ID Jacobian Matrix Plot of one row of the Jacobian Matrix: indicating the sensitivity of the field data due to a perturbations at each parameter node Derivative with respect to the 1 st parameter node

Adjoint Method JsJs Perturbations At Node n Source Receiver Original Method Evaluated Each Element in the Jacobian J 1 E 2 = J 2 E 1 J1J1 J2J2 E2E2 E1E1 Can be regarded as an equivalent Source. Placing an auxiliary source J r at receiver, and applying the reciprocity relationship Finally: Perturbation Current: Simple inner product of two field distributions with a weight

Total: Solving [A]{x}={b} per iteration To build Jacobian Solving [A]{x}={b} per iteration Real Calculation Results(10 iterations) [A] is 10571X10571 Sensitivity Equation Method Ns+Ns*NcNs*Nc5:37:05’’ (Using Parallel Solver 5 SGI machines) Adjoint Method NsOnly Vector Multiplication 13:23’’ Efficiency of Adjoint Method Computational cost for Sensitivity Equation Method: For each iteration: Solving the AX=b for (Ns+Ns*Nc) times, where Ns= Source number Nc= Parameter node number Computational cost for Adjoin method For each iteration: Solving the AX=b for Ns times, where Ns= Source number This is 1/(Nc+1) of the time required by the Sensitivity Equation Method Example: Problem size: forward nodes:10571(full 3D) Reconstruction nodes: 126(2D) For Building Jacobian Matrix For Forward Solution

Reconstruction of Simple Object Permittivity Conductivity 1D Crosscut Semi-Infinite Cylindrical Object

Reconstruction of Large Object Permittivity Conductivity Semi-Infinite Cylindrical Object with Inclusion 1D Crosscut

Multi-RHS Iterative Solver This matrix equation is solved using an iterative multi-RHS QMR solver, with pre-conditioning of the LHS matrix. It is found that the solver converged faster when source locations were ordered sequentially verses randomly. By benchmarking, an optimal RHS package size was determined to provide minimum averaging solving time. In each iteration of the reconstruction process, a number of forward solutions must be completed,since the Left-Hand Side(LHS) matrix is identical for all sources, it is possible to decompose it once,and perform the back substitutions for a set of RHS’s. Best RHS number for solving once is 7

Conclusions Forward solvers Reconstruction Schemes Dual-meshAdjoint Model 1.The Dual-Mesh Scheme provides great flexibility in choosing the forward solver and reconstruction method. 2.The Adjoint method demonstrates dramatic improvement in speed and efficiency in construction of the Jacobian matrix. 3.3D wave propagation effects were reduced by using 3D forward solver when compared with the pure 2D counterpart. 4.The Phase unwrapping formulation provides the algorithm the ability to reconstruct large/high contrast objects. 5.The Dual-mesh and Adjoint techniques are both easily extended to the full. 6.Efficiencies gained using the Adjoint method have been applied to the full 2D approach where reconstructions can now be computed in about 2 minutes

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