Challenges in Modern Medical Image Reconstruction · Challenges in Modern Medical Image...
Transcript of Challenges in Modern Medical Image Reconstruction · Challenges in Modern Medical Image...
Challenges inModernMedical
Image Recon-struction
Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
Challenges in Modern Medical ImageReconstruction
Misha E. Kilmer
Department of MathematicsTufts University
SIAM ALA, June 2012
Thanks: NIH R01-CA154774
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Challenges inModernMedical
Image Recon-struction
Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
Outline
1 Forward & Inverse Problems
2 Reduced Order Image Models: Motivation
3 Parametric Level Sets
4 Reduced Order Forward Model
5 Trust Region Regularized GN
6 Results
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Challenges inModernMedical
Image Recon-struction
Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
Collaborators
PaLS and Diffuse Optical Tomography
• Prof. Eric Miller, ECE, Tufts
• Dr. Alireza Aghasi, ECE, GA Tech
• Mr. Fridrik Larusson, ECE, Tufts
• Prof. Sergio Fantini, BME, Tufts
ROM for Diffuse Optical Tomography
• Prof. Serkan Gugerin, Math, VA Tech
• Prof. Chris Beattie, Math, VA Tech
• Prof. Eric de Sturler, Math, VA Tech
• Dr. Saifon Chaturantabut, Math, VA Tech
Trust-region Regularized GN
• Prof. Eric de Sturler, Math, VA Tech
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Challenges inModernMedical
Image Recon-struction
Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
Forward Problem Description
The discrete forward problem for an observation vector m is
m = M(f) + η
where f is a discretized representation of a property, f(x) ofthe medium: e.g., f is a (vectorized) 2D or 3D “image” of
• electrical conductivity
• mass density
• optical absorption
• etc.
and M represents the appropriate (often nonlinear) model map.
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Challenges inModernMedical
Image Recon-struction
Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
Generic Inverse Problem Formulation
The inverse problem refers to the recovery of vectorized imagef ∈ Rm×n(×k) given the model M and noisy data m.
In many (medical) image reconstruction problems, surfacemeasurements ⇒ underdetermined, ill-posed.
minf∈Rm×n×k
‖m−M(f)‖2 + λ2Γ(f)
Regularization needed to to damp noise, force unique soln.
Question: Is a voxel-based reconstruction overkill and/or overlyambitious?
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Challenges inModernMedical
Image Recon-struction
Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
Motivation: Diffuse Optical Tomography
Find images of optical absorption/diffusion using measurementsof near infrared (frequency modulated) light on surface.
1
ν
∂
∂tη(x, t) = ∇ ·D(x)∇η(x, t)− µ(x)η(x, t) + bj(x)uj(t),
for x ∈ Ω
0 = η(x, t) + 2AD(x)∂
∂ξη(x, t), for x ∈ ∂Ω±
mi(t) =
∫∂Ωci(x)η(x, t) dx for i = 1, . . . , ndet
(see S. R. Arridge, Inverse Problems, 1999).
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Challenges inModernMedical
Image Recon-struction
Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
An Example: Breast Tissue Imaging
Breast tissue made up of adipose, fibroglandular, tumor. Each,a different optical contrast, but within each, fairly uniform.
Figure : Ben Brooksby, et. al Imaging breast adipose andfibroglandular tissue molecular signatures by using hybrid MRI-guidednear-infrared spectral tomography PNAS 2006 103 (23) 8828-8833 7 / 36
Challenges inModernMedical
Image Recon-struction
Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
Shape-based Approach
Sufficient to model unknown f as piecewise continuous.
χD(x) =
1 x ∈ D0 x ∈ Ω\D.
In a continuous setting, the unknown property f(x) can bedefined over Ω
f(x) = fi(x)χD(x) + fo(x)(1− χD(x))
Goal: find ∂D (and parameters defining fi, fo).
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Challenges inModernMedical
Image Recon-struction
Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
Traditional Level Sets
Traditional level set (Santosa, ‘96), let φ : Ω→ R.• ∂D := (x, y) ∈ Ω|z = φ(x, y) = 0.• Topologically quite flexible (no. regions not spec. a priori)• Evolve the 3D function to pick up right no. connected
components by minimizing a cost function
Figure : Thanks, Wikipedia!9 / 36
Challenges inModernMedical
Image Recon-struction
Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
Traditional Level Set Approach
If H(r) = 12(1 + sign(r)),
f(x) = fi(x)H(φ(x)− c) + fo(x)(1−H(φ(x)− c))
=
fi(x) φ(x) ≥ cfo(x) φ(x) < c
If fi, fo are known, the optimization problem is to recover φ.
Can be made to work for inverse problems but...
• Highly non-trivial implementation of evolution, speedfunction, initialization, etc.
• Rate of convergence
• Regularization
• Specialized optimization (e.g. van den Doel et al, Journalof Sci. Comp. 2010; van den Doel and Ascher, SISC 2012)
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Challenges inModernMedical
Image Recon-struction
Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
PaLS
Instead, φ function of m-length parameter vector p AND x:
φ(x,p) =
m0∑i=1
αiψi(x)
and p contains the expansion coefficients, and any parametersthat define the ith basis function.
Now, the property to be recovered is described
f(x,p) = fi(x)Hε(φ(x,p)− c) + fo(x) (1−Hε(φ(x,p)− c)) ,
where Hε differentiable surrogate for Heaviside function.
Solve for the parameter vector p that defines φ.
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Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
Choice of PaLS Function
φ(x,p) should be linear combination of “basis functions” orparametrically defined functions. Incomplete list:
• Polynomial basis [K., Miller, et al, SPIE Proceedings,‘04]
• Sinusiods, exponentials [Tarokh, NEU Ph.D. Thesis, ‘05]
• Multiquadric RBFs (topology optimization)[Wang & Wang, Int’l J. for Num. Mtds. Eng,‘06]
• CSRBFs (segmentation only) [Gelas et al, IEEE TIP,‘07]
• B-splines (model evolution application) [Bernard et al,IEEE TIP,‘09]
• CSRBFs for inverse problems [Aghasi, K., Miller,SIAM J. Imag. Sci, ‘11]
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Image Recon-struction
Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
CSRBF from Wendland, Cambridge Univ. Press,‘05
A CSRBF: ψ(r) = (max(0, 1− r))2 (2r + 1); r =√x2 + y2.
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Image Recon-struction
Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
Weighted CSRBFs
Let ψ : R+ → R denote a sufficiently smooth CSRBF.
φ(x,p) :=
m0∑j=1
αjψ(‖βj(x− χ(j))‖†),
where the χ(j) are the centers, βj are dilation factors, αj areexpansion coefficients and
‖x‖† :=√‖x‖22 + ν2
Desired parameter vector defining shape(s) p =
abχxχyχz
.
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Image Recon-struction
Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
CSRBFs
Usingψj(x) := ψ(‖βj(x− χ(j))‖†)
we can think of our level set function φ(x,p) as a weightedsum of CSRBFs or “bumps”.
Advantages:
• Summing appropriately paired/inverted bumps can giveedges
• Compact support can imply sparse updates in context ofoptimization
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Image Recon-struction
Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
Behavior - Geometry
Considering c-level sets with c ≈ 0 → representation of objectswith edges, complex geometries, with only a few CSRBF’s.
−1 1−1
1
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Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
Narrow Banding Illustration
Possible the supports of ∂∂φHε = δε(φ− c) and ∂φ
∂pjfor jth
parameter pj do not intersect, leading to efficiencies in theoptimization algorithm.
−1 1−1
1
supp(∂φ
∂µj)
supp(∂φ
∂µj0
)
supp(δ2,ε(φ− c))
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Image Recon-struction
Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
Optimization Revisited
f(x,p) = fi(x)Hε(φ(x,p)− c) + f0(x)(1−Hε(φ(x,p)− c))
Consider fi(x) = fi and fo(x) = fo, (if unknown, append) andf(p) the discretization of f(x,p) over a grid.
minp‖m−M(f(p))‖2
Nonlinear LS problem of relatively small dimension, often noadditional regularization except stopping criterion.
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Challenges inModernMedical
Image Recon-struction
Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
Related Work
PaLS for
• Hyperspectral diffuse optical tomography, linear forwardmodel [see Larusson et al, Biomed. Optics Expr., ’2012]
• Dual energy X-ray Computed Tomography [see Semerciand Miller, IEEE TIP ’2011]
• 2D Limited angle CT & ERT, [see Aghasi, K., Miller,SIAM J. Imag. Sci., ’2011]
• 3D joint recon using electrical impedance tomography(EIT) & hydrology data [Aghasi, et al, in review]
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Image Recon-struction
Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
When the Forward Model is Nonlinear ...
Remaining Concerns:
• When M is nonlinear, bottleneck still forward (adjoint)model evaluations.
• Ill-conditioned Jacobian, even if narrow banding exploitedto determine the search direction.
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Image Recon-struction
Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
Reduced Order (Forward) Model for DOT
DOT-PaLS model, dynamical system notation1:
1
νE y(t; p) = −A(p)y(t; p)+Bu(t) with m(t; p) = Cy(t; p)
where E,A(p)Rn×n and B ∈ Rn×m,C ∈ Rn×`.
If uj(ω) is FT of jth source uj ,
mj(ω; p) = Ψ(ıω; p) uj(ω)
where
Ψ(s; p) = C( sν
E + A(p))−1
B
maps from sources (inputs) to measurements (outputs).
1Alternative approach, see Arridge et al, Inverse Problems, 200621 / 36
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Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
Reduced Order (Forward) Model
To solve minp ‖m−M(f(p))‖2, must evaluate M(f(p(k))) ⇒compute mj(ω`; p
(k)), for j = 1, . . . , nsrc, ` = 1, . . . , nω.Many large-scale PDE solves2 per optimization step!
Use a cheaper-to-evaluate approx. Ψr(s,p) such that
Ψ(s,p) ≈ Ψr(s,p) or equivalently mj(s,p) ≈ mr,j(s,p)
mj(s,p) = Ψ(s,p) uj(s) Ψ(s,p) = C (sE−A(p))−1 B
mr,j(s,p) = Ψr(s,p) uj(s) Ψr(s,p) = Cr (sEr −Ar(p))−1 Br
2Alternative to MRHS, ‘simultaneous source’ plus SSA/SA; see, e.g.Haber, 2011 presentation
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Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
Reduced Order Model
• Choose r-dimensional right modeling subspace (the trialsubspace) Ra(Vr), where Vr ∈ Rn×r
• Choose r-dimensional left modeling subspace (testsubspace) Ra(Wr), where Wr ∈ Rn×r
• Approximate y(t) ≈ Vryr(t) by forcing yr(t) to satisfy
WTr (EVryr −A(p)Vryr −B u) = 0 (Petrov-Galerkin)
• Leads to a reduced order model:
Er = WTr EVr, Br = WT
r B,
Ar(p) = WTr A(p)Vr, Cr = CVr,
• Here, use fact A(p) = A[0] + A[1](p).
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Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
Reduced Order Model
• Enforce Ψ(s,p) ≈ Ψr(s,p) via Rational Interpolation• Choose π1,π2, . . . ,πJ ∈ Rν and σ1, σ2, . . . , σK ∈ C
Ψ(σk,πj) = Ψr(σk,πj)
Ψ′(σk,πj) = Ψ′r(σk,πj)
∇pΨ(σk,πj) = ∇pΨr(σk,πj)
for k = 1, . . . ,K and j = 1, . . . , J .• Per opt. step: r × r system solves ∀ src, det, freq.; eval
WTr A(p(k))Vr. Up front cost to produce Vr,Wr, but
can be reused!
Serkan Gugercin, 12:15-12:40 in MS 8, “Interpolatory modelreduction strategies for nonlinear parametric inversion”
Eric de Sturler, 11:25-11:50 in MS 6, “Updatingpreconditioners for parameterized systems”
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Image Recon-struction
Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
Trust Region Regularized GN for ParametricInversion3
The Gauss-Newton Model for solving the NLS problem:
mGN(p) =1
2rT r + rTJ(p− pc) +
1
2(p− pc)
TJTJ(p− pc),
where r = r(pc) and J = ∇r(pc).Let the reduced SVD of the m× n current Jacobian J withrank n be
J = UΣVT =
n∑i=1
σiuivTi ,
Then GN and LM have search steps that can be written
sΘ = −n∑i=1
viuTi r
σi· θi = −VΘΣ−1UT r,
where Θ = diag(θ1, . . . ,θn).3de Sturler and K., SISC, 2011.
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Misha E.Kilmer
Forward &InverseProblems
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Trust RegionRegularizedGN
Results
SVD Analysis
R(sΘ) := mGN(pc)−mGN(pc+ sΘ) =1
2
n∑i=1
(uTi r)2θi(2−θi).
gives the estimated reduction of the objective function.Study this for (dM)GN and LM, we observe:
• Overdamping of directions that could have reducedfunction rapidly
• Emphasize large(r) singular values BUT this may ignoreimportant directions: Must take rhs into account
• Look at relative sizes of |uTi r|
• Avoid for which |uTi r|/σi could have undue influence;
others critical directions• Some (possibly damped) step in all critical directions
• Use (dual) trust region approach to limit step size
• If full GN step fits in TR, take it.26 / 36
Challenges inModernMedical
Image Recon-struction
Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
TREGS in a Nutshell
Working with the reduced, compressed SVD
U = [UA,UB,UC ], V = [VA,VB,VC ],
Σ−1
= diag(Σ−1A ,Σ−1
B ,Σ−1C )
Θ = diag[θi], reordered θ’s
• group A: θi = 1
• group B: θi ∈ (0, 1)
• group C: θi = 0
Step is s = −VΘΣUT r.In combination with the GCV condition this leads to aguaranteed reduction proportional to‖JT r‖ ⇒ global convergence
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Image Recon-struction
Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
Results Summary
• Lab results to show potential of PaLS with real data
• 2D Synthetic results to show the power of using ROMwith PaLS approach
• 3D Synthetic results to show the ability to recover edgesand illustrate superiority of TREGS in optimizing for PaLSparameters
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Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
Experimental Results, DOT Data4
• black rod(s), .5cm diameter
• emersed in milk/water solution
• 2D slice-by-slice PaLS reconstructions, 7cm×6.5cm over7cm length
• 7 CSRBFs, randomly chosen to start
• 72 data points per 2D image, image resolution 64 x 71
4Results thanks to F. Larusson, E. Miller, S. Fantini29 / 36
Challenges inModernMedical
Image Recon-struction
Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
Experimental Results, DOT Data5
Single rod angled in the y-z plane. Excellent thicknessreconstruction, depth variation 5mm from truth
5Results thanks to F. Larusson, E. Miller, S. Fantini30 / 36
Challenges inModernMedical
Image Recon-struction
Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
Experimental Results, DOT Dat6
Dual rods, different depths, appear overlapping from thisperspective. Truth, dashed lines.
6Results thanks to F. Larusson, E. Miller, S. Fantini31 / 36
Challenges inModernMedical
Image Recon-struction
Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
ROM Example
32 sources & 32 detectors; heterogeneous noise + additiveOriginal System Size: n = 160,801Parametric Sample points: 5Up front cost: *) 10 n× n systems, 32 RHS per system
*) compressed SVD on n× 320ROM size: r = 250
No. Fun evals No. Jac evals blk systems sys sizeFOM 23 12 35 n× nROM 25 13 38 r × r
For other recons, reuse the V,W!
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Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
ROM Example, Con’t
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Challenges inModernMedical
Image Recon-struction
Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
3D TREGS vs. LM
15× 15 array src & dets 4cm × 4cm × 4cm, 32× 32× 21 gridm0 = 125 (CSRBFs in 5× 5× 5 grid) .05 percent Gaussiannoise; discrepancy stopping
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Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
TREGS: 44 Fev, 29 Jev; LM: 229 Fev, 52 Jev
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Misha E.Kilmer
Forward &InverseProblems
ReducedOrder ImageModels:Motivation
ParametricLevel Sets
ReducedOrderForwardModel
Trust RegionRegularizedGN
Results
Summary and Current, Future Work
• PaLS useful for medical imaging where pw cont. reducedmodeling makes sense. CSRBFs can capture shape, havenarrow-banding advantage
• Resulting nonlinear least squares problem solved efficientlyby TREGS
• For inverse problems involving nonlinear forward model,ROM-approach that exploits parameterized image model ispromising for reducing the computational bottleneck
• To Do: 3D nonlinear (hyperspectral) DOT. Merge withmeasurement sampling techniques [van den Doel &Ascher, 2011]?
• Other imaging modalities
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