computational methods for microwave medical

imagingPh.D. Thesis Defense

Qianqian FangThayer School of Engineering

Dartmouth College, Hanover, NH, 03755

Exam Committee:Professor Paul Meaney

Professor Keith PaulsenProfessor William Lotko

Professor Eric Miller

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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

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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

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Nonlinearity

Nonlinearity between the measurement and the property:

Forward problem is nonlinear Inverse problem is nonlinear

( , ) ( ( , ))E r t k r tF

1 2 1 2( ( , ) ( , )) ( ( , )) ( ( , ))

( ( , )) ( ( , ))

k r t k r t k r t k r t

k r t k r t

F F F

F Fa a

+ = +

=

?

?

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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

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Forward field modeling efficiency2D 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

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FDTD (Finite Difference-Time Domain) method in microwave tomography Conceptually straightforward, easy to progra

m Good absorption boundary condition Marching-On-Time feature (MF,initial field) Lower computational complexity Easy to parallelize

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2D FDTD dual-mesh

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Using FDTD forward modeling in an iterative reconstruction

Start

Set initial guess

Evaluate forwardsolution

Compare predicted fieldmeasured field

Evaluate Jacobian

Solve for parameter updates

Good enough?

FEM:

1. Assemble A

2. Assemble b

3. Apply BC

4. Solve Ax=b

FDTD:

1. Compute update coeff.

2. do t=1:timestep

1. Update E

2. Update H

3. If steady-state? break

3. enddo

4. amp&phase extractionEnd

no

yes

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Computational efficiency comparison

FE/BE (direct method)Matrix size: Half-bandwidth:

Banded LU decomposition: flop=2np2+2npCholesky 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

2

4n Np=p Np=

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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

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Forward field accuracy

2D/3D scalar/3D vector in homogeneous and inhomogeneous cases

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Path to 3D imaging

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]

2D dielectric property distribution [not true] Infinitely long line source [not true] 2D TM wave [not true]

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3D FDTD

FDTD+UPML for lossy media

Computational efficiency

Yee-grid+PML layer

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I: High-order FDTD: 4-th order spatial difference Reduction in mesh size X1/8 (NN/2) FLOPiter count X6 Conclusion: computational enhancement is not significa

nt. 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

Optimizations of 3D FDTD

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3D microwave imaging system

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Reconstruction accuracy: appropriateness of parameter estimation model

( ) F k

OLS estimator

WLS estimator

ML estimator

MAP estimator?•Gaussian distribution•additive noise•zero mean•constant variance•….

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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

,2

1, , ( )( , , )

ˆ,s r

n

i j n s r

s r ni j n sr r

r rr

r rk

J rff j d

ff j

-ì -ïïï=¶

¶= íï -ïïî

A E

E E

Er r

r

rr r r sensitivity equation

adjoint method

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Nodal adjoint approximation

Non-conformal dual-meshes: evaluation of the integral is difficult

Node i

Node j

n

ee R

n

V

VM

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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 in

dependent information about the target

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SVD analysis of Jacobian Linear approximation to the inverse of the imaging

operator Nodal adjoint form of the Jacobian matrix:

,

( , ) ( , ) ( , ) ( )ns r m m s r

mn

g r r g r r r r rl f j= å

1 1: 1: 2 1: 1: 1: 1:

1 2: 2: 2 2: 2: 2: 2:

1 : : 2 : : : :

( , ) ( , ) ( , )

( , ) ( , ) ( , )

( , ) ( , ) ( , )

s r s r K s r

s r s r K s r

Q s Q r Q s Q r K Q s Q r

r r r r r r

r r r r r r

r r r r r r

ff f

ff f

ff f

æ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷ç ÷ç ÷ç ÷ç ÷çè ø

1

2

K

l

l

l

æ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷ç ÷ç ÷ç ÷ç ÷÷çè ø

1 1 2 1 1

1 2 2 2 2

1 2

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

TK

K

Q Q K Q

r r r

r r r

r r r

j j j

j j j

j j j

æ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷ç ÷ç ÷ç ÷ç ÷çè ø

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Singular vectors: basis functions basis of the image: linear combination of basis of RHS: linear combination of

( )rj( , )s rr rf

Zernike polynomials

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Singular values: degree of ill-posedness singular spectrum: measure the information redun

dancy & the difficulty of solving the problem

singular spectrum

degree-of-illposednessslope

effective rank

maximum angular/radial modes

image resolution

measurement noiseill-posed nature

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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

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3D scattering nulls

in 3, the equal-amplitude and out-of-phase point set are 2D surfaces, their intersection is 1D curve.

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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

C

2 Ind( ')p× G

12 Lk( , (0))Wp -× G

: nW ®R C

( ( ), )W r GU

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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.

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varying frequency from 600MHz-2.5G varying contrast of the object

Migration of scattering nulls

out-of-phase curvesequal-amplitude curves

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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.

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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

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Acknowledgement

Professor Paul Meaney Professor Keith Paulse

n Professor William Lotk

o Professor Eric Miller Professor Eugene Demi

denco Professor Brian Pogue Professor Vladimir Che

rnov

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

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Thanks!

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Questions?