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Numerical Modeling for Image Reconstruction

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).

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Numerical Modeling for Image Reconstruction

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  1. Numerical Modeling for Image Reconstruction SubhaSrinivasan 11/2/2009

  2. 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)

  3. Ways of Solving Inverse Problems • Back-projection methods • Perturbation methods • Non-linear optimization methods

  4. 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]

  5. 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

  6. Structure of Jacobian Ф = I e-i(ωt+θ) I = signal amplitude θ = signal phase m = [ma, D] Absorption coeff. Diffusion Coeff.

  7. Shape of Jacobian Calculated by: 1)Perturbation Method 2)Direct Analytic Jacobian 3) Adjoint method

  8. Adjoint method for Jacobian Calculation Dehghani notes

  9. 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

  10. 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.

  11. Deriving Update Equation using Least Squares Minimization • Minimizing error functional: • Setting derivative to zero: • Taylor’s approximation • Rewriting: • Substituting: • Update equation:

  12. 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

  13. 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

  14. 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)

  15. Flow-chart for Iterative Image Reconstruction

  16. 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’

  17. Reconstruction Results • Simulated Measurements, 5% Noise

  18. Recovery of Absorption

  19. Recovery of Scattering

  20. Spectral Image Reconstruction Data from Boulnois et al, Hale & Quarry, figure from thesis Srinivasan et al

  21. Spectral Image Reconstruction Relationships between Jsp & J can be obtained Details, refer to Srinivasan et al, AO, 2005

  22. 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

  23. Results from Image Reconstruction:Experimental Data Brooksby, Srinivasan et al, Opt Lett, 2005

  24. 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!

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