Transcript Computational methods for microwave medical imaging

THAYER
SCHOOL OF
ENGINEERING
DARTMOUTH
COLLEGE
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
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
2004-12-22
Qianqian's PhD Thesis Defense
2
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
2004-12-22
Qianqian's PhD Thesis Defense
3
Nonlinearity
 Nonlinearity between the measurement and the
property:
E (r , t )  F (k (r , t ))
F (k1(r , t ) + k2 (r , t )) =? F (k1(r , t )) + F (k2 (r , t ))
F (a k (r , t )) =? a F (k (r , t ))
 Forward problem is nonlinear
 Inverse problem is nonlinear
2004-12-22
Qianqian's PhD Thesis Defense
4
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
2004-12-22
Qianqian's PhD Thesis Defense
5
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
2004-12-22
Qianqian's PhD Thesis Defense
6
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
2004-12-22
Qianqian's PhD Thesis Defense
7
2D FDTD dual-mesh
2004-12-22
Qianqian's PhD Thesis Defense
8
Using FDTD forward modeling in an
iterative reconstruction
Start
Set initial guess
Solve for
parameter updates
Evaluate forward
solution
Compare
predicted field
measured field
Evaluate
Jacobian
no
FEM:
FDTD:
1.
Assemble A
1.
Compute update coeff.
2.
Assemble b
2.
do t=1:timestep
3.
Apply BC
1.
Update E
4.
Solve Ax=b
2.
Update H
3.
If steady-state? break
Good enough?
yes
3.
enddo
4.
amp&phase extraction
End
2004-12-22
Qianqian's PhD Thesis Defense
9
Computational efficiency comparison
FE/BE (direct method)
n = p 4N 2
Matrix size:
Half-bandwidth: p = p N
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
2004-12-22
Qianqian's PhD Thesis Defense
10
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
2004-12-22
Qianqian's PhD Thesis Defense
11
Forward field accuracy
 2D/3D scalar/3D vector in homogeneous and
inhomogeneous cases
2004-12-22
Qianqian's PhD Thesis Defense
12
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]
2004-12-22
Qianqian's PhD Thesis Defense
13
3D FDTD
 FDTD+UPML for lossy
media
 Computational efficiency
Yee-grid
+PML layer
2004-12-22
Qianqian's PhD Thesis Defense
14
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
2004-12-22
Qianqian's PhD Thesis Defense
15
3D microwave imaging system
2004-12-22
Qianqian's PhD Thesis Defense
16
Reconstruction accuracy:
appropriateness of parameter estimation model
WLS estimator
•Gaussian distribution
•additive noise
•zero mean
•constant variance
•….
2004-12-22
?
MAP estimator
  F (k )
Qianqian's PhD Thesis Defense
17
Reconstruction efficiency
¶ Errs ,rrr
r r r
J (rs , rr , rn ) =
¶ krrn 2
ìï ï
= í
ïï ïî
A- 1, f i f j j
f if jj
n
n
ˆr
Es , E
r
Es , d(rr )
sensitivity equation
adjoint method
 The sensitivity equation method:
need to perform forward equation back substitution for (ns X
np) times
 The adjoint method:
only matrix-vector multiplications
2004-12-22
Qianqian's PhD Thesis Defense
18
Nodal adjoint approximation
 Non-conformal dual-meshes:
evaluation of the integral is
difficult
Node i
Node j
Vn 
2004-12-22
V
eR n
e
M
Qianqian's PhD Thesis Defense
19
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
2004-12-22
Qianqian's PhD Thesis Defense
20
SVD analysis of Jacobian
 Linear approximation to the inverse of the imaging operator
 Nodal adjoint form of the Jacobian matrix:
g(rs , r )g(rr , r ) =
å
l mn f m (rs , rr )j (r )
m ,n
æf 1(r1:s , r1:r ) f 2 (r1:s , r1:r )
çç
çç f (r , r ) f (r , r )
2 2:s 2:r
çç 1 2:s 2:r
çç
çç
ççf (r , r ) f (r , r )
2 Q :s Q :r
è 1 Q :s Q :r
2004-12-22
ö
f K (r1:s , r1:r ) ÷
÷
÷
f K (r2:s , r2:r ) ÷
÷
÷
÷
÷
÷
÷
÷
÷
÷
f K (rQ :s , rQ :r )ø
÷
æl 1
çç
çç
l2
çç
çç
çç
çç
çè
ö
÷
÷
÷
÷
÷
÷
÷
÷
÷
÷
÷
÷
÷
÷
lKø
Qianqian's PhD Thesis Defense
æj 1(r1 ) j 2 (r1 )
çç
ççj (r ) j (r )
2 2
çç 1 2
çç
çç
çj (r ) j (r )
çè 1 Q
2 Q
T
ö
j K (r1 ) ÷
÷
÷
j K (r2 ) ÷
÷
÷
÷
÷
÷
÷
÷
÷
j K (rQ )÷
÷
ø
21
Singular vectors: basis functions
 basis of the image: linear combination of j (r )
 basis of RHS: linear combination of f (rs , rr )
Zernike polynomials
2004-12-22
Qianqian's PhD Thesis Defense
22
Singular values:
degree of ill-posedness
 singular spectrum: measure the information redundancy & the
difficulty of solving the problem
degree-of-illposedness
measurement noise
ill-posed nature
slope
effective rank
maximum angular/radial modes
singular spectrum
image resolution
2004-12-22
Qianqian's PhD Thesis Defense
23
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

2004-12-22
Qianqian's PhD Thesis Defense
24
3D scattering nulls
 in R3, the equal-amplitude and out-of-phase point set
are 2D surfaces, their intersection is 1D curve.
2004-12-22
Qianqian's PhD Thesis Defense
25
Phase unwrapping with the presence
of phase singularities
 Theorem 1: Let W : R n ® C be a continuously realdifferentiable function; let  be a path, then the value
of phase unwrapping integral U(W (r ), G) is unique.
 Theorem 2: If the image of a close path  in C plane
is ’, then, the value of close-path phase unwrapping
integral equals to
2p ×Ind( G')
 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
2p ×Lk(G,W - 1(0))
2004-12-22
Qianqian's PhD Thesis Defense
26
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.
2004-12-22
Qianqian's PhD Thesis Defense
27
Migration of scattering nulls
varying frequency from 600MHz-2.5G
varying contrast of the object
out-of-phase curves
equal-amplitude curves
2004-12-22
Qianqian's PhD Thesis Defense
28
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.
2004-12-22
Qianqian's PhD Thesis Defense
29
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
2004-12-22
Qianqian's PhD Thesis Defense
30
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
2004-12-22
Qianqian's PhD Thesis Defense
31
Thanks!
2004-12-22
Qianqian's PhD Thesis Defense
32
Questions?
2004-12-22
Qianqian's PhD Thesis Defense
33