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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).

Numerical Modeling for Image Reconstruction

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Numerical Modeling for Image Reconstruction

SubhaSrinivasan

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

fan beam

detectors

patient

x-rays

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

Structure of Jacobian

Ф = I e-i(ωt+θ)

I = signal amplitude

θ = signal phase

m = [ma, D]

Absorption coeff.

Diffusion Coeff.

Shape of Jacobian

Calculated by:

1)Perturbation

Method

2)Direct Analytic

Jacobian

Dehghani notes

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:

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

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

• parameters within the minimization scheme => stability

Limitation:

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

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

Spectral Image Reconstruction

Data from Boulnois et al, Hale & Quarry,

figure from thesis Srinivasan et al

Spectral Image Reconstruction

Relationships between Jsp & J can be obtained

Details, refer to Srinivasan et al, AO, 2005

Simulations show Reduced Cross-talk in spectral images

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

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!