Geometric Reconstruction in Bioluminescence Tomography...

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


The (stationary) radiative transfer equation (RTE)(Boltzmann transport equation)


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


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


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


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

⊲



stabilization



Summary


Inverse problem of BLT (in the diffusive regime)


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


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)


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?


Consider, for instance,

q = λχS where λ ≥ 0 is a constant and S ⊂ Ω.

Can we restore uniqueness by a priori information?


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


⊲


stabilization

ReformulationTikhonov-likeregularizationExistence of aminimizer & stabilityRegularizationproperty



Summary


Reformulation


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


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


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


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



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)


Summary


Domain derivative: general definition


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)


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


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


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)


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.




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



stabilization


⊲

Numericalexperiments in2D

Star-shaped do-mainsAlgorithm: ProjectedGradient Method

The model

H3-Reconstructions

L2-Reconstructions

Summary


Star-shaped domains


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


(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


lung lungheart

bone

muscle

source

H3-Reconstructions


0.5

1

1.5

2

2.5

3

30

210

60

240

90

270

120

300

150

330

180 0

λ = 0.97843 for α = 0.00763

0.5

1

1.5

2

2.5

3

30

210

60

240

90

270

120

300

150

330

180 0

λ = 0.702 for α = 0.00762

Reconstructions (blue) and source (red).Left: 69 iterations, right: k = 17 iterations

L2-Reconstructions


0.5

1

1.5

2

2.5

3

30

210

60

240

90

270

120

300

150

330

180 0

λ = 0.84033 for α = 0.0079

0.5

1

1.5

2

2.5

3

30

210

60

240

90

270

120

300

150

330

180 0

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


GeoRecBioLum.aviMedia File (video/avi)

Summary

Mathematical model



stabilization



⊲ SummaryWhat to rememberfrom this talk


What to remember from this talk


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

Date post:	12-Jul-2020
Category:	Documents
Upload:	others
View:	0 times
Download:	0 times

Geometric Reconstruction in Bioluminescence Tomography...

Documents