On the identification of piecewise constant coefficients in optical - - PowerPoint PPT Presentation
Introduction Inverse Problem Numerical Examples Conclusion On the identification of piecewise constant coefficients in optical diffusion tomography by level set. Juan Pablo Agnelli Universidad Nacional de C ordoba and CONICET. New
Introduction Inverse Problem Numerical Examples Conclusion
Juan Pablo Agnelli
Universidad Nacional de C´
and CONICET.
New Trends in Parameter Identification for Mathematical Model, IMPA 2017
Introduction Inverse Problem Numerical Examples Conclusion
Adriano De Cezaro: Univ. Fed. Rio Grande. Antonio Leit˜ ao: Univ. Fed. Santa Catarina. Maicon Marques Alves: Univ. Fed. Santa Catarina.
Introduction Inverse Problem Numerical Examples Conclusion
1
Introduction
2
Inverse Problem
3
Numerical Examples
4
Conclusion
Introduction Inverse Problem Numerical Examples Conclusion Diffuse Optical Tomography
DOT is a non-invasive technique that utilize light in the near infrared spectral region to measure the optical properties of a physical body. The object under study has to be light-transmitting or translucent, so it works best on soft tissues such as breast and brain tissue. By monitoring variations in the light absorption and scattering of the tissue, spatial maps of properties such as total hemoglobin concentration, blood oxygen saturation and scattering can be obtained. DOT has been applied in breast cancer imaging, brain functional imaging, stroke detection, muscle functional studies, etc.
Introduction Inverse Problem Numerical Examples Conclusion Diffuse Optical Tomography
DOT is a non-invasive technique that utilize light in the near infrared spectral region to measure the optical properties of a physical body. The object under study has to be light-transmitting or translucent, so it works best on soft tissues such as breast and brain tissue. By monitoring variations in the light absorption and scattering of the tissue, spatial maps of properties such as total hemoglobin concentration, blood oxygen saturation and scattering can be obtained. DOT has been applied in breast cancer imaging, brain functional imaging, stroke detection, muscle functional studies, etc.
Introduction Inverse Problem Numerical Examples Conclusion Diffuse Optical Tomography
A simplified equation to model the light propagation is the following:
in Ω,
Introduction Inverse Problem Numerical Examples Conclusion Diffuse Optical Tomography
Parameter-to-measurement (forward) map
where u = u(g) is the unique solution of (DP) given the boundary data
where a, a, c and c are known non negative real numbers.
Introduction Inverse Problem Numerical Examples Conclusion Diffuse Optical Tomography
Theorem For each g ∈ H−1/2(Γ), the corresponding forward map
The proof is based on a generalization of Meyers’ Theorem which prove that the solution u of (DP) belongs to W 1,p(Ω) for some p > 2 (therefore better than the standard regularity u ∈ H1(Ω)).
Introduction Inverse Problem Numerical Examples Conclusion Diffuse Optical Tomography
Since the optical properties within tissue are determined by the values
DOT is the simultaneous identification of both coefficients from measurements of near-infrared diffusive light along the tissue boundary. Given a finite number of measurements hm, corresponding to inputs
∂ν .
Find (a,c) ∈ D(F) such that
for m = 1,...,M. (1)
Introduction Inverse Problem Numerical Examples Conclusion Diffuse Optical Tomography
Given the nature of the measurements, we can not expect that exact data hm are available. Instead, one disposes only an approximate measured data hδ
m satisfying
m
for m = 1,...,M where δ > 0 is the noise level. Find (a,c) ∈ D(F) such that
m,
for m = 1,...,M. (2)
Introduction Inverse Problem Numerical Examples Conclusion Level set approach Level set approach: convergence analysis Level set approach: numerical realization
Level set functions φa,φc ∈ H1(Ω) are chosen in such a way that discontinuities of the coefficients (a,c) are located “along” its zero level sets Γφi := {x ∈ Ω|φi(x) = 0}. The diffusion and absorption coefficients can be written as
Inverse problem: Find (φa,φc) ∈ [H1(Ω)]2 such that
m,
for
Introduction Inverse Problem Numerical Examples Conclusion Level set approach Level set approach: convergence analysis Level set approach: numerical realization
Level set functions φa,φc ∈ H1(Ω) are chosen in such a way that discontinuities of the coefficients (a,c) are located “along” its zero level sets Γφi := {x ∈ Ω|φi(x) = 0}. The diffusion and absorption coefficients can be written as
Inverse problem: Find (φa,φc) ∈ [H1(Ω)]2 such that
m,
for
Introduction Inverse Problem Numerical Examples Conclusion Level set approach Level set approach: convergence analysis Level set approach: numerical realization
A natural alternative to obtain stable solutions is to use a least-square approach combined with a Tikhonov-type regularization
M
m=1
m2 L2(Γ) +αR(φa,φc)
(3) where
02 H1(Ω)+φc−φc 02 H1(Ω)+βa|H(φa)|BV(Ω)+βc|H(φc)|BV(Ω)
factors. The H1(Ω)-terms act as a control on the size of the norm of the level set function (key role to prove existence of minimizers). The BV(Ω)-seminorm terms penalize the length of the Hausdorff measure of the boundary of the sets Γφi
0 .
Introduction Inverse Problem Numerical Examples Conclusion Level set approach Level set approach: convergence analysis Level set approach: numerical realization
A natural alternative to obtain stable solutions is to use a least-square approach combined with a Tikhonov-type regularization
M
m=1
m2 L2(Γ) +αR(φa,φc)
(3) where
02 H1(Ω)+φc−φc 02 H1(Ω)+βa|H(φa)|BV(Ω)+βc|H(φc)|BV(Ω)
factors. The H1(Ω)-terms act as a control on the size of the norm of the level set function (key role to prove existence of minimizers). The BV(Ω)-seminorm terms penalize the length of the Hausdorff measure of the boundary of the sets Γφi
0 .
Introduction Inverse Problem Numerical Examples Conclusion Level set approach Level set approach: convergence analysis Level set approach: numerical realization
A natural alternative to obtain stable solutions is to use a least-square approach combined with a Tikhonov-type regularization
M
m=1
m2 L2(Γ) +αR(φa,φc)
(3) where
02 H1(Ω)+φc−φc 02 H1(Ω)+βa|H(φa)|BV(Ω)+βc|H(φc)|BV(Ω)
factors. The H1(Ω)-terms act as a control on the size of the norm of the level set function (key role to prove existence of minimizers). The BV(Ω)-seminorm terms penalize the length of the Hausdorff measure of the boundary of the sets Γφi
0 .
Introduction Inverse Problem Numerical Examples Conclusion Level set approach Level set approach: convergence analysis Level set approach: numerical realization
A natural alternative to obtain stable solutions is to use a least-square approach combined with a Tikhonov-type regularization
M
m=1
m2 L2(Γ) +αR(φa,φc)
(3) where
02 H1(Ω)+φc−φc 02 H1(Ω)+βa|H(φa)|BV(Ω)+βc|H(φc)|BV(Ω)
factors. The H1(Ω)-terms act as a control on the size of the norm of the level set function (key role to prove existence of minimizers). The BV(Ω)-seminorm terms penalize the length of the Hausdorff measure of the boundary of the sets Γφi
0 .
Introduction Inverse Problem Numerical Examples Conclusion Level set approach Level set approach: convergence analysis Level set approach: numerical realization
In general, variational minimization techniques involve compact embedding arguments and continuity of the operator on the set of admissible functions to guarantee the existence of minimizers. We are dealing with the Heaviside operator H and consequently the
For each ε > 0, we consider the smooth approximations Hε(t) := 1+t/ε
for t ∈ [−ε,0]
H(t)
for t ∈ R\[−ε,0]
Pε(φa,φc) :=
Introduction Inverse Problem Numerical Examples Conclusion Level set approach Level set approach: convergence analysis Level set approach: numerical realization
In general, variational minimization techniques involve compact embedding arguments and continuity of the operator on the set of admissible functions to guarantee the existence of minimizers. We are dealing with the Heaviside operator H and consequently the
For each ε > 0, we consider the smooth approximations Hε(t) := 1+t/ε
for t ∈ [−ε,0]
H(t)
for t ∈ R\[−ε,0]
Pε(φa,φc) :=
Introduction Inverse Problem Numerical Examples Conclusion Level set approach Level set approach: convergence analysis Level set approach: numerical realization
A vector (φa,φc,za,zc) ∈ [H1(Ω)]2 ×[L∞(Ω)]2 is called admissible if there exist sequences {φj
k} of H1-functions and a sequence {εk} ∈ R+
converging to zero such that
k→∞φj k −φjL2(Ω) = 0
and
k→∞Hεk(φj k)−zjL1(Ω) = 0.
A generalized minimizer of the functional Fα in (3) is any admissible vector (φa,φc,za,zc) minimizing
M
m=1
m2 L2(Γ) +αρ(φa,φc,za,zc),
Q(za,zc) := (a2 +(a1 −a2)za, c2 +(c1 −c2)zc), ρ(φa,φc,za,zc) := inf
j=1
k)|BV +φ j k −φj 02 H1
Introduction Inverse Problem Numerical Examples Conclusion Level set approach Level set approach: convergence analysis Level set approach: numerical realization
A vector (φa,φc,za,zc) ∈ [H1(Ω)]2 ×[L∞(Ω)]2 is called admissible if there exist sequences {φj
k} of H1-functions and a sequence {εk} ∈ R+
converging to zero such that
k→∞φj k −φjL2(Ω) = 0
and
k→∞Hεk(φj k)−zjL1(Ω) = 0.
A generalized minimizer of the functional Fα in (3) is any admissible vector (φa,φc,za,zc) minimizing
M
m=1
m2 L2(Γ) +αρ(φa,φc,za,zc),
Q(za,zc) := (a2 +(a1 −a2)za, c2 +(c1 −c2)zc), ρ(φa,φc,za,zc) := inf
j=1
k)|BV +φ j k −φj 02 H1
Introduction Inverse Problem Numerical Examples Conclusion Level set approach Level set approach: convergence analysis Level set approach: numerical realization
Theorem
1
[Well-posedness] ˆ
2
[Convergence for exact data] Assume that hδ = h. For every α > 0 denote by
α,φc α,za α,zc α) a minimizer of ˆ
such that (φa
αk,φc αk,za αk,zc αk) is strongly convergent in [L2(Ω)]2 ×[L1(Ω)]2.
Moreover, the limit is a solution of (1), that is Fm(Q(¯
3
[Convergence for noisy data] Let α = α(δ) be a function satisfying
δ→0α(δ) = 0
and
δ→0δ2α(δ)−1 = 0.
Moreover, let {δk} be a sequence of positive numbers converging to zero and
again by {δk}, and a sequence {αk := α(δk)} such that (φa
αk,φc αk,za αk,zc αk)
converges in [L2(Ω)]2 ×[L1(Ω)]2 to a solution of (1).
Introduction Inverse Problem Numerical Examples Conclusion Level set approach Level set approach: convergence analysis Level set approach: numerical realization
Theorem
1
[Well-posedness] ˆ
2
[Convergence for exact data] Assume that hδ = h. For every α > 0 denote by
α,φc α,za α,zc α) a minimizer of ˆ
such that (φa
αk,φc αk,za αk,zc αk) is strongly convergent in [L2(Ω)]2 ×[L1(Ω)]2.
Moreover, the limit is a solution of (1), that is Fm(Q(¯
3
[Convergence for noisy data] Let α = α(δ) be a function satisfying
δ→0α(δ) = 0
and
δ→0δ2α(δ)−1 = 0.
Moreover, let {δk} be a sequence of positive numbers converging to zero and
again by {δk}, and a sequence {αk := α(δk)} such that (φa
αk,φc αk,za αk,zc αk)
converges in [L2(Ω)]2 ×[L1(Ω)]2 to a solution of (1).
Introduction Inverse Problem Numerical Examples Conclusion Level set approach Level set approach: convergence analysis Level set approach: numerical realization
Theorem
1
[Well-posedness] ˆ
2
[Convergence for exact data] Assume that hδ = h. For every α > 0 denote by
α,φc α,za α,zc α) a minimizer of ˆ
such that (φa
αk,φc αk,za αk,zc αk) is strongly convergent in [L2(Ω)]2 ×[L1(Ω)]2.
Moreover, the limit is a solution of (1), that is Fm(Q(¯
3
[Convergence for noisy data] Let α = α(δ) be a function satisfying
δ→0α(δ) = 0
and
δ→0δ2α(δ)−1 = 0.
Moreover, let {δk} be a sequence of positive numbers converging to zero and
again by {δk}, and a sequence {αk := α(δk)} such that (φa
αk,φc αk,za αk,zc αk)
converges in [L2(Ω)]2 ×[L1(Ω)]2 to a solution of (1).
Introduction Inverse Problem Numerical Examples Conclusion Level set approach Level set approach: convergence analysis Level set approach: numerical realization
In this case, the energy functional is:
M
m=1
m2 L2(Γ) +αRε(φa,φc)
where
02 H1(Ω)+φc−φc 02 H1(Ω)
Introduction Inverse Problem Numerical Examples Conclusion Level set approach Level set approach: convergence analysis Level set approach: numerical realization
Theorem
1
Given α, ε > 0 and φi
0 ∈ H1, the functional Fα,ε attains a minimizer on [H1(Ω)]2.
2
Let α be given. For each ε > 0 denote by (φa
ε,α,φc ε,α) a minimizer of Fα,ε. There
exists a sequence of positive numbers {εk} converging to zero such that
εk,α,φc εk,α,Hεk(φa εk,α),Hεk(φc εk,α)) converges strongly in [L2(Ω)]2 ×[L1(Ω)]2
and the limit is a generalized minimizer of Fα. Differently from Fα, the minimizers of Fα,ε can be computed. Derive the first order optimality condition for a minimizer of Fα,ε.
Introduction Inverse Problem Numerical Examples Conclusion Level set approach Level set approach: convergence analysis Level set approach: numerical realization
Theorem
1
Given α, ε > 0 and φi
0 ∈ H1, the functional Fα,ε attains a minimizer on [H1(Ω)]2.
2
Let α be given. For each ε > 0 denote by (φa
ε,α,φc ε,α) a minimizer of Fα,ε. There
exists a sequence of positive numbers {εk} converging to zero such that
εk,α,φc εk,α,Hεk(φa εk,α),Hεk(φc εk,α)) converges strongly in [L2(Ω)]2 ×[L1(Ω)]2
and the limit is a generalized minimizer of Fα. Differently from Fα, the minimizers of Fα,ε can be computed. Derive the first order optimality condition for a minimizer of Fα,ε.
Introduction Inverse Problem Numerical Examples Conclusion Level set approach Level set approach: convergence analysis Level set approach: numerical realization
Theorem
1
Given α, ε > 0 and φi
0 ∈ H1, the functional Fα,ε attains a minimizer on [H1(Ω)]2.
2
Let α be given. For each ε > 0 denote by (φa
ε,α,φc ε,α) a minimizer of Fα,ε. There
exists a sequence of positive numbers {εk} converging to zero such that
εk,α,φc εk,α,Hεk(φa εk,α),Hεk(φc εk,α)) converges strongly in [L2(Ω)]2 ×[L1(Ω)]2
and the limit is a generalized minimizer of Fα. Differently from Fα, the minimizers of Fα,ε can be computed. Derive the first order optimality condition for a minimizer of Fα,ε.
Introduction Inverse Problem Numerical Examples Conclusion Level set approach Level set approach: convergence analysis Level set approach: numerical realization
First order optimality condition: ∂Fα,ε
∂φ j (h) = 0
0) =
ε,α(φa,φc)
in Ω
∂ ∂ν(φ j −φj 0) =
La
ε,α(φa,φc)
= (a1 −a2)H′
ε(φa)
m=1
∂Fm(Pε(φa,φc)) ∂φa ∗ (Fm(Pε(φa,φc))−hδ
m)
ε(φa)∇·
∇Hε(φa) |∇Hε(φa)|
Introduction Inverse Problem Numerical Examples Conclusion Level set approach Level set approach: convergence analysis Level set approach: numerical realization
k,φc k))−hm = uk,m|Γ −hm,
k,φc k))
∂φa
k
k,φc k))
∂φc
k
k and δφc k solutions of the BVP
k = Li ε,α(φa k,φc k)
∂δφi
k
∂ν = 0
k+1 = φa k +δφa k
k+1 = φc k +δφc k
Introduction Inverse Problem Numerical Examples Conclusion Identification of the absorption Identification of the difussion Identification of absorption and diffusion coefficients
X Y 0.5 1 0.5 1
Introduction Inverse Problem Numerical Examples Conclusion Identification of the absorption Identification of the difussion Identification of absorption and diffusion coefficients
Four (M = 4) distinct functions gm, each one supported at each side of
4, 3 4)×{0}
In order to avoid inverse crimes, the direct problem was solved using FEM in an uniform grid with 100 nodes at each boundary side. Alternatively, in the iterative process, all boundary value problems were solved on a uniform grid with 50 nodes at each boundary side. In all cases the initial level set function φj
0 was a paraboloid but with
different minima.
Introduction Inverse Problem Numerical Examples Conclusion Identification of the absorption Identification of the difussion Identification of absorption and diffusion coefficients
500 1000 1500 2000 2500 0.2 0.4 0.6 0.8 1 Iteration k L2 error || c* − ck ||L
2X Y Difference c* − ck −− Iteration k = 0 0.25 0.5 0.75 1 0.25 0.5 0.75 1 X Y Difference c* − ck −− Iteration k = 2500 0.25 0.5 0.75 1 0.25 0.5 0.75 1
500 1000 1500 2000 2500 0.2 0.4 0.6 0.8 1 Iteration k L2 error || c* − ck ||L
2X Y Difference c* − ck −− Iteration k = 0 0.25 0.5 0.75 1 0.25 0.5 0.75 1 X Y Difference c* − ck −− Iteration k = 2500 0.25 0.5 0.75 1 0.25 0.5 0.75 1
Introduction Inverse Problem Numerical Examples Conclusion Identification of the absorption Identification of the difussion Identification of absorption and diffusion coefficients
1000 2000 3000 4000 5000 0.2 0.4 0.6 0.8 1 Iteration k L2 error || a* − ak ||L
2X Y Difference a* − ak −− Iteration k = 0 0.25 0.5 0.75 1 0.25 0.5 0.75 1 X Y Difference a* − ak −− Iteration k = 5000 0.25 0.5 0.75 1 0.25 0.5 0.75 1
1000 2000 3000 4000 5000 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Iteration k L2 error || a* − ak ||L
2X Y Difference a* − ak −− Iteration k = 0 0.25 0.5 0.75 1 0.25 0.5 0.75 1 X Y Difference a* − ak −− Iteration k = 5000 0.25 0.5 0.75 1 0.25 0.5 0.75 1
Introduction Inverse Problem Numerical Examples Conclusion Identification of the absorption Identification of the difussion Identification of absorption and diffusion coefficients
Some facts to take into account:
1
The method for identifying c∗ performs well, even if a good approximation
2
On the other hand, the method may generate a sequence ak that does not approximate a∗ if ck −c∗ is large.
3
For simultaneous identification of (a∗,c∗) we observed that the error
ck −c∗ decreases from the very first iteration. However, the error ak −a∗ only starts improving when ck −c∗ is sufficiently small. Split strategy:
1
Set ak(x) ≡ 1 and iterate w.r.t. ck until the sequenece ck stagnates (ck −c∗ is small).
2
Set ck(x) ≡ ck1 and iterate w.r.t. ak until the sequenece ak stagnates (ak −a∗ is small).
3
Each iteration step consist in one iteration w.r.t. ck and two iterations w.r.t
ak.
Introduction Inverse Problem Numerical Examples Conclusion Identification of the absorption Identification of the difussion Identification of absorption and diffusion coefficients
Some facts to take into account:
1
The method for identifying c∗ performs well, even if a good approximation
2
On the other hand, the method may generate a sequence ak that does not approximate a∗ if ck −c∗ is large.
3
For simultaneous identification of (a∗,c∗) we observed that the error
ck −c∗ decreases from the very first iteration. However, the error ak −a∗ only starts improving when ck −c∗ is sufficiently small. Split strategy:
1
Set ak(x) ≡ 1 and iterate w.r.t. ck until the sequenece ck stagnates (ck −c∗ is small).
2
Set ck(x) ≡ ck1 and iterate w.r.t. ak until the sequenece ak stagnates (ak −a∗ is small).
3
Each iteration step consist in one iteration w.r.t. ck and two iterations w.r.t
ak.
Introduction Inverse Problem Numerical Examples Conclusion Identification of the absorption Identification of the difussion Identification of absorption and diffusion coefficients
Diffusion coefficient a(x)
500 1000 1500 2000 2500 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Iteration k L2 error || a* − ak ||L
2X Y Difference a* − ak −− Iteration k = 0 0.25 0.5 0.75 1 0.25 0.5 0.75 1 X Y Difference a* − ak −− Iteration k = 750 0.25 0.5 0.75 1 0.25 0.5 0.75 1 X Y Difference a* − ak −− Iteration k = 2500 0.25 0.5 0.75 1 0.25 0.5 0.75 1
Absorption coefficient c(x)
500 1000 1500 2000 2500 0.2 0.4 0.6 0.8 1 Iteration k L2 error || c* − ck ||L
2X Y Difference c* − ck −− Iteration k = 0 0.25 0.5 0.75 1 0.25 0.5 0.75 1 X Y Difference c* − ck −− Iteration k = 250 0.25 0.5 0.75 1 0.25 0.5 0.75 1 X Y Difference c* − ck −− Iteration k = 2500 0.25 0.5 0.75 1 0.25 0.5 0.75 1
Introduction Inverse Problem Numerical Examples Conclusion Identification of the absorption Identification of the difussion Identification of absorption and diffusion coefficients
Diffusion coefficient a(x)
250 500 750 1000 1250 1500 0.2 0.4 0.6 0.8 1 Iteration k L2 error || a* − ak ||L
2X Y Difference a* − ak −− Iteration k = 0 0.25 0.5 0.75 1 0.25 0.5 0.75 1 X Y Difference a* − ak −− Iteration k = 750 0.25 0.5 0.75 1 0.25 0.5 0.75 1 X Y Difference a* − ak −− Iteration k = 1500 0.25 0.5 0.75 1 0.25 0.5 0.75 1
Absorption coefficient c(x)
250 500 750 1000 1250 1500 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Iteration k L2 error || c* − ck ||L
2X Y Difference c* − ck −− Iteration k = 0 0.25 0.5 0.75 1 0.25 0.5 0.75 1 X Y Difference c* − ck −− Iteration k = 250 0.25 0.5 0.75 1 0.25 0.5 0.75 1 X Y Difference c* − ck −− Iteration k = 1500 0.25 0.5 0.75 1 0.25 0.5 0.75 1
Introduction Inverse Problem Numerical Examples Conclusion
We developed a level set approach for simultaneous reconstruction of the piecewise constant coefficients (a,c) from a finite set of boundary measurements of optical tomography in the diffusive regime. We proved that the forward map F is continuous in the L1-topology. Hence, by previous results, the presented level set approach is a regularization method. We proposed a split strategy for the simultaneous identification of the diffusion and absorption coefficient. This numerical strategy has not only demonstrated that gives very good results but also reduces significantly the computational time.
Introduction Inverse Problem Numerical Examples Conclusion
We developed a level set approach for simultaneous reconstruction of the piecewise constant coefficients (a,c) from a finite set of boundary measurements of optical tomography in the diffusive regime. We proved that the forward map F is continuous in the L1-topology. Hence, by previous results, the presented level set approach is a regularization method. We proposed a split strategy for the simultaneous identification of the diffusion and absorption coefficient. This numerical strategy has not only demonstrated that gives very good results but also reduces significantly the computational time.
Introduction Inverse Problem Numerical Examples Conclusion