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
Transcript
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.
2D Newton Method w/ LM2.5D Newton Method w/LM/SVD/CODFull 3D reconstructorSA/GA ReconstructorGlobal Optimization methods……
Property Distribution
Structure of Microwave Imaging
Current Status
Goal: A Fast, Accurate, Global Convergence, High Efficiency Iterative Approach
Initial Property Estimation
2
2 2
2 2T T1
Forward Solution
: ( , , ){ } ,
: [A]{ } { }
Building Jacobian Matrix
A[A] { }
Multi-Variable Gauss-Newton Formular
{ }
s s
s s
ss
n n
t t
FEM Weak Form k
Matrix Form b
k k
E J
E
EE
J J k k J
2 21 1 1 T1
{ }
Levenberg-Marquart Regularization
{ } { }
m ct
m ct t t
E E
D H D I D k k D J E E
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.
Monopole Antenna
Forward 3D Mesh
2D Reconstruction Mesh
Antenna Array Configurations
Forward Solutioncalculated over 3D Forward Mesh
Properties Reconstructed on 2D Grid
Radiation Boundary Condition
Mapping Property Distribution to 3D Mesh,for Forward Solver at Next Iteration
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 MeshReconstruction -> 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.
Timing for multi-RHS solver(opposite antenna position), s/RHS
Opposite AntennaRotate Antenna
Multi-RHS Iterative Solver
1 2 3 4[ ]{ } { | | | | }A E b b b b
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-mesh Adjoint 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
References
Comparison of the adjoint and influence coefficient methods for solving the inverse hyperthermia problem. Liauh C T; Hills R G; Roemer R B,JOURNAL OF BIOMECHANICAL ENGINEERING vol115(1), pp63-71,1993
Microwave Image Reconstruction Utilizing Log-Magnitude and Unwrapped Phase to Improve High-Contrast Object Recovery, P.M.Meaney.K.D.Paulsen,etc,IEEE TRANS. ON MEDICAL IMAGING,vol20,pp104-116,2001
Acknowledgement
This work was sponsored by NIH/NCI grant number R01 CA55034-09
Quantification of 3D field effects during 2D microwave imagingP.M.Meaney.K.D.Paulsen,etc,IEEE TRANS. ON MEDICAL IMAGING,2002(in press)
A numerical solution to full-vector electromagnetic scatterig by three-dimensional nonlinear bounded dielectricsCaorsi S,Massa A,Pastorino M, IEEE TRANS. ON MTT,vol. 43,pp.428-436,1995
