computational methods for microwave medical imaging

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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

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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

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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 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

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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

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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 forward solution
Solve for parameter updates

• FEM
• Assemble A
• Assemble b
• Apply BC
• Solve Axb

• FDTD
• Compute update coeff.
• do t1timestep
• Update E
• Update H
• If steady-state? break
• enddo
• ampphase extraction

Compare predicted field measured field
Evaluate Jacobian
no
Good enough?
yes
End
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Computational efficiency comparison
FE/BE (direct method) Matrix size
Half-bandwidth Banded LU decomposition
flop2np22np Cholesky decomposition
flopnp27np2nnflop(sqrt) LDLT decomposition
flopnp28npn
FDTD flopNsteadyflopiter 56sqrt(2)N(N2NPML
)2cmax/cbk
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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 volumeand 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
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3D FDTD

• FDTDUPML for lossy media
• Computational efficiency

Yee-grid PML layer
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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 FDTDinitial fields
• for high-resolution mesh, it may speed up
  computation by a factor of (3/6)CLFNADI /
  CLFNYEE

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3D microwave imaging system
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Reconstruction accuracy appropriateness of
parameter estimation model
WLS estimator
ML estimator
OLS estimator
?
MAP estimator

• Gaussian distribution
• additive noise
• zero mean
• constant variance
• .

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Reconstruction efficiency

• The sensitivity equation methodneed to perform
  forward equation back substitution for (ns X np)
  times
• The adjoint method only matrix-vector
  multiplications

sensitivity equation
adjoint method
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Nodal adjoint approximation

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

Node i
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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
  independent 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

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Singular vectors basis functions

• basis of the image linear combination of
• basis of RHS linear combination of

Zernike polynomials
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Singular values degree of ill-posedness

• singular spectrum measure the information
  redundancy the difficulty of solving the
  problem

measurement noise ill-posed nature
effective rank
maximum angular/radial modes
image resolution
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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

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3D scattering nulls

• in R3, 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 z0, then the close-path
  phase unwrapping integral equals to

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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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Migration of scattering nulls
varying frequency from 600MHz-2.5G
varying contrast of the object
out-of-phase curves equal-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 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

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Thanks!
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Questions?