. Coupled Physics Inverse Problems: EIT meets MRI Carlos Montalto - - PowerPoint PPT Presentation

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. .Coupled Physics Inverse Problems: EIT meets MRI

Carlos Montalto

Department of Mathematics

cmontalto@math.purdue.edu

November 14, 2014

Carlos Montalto Coupled Physics Inverse Problems

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. . Table of contents

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1 Introduction

Electrical Impedance Tomography (EIT) Applications and Limitations Magnetic Resonance Imaging . .

2 Coupled-Physics Inverse Problems

Current Density Impedance Imaging History of CDII . .

3 Stability Result

Approach . .

4 Final Remarks

Carlos Montalto Coupled Physics Inverse Problems

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. . Electrical Impedance Tomography

Electrical Impedance Tomography (EIT) is an imaging technique that uses electrical measurements on the surface of a body Ω to obtain the electrical conductivity σ at the interior of the body.

Carlos Montalto Coupled Physics Inverse Problems

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In EIT, an electric potential u is generated inside a body Ω while maintaining a voltage f at the boundary. Assuming the electrostatic approximation of Maxwell’s equations, the potential solves the following Dirichlet problem ∇ · σ∇u = 0 in Ω, u|∂Ω = f , (1) for isotropic electrical conductivity σ. The Dirichlet to Neumann map, or voltage to current map, is given by Λσ : f → (σ∂u/∂ν)|∂Ω, where ν denotes the unit outer normal to ∂Ω. The inverse EIT problem is to recover σ from knowledge of Λσ.

Carlos Montalto Coupled Physics Inverse Problems

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Figure 1: Electrical impedance tomography inverse problem.

Carlos Montalto Coupled Physics Inverse Problems

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There are four fundamental question in any inverse problem: . .

1 Existence: Given the DN-map is there any σ that actually

yields those observations? . .

2 Uniqueness: Can we determine σ uniquely from the

DN-map? (Sylvester-Uhlmann, 1986) - Good news. . .

3 Stability: How are the errors in the measuring the DN-map

amplified in the reconstruction of σ? (Alessandrini, 1988 - Logarithmic stability) - Bad news. . .

4 Reconstruction: Is there a computational efficient formula or

procedure to recover σ from the DN-map?

Carlos Montalto Coupled Physics Inverse Problems

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. . Electrical Resistivity Tomography

Electrical Tomography is successfully used for Geophysical exploration, for imaging sub-surface structures from electrical resistivity measurements from the surface. In such applications, the problem is known as Electrical Resistivity Tomography (ERT). Mathematically ERT and EIT are described by the same inverse problem, in ERT the interest is on recovering the interior resistivity

• f materials denoted by ρ and defined as

ρ(x) = 1 σ(x).

Carlos Montalto Coupled Physics Inverse Problems

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. . Geophysical Applications

Figure 2: Surface of the earth using Electrical Resistivity Tomography.

(Pierce et al., 2012)

Carlos Montalto Coupled Physics Inverse Problems

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. . EIT in Geophysics and Imaging

Table 1: Conductivity of different types of tissue or materials at 1 kHz

Tissue type/material Conductivity σ (S/m) copper 6 · 107 drinking water 5 · 10−2 granite (dry) 10−8

(Widlak and Scherzer, 2012)

Carlos Montalto Coupled Physics Inverse Problems

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. . Geophysical Applications

Figure 3: Electrical Resistivity Tomography used for water exploration.

(Pierce et al., 2012)

Carlos Montalto Coupled Physics Inverse Problems

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. . Medical Applications

There are two important reasons for using EIT in medical applications: . .

1 Monitoring: Can be applied at bedside as a continuous

monitoring technique (relatively inexpensive).

Carlos Montalto Coupled Physics Inverse Problems

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. . Monitoring Applications

Figure 4: EIT used for regional ventilation monitoring.

(Teschner and Imhoff, 1998)

Carlos Montalto Coupled Physics Inverse Problems

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. . Monitoring Lungs Ventilation

Changes in body position of healthy, spontaneously breathing individuals are associated with a major redistribution of regional

• ventilation. In the last few years prone positioning has been used

increasingly in the treatment of acute respiratory distress syndrome

• patients. EIT may help monitor ventilation change due to such

changes of position. It may help identify to responders to this kind

• f treatment

Carlos Montalto Coupled Physics Inverse Problems

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. . Monitoring Examples

Figure 5: Changes of End-Expiratory Lung Volume (EELV) measured with an EIT machine, in a chest trauma patient in a rotation bed, while being turned from a 60 right lateral (1st image taken at cursor position C1, representing the reference status) to supine position. The images depict how EELV increased in the right (initially dependent) lung during the rotation (blue color), while EELV decreased by a similar magnitude in the left lung (orange color). Despite the large changes in EELV, the distribution of ventilation in this patient did not change significantly during the rotation. .

(Teschner and Imhoff, 1998)

Carlos Montalto Coupled Physics Inverse Problems

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. . Medical Applications

There are two important reasons for using EIT in medical applications: . .

1 Monitoring: Can be applied at bedside as a continuous

monitoring technique (relatively inexpensive). . .

2 Diagnosis: Provides images based on new and different

information, such as electrical tissue properties. High quality images could provide better differentiation of tissue or organs, resulting in enhanced diagnosis and treatment of numerous diseases.

Carlos Montalto Coupled Physics Inverse Problems

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. . Diagnosis Applications

Figure 6: Contrast of conductivity in biological tissue at frequencies ranging from 50Hz to 500KHz.

(Widlak and Scherzer 2012)

Carlos Montalto Coupled Physics Inverse Problems

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. . Diagnosis Applications

Figure 7: Contrast of conductivity in biological tissue at frequencies ranging from 50MHz to 500MHz.

(Widlak and Scherzer 2012)

Carlos Montalto Coupled Physics Inverse Problems

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. . Examples of EIT in Medical Imaging

Figure 8: Reconstruction of a phantom of a heart and lungs using D-bar method D-bar in 2D.

(Motoya-Vallejo, 2012)

Carlos Montalto Coupled Physics Inverse Problems

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. . Obstacles of EIT for Medical Diagnosis

Unfortunately there are two obstacles on using EIT for medical diagnosis. Difference in conductivity : The conductivity differences in human and biological tissue are smaller compared to the material in geophysical exploration.

Carlos Montalto Coupled Physics Inverse Problems

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. . EIT in Geophysics and Imaging

Table 2: Conductivity of different types of tissue or materials at 1 kHz

Tissue type/material Conductivity σ (S/m) Application copper 6 · 107 geophysics drinking water 5 · 10−2 geophysics granite (dry) 10−8 geophysics skin (wet) 3 · 10−3 medical blood 7 · 10−1 medical fat 2 · 10−2 medical liver 5 · 10−2 medical

(Widlak and Scherzer, 2012)

Carlos Montalto Coupled Physics Inverse Problems

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. . Obstacles of EIT for Medical Diagnosis

Unfortunately there are two obstacles on using EIT for medical diagnosis. Difference in conductivity : The conductivity differences in human and biological tissue are smaller compared to the material in geophysical exploration. Logarithmic Stability : The EIT has logarithmic stability that only guarantees very low resolution. This type of stability is sometimes refer as ’instability’

Carlos Montalto Coupled Physics Inverse Problems

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. . EIT inverse problem

EIT model: Find the conductivity σ in ∇ · σ∇u = 0, u|∂Ω = f , from knowledge of the DN map Λσ = {(f , σ∂u/∂ν) : for all f }

Figure 9: Illustration of EIT experiment.

Carlos Montalto Coupled Physics Inverse Problems

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. . Instability of EIT

For dimension n ≥ 3 Alessandrini [1988] showed that for smooth, positive and bounded conductivities the stability of the EIT problem is of logarithmic type ||σ − ˜ σ||L∞(Ω) ≤ C(| ln ||Λσ − Λ˜

σ|||−µ + ||Λσ − Λ˜ σ||)

for 0 < µ < 1. For example, if the error in measurements = ||Λσ − Λ˜

σ|| ∼ 1 × 10−10

then error in reconstruction ∼ log 10−10 = 1 × 10−1. Note: log = ln in the example.

Carlos Montalto Coupled Physics Inverse Problems

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. . MRI

Magnetic Resonance Imaging uses magnetic fields to detect the radio frequency signal emitted by excited hydrogen atoms by using the fact that their protons are spin 1/2 particles. Usual MRI images can achieve images with spatial resolution of about 1 mm (New MRI, INUMAC (Imaging of Neuro disease Using high-field MR And Contrastophores) 11.75-Tesla resolves up to 0.1mm).

Carlos Montalto Coupled Physics Inverse Problems

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. . Constrast Problem in MRI

Figure 10: Defect of blood-brain barrier after stroke in MRI. (Wikipedia)

Carlos Montalto Coupled Physics Inverse Problems

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. . Coupled Physics Inverse Problems

Coupled-Physics Inverse Problems are new medical imaging modalities that combine the best imaging properties of different type of ’waves’ to generate high contrast and high resolution images.

Carlos Montalto Coupled Physics Inverse Problems

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. . High contrast modalities

Example of high contrast techniques: Optical tomography - OT: Uses optical waves to recover dielectric permittivity and optical absorption. Electrical impedance tomography - EIT: Uses low-frequency electromagnetic waves to recover electrical impedance (conductivity). Elastic tomography - ET: Uses sonic shear waves to recover shear modulus (viscosity).

Carlos Montalto Coupled Physics Inverse Problems

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. . High resolution modalities

Example of high spatial resolution techniques: Magnetic resonance imaging - MRI: Uses magnetic fields to detect the radio frequency signal emitted by excited hydrogen atoms. Ultrasound tomography - UT: Uses ultrasound waves to recover bulk compressibility.

Carlos Montalto Coupled Physics Inverse Problems

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. . Hybrid Inverse Problems Classification

Carlos Montalto Coupled Physics Inverse Problems

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. . Types of Coupling

These physical couplings can be explained by three potential interactions: Generation: the interaction of the first wave with the tissue can generate a second kind of wave (photo-acoustic effect or thermo-acoustic effect). Tagged: the first wave is tagged locally by a the second type

• f wave.

Movie: the first wave travels much faster than the second type of wave, this difference is used to produce a movie of the slow wave propagation.

Carlos Montalto Coupled Physics Inverse Problems

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. . MREIT and CDII

Magnetic Resonance Electrical Impedance Imaging (MREIT) and Current Density Impedance Imaging (CDII) are two examples of coupled-physics inverse problems. In these models MRI measurements are combined with EIT information to

• vercome the poor spatial resolution of EIT while taking advantage
• f its contrast.

Carlos Montalto Coupled Physics Inverse Problems

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. . CDII Experiment

Figure 11: An electric potential u is generated, inside a body Ω, using a boundary voltage f . With an MRI machine, we measure the magnetic flux density B that by Amp` ere’s law gives the current density J inside Ω. The problem in CDII becomes to recover σ in ∇ · σ∇u = 0, u|∂Ω = f with the additional internal information

• f the current density

J = −σ∇u

(Picture from Prof. Tamasan’s web page)

Carlos Montalto Coupled Physics Inverse Problems

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. . Difference with MREIT

The difference of CDII with MREIT is that in the second we only have the z-component of the magnetic field magnetic field B, and is related to σ as follows, Bz(r) = ∫

Ω

−[σ∇u × (r − r′)]z |r − r′|3 dr′. To obtain B (i.e., Bx and By) and hence J, as in CDII, two rotations of the body are necessary. We sometimes refer to them as J-based MREIT and Bz-based MREIT.

Carlos Montalto Coupled Physics Inverse Problems

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. . Summary

Table 3: Coupled physics inverse problems based on EIT and MRI.

Technique Equation in Ω Boundary data Interior data CDII ∇ · σ∇u = 0 u, σ∂νu J = −σ∇u MREIT ∇ · σ∇u = 0 u, σ∂νu Bz = F(σ∇u) There are four fundamental question in all inverse problems: Existence. Uniqueness. Stability. Reconstruction.

Carlos Montalto Coupled Physics Inverse Problems

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. . History of CDII

1991 Scott G. C., Joy L. G. and Nanping Z. Used MRI to measure the current density J. 1994 Woo EJ., Lee SY. and Mun CW. Minimization algorithm to obtain conductivity σ from J 2002-2003 Kwon O., Woo EJ., Yoon JR. and Seo JK. Introduced J-substitution with the Neumann boundary data. Uniqueness and reconstruction of CDII with two measurements.

Carlos Montalto Coupled Physics Inverse Problems

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Table 4: CDII with J-substitution.

Technique Equation in Ω Boundary data Interior data CDII ∇ · σ∇u = 0 u, σ∂νu |J| = |σ∇u| The J-substitution algorithm consist in substitute J = −σ∇u on ∇ · σ∇u = 0, u|∂Ω = f to get ∇ · ( |J| |∇u|∇u ) = 0, u|∂Ω = f .

Carlos Montalto Coupled Physics Inverse Problems

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2007-2010 Nachman A., Tamasan A. and Timonov A. For n = 2, uniqueness of CDII for planar domains using

• nly one |J| and also with partial illumination.

Sufficient conditions on Dirichlet boundary data to guarantee uniqueness. Reconstruction with partial illumination and conditional stability. For n >≥ 3, uniqueness using only |J| and Dirichlet data. Nice geometric variational approach for recovering the potential.

Carlos Montalto Coupled Physics Inverse Problems

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They show that u0 solves ∇ · ( |J| |∇u0|∇u0 ) = 0, u0|∂Ω = f , if and only if, u0 minimizes F[u] = ∫

Ω

|J(x)| · |∇u(x)| dx restricted to u|∂Ω = f .

Carlos Montalto Coupled Physics Inverse Problems

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2008 Hasanov K., Ma A., Nachman A. and Joy M. Reconstruction algorithm using two current density and in-vitro experiment. 2012 Moradifam A., Nachman A. and Tamasan A. Proved uniqueness in the case of perfectly conducting and insulating inclusions. 2012 Tamasan A. and Veras J. Reconstruction method for planar conductivities with partial data and stability of the potential. 2012 Kuchment P. Steinhauer D. Proved that the linearization is elliptic using two internal measurements. 2012 Monard F. and Bal G. Proved Lipschitz stability of CDII using n + 1 measurements. 2013 Joint work with P. Stefanov Proved H¨

• lder

stability of CDII using only one measurement.

Carlos Montalto Coupled Physics Inverse Problems

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. . Main Result

. Theorem (Local H¨

• lder Stability for the CDII Problem)

. . For σ0 ∈ C 2(Ω) and f ∈ C 2,α(∂Ω) for 0 < α < 1, let u0 such that ∇ · σ0∇u0 = 0, u0|∂Ω = f , with ∇u0 = 0 in Ω. For any 0 < θ < 1, there exist s > 0 so that if σHs(Ω) < L for some L > 0, there exist ǫ > 0 such that if σ − σ0C 2(¯

Ω) < ǫ and (σ − σ0)|∂Ω = 0 then

σ − σ0L2(Ω) < CF(σ) − F(σ0)θ

L2(Ω).

(2)

• Remark. The main difference is that we do not use the

J-substitution to deal with the problem of CDII.

Carlos Montalto Coupled Physics Inverse Problems

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. . Strategy

Use linearization for the CDII problem. Analyze as the linearization as the decomposition of invertible

• perators.

Use SU paper Linearizing non-linear inverse problems and an application to inverse backscattering, 09 to get conditional stability for the non-linear problem from stability of the linearization.

Carlos Montalto Coupled Physics Inverse Problems

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. . Linearization

Comment: We use a more general functional of the from F(σ) = σ|∇u|p for 0 < p ≤ 1. the CDII case is when p = 1. The derivative of F at some fixed σ0 is given by dFσ0(ρ) = σ0|∇u0|p ( ρ + p∇u0 · ∇v(ρ) |∇u0|2 ) , ρ := δσ/σ0, (3) where v solves ∇ · σ0∇v = −σ0∇u0 · ∇ρ in Ω, v|∂Ω = 0. (4) Notice that (3) makes sense as long as ∇u0 = 0 in Ω.

Carlos Montalto Coupled Physics Inverse Problems

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. . Linearization

Solve (3) for the free ρ term and plug that into (4) to get −∇ · σ0∇v + p∇ · ( σ0 ∇u0 · ∇v |∇u0|2 ∇u0 ) = ∇ · (dFσ0(ρ) |∇u0|p ∇u0 ) = σ0∇u0 · ∇ ( dFσ0(ρ) σ0|∇u0|p ) . in Ω, with v|∂Ω = 0.

Carlos Montalto Coupled Physics Inverse Problems

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. . Representation of dFσ0

. Proposition . . Let u0 be σ0-harmonic, with ∇u0 = 0 in Ω. Then σ0T0 dFσ0(ρ) σ0|∇u0|p = −L∆−1

σ0,Dσ0T0ρ,

where T0 = ∇u0 · ∇ is a transport operator along the gradient field

• f u0, ∆σ,D is the Dirichlet realization of ∆σ := ∇ · σ∇ and

Lv := −∇ · σ0∇v + p∇ · ( σ0 ∇u0 · ∇v |∇u0|2 ∇u0 ) . Remark: The operator L is the only interesting object to study.

Carlos Montalto Coupled Physics Inverse Problems

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. . Key Example

Example: Consider the particular case when σ0 = 1, f = xn. Then u0 = xn and L = −∆x′ − (1 − p)∂2

xn, where x = (x′, xn).

Notice that: For 0 ≤ p < 1, L is an elliptic operator. For p = 1, L becomes the restriction of the Laplacian over xn = 0. For p > 1, L is a hyperbolic operator. This characterization remains true in the general case.

Carlos Montalto Coupled Physics Inverse Problems

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. Proposition . . Let u0 be σ0-harmonic, with ∇u0 = 0 in Ω. There exist local coordinates (y′, yn) such that dx2 = c2(dyn)2 + gαβdyαdyβ, gαβ := ∑

i

∂xi ∂yα ∂xi ∂yβ (5) where c = |∇u0|−1. In this coordinates L = Q + − 1 − p √det g ∂ ∂yn c−2σ0 √ det g ∂ ∂yn , where Q is a second order elliptic positively defined differential

• perator in the variables y′ smoothly dependent on y n. In

dimension two this representation is global.

Carlos Montalto Coupled Physics Inverse Problems

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• Proof. Recall

Lv = −∇ · σ0∇v + p∇ · ( σ0 ∇u0 · ∇v |∇u0|2 ∇u0 ) Let

• Π0ω the the orthogonal projection of the covector ω onto ∇u0.
• Π⊥ := Id − Π0.

Take a test function φ ∈ C ∞

0 (Ω), and compute

(Lv, φ) = (σ0∇v, ∇φ) − p(σ0Π0∇v, ∇φ), = (σ0Π⊥∇v, Π⊥∇φ) + (1 − p) (σ0Π0∇v, Π0∇φ) . (6)

Carlos Montalto Coupled Physics Inverse Problems

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Hence L = −(Π⊥∇)′ · σ0(Π⊥∇) − (1 − p)(Π0∇)′ · σ0(Π0∇), (7) where the prime stands for transpose. Notice that trivially c2|∇u0|2 = 1 for c = |∇u0|−1. Near x0 choose boundary local coordinates to the level surface u0(x) = u0(x0), with yn = u0 then c−2dx2 = (dyn)2 + c−2gαβdyαdyβ and then dx2 = c2(dyn)2 + gαβdyαdyβ, gαβ := ∑

i

∂xi ∂yα ∂xi ∂yβ

Carlos Montalto Coupled Physics Inverse Problems

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In this coordinates, Π0v = (0, . . . , 0, ∂v/∂yn) Π⊥v = (∂v/∂y1, . . . , ∂v/∂yn−1, 0) (Lv, φ) = ∫ σ0 ( gαβ ∂v ∂yα ∂ ¯ φ ∂yβ + (1 − p)c−2 ∂v ∂yn ∂ ¯ φ ∂yn ) √ det g dy Hence L = − 1 √det g ( ∂ ∂yβ σ0gαβ√ det g ∂ ∂yα + (1 − p) ∂ ∂yn c−2σ0 √ det g ∂ ∂yn )

Carlos Montalto Coupled Physics Inverse Problems

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. . Images with Real Data

Figure 12: Postmortem animal imaging of a swine leg using a 3 T MRI scanner.

(Minhas et al., 2011)

Carlos Montalto Coupled Physics Inverse Problems

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. . In Vivo Images

Figure 13: In vivo MREIT imaging experiment of a human leg.

(Seo et al., 2011)

Carlos Montalto Coupled Physics Inverse Problems

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. . Open Problems

Critical points (easier in n=2). General approach to get stability for a non-linear problem (allow stability in the partial data setup). Case p = 2 has application to Ultrasound Modulates Optical Tomography.

Carlos Montalto Coupled Physics Inverse Problems

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Thank you!

Carlos Montalto Coupled Physics Inverse Problems