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Geometric Reconstruction inBioluminescence Tomography (BLT)
Andreas Rieder jointly with Tim Kreutzmann
KIT – University of the State of Baden-Württemberg andNational Research Center of the Helmholtz Association
FAKULTÄT FÜR MATHEMATIK – INSTITUT FÜR ANGEWANDTE UND NUMERISCHE MATHEMATIK
www.kit.edu
(Wang et al. ’06)
Overview
2 / 32 c©Andreas Rieder – Geometric Reconstruction in Bioluminescence Tomography CBM 2013, Rio de Janeiro
Mathematical model
Inverse problem: formulation & uniqueness
Inverse problem: reformulation & stabilization
Gradient of the minimization functional
Numerical experiments in 2D
Summary
Mathematical model
⊲Mathematicalmodel
Inverse problem:formulation &uniqueness
Inverse problem:reformulation &
stabilization
Gradient of the mini-mization functional
Numerical experi-ments in 2D
Summary
3 / 32 c©Andreas Rieder – Geometric Reconstruction in Bioluminescence Tomography CBM 2013, Rio de Janeiro
The (stationary) radiative transfer equation (RTE)(Boltzmann transport equation)
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Let u(x, θ) be the photon flux (radiance) in direction θ ∈ S2 about x ∈ Ω ⊂R
3. Then,
θ · ∇u(x, θ) + µ(x)u(x, θ) = µs(x)
∫
S2
η(θ · θ′)u(x, θ′)dθ′ + q(x, θ)
u(x, θ) = g−(x, θ), x ∈ ∂Ω, n(x) · θ ≤ 0
g(x) =1
4π
∫
S2
n(x) · θu(x, θ)dθ, x ∈ ∂Ω
where µ = µs + µa and
µs / µa scattering/absorption coefficients
η scattering kernel (∫S2
η(θ · θ′)dθ′ = 1)
q source term
µs = 0: RTE yields integral eqs. of transmission and emission tomography(F. Natterer & F. Wübbeling, Math. Methods in Image Reconstr., SIAM, ’01)
Diffusion approximation: setting
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Assume thatu(x, θ) = u0(x) + 3θ · u1(x)
where
u0(x) =1
4π
∫
S2
u(x, θ)dθ ∈ R and u1(x) =1
4π
∫
S2
θu(x, θ)dθ ∈ R3.
By the Funk-Hecke theorem,∫
S2
θη(θ · θ′)dθ = η θ′
where η =∫
S2
θ′ · θ η(θ · θ′)dθ is the scattering anisotropy.
Diffusion approximation: derivation
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Integrate RTE over S2,multiply RTE by θ, integrate again, andassume g−(x, θ) = g−(x).
Then,
−∇ · (D∇u0) + µau0 = q0 :=1
4π
∫
S2
q(·, θ)dθ,
u0 + 2D∂nu0 = g− on ∂Ω,
D∂nu0 = −g on ∂Ω,
whereD =
1
3(µ − ηµs)
is the diffusion coefficient (reduced scattering coefficient).
Diffusion approximation: final equation
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Change of notation: u = u0, q = q0, µ = µa, and g = −g.
The photon density u obeys the BVP
−∇ · (D∇u) + µu = q in Ω,
u + 2D∂nu = g− on ∂Ω.
The measurements are given by
D∂nu = g on ∂Ω.
Assume g− = 0 (no photons penetrate the object from outside).
Inverse problem: formulation & uniqueness
Mathematical model
⊲
Inverse problem:formulation &uniqueness
Inverse problem:reformulation &
stabilization
Gradient of the mini-mization functional
Numerical experi-ments in 2D
Summary
8 / 32 c©Andreas Rieder – Geometric Reconstruction in Bioluminescence Tomography CBM 2013, Rio de Janeiro
Inverse problem of BLT (in the diffusive regime)
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Define the (linear) forward operator
A : L2(Ω) → H−12 (∂Ω),
q 7→ D∂nu ,
where u solves the BVP with g− = 0:
−∇ · (D∇u) + µu = q in Ω,
u + 2D∂nu = 0 on ∂Ω.
BLT Problem : Given g ∈ R(A), find a source q ∈ L2(Ω) satisfying
Aq = g.
Null Space of A
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Lemma (Wang, Li & Jiang ’04, Kreutzmann ’13):There is an isomorphism Φ: H1(Ω) → H1(Ω)′ such that
N(A) = Φ(H10 (Ω)
)∩ L2(Ω).
If D ∈ W 1,∞ then
N(A) = Φ(H10 (Ω) ∩ H
2(Ω)).
Proof: Define
Φ: H1(Ω) → H1(Ω)′, u 7→ (Φu)(v) = a(u, v)
where
a(u, v) =
∫
Ω
(D∇u · ∇v + µuv
)dx +
1
2
∫
∂Ωuv ds.
Singular Functions of A : L2(Ω0) → L2(∂Ω)
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H LL
M
B
−8 −6 −4
−5
0
5
σ1 = 0.34221
−0.4
−0.3
−0.2
−0.1
−8 −6 −4
−5
0
5
σ2 = 0.2932
−0.2
0
0.2
−8 −6 −4
−5
0
5
σ3 = 0.23248
−0.2
0
0.2
−8 −6 −4
−5
0
5
σ4 = 0.17673
−0.4
−0.2
0
0.2
−8 −6 −4
−5
0
5
σ5 = 0.1296
−0.2
0
0.2
0.4
−8 −6 −4
−5
0
5
σ6 = 0.096659
−0.4
−0.2
0
0.2
−8 −6 −4
−5
0
5
σ7 = 0.070445
−0.2
0
0.2
−8 −6 −4
−5
0
5
σ8 = 0.04619
−0.2
0
0.2
−8 −6 −4
−5
0
5
σ9 = 0.035356
−0.2
0
0.2
0.4
−8 −6 −4
−5
0
5
σ10
= 0.021841
−0.2
0
0.2
0.4
−8 −6 −4
−5
0
5
σ11
= 0.013265
−0.4
−0.2
0
0.2
0.4
−8 −6 −4
−5
0
5
σ12
= 0.0089158
−0.4
−0.2
0
0.2
Can we restore uniqueness by a priori information?
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Consider, for instance,
q = λχS where λ ≥ 0 is a constant and S ⊂ Ω.
Can we restore uniqueness by a priori information?
12 / 32 c©Andreas Rieder – Geometric Reconstruction in Bioluminescence Tomography CBM 2013, Rio de Janeiro
Consider, for instance,
q = λχS where λ ≥ 0 is a constant and S ⊂ Ω.
Lemma (Wang, Li & Jiang ’04):
There exist z ∈ Ω, λ1 6= λ2 and r1 6= r2 such that
A(λ1χB1) = A(λ2χB2)
with Bk = Brk(z).
Inverse problem:reformulation & stabilization
Mathematical model
Inverse problem:formulation &uniqueness
⊲
Inverse problem:reformulation &
stabilization
ReformulationTikhonov-likeregularizationExistence of aminimizer & stabilityRegularizationproperty
Gradient of the mini-mization functional
Numerical experi-ments in 2D
Summary
13 / 32 c©Andreas Rieder – Geometric Reconstruction in Bioluminescence Tomography CBM 2013, Rio de Janeiro
Reformulation
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Ansatz: q =I∑
i=1
λiχSi where Si ⊂ Ω, λi ∈ [λi, λi] = Λi, and I ∈ N.
For the ease of presentation: I = 1.
Define the nonlinear operator
F : Λ × L → L2(∂Ω),
(λ, S) 7→ D∂nu|∂Ω
where L is the set of all measurable subsets of Ω.
Note: F (λ, S) = λAχS
BLT Problem : Given measurements g, find an intensity λ ∈ Λ anda domain S ∈ L such that
F (λ, S) = g.
Tikhonov-like regularization
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Minimize Jα(λ, S) =1
2‖F (λ, S) − g‖2L2 + αPer(S) over Λ × L
where α > 0 is the regularization parameter and Per(S) is the perimeterof S:
Per(S) = |D(χS)|,
with |D(·)| denoting the BV-semi-norm (Ramlau & Ring ’07, ’10).
AG Sahin, Univ. Mainz
Existence of a minimizer & stability
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Theorem: For all α > 0 and g ∈ L2(∂Ω) there exists a minimizer(λ∗, S∗) ∈ Λ × L, that is,
Jα(λ∗, S∗) ≤ Jα(λ, S) for all (λ, S) ∈ Λ × L.
Theorem: Let gn → g in L2 as n → ∞ and let (λn, Sn) minimize
Jnα(λ, S) =12‖F (λ, S) − gn‖
2L2
+ αPer(S) over Λ × L.
Then there exists a subsequence {(λnk , Snk)}k converging to aminimizer (λ∗, S∗) ∈ Λ × L of Jα in the sense that
‖λnkχSnk − λ∗χS∗‖L2 → 0 as k → ∞.
Furthermore, every convergent subsequence of {(λn, Sn)}nconverges to a minimizer of Jα.
Regularization property
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Theorem: Let g be in range(F ) and let δ 7→ α(δ) where
α(δ) → 0 andδ2
α(δ)→ 0 as δ → 0.
In addition, let {δn}n be a positive null sequence and {gn}n such that
‖gn − g‖L2 ≤ δn.
Then, the sequence {(λn, Sn)} of minimizers of Jnα(δn)
possesses a sub-sequence converging to a solution (λ+, S+) where
S+ = arg min{Per(S) : S ∈ L s.t. ∃λ ∈ Λ with F (λ, S) = g}.
Furthermore, every convergent subsequence of {(λn, Sn)}n converges toa pair (λ†, S†) with above property.
Gradient of the minimization functional
Mathematical model
Inverse problem:formulation &uniqueness
Inverse problem:reformulation &
stabilization
⊲
Gradient of theminimizationfunctional
Domain derivative:general definitionDomain derivativeof F (λ, ·) : S →L2(∂Ω)
Domain derivative ofPer : S → RDerivative ofJα : Λ × S → R
Approximate vari-ational principle(Ekeland 1974)
Numerical experi-ments in 2D
Summary
18 / 32 c©Andreas Rieder – Geometric Reconstruction in Bioluminescence Tomography CBM 2013, Rio de Janeiro
Domain derivative: general definition
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Let Γ ∈ S = {Γ̃ ⊂ Ω : ∂Γ̃ ∈ C2} and let h ∈ C20 (Ω, Rd). Define
Γh = {x + h(x) : x ∈ Γ}.
If h is small enough, say if ‖h‖C2 < 1/2, then Γh ∈ S.
By the domain derivative of Φ: S → Y about Γ we understand Φ′(Γ) ∈L(C2, Y ) satisfying
‖Φ(Γh) − Φ(Γ) − Φ′(Γ)h‖Y = o(‖h‖C2)
where Y is a normed space.
Domain derivative of F (λ, ·) : S → L2(∂Ω)
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Reminder: F (λ, S) = λAχS
Lemma: We have that∂SF (λ, S)h = u
′|∂Ω
where u′ ∈ H1(Ω\∂S) solves the transmission bvp
−∇ · (D∇u′) + µu′ = 0 in Ω\∂S,
2D∂nu′ + u′ = 0 on ∂Ω,
[u′]± = 0 on ∂S,[D∂nu
′]±
= −λh · n on ∂S.
Proof: similar to Hettlich’s habilitation thesis 1999.
Domain derivative of Per: S → R
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Lemma (Simon 1980):
We have that
∂SPer(S)h =
∫
∂S
H∂S(h · n) ds
where H∂S denotes the mean curvature of ∂S.
Derivative of Jα : Λ × S → R
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Jα(λ, S) =1
2‖F (λ, S) − g‖2L2 + αPer(S)
∂λF (λ, S)k = kAχS = F (k, S)
Theorem: We have that
J ′α(λ, S)(k, h) =〈F (λ, S) − g, F (k, S) + u′
〉L2(∂Ω)
+ α
∫
∂S
H∂S(h · n) ds
for k ∈ R, h ∈ C20(Ω, R3).
Proof:J ′α(λ, S)(k, h) = ∂λJα(λ, S)k + ∂SJα(λ, S)h
Approximate variational principle (Ekeland 1974)
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There exist smooth almost critical points of Jα near to any of itsminimizers.
Approximate variational principle (Ekeland 1974)
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There exist smooth almost critical points of Jα near to any of itsminimizers.
Theorem: Let (λ∗, S∗) be a minimizer of Jα where λ∗ is an inner point ofΛ. Then, for any ε > 0 sufficiently small there is a (λε, Sε) ∈ Λ × S with
Jα(λε, Sε) − Jα(λ
∗, S∗) ≤ ε,
‖λεχSε − λ∗χS∗‖L1 ≤ ε,
‖J ′α(λε, Sε)‖R×C2→R ≤ ε.
Approximate variational principle (Ekeland 1974)
23 / 32 c©Andreas Rieder – Geometric Reconstruction in Bioluminescence Tomography CBM 2013, Rio de Janeiro
There exist smooth almost critical points of Jα near to any of itsminimizers.
Theorem: Let (λ∗, S∗) be a minimizer of Jα where λ∗ is an inner point ofΛ. Then, for any ε > 0 sufficiently small there is a (λε, Sε) ∈ Λ × S with
Jα(λε, Sε) − Jα(λ
∗, S∗) ≤ ε,
‖λεχSε − λ∗χS∗‖L1 ≤ ε,
‖J ′α(λε, Sε)‖R×C2→R ≤ ε.
Proof: Key ingredient isTo any bounded measurable Γ ⊂ Rd with finite perimeter exists a se-quence {Γn}n of C∞-domains such that
∫
Rd
|χΓn − χΓ|dx → 0 and Per(Γn) → Per(Γ) as n → ∞.
Numerical experiments in 2D
Mathematical model
Inverse problem:formulation &uniqueness
Inverse problem:reformulation &
stabilization
Gradient of the mini-mization functional
⊲
Numericalexperiments in2D
Star-shaped do-mainsAlgorithm: ProjectedGradient Method
The model
H3-Reconstructions
L2-Reconstructions
Summary
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Star-shaped domains
25 / 32 c©Andreas Rieder – Geometric Reconstruction in Bioluminescence Tomography CBM 2013, Rio de Janeiro
For the numerical experiments we consider a star-shaped domain only:
S = {x ∈ R2 : x = m + t θ(ϑ)r(ϑ), 0 ≤ t ≤ 1, 0 ≤ ϑ ≤ 2π}
where m is the center (assumed to be known) and r : [0, 2π] → [0,∞[parameterizes the boundary of S.
All previous results hold in this setting as well if we work in a space ofsmooth parameterizations, say, r ∈ H3p(0, 2π) ⊂ C
2p(0, 2π).
(λ, S) (λ, r) ∈ Λ × Rad where Rad ={r ∈ H3p(0, 2π) : r ≥ 0
}.
Gradient equation:〈gradJα(λ, r), (k, h)
〉R×H3
= J ′α(λ, r)(k, h).
We have implemented star-shaped domains using trigonometric poly-nomials.
Algorithm: Projected Gradient Method
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(S0) Choose (λ0, r0) ∈ C := Λ × Rad, k := 0
(S1) Iterate (S2)-(S4) until(S2) Set ∇k := −gradJα(λk, rk).(S3) Choose σk by a projected step size rule such that
Jα
(PC
((λk, rk) + σk∇k
))< Jα(λ
k, rk).
(S4) Set (λk+1, rk+1) := PC((λk, rk) + σk∇k
).
The model
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lung lungheart
bone
muscle
source
H3-Reconstructions
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λ = 0.97843 for α = 0.00763
0.5
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1.5
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λ = 0.702 for α = 0.00762
Reconstructions (blue) and source (red).Left: 69 iterations, right: k = 17 iterations
L2-Reconstructions
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λ = 0.84033 for α = 0.0079
0.5
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150
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λ = 0.77183 for α = 0.008
Reconstructions (blue) and source (red).Left: 37 iterations, right: noisy data (rel. 3%), 24 iterations
L2-Reconstruction with variable center
30 / 32 c©Andreas Rieder – Geometric Reconstruction in Bioluminescence Tomography CBM 2013, Rio de Janeiro
GeoRecBioLum.aviMedia File (video/avi)
Summary
Mathematical model
Inverse problem:formulation &uniqueness
Inverse problem:reformulation &
stabilization
Gradient of the mini-mization functional
Numerical experi-ments in 2D
⊲ SummaryWhat to rememberfrom this talk
31 / 32 c©Andreas Rieder – Geometric Reconstruction in Bioluminescence Tomography CBM 2013, Rio de Janeiro
What to remember from this talk
32 / 32 c©Andreas Rieder – Geometric Reconstruction in Bioluminescence Tomography CBM 2013, Rio de Janeiro
Bioluminescence tomography images cells in vivo. From a mathemati-cal point of view it is an inverse source problem which suffers from non-uniqueness (diffusion approximation) and ill-posedness.
To overcome these difficulties the sources are modeled as ”hot spots”leading to a nonlinear problem which is stabilized by a Tikhonov-likeregularization penalizing the perimeter of the hot spots.
The approximate variational principle justifies the restriction to hot spotswith smooth boundaries.
For star-shaped domains in 2D a projected steepest decent solver hasbeen implemented and tested.
Mathematical model12cm [-0.7cm]The (stationary) radiative transfer equation (RTE) (Boltzmann transport equation)Diffusion approximation: settingDiffusion approximation: derivationDiffusion approximation: final equation
Inverse problem: formulation & uniquenessInverse problem of BLT (in the diffusive regime)Null Space of ASingular Functions of A2mu-:6muplus1muL2(0)L2()Can we restore uniqueness by a priori information?
Inverse problem: reformulation & stabilizationReformulationTikhonov-like regularizationExistence of a minimizer & stabilityRegularization property
Gradient of the minimization functionalDomain derivative: general definitionDomain derivative of F(,)2mu-:6muplus1muSL2()Domain derivative of Per2mu-:6muplus1muSRDerivative of J2mu-:6muplus1muSRApproximate variational principle (Ekeland 1974)
Numerical experiments in 2DStar-shaped domainsAlgorithm: Projected Gradient MethodThe modelH3-ReconstructionsL2-ReconstructionsL2-Reconstruction with variable center
SummaryWhat to remember from this talk