1. computational methods for microwave medical imaging
Ph.D. Thesis Defense
Qianqian Fang
Thayer School of Engineering
Dartmouth College,
Hanover, NH, 03755
Exam Committee:
Professor Paul Meaney
Professor Keith Paulsen
Professor William Lotko
Professor Eric Miller

2. Qianqian's PhD Thesis Defense
Outline
Overview
Forward field modeling accuracy and efficiency
Implementation of the FDTD method
3D microwave imaging
System and results
Reconstruction efficiency
Estimation model
The adjoint method and the nodal adjoint approximation
SVD analysis of the Jacobian matrix
Phase singularity and phase unwrapping
Scattering nulls
Dynamic phase unwrapping in image reconstruction
Conclusions
Qianqian's PhD Thesis Defense

3. Characteristics of Dartmouth Microwave imaging system
Tomography, wide-band operating frequency, small target, lossy background, simple antenna
Modeling nonlinear scattered field, nonlinear (iterative) parameter estimation
Advantage of accessing in vivo data (small animal/patient breast imaging), first clinical microwave imaging system in the US
Qianqian's PhD Thesis Defense

4. Qianqian's PhD Thesis Defense
Nonlinearity
Nonlinearity between the measurement and the property:
Forward problem is nonlinear
Inverse problem is nonlinear ? ?
Qianqian's PhD Thesis Defense

5. Qianqian's PhD Thesis Defense
Specific aims
Improving image reconstruction performance:
forward modeling accuracy (3D imaging) and efficiency, explore the balance point, generalized dual-mesh
reconstruction quality/efficiency improvement: correctness of the estimation model, multi-frequency measurement data, adjoint method and nodal adjoint approximation
In-depth understanding of nonlinear tomography
impact of noise, resolution limit, optimization of system configuration
Scattering nulls and math of phase unwrapping
Qianqian's PhD Thesis Defense

6. Forward field modeling efficiency
2D scalar FE/BE method:
2D scalar model requires approximations
The coupling between the FE/BE equations increases the programming complexity,
BE method: accurate (compared with approximated BC), but enlarges the bandwidth of the combined system
Qianqian's PhD Thesis Defense

7. FDTD (Finite Difference-Time Domain) method in microwave tomography
Conceptually straightforward, easy to program
Good absorption boundary condition
Marching-On-Time feature (MF,initial field)
Lower computational complexity
Easy to parallelize
Qianqian's PhD Thesis Defense

8. Qianqian's PhD Thesis Defense
2D FDTD dual-mesh
Qianqian's PhD Thesis Defense

9. Using FDTD forward modeling in an iterative reconstruction
Start
Set initial guess
Evaluate forward
solution
Solve for
parameter updates
FEM:
Assemble A
Assemble b
Apply BC
Solve Ax=b
FDTD:
Compute update coeff.
do t=1:timestep
Update E
Update H
If steady-state? break
enddo
amp&phase extraction
Compare
predicted field
measured field
Evaluate
Jacobian no
Good enough?
yes
End
Qianqian's PhD Thesis Defense

10. Computational efficiency comparison
FE/BE (direct method)
Matrix size:
Half-bandwidth:
Banded LU decomposition:
flop=2np2+2np
Cholesky decomposition:
flop=np2+7np+2n+n*flop(sqrt)
LDLT decomposition
flop=np2+8np+n
FDTD:
flop=Nsteady*flopiter
=56sqrt(2)N(N+2NPML)2cmax/cbk
Qianqian's PhD Thesis Defense

11. Qianqian's PhD Thesis Defense
FLOP count vs. mesh size
The result may be different if : FE
uses an iterative solver
uses approximated BC
FDTD
use polar coordinate
separate working volume and PML layer
Qianqian's PhD Thesis Defense

12. Forward field accuracy
2D/3D scalar/3D vector in homogeneous and inhomogeneous cases
Qianqian's PhD Thesis Defense

13. Qianqian's PhD Thesis Defense
Path to 3D imaging
2D dielectric property distribution [not true]
Infinitely long line source [not true]
2D TM wave [not true]
2D dielectric property distribution [not true]
Infinitely long line source
2D TM wave  3D scalar field [not true]
2D dielectric property distribution [not true]
Infinitely long line source
2D TM wave  3D scalar field [not true]
Qianqian's PhD Thesis Defense

14. Qianqian's PhD Thesis Defense
3D FDTD
FDTD+UPML for lossy media
Computational efficiency
Yee-grid
+PML layer
Qianqian's PhD Thesis Defense

15. Optimizations of 3D FDTD
I: High-order FDTD: 4-th order spatial difference
Reduction in mesh size  X1/8 (NN/2)
FLOPiter count  X6
Conclusion: computational enhancement is not significant.
II: Setting initial fields
start FDTD time-stepping from the final field of last iteration can reduce steady-state time step to 1/2 or 1/3
III: ADI FDTD+initial fields
for high-resolution mesh, it may speed up computation by a factor of (3/6)*CLFNADI / CLFNYEE
Qianqian's PhD Thesis Defense

16. 3D microwave imaging system
Qianqian's PhD Thesis Defense

17. Reconstruction accuracy: appropriateness of parameter estimation model
WLS estimator
ML estimator
OLS estimator ?
MAP estimator
Gaussian distribution
additive noise
zero mean
constant variance ….
Qianqian's PhD Thesis Defense

18. Reconstruction efficiency
The sensitivity equation method: need to perform forward equation back substitution for (ns X np) times
The adjoint method: only matrix-vector multiplications
sensitivity equation
adjoint method
Qianqian's PhD Thesis Defense

19. Nodal adjoint approximation
Non-conformal dual-meshes: evaluation of the integral is difficult
Node i
Node j
Qianqian's PhD Thesis Defense

20. Multi-frequency reconstruction
Trade-off in operating frequency:
Low High
Frequency
Ill-posedness
Nonlinearity
Assumptions:
Known (simple) dispersion relationships
Measurements at different freq. provide linearly independent information about the target
Qianqian's PhD Thesis Defense

21. SVD analysis of Jacobian
Linear approximation to the inverse of the imaging operator
Nodal adjoint form of the Jacobian matrix:
Qianqian's PhD Thesis Defense

22. Singular vectors: basis functions
basis of the image: linear combination of
basis of RHS: linear combination of
Zernike polynomials
Qianqian's PhD Thesis Defense

23. Singular values: degree of ill-posedness
singular spectrum: measure the information redundancy & the difficulty of solving the problem
singular spectrum
degree-of-illposedness
slope
measurement noise
ill-posed nature
effective rank
maximum angular/radial modes
image resolution
Qianqian's PhD Thesis Defense

24. Qianqian's PhD Thesis Defense
Scattering nulls
Definition: the interference between the incidence wave and scattered wave creates null field at certain spatial locations (such as points or curves).
Properties: field amplitude is zero, phase is uncertain  ambiguity in phase unwrapping
plane wave scattered by cylindrical object at 700MHz
Qianqian's PhD Thesis Defense

25. Qianqian's PhD Thesis Defense
3D scattering nulls
in R3, the equal-amplitude and out-of-phase point set are 2D surfaces, their intersection is 1D curve.
Qianqian's PhD Thesis Defense

26. Phase unwrapping with the presence of phase singularities
Theorem 1: Let be a continuously real-differentiable function; let  be a path, then the value of phase unwrapping integral is unique.
Theorem 2: If the image of a close path  in plane is ’, then, the value of close-path phase unwrapping integral equals to
Theorem 3: If W has full rank at every point in the inverse image of z=0, then the close-path phase unwrapping integral equals to
Qianqian's PhD Thesis Defense

27. Static and dynamic phase unwrapping problems
Static phase unwrapping: evaluate the line-integral along a selected unwrapping path over a static phase map;
Dynamic phase unwrapping: evaluate static phase unwrapping at a series of phase map frames, the results should satisfy continuation condition.
Qianqian's PhD Thesis Defense

28. Migration of scattering nulls
varying frequency from 600MHz-2.5G
varying contrast of the object
out-of-phase curves
equal-amplitude curves
Qianqian's PhD Thesis Defense

29. Implementation of phase unwrapping in image reconstruction
LMPF algorithm: log-magnitude and unwrapped phase  faster convergence behavior, less artifacts
Break down of LMPF algorithm for high-contrast object reconstruction (scattering nulls, intermediate nulls)
Dynamic phase unwrapping problem: detect the trajectory of scattering null and adjust the result to satisfy continuation condition.
Qianqian's PhD Thesis Defense

30. Qianqian's PhD Thesis Defense
Conclusion
FDTD method shows promise
3D imaging is viable with current computational power
Adjoint method is critical
SVD analysis is useful to show insight about image formation and correlates the important system parameters
The phenomenon of scattering null has both theoretical and practical value for both electromagnetics and mathematics
Investigation of nonlinear phenomena for imaging is important for
Qianqian's PhD Thesis Defense

31. Qianqian's PhD Thesis Defense
Acknowledgement
Professor Paul Meaney
Professor Keith Paulsen
Professor William Lotko
Professor Eric Miller
Professor Eugene Demidenco
Professor Brian Pogue
Professor Vladimir Chernov
Margaret Fanning
Dun Li
Sarah Pendergrass
Colleen Fox
Timothy Raynolds
Navin Yagnamurthy
Xiaomei Song, Qing Feng, Heng Xu, Chao Sheng, Nirmal Soni, Subhadra Srinivasan, Kyung Park
My parents and my girl friend Yinghua Shen
Qianqian's PhD Thesis Defense

32. Qianqian's PhD Thesis Defense
Thanks!
Qianqian's PhD Thesis Defense

33. Qianqian's PhD Thesis Defense
Questions?
Qianqian's PhD Thesis Defense