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Mathematical imaging
Habib Ammari
Department of Mathematics, ETH Zürich
Mathematical imaging
Habib Ammari
Mathematical imaging
• Nanophotonics:
• Control, manipulate, reshape, guide, focus electromagnetic
waves at nanometer length scales (beyond the resolution limit).
• Push the resolution limit by reducing the focal spot size;
confine waves to a length scale significantly smaller than half
the wavelength.
• Biomedicine:
• Image electrical, optical, and mechanical tissue properties
using electromagnetic and elastic waves at single or multiple
frequencies.
• Enhance the resolution, the stability, and the specificity.
• Direct, inverse, and optimal design problems for wave propagation in
complex media.
• Build mathematical frameworks and develop effective numerical
algorithms for photonic and biomedical imaging applications.
Mathematical imaging
Habib Ammari
Mathematical imaging
• Key concepts:
• Resolution: smallest detail that can be resolved.
• Robustness: stability of the image formation with respect to
model uncertainty and electronic noise.
• Specificity: physical nature (benign or malignant for tumors).
Mathematical imaging
Habib Ammari
Mathematical imaging
• Recent works in nanophotonics:
• Plasmonic resonant nanoparticles (with P. Millien, M. Ruiz,
S. Yu, H. Zhang, Arch. Rat. Mech. Anal., 2015).
• Super-resolution using resonant media (with H. Zhang, Comm.
Math. Phys., 2015).
• Metamaterials for cloaking and wave-field shaping (with
M. Lim, H. Kang, H. Lee, Comm. Math. Phys., 2014.)
Mathematical imaging
Habib Ammari
Plasmonic resonances for nanoparticles
• Gold nano-particles: accumulate selectively in tumor cells; bio-compatible;
reduced toxicity.
• Detection: localized enhancement in radiation dose (strong scattering).
• Ablation: localized damage (strong absorption).
• Functionalization: targeted drugs.
M.A. El-Sayed et al.
Mathematical imaging
Habib Ammari
Plasmonic resonances for nanoparticles
• D: nanoparticle; ν: normal to ∂D; ε(ω): complex permittivity contrast;
λ(ω) = (ε(ω) + 1)/(2(ε(ω) − 1)).
• Neumann-Poincaré operator KD∗ :
∗
KD
[ϕ](x) =
1
2π
Z
∂D
hx − y , νx i
ϕ(y ) ds(y ) ,
|x − y |2
x ∈ ∂D.
• Symmetrization technique (Calderón’s identity): Discrete spectrum
∗
σ(KD
) in ] − 1/2, 1/2[.
• Quasi-static plasmonic resonance: dist(λ(ω), σ(KD∗ )) minimal
(<e ε(ω) < 0).
• Enhancement of the absorption and scattering cross-sections Q a and Q s
at plasmonic resonances:
2
Q a + Q s ∝ =m Trace(M(λ(ω), D)); Q s ∝ Trace(M(λ(ω), D)) .
• Polarization tensor: M(λ(ω), D) :=
Z
∗ −1
x(λ(ω)I − KD
) [ν](x) ds(x).
∂D
Mathematical imaging
Habib Ammari
Plasmonic resonances for nanoparticles
• KD∗ : scale invariant ⇒ Quasi-static plasmonic resonances: size
independent.
• Analytic formula for the first-order correction to quasi-static plasmonic
resonances in terms of the particle’s characteristic size δ:
M.A. El-Sayed et al.
• Operator-Valued function δ 7→ Aδ (ω):
A0 (ω)
z
}|
{
∗
Aδ (ω) = (λ(ω)I − KD
) +(ωδ)2 A1 (ω) + O((ωδ)3 ).
Mathematical imaging
Habib Ammari
Resonant media for super-resolution
• Resolution: determined by the behavior of the imaginary part of the
Green function. Helmholtz-Kirchhoff identity:
Z
=m G (x, x0 , ω) = ω
G (y , x0 , ω)G (x, y , ω)ds(y ),
R → +∞.
|y |=R
• The sharper is =m G , the better is the resolution.
• Local resonant media used to make shape peaks of =m G .
• Mechanism of super-resolution in resonant media:
• Interaction of the point source x0 with the resonant structure
excites high-modes.
• Resonant modes encode the information about the point
source and can propagate into the far-field.
• Super-resolution: only limited by the resonant structure and
the signal-to-noise ratio in the data.
Mathematical imaging
Habib Ammari
Resonant media for super-resolution
• Helmholtz resonators:
M. Fink et al.
• Asymptotic expansion of the Green function (: size of the resonator
openings; zj : center of aperture for jth resonator; M: number of
√
resonators; ω = O( )):
M
=m G (x, x0 , ω) ≈
Mathematical imaging
sin ω|x − x0 | √ X
cj
+ .
2π|x − x0 |
|x
−
z
j | |x0 − zj |
j=1
Habib Ammari
Resonant media for super-resolution
• Similar mechanism for super-resolution for plasmonic nanoparticles:
S. Nicosia & C. Ciraci, Cover, Science 2012
Mathematical imaging
Habib Ammari
Electromagnetic invisibility
• Make a target invisible when probed by electromagnetic waves.
• u i : plane wave; ε: permittivity; µ: permeability. Helmholtz equation:

1

 ∇· χ(R2 \ D̄) + χ(D) ∇u + ω 2 χ(R2 \ D̄) + εχ(D) u = 0,
µ

 s
i
u := u − u satisfies the outgoing radiation condition.
• Scattering coefficients:
Z
ψm (y )Jn (ω|y |)e −inθy ds(y ).
Wmn (D, ε, µ, ω) =
∂D
• ψm : electric current density on ∂D induced by the cylindrical wave
Jm (ω|x|)e imθx .
• Jn : Bessel function.
Mathematical imaging
Habib Ammari
Electromagnetic invisibility
Properties of the scattering coefficients:
• Wmn decays rapidly:
|Wmn | .
C |m|+|n|
, m, n ∈ Z.
|m||m| |n||n|
• For any z ∈ R2 , θ ∈ [0, 2π), s > 0,
Wmn (D z ) =
X
0
0
Jn0 (ω|z|)Jm0 (ω|z|)e i(m −n )θz Wm−m0 ,n−n0 (D),
m0 ,n0 ∈Z
Wmn (D θ ) = e i(m−n)θ Wmn (D),
Wmn (D s , ω) = Wmn (D, sω).
Mathematical imaging
Habib Ammari
Electromagnetic invisibility
• Scattering amplitude:
πi
1
e iω|x|
u s (x) = −ie − 4 p
A∞ [D, ε, µ, ω](θ, θ0 ) + o(|x|− 2 ),
8πω|x|
|x| → ∞; θ, θ0 : incident and scattered directions.
• Graf’s formula:
X
A∞ [D, ε, µ, ω](θ, θ0 ) =
0
(−i)n i m e inθ Wnm (D, ε, µ, ω)e −imθ .
n,m∈Z
• Scattering cross-section:
s
0
Z
Q [D, ε, µ, ω](θ ) :=
0
Mathematical imaging
2π
2
A∞ [D, ε, µ, ω](θ, θ0 ) dθ.
Habib Ammari
Electromagnetic invisibility
• Scattering coefficient cancellation technique:
• Small layered object with vanishing first-order scattering
coefficients.
• Transformation optics:
(Fρ )∗ [φ](y ) =
DFρ (x)φ(x)DFρ (x)t
,
det(DFρ (x))
x = Fρ−1 (y ).
• Change of variables Fρ sends the annulus [ρ, 2ρ] onto a fixed
annulus.
• Scattering coefficients vanishing structures of order N:
i
h
Q s D, (Fρ )∗ (µ ◦ Ψ 1 ), (Fρ )∗ (ε ◦ Ψ 1 ), ω (θ0 ) = o(ρ4N ).
ρ
ρ
ρ: size of the small object; N: number of layers;
Ψ1/ρ (x) = (1/ρ)x.
• Anisotropic permittivity and permeability distributions.
• Invisibility at ω ⇒ invisibility at all frequencies ≤ ω.
Mathematical imaging
Habib Ammari
Electromagnetic invisibility
Cancellation of the scattered field and the scattering cross-section: 4 orders of
magnitude (with wavelength of order 1, ρ = 10−1 , and N = 1).
Mathematical imaging
Habib Ammari
Mathematics for biomedical imaging
• Waves play a key role in biomedical imaging techniques.
• Visualize contrast information on the electrical, optical, mechanical
properties of tissues.
• Tissue contrasts:
• Highly sensitive to physiological and pathological tissue status.
• Depend on the cell organization and composition.
• Overall parameters, averaged in space over many cells.
• Recognize the microscopic cell organization and composition from
measurements at the macroscopic level.
Mathematical imaging
Habib Ammari
Mathematics for biomedical imaging
• Diagnosis and staging of cancer disease.
• Help surgeons to make sure they removed everything unwanted around
the margin of the cancer tumor.
• Perform biopsy in the operating room.
Mathematical imaging
Habib Ammari
Mathematics for biomedical imaging
• Recent works in biomedical imaging:
• Dictionary matching based approach (with T. Boulier,
J. Garnier, and H. Wang, Proc. Natl. Acad. Sci., 2014).
• Spectroscopic electrical tissue property imaging (with
J. Garnier, L. Giovangigli, W. Jing, and J.K. Seo, J. Math.
Pures Appl., 2015).
• Differential approach for multi-wave imaging:
• Ultrasound-modulated optical tomography (with L. Nguyen
and L. Seppecher, J. Funct. Anal., 2014);
• Ultrasonically-induced Lorentz force electrical impedance
tomography (with P. Grasland-Mongrain, P. Millien, J.K. Seo,
and L. Seppecher, J. Math. Pures Appl., 2015).
Mathematical imaging
Habib Ammari
Bio-inspired dictionary matching based approach
• Electrolocation for weakly electric fish:
• Electric organ: generate a stable, high-frequency, weak electric
field.
• Electroreceptors: measure the transdermal potential
modulations caused by a nearby target.
• Nervous system: perceive target’s shape.
Mathematical imaging
Habib Ammari
Shape perception
Mechanism for mimicking shape perception:
• Form an image from the perturbations of the field due to targets.
• Identify and classify the target, knowing by advance that it belongs to a
learned dictionary of shapes.
• Extract the features from the data.
• Construct invariants with respect to rigid transformations and
scaling.
• Compare the invariants with precomputed ones for the
dictionary.
• Biological targets: frequency dependent electromagnetic properties
(capacitive effect generated by the cell membrane structure).
• Spectroscopic measurements of the target’s polarization tensor.
Mathematical imaging
Habib Ammari
Weakly electric fish
• Wave-type electric signal: f (x, t) = f (x)
P
n
an e inω0 t ; ω0 : fundamental
frequency.
• Skin: very thin (δ ∼ 100µm) and highly resistive (σs /σ0 ∼ 10−2 );
σb /σ0 ∼ 102 (highly conductive).
Mathematical imaging
Habib Ammari
Weakly electric fish
• Target D = z + δ 0 B; z: location; δ 0 : characteristic size of the target;
k(ω) = (σ(ω) + iωε(ω))/σ0 ; k, σ, and ε: the admittivity, the
conductivity, and the permittivity of the target; ωn = nω0 : the probing
frequency.
• un : the electric potential field generated by the fish:








∆un = f ,
x ∈ Ω,
∇ · (1 + (k − 1)χ(D))∇un = 0,
x ∈ R2 \ Ω,
∂un ∂un 
= 0, [un ] = ξ
x ∈ ∂Ω,


∂ν −
∂ν +




|un (x)| = O(|x|−1 ), |x| → ∞.
• ξ := δσ0 /σs effective thickness.
• λ(ω) = (k(ω) + 1)/(2(k(ω) − 1)).
Mathematical imaging
Habib Ammari
Weakly electric fish
• Dipole approximation: un (x) − U(x) ' p · ∇G (x − z).
• G : Green’s function associated to Robin boundary conditions.
• Dipole moment p = −
M(λ(ω), D) ∇U(z).
Z
• M(λ(ω), D) =
|
{z
}
Polarization tensor
∗ −1
x(λI − KD
) [ν](x) ds(x).
∂D
1.5
0.06
1
0.04
0.5
0.02
0
0
−0.02
−0.5
−0.04
−1
−0.06
−1.5
−1.5
Mathematical imaging
−1
−0.5
0
0.5
1
1.5
Habib Ammari
Dictionary matching based approach
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
0%
Ellipse
Disk
A
E
Square
Rectangle
Triangle
Different ellipse
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Probability of detection in terms of the noise level. Stability of classification
based on differences between ratios of eigenvalues of =m M(λ(ω), D).
Mathematical imaging
Habib Ammari
Weakly electric fish
Nonbiological targets (frequency-independent material parameters):
F. Boyer
• Use multipolar approximation:
un (x) − U(x) '
X
∂ α G (x − z)Mαβ (λ, D)∂ β U(z).
α,β
• Mαβ (λ, D): high-order polarization tensors.
Z
Mαβ (λ, D) :=
∗ −1
x β (λI − KD
) [∂x α /∂ν](x) ds(x).
∂D
Mathematical imaging
Habib Ammari
Weakly electric fish
Properties of high-order polarization tensors:
1
0
• Recover high-frequency information on the
−1
shape;
• Separate topology;
• Determine uniquely the shape and the
−1
0
1
−1
0
1
1
material parameter.
0
−1
Mathematical imaging
Habib Ammari
Weakly electric fish
• Positivity and symmetry properties on
harmonic coefficients; optimal bounds.
• Harmonic coefficients:
(x1 + ix2 )m =
X
|α|=m
m α
aα
x +i
X
bβm x β .
|β|=m
• Translation, rotation, and scaling formulas.
• Construct shape descriptors invariant with
respect to translation, rotation, and scaling .
Mathematical imaging
Habib Ammari
Weakly electric fish
100%
90%
Probability of detection
80%
70%
60%
50%
40% Ellipse
Disk
30% A
E
20% Square
Rectangle
10% Triangle
Different ellipse
0%
0%
0,5%
1%
1,5%
2%
2,5%
3%
Strength of noise
3,5%
4%
4,5%
5%
Stability of classification based on Shape Descriptors.
Mathematical imaging
Habib Ammari
Spectroscopic electrical tissue property imaging
• Differentiate between normal, pre-cancerous and cancerous tissues from
electrical measurements at tissue level.
• Frequency dependence of the (anisotropic) homogenized admittivity:
ω 7→ K ∗ (ω).
• Relaxation times:
• 1/ arg maxω eigenvalues of =m K ∗ (ω);
• Classification: invariance properties;
• Measure of anisotropy: ratios of the eigenvalues of =m K ∗ (ω).
0.1
0.09
0.08
0.07
λ1 (C )
0.06
0.05
0.04
0.03
0.02
0.01
0
104
105
106
107
108
109
ω
Mathematical imaging
Habib Ammari
Spectroscopic electrical tissue property imaging
The effective admittivity of a periodic dilute suspension:
−1 !
f
∗
K = k0 I + f M I − M
+ o(f 2 ).
2
• f : volume fraction; ξ:
√ effective thickness of the membrane; ∂D: cell
e = D/ f : rescaled cell.
membrane; D
• M: membrane polarization tensor
Z
M=− ξ
e
∂D
• LDe [ϕ](x) =
Mathematical imaging
1
p.v.
2π
Z
e
∂D
νj I + ξLDe
−1
[νi ](y )ds(y )
∂ 2 ln |x − y |
ϕ(y )ds(y ),
∂ν(x)∂ν(y )
.
i,j=1,2
e
x ∈ ∂ D.
Habib Ammari
Spectroscopic electrical tissue property imaging
• Properties of the membrane polarization tensor:
•
•
•
•
M: symmetric; invariant by translation;
M(sC , ξ) = s 2 M(C , ξs ) for any scaling parameter s > 0.
M(RC , ξ) = RM(C , ξ)Rt for any rotation R.
=m M is positive and its eigenvalues, λ1 ≥ λ2 , have one
maximum with respect to ω.
• Relaxation times for the arbitrary-shaped cells:
1
:= arg max λi (ω).
ω
τi
• (τi )i=1,2 : invariant by translation, rotation and scaling.
• Concentric circular-shaped cells: Maxwell-Wagner-Fricke formula
(λ1 = λ2 ).
• Nondilute regime: Assume f known ⇒ Classification based on relaxation
times.
Mathematical imaging
Habib Ammari
Differential imaging
• Acoustically modulated optical tomography:
Acoustic source
Light source
Focused acoustic beam
Ω
y
Spherical acoustic
pulses
Light detectors
6
Contrasted anomaly
• Record the variations of the light intensity on the boundary due to the
propagation of the acoustic pulses.
Mathematical imaging
Habib Ammari
Differential imaging
• g : the light illumination; a: optical absorption coefficient; l: extrapolation
length. Fluence Φ (in the unperturbed domain):

 −∆Φ + aΦ = 0 in Ω,

l
∂Φ
+ Φ = g on ∂Ω.
∂ν
• Acoustic pulse propagation: a → au (x) = a(x + u(x)).
• Fluence Φu (in the displaced domain):

 −∆Φu + au Φu = 0 in Ω,
∂Φu

l
+ Φu = g on ∂Ω.
∂ν
• u: thin spherical shell growing at a constant speed; y : source point; r : radius.
• Cross-correlation formula:
Z
M(y , r ) :=
∂Ω
Z
Z
∂Φu
∂Φ
Φu −
Φ =
(au − a)ΦΦu ≈
u · ∇a |Φ|2 .
∂ν
∂ν
Ω
Ω
|
{z
}
Taylor +Born
Mathematical imaging
Habib Ammari
Differential imaging
• Helmholtz decomposition: Φ2 ∇a = ∇ψ + ∇ × A.
1
c
• Spherical Radon transform: ∇ψ = − ∇R−1
r
Z
0
M(y , ρ)
dρ
.
ρd−2
• System of nonlinearly coupled elliptic equations: ∇ · Φ2 ∇a = ∆ψ and
∆Φ + aΦ = 0.
• Fixed point and Optimal control algorithms.
• Reconstruction for a realistic absorption map.
• Proofs of convergence for highly discontinuous absorption maps (bounded
variation).
Mathematical imaging
Habib Ammari
Lorentz force electrical impedance tomography
transducer (500 kHz)
sample with electrodes
magnet (300 mT)
oil tank
degassed water
absorber
Example of the imaging device. A transducer is emitting ultrasound in a sample
placed in a constant magnetic field. The induced electrical current is collected
by two electrodes.
Mathematical imaging
Habib Ammari
Lorentz force electrical impedance tomography
support of x 7→ v (x, t)
Be3
Γ1
support of the acoustic beam
Ω
y
Γ0
I
τ
Γ0
e2
e1
ξ
Γ2
e3
• Interaction between v (x, t)ξ and Be3 : induces Lorentz’ force on the ions
in Ω ⇒ separation of charges ≡ source of current and potential:
jS (x, t) = eB+ σ(x)v (x, t)τ ; e + : elementary charge.
• Voltage potential u:

−∇ · (σ∇u) = ∇ · jS in Ω,

 u = 0 on Γ1 ∪ Γ2 , ∂u = 0 on Γ0 .
∂ν
Z
∂u
• Measured intensity: I (y , ξ) =
σ .
∂ν
Γ2
Mathematical imaging
Habib Ammari
Lorentz force electrical impedance tomography
• Virtual potential:

−∇ · (σ∇U) = 0




U=0
U := F [σ] =

U=1



∂ν U = 0
in Ω,
on Γ1 ,
on Γ2 ,
on Γ0 .
• Wiener deconvolution filter: recover J(x) = (σ∇U)(x) from measured
intensities I (y , ξ).
• Recover σ from J = σ∇U.
• Optimal control algorithm:
Z
• min
|σ∇F [σ] − J|2 + regularization term (a prior).
σ
Ω
• Nonconvexity (numerically); high sensitivity to noise.
Mathematical imaging
Habib Ammari
Lorentz force electrical impedance tomography
Direct method
• Viscosity-type regularization method:
(
∇ · (εI + (J ⊥ (J ⊥ )T )∇Uε = 0
Uε = x2
in Ω,
on ∂Ω.
• Reconstructed image:
1
J ⊥ · ∇Uε
1
:=
→ in L2
σε
|J|2
σ
as the viscosity parameter ε → 0.
Mathematical imaging
Habib Ammari
Lorentz force electrical impedance tomography
10
1
0.8
8
0.6
6
0.4
4
0.2
2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
1
10
0.8
8
0.6
6
0.4
4
0.2
2
0
Mathematical imaging
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Habib Ammari
Lorentz force electrical impedance tomography
L2 norm of the relative error
102
101
100
10−1
10−2
10−3 −4
10
Mathematical imaging
10−3
10−2
noise level
10−1
100
Habib Ammari
Concluding remarks
Resolution, stability, and specificity enhancement:
• Hybrid imaging;
• Spectroscopic imaging;
• Physic-based learning approach.
Ultrafast ultrasound imaging:
Mathematical and computational framework for
superresolution imaging: signal separation techniques.
M. Tanter et al.
Mathematical imaging
Habib Ammari
Concluding remarks
Functional conductivity imaging:
Neuro-activity: localized conductivity change.
Inject a dc current through electrodes and measure the change in the current density.
Stem cells monitoring and tissue engineering:
Image each layer and measure noninvasively its
composition.
Mathematical imaging
Habib Ammari