Numerical Modeling for Image Reconstruction

Subha Srinivasan

11/2/2009

Definition of Inverse Problem..

• Definition: Given a distribution of sources and a distribution of measurements at the boundary dΩ, finding the tissue parameter distribution within domain Ω.

• Expressed as x = F-1(y)

Ways of Solving Inverse Problems

• Back-projection methods

• Perturbation methods

• Non-linear optimization methods

Back-projection methods

• Assumes that each projection provides a nearly independent measurement of the domain.

• Assumes that light travels in a straight line: not true with tissue unless scattering is isolated

x-rays

fan beamdetectorspatientpatient

Filtered Back-projection method:[measurements] = [attenuation op.] [object ]

[image] = [attenuation op.]T [filter] [measurements]

Linear reconstruction for Change in Optical Properties

• x = F-1(y) is a non-linear problem: can be linearized using Taylor’s series expansion if initial estimate is close to actual values:

Jacobian matrix

Reconstructing for changes rather than absolute values

y = f(μ0 ) + f '(μ0 )(μ −μ0 ) + ...

J =∂f(μ0 )

∂μ

Ф = I e-i(ωt+θ)

I = signal amplitudeθ = signal phase

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C

Jμ

μ = [μa, D]Absorption coeff.Diffusion Coeff.

Structure of Jacobian

Calculated by:

1)PerturbationMethod

2)Direct Analytic Jacobian

3) Adjoint method

Shape of Jacobian

Adjoint method for Jacobian Calculation

Dehghani notes

−∇.κ∇Φadj + μ a −iω

c⎛⎝⎜

⎞⎠⎟

ΦAdj = qAdj

PDIRECT ×PADJOINT

Solving

• Linearizing change in intensity: born approximation• Linearizing change in log intensity: Rytov

approximation• Inverting J: large, under-determined and ill-posed:

some standard methods can be used• Truncated SVD, Tikhonov regularization, Algebriac

reconstruction techniques (ART) & Conjugate Gradient methods are commonly used

Terminology: Inverse Problem

• Ill-posed–Small changes in the data can cause large changes in the parameters.

• Ill-conditioned–The condition number (ratio of largest singular value to smallest singular value) is large, which implies the inverse solution would not be unique.

• Ill-determined–(or under-determined) The number of independent equations are smaller than number of unknowns.

Deriving Update Equation using Least Squares Minimization

• Minimizing error functional:

• Setting derivative to zero:

• Taylor’s approximation

• Rewriting:

• Substituting:

• Update equation:

Ω = y − f (μ )2

∂Ω∂μ

=J Tδ = 0

f (μi ) = f(μi−1) + J Δμi + ...

i = y − f (μ i ) = y − f (μ i−1) − JΔμ i = δ i−1 − JΔμ i

J T i =0

⇒ J T ( i−1 −J Δμ) =0

[J T J ]Δμi =J T i−1

[J T J + λI ]Δμi =J T i−1

Assumptions of Levenberg-Marquardt Minimization:

• JTJ is positive-definite

• Initial guess must be close to actual solution

• Update equation does not solve first-order conditions unless α = 0

*Yalavarthy et. al., Medical Physics, 2007

L is dimensionless

Tikhonov Minimization:

*Tikhonov et. al, 1977; Tarantola SIAM 2004.

common choice: L = I(the identity matrix)

*Yalavarthy et. al., Medical Physics, 2007

Key idea is to introduce apriori assumptions about size and smoothness of desired solution:

Tikhonov Minimization

Advantage:

• parameters within the minimization scheme => stability

Limitation:

• it requires a prior opinion about the noise characteristics of the parameter and data spaces (for λ)

Flow-chart for Iterative Image Reconstruction

Choosing Regularization: L-curve criterion

• Convenient graphical tool for displaying trade-off between size of solution and its fit to the given data as λ varies.

• λ can also be chosen empirically or based on parameter/data values.

Hansen, ‘L-curve and its use in numerical treatment of inverse problems’

Reconstruction Results

• Simulated Measurements, 5% Noise

Recovery of Absorption

Recovery of Scattering

Spectral Image Reconstruction

μa (λ ) = ε i (λ )cii=1

nc

∑

μs '(λ ) = Aλ −b

Data from Boulnois et al, Hale & Quarry,figure from thesis Srinivasan et al

Spectral Image Reconstruction

Δc = (JspT Jsp + α spI )−1 Jsp

T ΔΦ sp

Δsp =

ΔΦλ 1

ΔΦλ 2

MΔΦλ k

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c =

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Jsp =

J c1,λ1 J c2,λ1 K J cn,λ1 J A,λ1 J b,λ1

J c1,λ2 J c2,λ2 K J cn,λ2 J A,λ2 J b,λ2

M M M M MJ c1,λk J c2,λk K J cn,λk J A,λk J b,λk

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Relationships between Jsp & J can be obtained

Details, refer to Srinivasan et al, AO, 2005

Simulations show Reduced Cross-talk in spectral images• Data generated from a tumor-simulating phantom using FEM forward model,

with 1% random-Gaussian noise added.

HbT(μM) StO2(%) Water (%) Scatt Ampl. Scatt Power

True

Spectral

Conv.

Srinivasan et al, PhD thesis, 2005

• Spectral Method: Smoother Images; 15.3 % mean error compared to 43% (conv. Method).

• Reduced Cross-talk between HbO2 and water: from 30% (conv.) to 7% (spectral).

•Accuracy in StO2 accurate (<1% error)

Results from Image Reconstruction:Experimental Data

Brooksby, Srinivasan et al, Opt Lett, 2005

References

• Gibson et al, Phy Med Bio: 50 : 2005: A review paper

• Paulsen et al, Med Phy: 22(6): 1995: first results from image-reconstruction in DOT

• Yalavarthy et al, Med Phy: 34(6): 2007: good explanation of math

• Brooksby et al, IEEE Journal of selected topics in quantum electronics: 9(2): 2003: good reference for spatial priors

• Hansen: ‘Rank deficient and discrete ill-posed problems’: SIAM: 1998: good reference for tikhonov/l-curve

• Srinivasan et al, Appl Optics: 44(10): 2005: reference for spectral priors

• Press et al: ‘Numerical Recipes in Fortran 77’: II edition: 1992: great book for numerical folks!