Single-Step

Image

Reconstruction

in

Magnetic

Induction

Tomography

Theoretical

Limits

of

Spatial

Resolution

and

Contrast

to

Noise

Ratio

Magnetic

induction

tomography

-LRB-

MIT

-RRB-

is

a

low-resolution

imaging

modality

for

reconstructing

the

changes

of

the

complex

conductivity

in

an

object

MIT

is

based

on

determining

the

perturbation

of

an

alternating

magnetic

field

which

is

coupled

from

several

excitation

coils

to

the

object

The

conductivity

distribution

is

reconstructed

from

the

corresponding

voltage

changes

induced

in

several

receiver

coils

Potential

medical

applications

comprise

the

continuous

non-invasive

monitoring

of

tissue

alterations

which

are

reflected

in

the

change

of

the

conductivity

e.g.

edema

ventilation

disorders

wound

healing

and

ischemic

processes

MIT

requires

the

solution

of

an

ill-posed

inverse

eddy

current

problem

A

linearized

version

of

this

problem

was

solved

for

excitation

coils

and

receiver

coils

with

a

model

of

two

spherical

perturbations

within

a

cylindrical

phantom

The

method

was

tested

with

simulated

measurement

data

Images

were

reconstructed

with

a

regularized

single-step

Gauss

Newton

approach

Theoretical

limits

for

spatial

resolution

and

contrast/noise

ratio

were

calculated

and

compared

with

the

empirical

results

from

a

Monte-Carlo

study

The

conductivity

perturbations

inside

a

homogeneous

cylinder

were

localized

for

a

SNR

between

and

dB

The

results

prove

the

feasibility

of

difference

imaging

with

MIT

and

give

some

quantitative

data

on

the

limitations

of

the

method

Introduction

Magnetic

induction

tomography

-LRB-

MIT

-RRB-

is

a

non-invasive

and

contact-less

imaging

modality

for

reconstructing

the

changes

Δκ

of

the

complex

conductivity

distribution

κ

σ

jω

ɛ0ɛr

in

a

target

object

MIT

requires

an

array

of

excitation

-LRB-

EXC

-RRB-

and

receiving

coils

Each

EXC

couples

an

alternating

magnetic

field

to

the

object

under

investigation

-LRB-

see

Fig.

-RRB-

Changes

Δκ

of

the

complex

conductivity

cause

a

field

perturbation

ΔB

due

to

the

induction

of

eddy

currents

The

perturbation

induces

voltage

changes

ΔV

in

the

receiver

coils

It

is

convenient

to

normalize

ΔV

to

the

voltage

which

is

induced

by

the

unperturbed

field

Schematic

of

a

possible

coil

system

for

MIT

with

excitation

coils

and

receiver

coils

Previous

reviews

of

MIT

have

been

given

The

method

has

been

developed

for

industrial

process

tomography

already

more

than

years

ago

but

is

comparatively

new

in

medical

imaging

Potential

medical

applications

usually

aim

at

the

characterization

of

biological

tissues

by

means

of

their

complex

conductivity

The

motivation

for

measuring

the

electrical

properties

is

their

characteristic

dependence

on

the

-LRB-

patho

-RRB-

physiological

state

of

tissues

especially

hydration

and

membrane

disorders

Medical

applications

so

far

suggested

are

imaging

of

limbs

imaging

of

the

brain

e.g.

for

the

monitoring

of

brain

edema

measurement

of

human

body

composition

monitoring

of

wound

healing

In

contrast

to

electrical

impedance

tomography

-LRB-

EIT

-RRB-

MIT

avoids

the

ill-defined

electrode-skin

interface

due

to

its

inherently

contact-less

operation

Figure

shows

a

schematic

MIT

coil

configuration

with

rectangular

coils

as

receivers

and

a

cylindrical

object

space

The

solenoid

excitation

coils

are

distributed

on

two

different

rings

in

order

to

obtain

a

true

The

reconstruction

of

the

absolute

conductivity

in

a

target

region

Ω

requires

the

solution

of

a

complex

inverse

eddy

current

problem

Let

be

the

discretized

non-linear

forward

mapping

of

the

conductivity

vector

to

the

vector

of

induced

voltages

y.

y

contains

M

a

b

entries

a

being

the

number

of

EXC

and

b

that

of

receiving

coils

The

corresponding

inverse

problem

is

ill-posed

and

usually

underdetermined

Uniqueness

of

the

solution

for

this

inverse

boundary

value

problem

was

established

provided

the

angular

frequency

ω

of

the

AC

field

is

not

a

resonant

frequency

The

generic

approach

for

the

solution

of

this

type

of

non-linear

problem

is

the

application

of

an

iterative

scheme

such

as

the

regularized

Gauss

Newton

approach

including

an

appropriate

regularization

scheme

To

the

knowledge

of

the

authors

the

full

inverse

problem

of

medical

MIT

in

has

not

yet

been

solved

satisfactorily

although

some

approximate

solutions

especially

for

have

been

presented

Some

proposed

the

use

of

weighted

back-projection

similar

to

EIT

In

all

published

cases

the

back-projection

is

done

along

magnetic

flux

tubes

between

excitation

and

receiving

coils

the

weights

being

calculated

for

the

case

of

conducting

perturbations

in

the

empty

space

However

own

suggest

that

the

basic

requirements

for

the

applicability

of

this

kind

of

back-projection

are

not

fulfilled

in

realistic

anatomical

structures

so

that

a

more

appropriate

inverse

approach

is

necessary

This

paper

is

dedicated

to

demonstrating

the

feasibility

of

the

reconstruction

of

a

spherical

perturbation

within

a

cylindrical

conducting

body

by

means

of

a

regularized

one-step

Gauss

Newton

reconstructor

The

conductivities

were

chosen

in

the

physiological

range

of

human

tissues

Methods

The

solution

of

-LRB-

-RRB-

requires

the

target

region

to

be

discretized

into

N

voxels

Within

each

voxel

i

the

assigned

component

κi

of

the

vector

of

conductivity

is

assumed

to

be

constant

A

grid

of

tetrahedral

finite

elements

of

second

order

was

employed

In

a

general

setup

is

then

found

with

the

iterative

scheme

Define

the

forward

problem

with

an

initial

parameter

vector

Measure

the

data

vector

ym.Solve

iteratively

for

the

estimated

true

parameters

whereby

means

the

estimated

true

parameter

vector

R

and

λ

are

a

regularization

matrix

and

a

regularization

parameter

respectively

which

are

required

to

stabilize

the

iteration

When

applying

Newton

's

method

starting

from

an

initial

guess

the

parameter

vector

is

updated

by

an

increment

pk

in

each

iteration

step

k

with

the

update

step

with

The

Jacobian

also

called

sensitivity

matrix

must

be

recalculated

in

each

iteration

step

This

procedure

is

very

time

consuming

hence

a

complete

iterative

identification

run

requires

significant

computing

power

the

bottleneck

being

the

solution

of

the

forward

model

However

in

EIT

it

could

be

shown

that

in

practice

most

features

of

the

image

can

already

be

recognized

very

satisfactorily

after

the

first

iteration

This

fact

led

to

the

development

of

the

so-called

Newton-one-step

reconstructor

-LRB-

-RRB-

NOSER

is

especially

appropriate

for

so-called

dynamical

imaging

where

only

the

change

in

the

conductivity

between

two

different

states

of

the

object

under

investigation

-LRB-

e.g.

lung

ventilation

-RRB-

are

of

interest

In

this

case

the

first

Newton

step

corresponds

to

the

solution

of

the

linearized

forward

problem

whereby

is

the

change

of

the

conductivity

between

two

states

of

the

observed

object

and

Δym

is

the

corresponding

change

of

the

measured

data

In

the

case

of

comparatively

small

changes

the

inversion

of

-LRB-

-RRB-

according

to

-LRB-

-RRB-

yields

a

fairly

correct

localization

of

the

perturbed

regions

We

evaluated

the

feasibility

of

this

kind

of

reconstruction

in

MIT

by

implementing

a

NOSER-approach

according

to

-LRB-

-RRB-

under

consideration

of

four

different

regularization

methods

Calculation

of

the

Forward

Solution

and

the

Sensitivity

Matrix

The

forward

mapping

is

given

by

Maxwell

's

equations

for

harmonic

excitation

with

H

magnetic

field

intensity

B

magnetic

flux

density

E

electric

field

strength

J

current

density

ɛ

dielectric

constant

μ

magnetic

permeability

σ

real

conductivity

κ

complex

conductivity

ω

angular

frequency

Ω

denotes

the

interior

of

the

object

under

investigation

This

forward

problem

is

solved

with

a

previously

published

finite

element

program

which

employs

an

Ar

V

Ar

formulation

with

edge

elements

of

second

order

for

the

reduced

magnetic

vector

potential

Ar

and

nodal

elements

of

second

order

for

the

electric

scalar

potential

V.

Boundary

conditions

on

the

far

boundary

-LRB-

normal

component

of

B

vanishes

-RRB-

were

prescribed

on

a

spherical

surface

with

a

radius

sufficiently

large

such

that

a

change

of

this

radius

by

resulted

in

a

change

of

the

induced

voltages

by

less

than

Special

attention

must

be

paid

to

the

efficient

calculation

of

the

Jacobian

A

mathematically

rigorous

treatment

of

this

topic

has

been

given

in

In

our

implementation

we

exploited

the

integral

formulation

published

by

which

is

based

on

a

physical

mutual

energy

concept

With

this

approach

the

absolute

sensitivity

dy/d

for

a

certain

pair

of

coils

is

calculated

according

to

-LRB-

-RRB-

with

Aϕ

AΨ

Vϕ

and

Vψ

denote

the

total

magnetic

vector

potential

and

the

electric

scalar

potential

in

the

region

Ω

due

to

currents

Iϕ

and

Iψ

in

the

excitation

and

receiver

coils

respectively

The

sensitivity

matrix

is

then

obtained

by

evaluating

-LRB-

-RRB-

for

all

individual

elements

and

all

coil

pairs

The

exact

numerical

implementation

of

-LRB-

-RRB-

was

described

in

detail

The

calculation

of

the

sensitivity

map

for

one

pair

of

coils

requires

only

two

forward

solutions

of

the

eddy

current

problem

for

generating

Lϕ

and

LΨ

in

-LRB-

-RRB-

Regularization

In

EIT

the

regularization

matrix

RTR

most

frequently

used

is

either

the

identity

matrix

I

or

a

discrete

spatial

derivative

operator

of

first

or

second

order

Such

approaches

have

been

discussed

extensively

in

the

literature

for

a

good

review

see

e.g.

Several

regularization

matrices

can

be

regarded

as

simple

smoothness

criteria

for

the

solution

but

they

have

also

a

more

general

statistical

meaning

in

the

framework

of

Bayesian

estimation

theory

-LRB-

see

e.g.

-RRB-

In

the

case

of

uncorrelated

noise

with

equal

variance

for

all

measurement

data

the

estimator

in

-LRB-

-RRB-

is

a

maximum

a

posteriori

-LRB-

MAP

-RRB-

estimator

with

RTR

being

the

inverse

of

the

expected

covariance

matrix

of

the

image

E

-LSB-

ppT

-RSB-

In

that

sense

e.g.

the

neighbouring

matrix

accounts

for

the

case

that

all

image

values

are

de-correlated

at

borders

between

homogeneous

regions

with

different

mean

values

According

to

our

own

observations

good

results

can

be

achieved

with

variance

which

imposes

a

special

assumption

of

the

prior

distribution

The

objective

here

is

to

uniformize

the

expected

variance

of

the

reconstructed

conductivity

changes

over

the

region

Ω

thus

providing

approximately

equal

image

noise

in

the

center

as

in

the

periphery

The

algorithm

has

been

described

in

detail

and

requires

singular

value

decomposition

of

G

according

to

G

UΣVT

Then

the

regularization

term

is

expressed

as

λ

RTR

VDVT

with

D

a

diagonal

matrix

with

the

entries

di

whereby

σi

is

the

Ith

singular

value

c

is

a

free

scalar

tuning

parameter

Alternatively

truncated

singular

value

decomposition

-LRB-

TSVD

-RRB-

has

been

applied

in

EIT-reconstruction

hence

this

approach

was

also

implemented

for

MIT

In

this

case

the

inverse

solution

becomes

whereby

t

denotes

the

truncation

level

of

the

original

matrices

V

Σ

and

U

thus

removing

the

contributions

of

singular

values

with

index

t

In

this

paper

the

results

obtained

with

four

different

regularization

schemes

were

compared

RTR

I.

Using

the

identity

matrix

is

the

most

simple

Tikhonov-regularization

method

penalizing

high

values

of

the

reconstructed

conductivity

changes

In

the

following

this

method

will

be

abbreviated

as

IM

RTR

N

with

N

the

neighbouring

matrix

defined

as

nn

is

the

number

of

neighbouring

elements

for

element

i

whereby

only

elements

with

common

facets

are

considered

as

neighbours

N

is

an

approximation

of

the

spatial

derivative

operator

of

second

order

Due

to

the

irregular

structure

of

the

grid

this

filter

is

not

spatially

invariant

nevertheless

it

gives

good

results

and

is

common

practice

for

this

type

of

inverse

problem

In

the

following

this

method

will

be

abbreviated

as

NM

λ

RTR

VDVT

according

to

the

variance

uniformization

approach

In

the

following

this

method

will

be

abbreviated

as

VU

TSVD

while

choosing

the

truncation

level

t

such

as

to

remove

all

singular

values

below

an

appropriately

chosen

threshold

Methods

and

require

the

regularization

parameter

to

be

chosen

optimally

while

method

in

addition

implies

the

choice

of

the

tuning

parameter

c

In

practice

it

turns

out

that

the

value

of

c

is

not

critical

over

a

very

wide

range

of

values

because

the

optimal

λ

depends

on

c

That

means

that

c

can

be

fixed

at

a

more

or

less

arbitrary

value

if

a

method

for

the

automatic

determination

of

λ

is

applied

In

our

case

c

was

set

to

The

regularization

parameter

λ

accounts

for

the

degree

of

smoothness

of

the

reconstructed

image

and

determines

the

condition

number

of

the

term

-LRB-

GTG

λRTR

-RRB-

in

-LRB-

-RRB-

Several

methods

for

the

optimal

choice

of

this

parameter

have

been

published

in

the

past

the

most

well-known

ones

being

L-curves

Generalized

Cross-Validation

and

the

Because

of

its

clear

physical

interpretation

we

choose

the

latter

method

In

this

case

the

optimal

λ

is

the

one

where

the

estimated

residuals

Gp

Δym

have

the

same

variance

as

the

measurement

noise

The

motivation

for

this

criterion

is

that

it

is

obviously

meaningless

to

make

the

residuals

lower

than

expected

from

the

statistics

of

the

data

This

method

provided

always

stable

images

independently

of

the

regularization

matrix

and

was

considered

as

a

good

basis

for

a

fair

comparison

between

the

different

regularization

methods

Modeling

Setup

The

inverse

solver

was

tested

with

the

simple

model

comprising

a

cylindrical

conductor

with

two

spherical

inhomogeneities

placed

with

their

centers

at

-LRB-

x

y

z

-RRB-

-LRB-

mm

-RRB-

and

-LRB-

x

y

and

z

-RRB-

-LRB-

mm

-RRB-

and

the

array

of

excitation

coils

and

receiving

coils

shown

in

Fig.

The

exact

geometry

is

illustrated

in

Fig.

The

cylinder

had

a

radius

and

a

height

of

mm

the

perturbing

sphere

had

a

radius

of

mm

The

solenoid

excitation

coils

were

modeled

by

cylindrical

rings

with

an

inner

diameter

of

mm

a

thickness

of

mm

and

a

width

of

mm

The

exciters

were

placed

in

groups

of

with

their

centers

on

two

rings

with

radii

of

mm

in

two

transversal

planes

of

the

cylinder

The

lower

ring

is

rotated

by

degrees

versus

the

upper

one

in

order

to

achieve

a

lower

degree

of

symmetry

The

square

receiver

coils

with

an

edge

length

of

mm

were

placed

with

their

centers

on

two

symmetrically

arranged

parallel

rings

with

a

radius

of

mm

each

comprising

evenly

spaced

coils

The

orientation

of

the

windings

were

opposite

in

both

rings

so

that

the

vertical

pairs

can

be

combined

to

planar

gradiometers

as

published

The

measured

data

were

simulated

in

terms

of

induced

voltage

changes

when

changing

the

conductivity

of

the

spherical

perturbation

from

S/m

-LRB-

background

conductivity

homogeneous

cylinder

-RRB-

to

S/m

The

relative

permittivity

was

kept

constant

at

in

all

compartments

The

excitation

frequency

was

kHz.Figure

Schematic

of

the

simulation

model

excitation

coils

and

receiver

coils

are

placed

on

two

concentric

rings

around

the

tank

respectively

Two

spherical

perturbations

are

placed

at

the

shown

locations

with

-LRB-

x

y

z

-RRB-

and

-LRB-

x

y

z

-RRB-

All

measures

are

given

in

-LRB-

mm

-RRB-

This

arrangement

was

chosen

as

the

model

system

for

our

analysis

because

of

two

reasons

It

represents

a

true

which

delivers

theoretically

independent

measuring

combinations

i.e.

data

points

for

one

image

reconstruction.It

is

similar

to

our

experimental

system

which

employs

excitation

sites

in

one

plane

and

planar

gradiometers

which

are

formed

by

connecting

in

counter-phase

the

coils

in

the

upper

and

in

the

lower

receiver

plane

Two

different

meshes

A

and

B

were

used

for

the

generation

of

an

artificial

dataset

and

for

the

reconstruction

Mesh

A

comprised

elements

within

the

cylinder

and

approximately

in

the

spheres

while

mesh

B

comprised

elements

for

the

homogeneous

cylinder

without

perturbation

In

both

cases

the

diameter

of

the

surrounding

spherical

surface

which

approximated

the

far

boundary

was

chosen

as

requiring

approximately

elements

-LRB-

mesh

A

-RRB-

and

elements

-LRB-

mesh

B

-RRB-

respectively

Uncorrelated

Gaussian

noise

was

added

to

the

voltage

data

in

order

to

simulate

the

noise

of

the

receiver

channels

This

type

of

noise

although

common

practice

in

simulations

of

this

kind

is

not

entirely

valid

for

real

situations

In

addition

the

noise

of

the

excitation

coils

is

propagated

to

all

receiver

coils

thus

resulting

in

a

certain

amount

of

correlated

noise

in

all

receiver

channels

This

phenomenon

has

been

studied

in

detail

for

EIT

but

should

be

disregarded

here

for

simplicity

The

calculation

of

the

complete

sensitivity

matrix

required

forward

solutions

according

to

the

Mortarelli-approach

Theoretical

Limits

of

Resolution

and

Contrast/Noise

Ratio

For

EIT

the

theoretical

limits

of

image

quality

in

terms

of

contrast/noise

ratio

-LRB-

CNR

-RRB-

and

resolution

have

been

studied

carefully

while

no

such

study

exists

for

MIT

A

similarly

rigorous

discussion

for

MIT

is

certainly

beyond

the

scope

of

this

paper

However

a

simplified

analysis

on

resolution

and

CNR

was

carried

out

for

our

linear

reconstruction

scheme

so

as

to

have

a

certain

theoretical

basis

for

interpreting

our

empirical

results

There

is

a

fundamental

limit

for

the

resolution

which

depends

on

the

amount

of

available

information

in

the

data

This

information

depends

on

the

number

of

data

points

i.e.

the

number

of

possible

sensor

detector-combinations

and

on

the

degree

of

independence

between

these

data

points

In

the

case

of

EIT

and

MIT

the

number

of

independent

data

points

is

usually

much

lower

than

the

number

of

voxels

so

that

the

system

is

under-determined

Moreover

the

different

data

are

correlated

to

a

certain

degree

so

that

the

effective

rank

is

comparatively

low

In

EIT

e.g.

electrodes

provide

independent

data

points

so

that

the

information

is

no

more

than

effective

pixels

Including

some

a

priori-information

in

the

form

of

the

regularization

terms

leads

to

a

defined

smearing

of

this

information

over

the

imaging

plane

and

provides

the

typical

diffuse

images

known

from

EIT

We

characterized

the

resolution

of

MIT

with

the

Raleigh

criterion

Accordingly

two

point-shaped

perturbations

are

still

separable

if

their

point

spread

functions

-LRB-

PSF

-RRB-

overlap

in

such

a

way

that

the

peak

of

the

first

one

coincides

with

the

first

zero

crossing

of

the

second

one

In

the

case

of

a

sinc-shaped

PSF

the

lowest

separable

distance

is

equivalent

to

the

width

of

the

PSF

In

contrast

to

e.g.

X-ray

CT

in

MIT

the

PSF

depends

on

the

location

of

the

perturbation

and

on

the

geometry

of

the

object

under

investigation

In

this

paper

the

object

is

the

model

cylinder

which

is

also

used

for

the

numerical

reconstruction

examples

and

for

the

phantoms

The

PSF

is

calculated

by

mapping

the

true

parameter

values

p

to

the

reconstructed

ones

p

via

the

reconstruction

equation

A

means

the

expression

A

-LRB-

GTG

λ

RTR

-RRB-

for

regularized

methods

or

A

Vt

Σt

for

TSVD

The

ith

column

of

M

in

Eq

is

then

the

shifted

PSF

for

the

ith

voxel

The

theoretical

limit

was

approximated

by

applying

TSVD

with

the

full

set

of

non-zero

singular

values

i.e.

We

chose

TSVD

for

the

estimation

of

the

theoretical

limit

because

it

requires

the

least

explicit

assumptions

about

the

a-priori

distribution

of

p

In

MIT

an

additional

difficulty

is

that

in

general

the

boundary

of

the

three-dimensional

PSF

is

not

spherical

and

thus

the

resolution

is

anisotropic

In

our

simplified

analysis

we

define

as

Raleigh-width

the

largest

axis

of

the

ellipsoid

which

best

approximates

the

boundary

As

CNR

at

the

point

x

we

defined

whereby

Δp

is

the

reconstructed

difference

between

perturbation

and

background

and

std

-LRB-

np

-RRB-

is

the

standard

deviation

of

the

image

noise

We

recognize

that

Δp

is

not

a

contrast

in

the

classical

sense

Usually

contrast

is

the

difference

between

p

in

the

perturbation

and

background

divided

by

the

background

value

However

as

our

method

is

a

differential

one

the

background

value

of

the

image

is

always

zero

and

the

classical

contrast

definition

is

meaningless

Moreover

the

determinant

for

the

detectability

of

a

perturbation

is

the

difference

Δp

rather

than

the

contrast

The

CNR

depends

on

the

size

and

location

of

the

perturbation

as

well

as

on

the

noise

level

of

the

measured

data

In

contrast

to

resolution

there

is

no

theoretical

lower

limit

in

the

case

of

noise-free

data

and

a

perfect

reconstruction

method

We

estimated

std

-LRB-

np

-LRB-

x

-RRB-

-RRB-

by

calculating

the

Cramer-Rao

lower

bound

of

the

covariance

of

the

parameters

Given

the

covariance

matrix

X

of

the

voltage

changes

Δym

the

Cramer-Rao

lower

bound

of

the

covariance

matrix

of

the

image

noise

vector

np

is

From

the

diagonal

elements

of

this

matrix

the

expected

lower

bound

of

the

coefficient

of

variation

-LRB-

CV

-RRB-

of

the

reconstructed

conductivity

in

the

perturbation

can

be

calculated

for

any

voxel

For

the

evaluation

of

Eq

the

reconstructed

value

Δp

in

voxel

i

follows

from

Eq

whereby

P

is

the

set

of

the

indices

of

all

voxels

inside

the

true

perturbation

This

equation

clearly

shows

that

the

CNR

must

decrease

when

the

size

of

the

perturbation

shrinks

Assuming

a

small

perturbation

and

an

approximately

constant

PSF

inside

this

region

the

CNR

is

approximately

inversely

proportional

to

the

volume

of

the

perturbation

The

detectability

limit

can

then

be

defined

as

the

one

where

the

CNR

equals

The

shown

approach

is

simple

and

allows

the

calculation

of

detectability

limits

for

perturbations

with

different

radii

and

locations

but

it

is

only

valid

in

the

linear

case

Alternatively

Monte-Carlo

studies

can

be

carried

out

for

different

perturbations

varying

in

size

and

contrast

Such

a

study

has

been

published

previously

for

a

spherical

perturbation

in

the

center

of

a

brain

model

For

simplicity

we

assumed

the

noise

to

be

Gaussian

with

zero

mean

although

in

the

general

non-linear

case

this

may

not

be

entirely

valid

For

characterizing

the

noise

level

we

defined

as

SNR

the

ratio

max

-LRB-

ΔV

-RRB-

std

-LRB-

V

-RRB-

whereby

std

-LRB-

V

-RRB-

is

the

standard

deviation

of

the

noise

voltage

ΔV

is

the

vector

of

voltage

changes

in

all

excitation/sensor

combinations

when

a

test

object

is

placed

into

the

empty

measurement

system

In

order

to

be

independent

on

size

and

location

of

the

perturbation

we

chose

as

the

test

object

the

homogeneous

background

cylinder

of

our

model

Results

The

PSF

was

evaluated

at

equally

spaced

points

along

the

x-axis

between

the

center

and

the

border

of

the

cylinder

This

set

includes

also

the

point

-LRB-

-RRB-

i.e.

the

center

of

one

of

the

two

perturbations

in

our

simulation

model

To

mitigate

discretization

errors

the

PSF

was

calculated

by

rotating

the

coordinate

system

times

about

and

averaging

the

data

assuming

radial

symmetry

of

the

true

model

Then

the

normalized

resolution

was

calculated

from

the

PSF

as

the

inverse

of

the

ratio

between

the

Raleigh-width

and

the

cylinder

radius

This

normalized

resolution

can

be

interpreted

as

the

number

of

points

which

can

be

resolved

per

cylinder

radius

The

resulting

data

were

plotted

in

Fig.

as

a

function

of

the

normalized

x-coordinate

in

the

xy-plane

Curves

are

depicted

for

noise-free

data

-LRB-

TSVD

with

truncation

level

-RRB-

and

TSVD

with

truncation

levels

corresponding

to

a

SNR

of

and

dB

respectively

The

truncation

levels

were

chosen

according

to

the

Morozov-criterion

and

are

listed

in

table

The

above

three

SNR

levels

were

chosen

because

they

correspond

to

the

range

in

which

our

current

measurement

system

operates

Figure

Dependence

of

the

theoretical

normalized

resolution

on

the

noise

level

Curves

are

depicted

for

noise-free

data

-LRB-

TSVD

with

truncation

level

-RRB-

and

TSVD

with

truncation

levels

corresponding

to

a

SNR

of

and

dB

respectively.TABLE

Tuning

parameters

for

the

regularization

chosen

according

to

the

Morozov-criterion

Regularization

methodTSVD

Truncation

levelIM

λNM

λVU

λSNR

E

E

E

E

E

E

E

E

E

parameter

c

for

VU

was

always

kept

at

A

clear

increase

of

the

resolution

with

the

distance

from

the

center

is

observable

The

theoretical

limit

for

a

point-shaped

perturbation

in

the

xy-plane

is

-LRB-

corresponding

to

cm

separation

-RRB-

in

the

center

and

a

maximum

resolution

of

more

than

-LRB-

corresponding

to

cm

separation

-RRB-

at

the

periphery

At

the

locations

of

the

test

spheres

the

resolution

is

in

the

range

of

i.e.

cm

separation

The

relative

loss

of

resolution

with

the

noise

level

is

stronger

in

the

center

than

in

the

periphery

At

a

SNR

of

dB

the

resolution

does

not

increase

any

more

continuously

but

levels

off

above

a

normalized

x

position

of

about

In

analogy

to

EIT

the

PSF

depends

strongly

on

the

location

showing

the

broadest

distribution

in

the

center

of

the

cylinder

This

is

reflected

by

increasing

resolution

when

moving

from

the

center

towards

the

periphery

Increasing

the

amount

of

regularization

or

decreasing

the

truncation

level

according

to

increasing

measurement

noise

the

PSF

broadens

and

its

center

of

gravity

is

shifted

towards

the

borders

of

the

cylinder

Moreover

increasingly

strong

ringing

in

form

of

star-like

patterns

becomes

observable

close

to

the

border

-LRB-

not

shown

explicitly

in

this

paper

-RRB-

In

Fig.

the

four

regularization

methods

are

compared

at

a

SNR

of

dB

Except

for

few

points

TSVD

performs

poorest

which

is

in

accordance

with

the

assumption

of

least

explicit

a-priori

information.Figure

Theoretical

normalized

resolution

for

the

four

regularization

methods

at

a

SNR

of

dB

Figure

shows

the

CNR

for

TSVD

as

a

function

of

the

normalized

x-coordinate

and

in

dependence

on

the

noise

level

while

Fig.

compares

the

four

methods

at

a

SNR

of

dB

As

expected

the

theoretical

CNR

depends

strongly

on

the

location

of

the

perturbation

increasing

from

values

around

in

the

center

up

to

about

at

the

cylinder

border

At

the

location

of

the

perturbing

spheres

the

CNR

drops

from

about

to

when

decreasing

the

SNR

from

dB

to

dB

In

the

simulated

images

the

CNR

drops

from

to

i.e.

remains

fairly

in

the

same

range

Figure

reveals

that

IM

and

NM

yield

the

highest

CNR

followed

by

TSVD

and

VU

whereby

VU

is

characterized

by

dramatically

lower

values

In

the

center

VU

yields

a

CNR

around

which

is

already

very

close

to

the

limit

of

detectability

When

comparing

the

theoretical

values

with

the

reconstructed

ones

-LRB-

see

table

-RRB-

the

reconstructions

always

produce

a

lower

CNR

than

expected

the

discrepancy

being

stronger

at

high

noise

levels.Figure

Panel

A

CNR

for

TSVD

as

a

function

of

the

normalized

x-coordinate

-LRB-

relative

to

the

cylinder

radius

-RRB-

and

in

dependence

on

the

noise

level

Panel

B

comparison

of

the

four

methods

at

a

SNR

of

dB

Figures

and

show

the

reconstructed

mean

images

from

a

Monte-Carlo

study

with

runs

for

each

of

the

four

methods

and

an

SNR

of

and

dB

respectively

Representative

cross-sections

in

the

xy-plane

and

in

the

xz-plane

were

selected

The

respective

regularization

parameters

are

listed

in

Table

Figure

Mean

images

of

the

Monte-Carlo

study

Reconstructed

Δ

σ

-LRB-

transversal

and

saggittal

section

through

the

origin

-RRB-

for

the

spherical

perturbations

with

four

different

regularization

matrices

and

a

SNR

of

dB.Figure

Mean

images

of

the

Monte-Carlo

study

Reconstructed

Δ

σ

-LRB-

transversal

and

saggittal

section

through

the

origin

-RRB-

for

the

spherical

perturbations

with

four

different

regularization

matrices

and

a

SNR

of

dB

In

all

cases

the

two

perturbations

can

be

recognized

as

diffuse

bright

disks

The

dotted

circles

in

the

figures

delineate

the

original

position

of

the

perturbing

spheres

A

number

of

performance

indices

were

calculated

in

order

to

quantitatively

assess

the

results

in

Figs.

and

They

are

summarized

in

Table

and

comprise

Mean

and

CNR

of

the

pixel

values

in

the

center

of

gravity

of

each

reconstructed

perturbation

These

parameters

quantify

the

correctness

of

the

reconstructed

values

as

well

as

their

uncertainty

The

theoretically

expected

values

are

listed

for

comparison

The

center

of

gravity

was

chosen

as

evaluation

point

because

the

reconstructed

perturbations

deviate

more

or

less

from

the

spherical

shape

and

show

significant

outward

shift

with

increasing

noise

level.Radial

outward

shift

of

the

spheres

in

the

reconstructed

image

-LRB-

fidelity

of

the

location

-RRB-

This

shift

was

determined

by

localizing

the

center

of

gravity

for

each

spot

within

a

wedge-shaped

search

region

with

a

height

of

times

the

sphere

's

radius

and

excluding

the

outermost

mm

as

well

as

the

innermost

mm

in

radial

direction

from

the

center

The

restriction

of

the

search

region

to

this

volume

prevented

spurious

contributions

from

outliers

and

negative

image

values

far

away

from

the

real

perturbing

regions

Also

for

this

parameter

we

present

theoretical

values

as

expected

from

the

PSF.TABLE

Summary

of

the

performance

indices

defined

in

the

text.SNR

dBSNR

dBSNR

dBMethodPerformance

indexSphere

realSphere

theoretBackgroundSphere

realSphere

theoretBackgroundSphere

realSphere

theoretBackgroundIMMean

.0220.0160.000560.0060.0040.000510.00320.0010.00035

.0240.0240.000550.010.00650.000430.00600.00360.00035

.0330.0250.000540.010.00740.000500.01100.0020.00055

IMNormalized

outward

shift

Performance

measures

for

the

comparison

of

the

reconstruction

methods

Normalized

to

the

cylinder

radius

In

addition

Table

lists

the

theoretical

resolution

limits

for

all

methods

and

noise

levels

at

the

position

of

the

perturbations

With

dB

SNR

noise

the

two

spheres

can

be

resolved

comparatively

easily

with

all

four

methods

With

dB

SNR

noise

the

resolution

is

theoretically

still

possible

for

all

methods

In

the

reconstruction

the

resolution

is

already

somewhat

below

the

limit

for

IM

and

VU

the

image

values

in

the

notch

between

the

two

peaks

being

around

of

the

peak

values

TSVD

and

NM

appear

to

separate

the

perturbations

even

worse

although

theoretically

this

should

not

be

the

case

At

higher

noise

all

algorithms

tend

to

shift

the

objects

towards

the

borders

of

the

cylinder

when

looking

at

the

parameter

normalized

outward

shift

in

Table

Here

again

the

VU

performs

best

by

producing

the

lowest

shift

At

dB

SNR

the

mean

images

-LRB-

not

shown

here

-RRB-

are

in

general

comparatively

poor

IM

and

NM

interestingly

still

allow

a

clear

separation

of

two

objects

but

their

localization

is

very

poor

the

outward

shift

being

extremely

large

-LRB-

see

Table

-RRB-

VU

still

provides

a

much

better

localization

but

only

at

the

cost

of

CNR

TSVD

failed

to

produce

a

clear

image

an

observation

which

was

not

expected

from

theory

Depending

on

the

regularization

method

the

central

voxel

value

of

the

perturbation

at

-LRB-

-RRB-

decreases

from

S/m

to

S/m

compared

to

the

true

value

of

S/m

This

means

that

even

under

nearly

ideal

conditions

-LRB-

dB

SNR

-RRB-

the

conductivity

changes

are

strongly

underestimated

NM

and

TSVD

yield

nearly

the

same

central

voxel

values

as

TSVD

while

VU

produces

much

lower

values

Figure

shows

single

reconstructions

for

IM

and

VU

at

all

three

noise

levels

Both

methods

allow

a

separation

of

the

perturbations

in

all

cases

but

the

poor

CNR

of

VU

implicates

a

very

noisy

image

at

dB

The

pronounced

difference

in

outward

shift

is

clearly

visible

at

dB

where

VU

still

allows

a

fair

localization

while

IM

fails

completely

to

reconstruct

the

perturbations

at

the

right

positions.Figure

Comparison

of

single-shot

images

for

VU

and

IM

at

the

three

different

noise

levels

IM

and

NM

perform

nearly

identically

also

their

optimal

regularization

parameters

are

almost

identical

VU

yields

in

general

larger

values

in

the

perturbed

regions

but

also

a

larger

STD

Discussion

The

results

demonstrate

the

feasibility

of

image

reconstruction

from

MIT-data

with

the

same

methods

as

suggested

for

EIT

This

finding

is

not

self-evident

as

the

sensitivity

distribution

is

significantly

different

in

EIT

and

MIT

In

EIT

the

region

of

maximum

sensitivity

is

located

between

the

equipotential

surfaces

which

meet

the

surface

at

the

detection

electrodes

i.e.

within

a

tube-shaped

region

which

connects

injection

and

detection

sites

As

shown

in

MIT

the

sensitivity

is

not

concentrated

within

a

field

tube

between

excitation

and

receiver

coil

but

increases

with

the

distance

from

the

tube

axis

according

to

the

increase

of

the

eddy

current

density

This

may

be

the

main

reason

why

the

reconstructed

solution

tends

to

be

displaced

towards

the

nearest

border

of

the

cylindrical

tank

especially

at

higher

noise

levels

An

extreme

case

for

this

effect

can

be

observed

if

the

perturbation

is

placed

exactly

in

the

origin

and

if

the

senders

and

receivers

are

all

in

the

same

plane

-LRB-

image

not

shown

due

to

space

restrictions

-RRB-

Instead

of

the

expected

spot

in

the

origin

two

widely

separated

spots

appear

on

the

cylinder

axis

close

to

the

top

and

bottom

of

the

cylinder

respectively

In

fact

such

a

coil

arrangement

can

not

distinguish

between

an

object

in

the

center

and

two

objects

on

the

cylinder

axis

placed

symmetrically

with

respect

to

the

origin

because

it

is

always

possible

to

find

two

corresponding

conductivitiy

changes

so

that

the

field

perturbations

in

the

median

plane

are

the

same

Obviously

in

this

ambiguous

situation

the

algorithm

favors

the

splitted

solution

according

to

the

sensitivity

distribution

A

similar

ambiguity

occurs

when

using

differential

sensors

such

as

the

gradiometers

employed

in

our

setup

For

getting

rid

of

such

artifacts

it

is

very

important

to

use

a

less

symmetric

transceiver

setup

which

provides

enough

spatial

information

in

The

theoretical

resolution

limit

was

calculated

from

the

PSF

as

derived

from

the

TSVD

method

This

limit

depends

on

the

chosen

regularization

method

the

geometry

of

the

object

and

on

the

location

within

the

object

The

respective

dependences

are

shown

in

Fig.

and

Table

for

some

selected

positions

inside

a

cylinder

TSVD

was

chosen

for

the

calculation

of

the

theoretical

limits

because

it

requires

no

explicit

assumptions

about

any

prior

distribution

of

p

In

this

sense

it

contains

less

a-priori

information

than

the

other

methods

and

thus

describes

the

worst

case

as

confirmed

by

Fig.

and

Table

The

calculated

PSF

shows

all

basic

features

of

the

reconstructed

images

The

PSF

is

a

similar

to

a

analog

of

the

sinc

function

This

means

that

most

of

its

energy

is

concentrated

in

a

diffuse

cloud

around

the

considered

point

but

that

there

exist

three-dimensional

side

lobes

which

decay

with

the

distance

and

show

some

kind

of

periodicity

The

bean

shaped

artifacts

which

are

visible

in

most

top

views

of

Figs.

and

are

typical

features

of

the

PSF

as

well

as

the

star-artifact

in

the

TSVD-images

Therefore

these

ringing

artifacts

do

not

stem

from

inaccuracies

of

the

reconstruction

method

or

measurement

errors

but

instead

are

inherent

in

the

PSF

The

resolution

clearly

also

depends

on

the

contrast

in

the

presence

of

noise

because

the

contrast

determines

the

SNR

Increased

noise

requires

more

regularization

and

hence

leads

to

a

broadening

of

the

PSF-distributions

Figure

shows

the

dependence

of

the

resolution

on

the

noise

in

the

case

of

TSVD

at

one

single

contrast

of

only

A

more

complete

analysis

similar

to

that

given

for

EIT

should

also

show

the

dependence

of

the

resolution

on

contrast

size

and

location

of

the

perturbation

at

a

given

noise

level

However

such

a

comprehensive

analysis

requires

a

separate

paper

and

should

not

be

given

here

The

CNR

depends

strongly

on

the

radius

and

to

a

less

extent

on

the

noise

level

Obviously

IM

and

NM

produce

very

similar

values

followed

by

TSVD

VU

in

general

yields

comparatively

small

CNR

but

higher

central

voxel

values

For

centrally

placed

spheres

with

cm

diameter

VU

yields

CNRs

close

to

the

detection

limit

When

comparing

the

theoretical

values

with

the

reconstructed

ones

-LRB-

see

Table

-RRB-

the

reconstruction

always

produces

a

lower

CNR

than

expected

the

discrepancy

being

stronger

at

high

noise

levels

One

surprising

detail

of

Fig.

is

that

at

higher

noise

levels

the

CNR-curves

cross

the

curve

for

dB

This

means

that

very

close

to

the

periphery

noisier

data

yield

higher

CNR

values

than

less

noisy

data

The

reason

for

this

counter-intuitive

effect

is

not

yet

entirely

clear

but

may

be

related

to

the

strong

outward

shift

of

the

PSF

at

higher

noise

In

those

cases

the

evaluation

of

the

CNR

at

the

original

position

of

the

perturbation

may

not

be

appropriate

any

more

and

should

be

interpreted

with

caution

In

the

case

of

weak

perturbation

we

can

assume

that

the

CNR

depends

approximately

linearly

on

the

conductivity

difference

Δσ

The

dependence

on

the

volume

of

the

perturbation

is

in

general

more

complicated

because

the

PSF

depends

on

the

location

and

is

therefore

not

constant

throughout

the

whole

perturbation

Only

in

the

case

of

small

spatial

extension

of

the

perturbation

an

approximately

linear

dependence

on

the

volume

can

be

assumed

The

low

number

of

significant

singular

values

even

at

comparatively

low

noise

-LRB-

dB

SNR

-RRB-

suggests

that

similar

as

in

EIT

a

significant

amount

of

sensor

combinations

does

not

provide

enough

independent

information

Intuitively

one

would

expect

this

finding

because

there

exist

pairs

of

excitation/receiving

coils

which

nearly

fulfill

the

reciprocity

condition

and

hence

reduce

the

amount

of

useful

combinations

to

about

half

of

the

number

of

possible

combinations

i.e.

to

in

our

case

Further

investigations

should

determine

the

maximum

useful

number

of

sensors

in

one

plane

i.e.

that

number

beyond

which

additional

sensors

do

not

increase

the

resolution

significantly

Adding

more

sensors

off-plane

may

add

more

and

hence

still

provide

improvement

This

possibility

should

be

studied

in

further

research

When

comparing

the

regularization

schemes

after

application

of

the

Morozov-criterion

the

IM

and

the

NM

approach

yield

the

smoothest

visual

appearance

and

the

highest

CNR

However

they

also

tend

to

displace

the

perturbations

towards

the

border

of

the

tank

The

best

localization

is

obtained

with

VU

probably

because

the

imposed

variance

counteracts

somewhat

the

lower

sensitivity

in

the

center

of

the

object

However

VU

yields

also

the

lowest

CNR

i.e.

the

less

homogeneous

images

and

more

pronounced

ghosts

The

failure

of

TSVD

at

a

SNR

of

dB

was

not

expected

theoretically

although

in

general

it

produces

the

poorest

theoretical

resolution

In

terms

of

separability

of

the

two

perturbations

VU

performs

best

especially

when

also

taking

into

account

the

correct

localization

In

neither

case

however

the

single-step

solution

provides

the

correct

values

for

Δσ

Even

at

a

SNR

of

dB

the

reconstructed

differences

are

too

low

by

a

factor

of

at

least

thus

demonstrating

that

the

method

yields

the

correct

search

direction

but

not

the

correct

step

size

The

highest

mean

voxel

values

are

provided

by

VU

and

TSVD

the

drop

with

the

noise

levels

being

lowest

However

on

the

other

hand

VU

yields

the

highest

standard

deviations

Moreover

VU

tends

to

produce

more

pronounced

ghost

objects

in

the

homogeneous

region

than

IM

and

NM

As

expected

from

the

PSF

TSVD

tends

to

produce

star-artifacts

at

the

cylinder

border

i.e.

a

periodic

pattern

with

peaks

close

to

the

centers

of

the

receiving

coils

This

artifact

gets

worse

at

increasing

noise

level

First

experiments

with

smaller

models

and

at

least

iterations

with

an

iterative

solver

show

that

the

solution

converges

towards

the

correct

voxel

values

Nevertheless

the

single-step

method

may

be

completely

justified

in

cases

where

only

qualitative

changes

are

sought

for

or

where

proportions

are

to

be

reconstructed

e.g.

in

frequency

differential

spectroscopic

imaging

Therefore

the

area

of

applicability

of

a

single-step

approach

has

to

be

analyzed

carefully

in

future

work

At

least

for

the

shown

examples

MIT

appears

relatively

robust

against

Gaussian

measurement

noise

A

SNR

of

dB

allows

for

a

stable

and

distinct

solution

Even

dB

allow

the

recognition

of

the

two

spheres

when

applying

the

correct

regularization

This

result

is

very

important

for

the

practical

implementation

because

due

to

technical

reasons

MIT

is

expected

to

yield

low

SNR

-LRB-

around

dB

-RRB-

at

frequencies

as

low

as

kHz

which

are

interesting

for

the

imaging

of

pathophysiological

processes

However

our

results

have

only

been

achieved

with

two

single

focal

perturbations

with

a

relatively

large

diameter

of

of

the

background

object

In

a

more

advanced

study

the

stability

and

the

resolution

of

the

images

should

be

investigated

for

a

series

of

perturbations

with

different

diameter

and

spacing

For

the

monitoring

of

brain

edema

which

usually

do

not

split

up

in

separate

sub-regions

our

approach

may

be

sufficiently

stable

This

hypothesis

has

to

be

tested

both

theoretically

and

empirically

for

centrally

placed

perturbations

-LRB-

worst

case

-RRB-

As

to

the

detectability

of

spherical

perturbations

in

a

human

brain

simulation

results

have

shown

that

a

sphere

with

a

diameter

of

about

mm

a

background

conductivity

of

S/m

a

contrast

of

yields

a

SNR

of

dB

at

kHz

when

applying

A

to

an

excitation

coil

with

turns

and

a

receiver

coil

with

turn

The

assumed

acquisition

time

was

ms

With

our

present

technology

single

shot

images

are

generated

with

an

acquisition

time

of

ms

an

excitation

coil

with

five

turns

and

a

current

up

to

A

The

receiver

coils

have

turns

with

otherwise

unchanged

geometry

This

means

an

overall

increase

in

SNR

by

dB

Extrapolating

the

analysis

given

in

an

improvement

of

the

SNR

by

a

factor

of

is

still

technically

possible

thus

reaching

dB

which

is

obviously

sufficient

for

producing

fairly

acceptable

difference

images

Another

open

question

is

the

influence

of

the

mesh

quality

on

the

reconstruction

results

We

used

a

comparatively

coarse

non-uniform

grid

for

the

reconstruction

Therefore

non-negligible

numerical

errors

are

to

be

expected

which

may

explain

the

discrepancies

between

theoretically

expected

and

the

reconstructed

values

for

CNR

and

radial

displacement

Also

the

apparently

somewhat

worse

spatial

resolution

in

the

reconstructed

images

than

theoretically

expected

may

be

due

to

such

numerical

problems

The

influence

of

the

mesh

and

the

optimization

of

mesh

quality

should

be

a

major

issue

for

further

developments

The

results

were

obtained

at

a

single

frequency

only

Future

work

should

concentrate

on

the

exploitation

of

the

frequency

dependence

of

the

tissue

conductivity

and

measurements

at

frequencies

up

to

several

MHz

A

multi-frequency

approach

is

expected

to

increase

significantly

the

available

information

and

thus

the

quality

of

the

images

Possible

applications

may

then

in

fact

be

the

same

as

for

EIT

-LRB-

lung

function

monitoring

lung

edema

monitoring

-RRB-

and

hydration

monitoring

in

the

brain

O O O O O B I I O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

magnetic induction tomography
inverse problem
regularization

passive electrical properties
conductivity imaging