numerical modeling for image reconstruction
DESCRIPTION
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). - PowerPoint PPT PresentationTRANSCRIPT
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!