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Numerical Modeling for Image Reconstruction

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Numerical Modeling for Image Reconstruction

SubhaSrinivasan

11/2/2009

- 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)

- Back-projection methods
- Perturbation methods
- Non-linear optimization 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

fan beam

detectors

patient

x-rays

Filtered Back-projection method:

[measurements] = [attenuation op.] [object ]

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

- 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

Ф = I e-i(ωt+θ)

I = signal amplitude

θ = signal phase

m = [ma, D]

Absorption coeff.

Diffusion Coeff.

Calculated by:

1)Perturbation

Method

2)Direct Analytic

Jacobian

3) Adjoint method

Dehghani notes

- 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

- 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.

- Minimizing error functional:
- Setting derivative to zero:
- Taylor’s approximation
- Rewriting:
- Substituting:
- Update equation:

- 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

Tikhonov Minimization:

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

L is dimensionless

common choice: L = I

(the identity matrix)

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

*Yalavarthy et. al., Medical Physics, 2007

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)

- 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’

- Simulated Measurements, 5% Noise

Data from Boulnois et al, Hale & Quarry,

figure from thesis Srinivasan et al

Relationships between Jsp & J can be obtained

Details, refer to Srinivasan et al, AO, 2005

HbT(μM)

StO2(%)

Water (%)

Scatt Ampl.

Scatt Power

True

- Data generated from a tumor-simulating phantom using FEM forward model, with 1% random-Gaussian noise added.

Spectral

Conv.

- Spectral Method: Smoother Images; 15.3 % mean error compared to 43% (conv. Method).
- Reduced Cross-talk between HbO2 and water: from30% (conv.) to 7% (spectral).
- Accuracy in StO2 accurate (<1% error)

Srinivasan et al, PhD thesis, 2005

Brooksby, Srinivasan et al, Opt Lett, 2005

- 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!