A non-iterative sampling approach for EIT using noise subspace - - PowerPoint PPT Presentation

A non-iterative sampling approach for EIT using noise subspace projection

Cédric Bellis1 Andrei Constantinescu2 Armin Lechleiter3 Thomas Coquet4 Thomas Jaravel4

1Dept Applied Physics & Applied Mathematics · Columbia University · USA 2Solid Mechanics Laboratory (UMR CNRS 7649) · École Polytechnique · France 3Center for Industrial Mathematics · University of Bremen · Germany 4Dept of Mechanics · École Polytechnique · France

PICOF’12 · Wednesday, April 4th

Context

• Let Ω ⊂ Rd, d = 2, 3 bounded connected background domain, ∂Ω Lipschitz
• Finite union I = K

j=1 Ωj of disjoint inclusions Ωj ⊂ Ω and Ω\I connected.

γ(ξ) = 1 if ξ ∈ Ω \ I γj(ξ) if ξ ∈ Ωj, j = 1, . . . , K. with γj ∈ L∞(Ω, R) and 0 < c ≤ γj < 1 (could be changed into 1 < γj ≤ C)

• Mean-free current density f ∈ L2(∂Ω) generates potential u solution of

∇ · (γ∇u) = 0 in Ω (γ∇u) · n = f

• n ∂Ω,

→ Determine γ from knowledge of NtD operator Λγ : f → u|∂Ω

• Geometric Prior: Inclusions embedded in known background medium
• C. Bellis & al.

Noise Subspace Projection for EIT PICOF’12 · 4/4/12 2 / 18

Context

• Let Ω ⊂ Rd, d = 2, 3 bounded connected background domain, ∂Ω Lipschitz
• Finite union I = K

j=1 Ωj of disjoint inclusions Ωj ⊂ Ω and Ω\I connected.

γ(ξ) = 1 if ξ ∈ Ω \ I γj(ξ) if ξ ∈ Ωj, j = 1, . . . , K. with γj ∈ L∞(Ω, R) and 0 < c ≤ γj < 1 (could be changed into 1 < γj ≤ C)

• Mean-free current density f ∈ L2(∂Ω) generates potential u solution of

∇ · (γ∇u) = 0 in Ω (γ∇u) · n = f

• n ∂Ω,

→ Determine γ from knowledge of NtD operator Λγ : f → u|∂Ω

• Geometric Prior: Inclusions embedded in known background medium
• C. Bellis & al.

Noise Subspace Projection for EIT PICOF’12 · 4/4/12 2 / 18

Motivations

• The map γ → Λγ is non-linear
• Severe ill-posedness:

Given γ, γ′ ∈ H2+s(Ω) with s > d/2 (dimension d = 2, 3)

• If γ, γ′ piecewise-constant and Ωj, Ω′

j known Lipschitz domains

• γ − γ′
• L∞(Ω) ≤ β
• Λ−1

γ

− Λ−1

γ′

• H1/2(∂Ω)→H−1/2(∂Ω)
• G. Alessandrini, S. Vessella, Adv. Appl. Math., 2005.
• Standard logarithmic stability for generic configurations
• γ − γ′
• L∞(Ω) ≤ β
• log
• Λ−1

γ

− Λ−1

γ′

• H1/2(∂Ω)→H−1/2(∂Ω)
• −α
• G. Alessandrini, Appl. Anal., 1988.

→ Qualitative sampling methods: computationally-effective approaches Factorization method, point source method, topological sensitivity, MUSIC, . . .

• C. Bellis & al.

Noise Subspace Projection for EIT PICOF’12 · 4/4/12 3 / 18

Motivations

• The map γ → Λγ is non-linear
• Severe ill-posedness:

Given γ, γ′ ∈ H2+s(Ω) with s > d/2 (dimension d = 2, 3)

• If γ, γ′ piecewise-constant and Ωj, Ω′

j known Lipschitz domains

• γ − γ′
• L∞(Ω) ≤ β
• Λ−1

γ

− Λ−1

γ′

• H1/2(∂Ω)→H−1/2(∂Ω)
• G. Alessandrini, S. Vessella, Adv. Appl. Math., 2005.
• Standard logarithmic stability for generic configurations
• γ − γ′
• L∞(Ω) ≤ β
• log
• Λ−1

γ

− Λ−1

γ′

• H1/2(∂Ω)→H−1/2(∂Ω)
• −α
• G. Alessandrini, Appl. Anal., 1988.

→ Qualitative sampling methods: computationally-effective approaches Factorization method, point source method, topological sensitivity, MUSIC, . . .

• C. Bellis & al.

Noise Subspace Projection for EIT PICOF’12 · 4/4/12 3 / 18

Outline

1 Sampling approaches and noise subspace projection 2 Finite dimensional approximation of NtD operators 3 Numerical implementations and examples

• C. Bellis & al.

Noise Subspace Projection for EIT PICOF’12 · 4/4/12 4 / 18

Identification framework

• Let L2

⋄(∂Ω) :=

• ϕ ∈ L2(∂Ω) :
• ∂Ω ϕ dS = 0
• → NtD map Λ : L2

⋄(∂Ω) → L2 ⋄(∂Ω) s.t. Λf = u|∂Ω

with u ∈ H1

⋄(Ω) solution of:

• Ω

γ∇u · ∇ϕ dV =

• ∂Ω

f ϕ dS, ∀ϕ ∈ H1

⋄(Ω)

• Reference homogeneous counterpart, i.e. γ = 1 in Ω: NtD map Λ1f = u1|∂Ω
• NtD operators Λ and Λ1 are compact on L2

⋄(∂Ω)

→ Measurement operator Π = Λ − Λ1

• Extract the informations synthesized in Π

→ Probe range R(Π) using a fundamental solution gz

gz has singular behavior at chosen sampling point z ∈ Ω

• Π is self-adjoint and compact ⇒ ∃{λj, ψj} with λj > 0 and ψj ∈ L2

⋄(∂Ω) s.t.

Πf =

∞

• j=1

λj(f , ψj)L2(∂Ω)ψj

• C. Bellis & al.

Noise Subspace Projection for EIT PICOF’12 · 4/4/12 5 / 18

Identification framework

• Let L2

⋄(∂Ω) :=

• ϕ ∈ L2(∂Ω) :
• ∂Ω ϕ dS = 0
• → NtD map Λ : L2

⋄(∂Ω) → L2 ⋄(∂Ω) s.t. Λf = u|∂Ω

with u ∈ H1

⋄(Ω) solution of:

• Ω

γ∇u · ∇ϕ dV =

• ∂Ω

f ϕ dS, ∀ϕ ∈ H1

⋄(Ω)

• Reference homogeneous counterpart, i.e. γ = 1 in Ω: NtD map Λ1f = u1|∂Ω
• NtD operators Λ and Λ1 are compact on L2

⋄(∂Ω)

→ Measurement operator Π = Λ − Λ1

• Extract the informations synthesized in Π

→ Probe range R(Π) using a fundamental solution gz

gz has singular behavior at chosen sampling point z ∈ Ω

• Π is self-adjoint and compact ⇒ ∃{λj, ψj} with λj > 0 and ψj ∈ L2

⋄(∂Ω) s.t.

Πf =

∞

• j=1

λj(f , ψj)L2(∂Ω)ψj

• C. Bellis & al.

Noise Subspace Projection for EIT PICOF’12 · 4/4/12 5 / 18

Existing sampling approach 1/2

• Low-dimensional parameterization: K inclusions of small volume

→ Ωj ≡ Ωε

j = zj + εˆ

Ωj

• Measurement operator Π ≡ Πε = Λε − Λ1 converges to finite-rank operator ˆ

Π

• Πε − εd ˆ

Π

• L2

⋄(∂Ω)→L2 ⋄(∂Ω) = O(εd+ 1 2 ).

with range: R(ˆ Π) = span{ek · ∇

zN(·, zj), k = 1, . . . , d ; j = 1, . . . , K}

using mean-free Green’s function N(·, z) of −∆, i.e. −∆ξN(ξ, z) = δ(ξ − z) in Ω ∇

ξN(ξ, z) · n(ξ) = −|∂Ω|−1

• n ∂Ω,
• Theorem: For any d ∈ Sd−1 and z ∈ Ω, let gz,d = d · ∇

zN(·, z)|∂Ω then

z ∈ {z1, . . . , zK} if and only if gz,d ∈ R(ˆ Π) → MUSIC algorithm

• M. Bruhl, M. Hanke, M. Vogelius, Numer. Math., 2003
• C. Bellis & al.

Noise Subspace Projection for EIT PICOF’12 · 4/4/12 6 / 18

Existing sampling approach 1/2

• Low-dimensional parameterization: K inclusions of small volume

→ Ωj ≡ Ωε

j = zj + εˆ

Ωj

• Measurement operator Π ≡ Πε = Λε − Λ1 converges to finite-rank operator ˆ

Π

• Πε − εd ˆ

Π

• L2

⋄(∂Ω)→L2 ⋄(∂Ω) = O(εd+ 1 2 ).

with range: R(ˆ Π) = span{ek · ∇

zN(·, zj), k = 1, . . . , d ; j = 1, . . . , K}

using mean-free Green’s function N(·, z) of −∆, i.e. −∆ξN(ξ, z) = δ(ξ − z) in Ω ∇

ξN(ξ, z) · n(ξ) = −|∂Ω|−1

• n ∂Ω,
• Theorem: For any d ∈ Sd−1 and z ∈ Ω, let gz,d = d · ∇

zN(·, z)|∂Ω then

z ∈ {z1, . . . , zK} if and only if gz,d ∈ R(ˆ Π) → MUSIC algorithm

• M. Bruhl, M. Hanke, M. Vogelius, Numer. Math., 2003
• C. Bellis & al.

Noise Subspace Projection for EIT PICOF’12 · 4/4/12 6 / 18

Existing sampling approach 2/2

• Extended inclusions:

Separate influence of unknown inclusion from that of known background Using that Πf = (Λ − Λ1)f = (u − u1)|∂Ω with

• Ω

γ∇(u − u1) · ∇ϕ dV =

• I

(1 − γ)∇u1 · ∇ϕ dV ∀ϕ ∈ H1

⋄(Ω),

→ Factorization of the type: Π = A∗T A

• Characterization of R(A∗) = R(Π1/2)

Functions harmonic in Ω\I with homog. Neumann B.C. on ∂Ω

• Theorem: For any d ∈ Sd−1 and z ∈ Ω, let gz,d = d · ∇

zN(·, z)|∂Ω then

z ∈ I if and only if gz,d ∈ R(Π

1/2)

• Picard criterion:

z ∈ I if and only if the series

∞

• j=1

|(gz,d, ψj)L2(∂Ω)|2 λj converges

Bruhl, Hanke, Kirsch, . . .

• C. Bellis & al.

Noise Subspace Projection for EIT PICOF’12 · 4/4/12 7 / 18

Existing sampling approach 2/2

• Extended inclusions:

Separate influence of unknown inclusion from that of known background Using that Πf = (Λ − Λ1)f = (u − u1)|∂Ω with

• Ω

γ∇(u − u1) · ∇ϕ dV =

• I

(1 − γ)∇u1 · ∇ϕ dV ∀ϕ ∈ H1

⋄(Ω),

→ Factorization of the type: Π = A∗T A

• Characterization of R(A∗) = R(Π1/2)

Functions harmonic in Ω\I with homog. Neumann B.C. on ∂Ω

• Theorem: For any d ∈ Sd−1 and z ∈ Ω, let gz,d = d · ∇

zN(·, z)|∂Ω then

z ∈ I if and only if gz,d ∈ R(Π

1/2)

• Picard criterion:

z ∈ I if and only if the series

∞

• j=1

|(gz,d, ψj)L2(∂Ω)|2 λj converges

Bruhl, Hanke, Kirsch, . . .

• C. Bellis & al.

Noise Subspace Projection for EIT PICOF’12 · 4/4/12 7 / 18

Implementation and issues

• Discrete setting:

Π →

• ΠM = U Σ V T ∈ RM×M
• Use of Picard series:

M′

• m=1

|gT

z,d um|2

|σm| , 1 ≤ M′ ≤ M.

• Questions:

1 Choice of truncation parameter M′ 2 Characterization of the “blow-up” of the series when z ∈ Ω\I

– Linear regression – Partial sum

(a) Exterior point z ∈ Ω\I (b) Interior point z ∈ I

• C. Bellis & al.

Noise Subspace Projection for EIT PICOF’12 · 4/4/12 8 / 18

Noise subspace projection approach

• Interpretation: MUSIC for extended inclusions
• Similar work in inverse scattering

Luke, Devaney, Arens, Lechleiter, . . .

• Signal and noise subspaces for given δ > 0: Sδ, Nδ

Let Mδ s.t. any j > Mδ verifies λj ≤ δ then

• Sδ = span{ψj, j = 1, . . . , Mδ}

Nδ = span{ψj, j ≥ Mδ + 1}

• Idea:

1 Construct h ∈ Nδ ⊂ L2

⋄(∂Ω)

2 (h, gz,d)L2(∂Ω) is arbitrarily small when z ∈ I and large when z ∈ Ω\I

→ Construction: Define h

M∗,M∗

z,d

=

M∗

• j=M∗

(gz,d, ψj)L2(∂Ω) λj

1/2

ψj together with h → ˆ h

M∗,M∗

z,d

=

• h

M∗,M∗

z,d

• −1

L2(∂Ω) M∗

• j=M∗

(gz,d, ψj)L2(∂Ω) λj ψj

• C. Bellis & al.

Noise Subspace Projection for EIT PICOF’12 · 4/4/12 9 / 18

Noise subspace projection approach

• Interpretation: MUSIC for extended inclusions
• Similar work in inverse scattering

Luke, Devaney, Arens, Lechleiter, . . .

• Signal and noise subspaces for given δ > 0: Sδ, Nδ

Let Mδ s.t. any j > Mδ verifies λj ≤ δ then

• Sδ = span{ψj, j = 1, . . . , Mδ}

Nδ = span{ψj, j ≥ Mδ + 1}

• Idea:

1 Construct h ∈ Nδ ⊂ L2

⋄(∂Ω)

2 (h, gz,d)L2(∂Ω) is arbitrarily small when z ∈ I and large when z ∈ Ω\I

→ Construction: Define h

M∗,M∗

z,d

=

M∗

• j=M∗

(gz,d, ψj)L2(∂Ω) λj

1/2

ψj together with h → ˆ h

M∗,M∗

z,d

=

• h

M∗,M∗

z,d

• −1

L2(∂Ω) M∗

• j=M∗

(gz,d, ψj)L2(∂Ω) λj ψj

• C. Bellis & al.

Noise Subspace Projection for EIT PICOF’12 · 4/4/12 9 / 18

Main result

• Theorem: Let ε > 0 and d ∈ Sd−1.

(a) For x ∈ I, there exists 0 < M∗ ∈ N s.t. for all M∗ ∈ N with M∗ >M∗

• (ˆ

h

M∗,M∗

z,d

, gx,d)L2(∂Ω)

• < ε if z ∈ Ω.

(b) For z ∈ Ω\I and M∗ ∈ N, there exist M∗ ∈ N with M∗ >M∗ >0 and α>0, s.t.

• (ˆ

h

M∗,M∗

z,d

, gx,d)L2(∂Ω)

• > 1

ε if x ∈ B(z, α).

• Interpretation using Green’s formulae: (ˆ

h

M∗,M∗

z,d

, gx,d)L2(∂Ω) = d · ∇u1(x)

Where u1 harmonic in Ω with ˆ hM∗,M∗

z,d

as imposed Neumann B.C.

→ Construction of a non-interacting background solution

• Corollary: For ε > 0, d ∈ Sd−1 there exist M∗ > M∗ > 0, s.t.

(a)

M∗

• j=M∗

|(gz,d, ψj)L2(∂Ω)|2 λj <ε if z ∈ KI (b)

M∗

• j=M∗

|(gz,d, ψj)L2(∂Ω)|2 λj > 1 ε if z ∈ KΩ\I

• C. Bellis & al.

Noise Subspace Projection for EIT PICOF’12 · 4/4/12 10 / 18

Main result

• Theorem: Let ε > 0 and d ∈ Sd−1.

(a) For x ∈ I, there exists 0 < M∗ ∈ N s.t. for all M∗ ∈ N with M∗ >M∗

• (ˆ

h

M∗,M∗

z,d

, gx,d)L2(∂Ω)

• < ε if z ∈ Ω.

(b) For z ∈ Ω\I and M∗ ∈ N, there exist M∗ ∈ N with M∗ >M∗ >0 and α>0, s.t.

• (ˆ

h

M∗,M∗

z,d

, gx,d)L2(∂Ω)

• > 1

ε if x ∈ B(z, α).

• Interpretation using Green’s formulae: (ˆ

h

M∗,M∗

z,d

, gx,d)L2(∂Ω) = d · ∇u1(x)

Where u1 harmonic in Ω with ˆ hM∗,M∗

z,d

as imposed Neumann B.C.

→ Construction of a non-interacting background solution

• Corollary: For ε > 0, d ∈ Sd−1 there exist M∗ > M∗ > 0, s.t.

(a)

M∗

• j=M∗

|(gz,d, ψj)L2(∂Ω)|2 λj <ε if z ∈ KI (b)

M∗

• j=M∗

|(gz,d, ψj)L2(∂Ω)|2 λj > 1 ε if z ∈ KΩ\I

• C. Bellis & al.

Noise Subspace Projection for EIT PICOF’12 · 4/4/12 10 / 18

Main result

• Theorem: Let ε > 0 and d ∈ Sd−1.

(a) For x ∈ I, there exists 0 < M∗ ∈ N s.t. for all M∗ ∈ N with M∗ >M∗

• (ˆ

h

M∗,M∗

z,d

, gx,d)L2(∂Ω)

• < ε if z ∈ Ω.

(b) For z ∈ Ω\I and M∗ ∈ N, there exist M∗ ∈ N with M∗ >M∗ >0 and α>0, s.t.

• (ˆ

h

M∗,M∗

z,d

, gx,d)L2(∂Ω)

• > 1

ε if x ∈ B(z, α).

• Interpretation using Green’s formulae: (ˆ

h

M∗,M∗

z,d

, gx,d)L2(∂Ω) = d · ∇u1(x)

Where u1 harmonic in Ω with ˆ hM∗,M∗

z,d

as imposed Neumann B.C.

→ Construction of a non-interacting background solution

• Corollary: For ε > 0, d ∈ Sd−1 there exist M∗ > M∗ > 0, s.t.

(a)

M∗

• j=M∗

|(gz,d, ψj)L2(∂Ω)|2 λj <ε if z ∈ KI (b)

M∗

• j=M∗

|(gz,d, ψj)L2(∂Ω)|2 λj > 1 ε if z ∈ KΩ\I

• C. Bellis & al.

Noise Subspace Projection for EIT PICOF’12 · 4/4/12 10 / 18

Finite dimensional approximation of NtD op.

• Q: Approximation quality of finite dimensional approximation of op. Π ?
• Subset FM = span {fm}M

m=1 of lin. indepdt current densities

• Associated potentials (um − u1m)|∂Ω are measured in GN = span{gn}N

n=1

→ Finite dim. lin. op. ΠNM = PGN Π PFM → ΠNM characterized by matrix ΠNM ∈ RN×M

• Approximation quality:

Assume I−PFML2

⋄(∂Ω)→H−s ⋄ (∂Ω) ≤ CM−s and I−PGNHs ⋄(∂Ω)→L2(∂Ω) ≤ CN−s

and Ω ⊂ R2 or R3 is a smooth domain, then ΠNM − ΠL2

⋄(∂Ω)→L2(∂Ω) ≤ C(s)(N−s + M−s)

for N, M ≥ N0. – Ex1: Ω unit circle and fm = gm trigonometric polynomials

Super-exponential convergence

– Ex2: Ω convex polygon and finite elements discretization

Typically Q1

h2 Π Pp h1 − ΠL2

⋄(∂Ω)→L2(∂Ω) ≤ C(hp+1

1

+ hmin(k−1/2,2)

2

) as h1,2 → 0.

• C. Bellis & al.

Noise Subspace Projection for EIT PICOF’12 · 4/4/12 11 / 18

Finite dimensional approximation of NtD op.

• Q: Approximation quality of finite dimensional approximation of op. Π ?
• Subset FM = span {fm}M

m=1 of lin. indepdt current densities

• Associated potentials (um − u1m)|∂Ω are measured in GN = span{gn}N

n=1

→ Finite dim. lin. op. ΠNM = PGN Π PFM → ΠNM characterized by matrix ΠNM ∈ RN×M

• Approximation quality:

Assume I−PFML2

⋄(∂Ω)→H−s ⋄ (∂Ω) ≤ CM−s and I−PGNHs ⋄(∂Ω)→L2(∂Ω) ≤ CN−s

and Ω ⊂ R2 or R3 is a smooth domain, then ΠNM − ΠL2

⋄(∂Ω)→L2(∂Ω) ≤ C(s)(N−s + M−s)

for N, M ≥ N0. – Ex1: Ω unit circle and fm = gm trigonometric polynomials

Super-exponential convergence

– Ex2: Ω convex polygon and finite elements discretization

Typically Q1

h2 Π Pp h1 − ΠL2

⋄(∂Ω)→L2(∂Ω) ≤ C(hp+1

1

+ hmin(k−1/2,2)

2

) as h1,2 → 0.

• C. Bellis & al.

Noise Subspace Projection for EIT PICOF’12 · 4/4/12 11 / 18

Spectrum approximation

• Q: Spectrum of matrix

ΠNM ↔ Spectrum of Π ?

• Approximation quality:

Under previous assumptions on PFM and PGN, there is N0 ∈ N s.t. distH(σ(Π), σ(ΠNM)) ≤ C(s)(N−s + M−s) N, M ≥ N0, For each J ∈ N, |λj − λNM

j

| ≤ C(J, s)(N−s + M−s) j ≤ J and N, M ≥ N0.

• If the basis functions of FM and GM coincide then

λ eigenvalue of op. ΠM ⇔ λ eigenvalue of matrix ΠM

• Remark:

In general, the matrix ΠM does not possess an eigenvalue decomposition

→ Singular Value Decomposition and associated perturbation results

• C. Bellis & al.

Noise Subspace Projection for EIT PICOF’12 · 4/4/12 12 / 18

Spectrum approximation

• Q: Spectrum of matrix

ΠNM ↔ Spectrum of Π ?

• Approximation quality:

Under previous assumptions on PFM and PGN, there is N0 ∈ N s.t. distH(σ(Π), σ(ΠNM)) ≤ C(s)(N−s + M−s) N, M ≥ N0, For each J ∈ N, |λj − λNM

j

| ≤ C(J, s)(N−s + M−s) j ≤ J and N, M ≥ N0.

• If the basis functions of FM and GM coincide then

λ eigenvalue of op. ΠM ⇔ λ eigenvalue of matrix ΠM

• Remark:

In general, the matrix ΠM does not possess an eigenvalue decomposition

→ Singular Value Decomposition and associated perturbation results

• C. Bellis & al.

Noise Subspace Projection for EIT PICOF’12 · 4/4/12 12 / 18

Numerical implementations

• Discrete setting:

Π →

• ΠM = U Σ V T ∈ RM×M

Using orthonormal basis {fm}M

m=1

• Noise subspace

→ N∗ = span{vm, m = M∗, . . . , M∗}

• Indicator function:

For a given d ∈ Sd−1 and M ≥ M∗ > M∗ > 0 IM∗,M∗(z) =

• M∗
• m=M∗

|gT

z,d vm|2

σm −1 → IM∗,M∗(zi) ≫ IM∗,M∗(ze) with zi ∈ KI and ze ∈ KΩ\I

• Remark:

Test function gz,d computed in Ω from known dipole potential Φz,d in Rd → gz,d = Φz,d|∂Ω − Λ1(∇Φz,d · n) + c

• C. Bellis & al.

Noise Subspace Projection for EIT PICOF’12 · 4/4/12 13 / 18

Numerical implementations

• Discrete setting:

Π →

• ΠM = U Σ V T ∈ RM×M

Using orthonormal basis {fm}M

m=1

• Noise subspace

→ N∗ = span{vm, m = M∗, . . . , M∗}

• Indicator function:

For a given d ∈ Sd−1 and M ≥ M∗ > M∗ > 0 IM∗,M∗(z) =

• M∗
• m=M∗

|gT

z,d vm|2

σm −1 → IM∗,M∗(zi) ≫ IM∗,M∗(ze) with zi ∈ KI and ze ∈ KΩ\I

• Remark:

Test function gz,d computed in Ω from known dipole potential Φz,d in Rd → gz,d = Φz,d|∂Ω − Λ1(∇Φz,d · n) + c

• C. Bellis & al.

Noise Subspace Projection for EIT PICOF’12 · 4/4/12 13 / 18

Numerical examples 1/2

• Standard finite elements-based computational platform
• Discretized background domain: Square Ωh = [0; 1] × [0; 1]
• Set of M = 144 equidistributed unit nodal current densities {fm}

→ Computation of unitary indicator function – Summation over 8 equidistributed dipole direction d k = (cos θk; sin θk) – Maximum as M∗ = 20, . . . , 40 and M∗ = 130, . . . , 140

1 Single inclusion

(a) r = 0.07, γI = 0.01 (b) r = 0.12, γI = 0.5 (c) L-shaped, γI = 0.01

• C. Bellis & al.

Noise Subspace Projection for EIT PICOF’12 · 4/4/12 14 / 18

Numerical examples 2/2

• Standard finite elements-based computational platform
• Discretized background domain: Square Ωh = [0; 1] × [0; 1]
• Set of M = 144 equidistributed unit nodal current densities {fm}

→ Computation of unitary indicator function – Summation over 8 equidistributed dipole direction d k = (cos θk; sin θk) – Maximum as M∗ = 20, . . . , 40 and M∗ = 130, . . . , 140

2 Multiple inclusions: r = 0.05, γI = 0.01

(a) (b) (c)

• C. Bellis & al.

Noise Subspace Projection for EIT PICOF’12 · 4/4/12 15 / 18

Effect of noisy data

• Identical numerical setup
• Noisy measurements
• Πδ

M =

ΠM + Λ/ √ M →

• Πδ

M −

ΠM2 = δ ΠM2

with Λ real-valued random iid Gaussian entries zero-mean and std dev. σn

3 Multiple inclusions: r = 0.05, γI = 0.01

(a) δ = 0.01 (b) δ = 0.05 (c) δ = 0.1

• A first interpretation: Signal subspace only slightly modified by noise

⇒ span{vδ

m, m = M∗, . . . , M} = span{vm, m = M∗, . . . , M}

→ Estimate on indicator function σδ

M

δ IM∗,M(z) ≤ I δ

M∗,M(z) ≤ δ

σM IM∗,M(z)

• C. Bellis & al.

Noise Subspace Projection for EIT PICOF’12 · 4/4/12 16 / 18

Effect of noisy data

• Identical numerical setup
• Noisy measurements
• Πδ

M =

ΠM + Λ/ √ M →

• Πδ

M −

ΠM2 = δ ΠM2

with Λ real-valued random iid Gaussian entries zero-mean and std dev. σn

3 Multiple inclusions: r = 0.05, γI = 0.01

(a) δ = 0.01 (b) δ = 0.05 (c) δ = 0.1

• A first interpretation: Signal subspace only slightly modified by noise

⇒ span{vδ

m, m = M∗, . . . , M} = span{vm, m = M∗, . . . , M}

→ Estimate on indicator function σδ

M

δ IM∗,M(z) ≤ I δ

M∗,M(z) ≤ δ

σM IM∗,M(z)

• C. Bellis & al.

Noise Subspace Projection for EIT PICOF’12 · 4/4/12 16 / 18

Experimental data

• Experimental results provided by V. Choquet, J. Alaterre
• 20 × 28cm carbon-paper sheet with circular cut of radius r = 3cm
• Set of M = 15 current densities with disjoint supports on ∂Ω

(a) Singular values in log scale (b) Indicator function (c) Singular values in log scale (d) Indicator function

• C. Bellis & al.

Noise Subspace Projection for EIT PICOF’12 · 4/4/12 17 / 18

Conlusion

• Sampling approach in EIT

→ Extraction of informations from noise subspace of data-to-mesurements op.

– Extension to extend inclusions of the MUSIC algorithm – Relies on the factorization method – Construction of non-interacting background solutions

• Avoids the question of determining behavior of Picard series
• Good stability results in noisy environment (!)
• Need for relevant perturbation theory for inverse problems

Random preturbation: Πδ

M =

ΠM + 1 √ M Λ ✓ rank( ΠM) < M ✓ rank( ΠM) = M and σj = O(1) for all j ✗ rank( ΠM) = M and σk = o(1) for N < k < M

• C. Bellis & al.

Noise Subspace Projection for EIT PICOF’12 · 4/4/12 18 / 18

Conlusion

• Sampling approach in EIT

→ Extraction of informations from noise subspace of data-to-mesurements op.

– Extension to extend inclusions of the MUSIC algorithm – Relies on the factorization method – Construction of non-interacting background solutions

• Avoids the question of determining behavior of Picard series
• Good stability results in noisy environment (!)
• Need for relevant perturbation theory for inverse problems

Random preturbation: Πδ

M =

ΠM + 1 √ M Λ ✓ rank( ΠM) < M ✓ rank( ΠM) = M and σj = O(1) for all j ✗ rank( ΠM) = M and σk = o(1) for N < k < M

• C. Bellis & al.

Noise Subspace Projection for EIT PICOF’12 · 4/4/12 18 / 18