electrical tomography lecture v3
TRANSCRIPT
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Industrial Tomography Slide 1
Tony PeytonManchester University
School of Electrical and Electronic Engineering
Industrial Tomography
Electrical Tomography for Industrial Applications
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Industrial Tomography Slide 2
Electrical TomographyOverview of the course:
• Introduction to tomography• Overview of sensing modalities• Hardware design• Image reconstruction techniques • Industrial applications • Conclusions
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Industrial Tomography Slide 3
Introduction
GEhttp://www.ge.com/medical/
Analogichttp://www.analogic.com/Level3/CT%20III.html
Siemenshttp://www.med.siemens.com/
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Industrial Tomography Slide 4
IntroductionIndustrial tomography
• High resolution (spatial or contrast) may not be essential• High imaging speeds may be required
(e.g. 100’s frames/sec for fast flow applications)• Rugged operating conditions
(temperature and pressure)• Safety considerations• Greater inhomogeniety• Wide range of material properties• Cost
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Industrial Tomography Slide 5
IntroductionExamples of industrial techniques 1
Microstructural characterisation Magnetic resonance imaging (MRI).of components, particles, pastes, Neutron tomography.foams, filters X-ray micro-tomography.(1—10000 μm) Optical tomography
Liquid mixing and Optical tomography. multi-phase flow Electrical resistive tomography. (0.01—0.5 m) Electrical capacitance tomography.
Ultra-sonic / acoustic tomography
Powder mixing, transport and Positron-emission tomography (PET).conveying Electrical capacitance tomography. (0.01—0.5 m) γ-ray transmission.
Electro-dynamic tomography.
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Industrial Tomography Slide 6
Fluidisation and trickle bed γ-Tomography.reactor studies X-ray tomography.(0.01—3.0 m) Positron-emission tomography (PET).
Electrical capacitance tomography.
IntroductionExamples of industrial techniques 2
Thermal mapping of reactors, Infra-red emission imaging.objects and atmospheres Electrical resistance tomography.(0.01 m to 5 km) Microwave tomography.
Groundwater monitoring Electrical impedance tomographyand soil remediation(1 m to 2 km)
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Industrial Tomography Slide 7
IntroductionExamples of industrial techniques 3
Atmospheric pollution Laser absorption imagingmonitoring(50 m to 10 km)
Ore deposit and oilfield Acoustic velocity imaging.reservoir exploration Acoustic diffraction tomography(50 m to 50 km)
Air traffic control RADAR(100 m to 50+ km)
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Industrial Tomography Slide 8
Sensing Techniques
• Electromagnetic (hard field)• Electromagnetic (soft field)• Particle• Other
• Hybrids• Multi-modal systems
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Industrial Tomography Slide 9
Sensing techniques:Basic principles
Excitation (array) Detection arrayProcess
Images are formed by projections:
Rotate
• Mechanical rotation• Excitation and detection array
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Industrial Tomography Slide 10
Sensing techniques:EM (hard field)
Type Comments
γ-ray - Radio-active sources. - Mechanically scanned or fixed
arrays. - Potentially fast.
X-ray - High resolution. - Mechanically scanned. - Radiation confinement.
UV Optical Infra red
- Fast. - Optical access. - Use spectrometry to give
component specificity. Millimeter wave
- System components emerging.
f
1010
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Industrial Tomography Slide 11
Sensing techniques:EM (soft field)
Type Comments Micro-wave - Hard or soft.
- Fast. - Moderate resolution
(wavelength dependant) - Attentuation, reflection,
defraction Electrical
- Capacitance (ECT)- Resistance (ERT) - Inductance (EMT)
- Low resolution - Fast - Low cost - Robust
f
1010
0
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Industrial Tomography Slide 12
Sensing techniques:Nuclear particle
Type Comments Positron emission (PET)
- Uses labelled particles. - Not on-line.
Neutron - High resolution. - Spectrometry (TOF) for
element specificity. - Pulse or radioactive
sources. - Radiation confinement.
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Industrial Tomography Slide 13
Sensing techniques:Others
Type Comments Nuclear magnetic resonance
- Fast - High performance. - Large high stabiltiy
magnet. - “Gold” standard
Ultra-sound (sonic) - High resolution. - Frame rate determined by
speed of sound. - Phased arrays for beam
steering. Thermal conduction (heat flux)
- Slow - Soft field
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Industrial Tomography Slide 14
Sensing techniques: PETPositron emission tomography (PET, positron emission computed tomography, PECT) a technique in nuclear medicine for cross-sectional imaging that enables a non-invasive assessment and localization of metabolic activity to be made. Emission of a positron by a radioisotope results in annihilation of the positron on collision with an electron, and the creation of two gamma rays of known energy travelling in exactly opposite directions. The PET scanner has detectors on each side of the patient to detect the simultaneous arrival of the gamma rays. Images are created using reconstruction algorithms similar to CT scanning. Fluorodeoxyglucose (FDG), using fluorine-18, is used to examine glucose metabolism, and ammonia, using nitrogen-13, gives information on perfusion. Carbon-11 and oxygen-15 can also be used as radioisotopes for PET scanning. Some diseases result in decreased uptake of the radio-labelled material due to decreased function; others, including many tumours, show increased glucose metabolism and concentrate the isotope avidly. In this way functional activity of the tissues can be compared with anatomical images obtained by CT or MRI scanning. Originally used to study activity in the brain, PET is now also used for investigating the chest and abdomen. See also tomography. Compare computerized tomography."positron emission tomography" Concise Medical Dictionary. Oxford University Press, 2002. Oxford Reference Online. Oxford University Press.
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Industrial Tomography Slide 15
Sensing techniques: MRIThis diagnostic imaging technique is based on nuclear magnetic resonance (NMR), in which protons interact with a strong magnetic field and with radio waves to generate electrical pulses that can be processed in a similar way to computerized tomography. Images produced by MRI are similar to those produced by computerized tomography using X-rays, but without the radiation hazard.
A major factor in the high costs of MRI is the need for a superconducting magnet to produce the very strong magnetic fields (0.1–2 tesla). Superimposed on this large magnetic field are smaller fields, with known gradients in two directions. These gradient fields produce a unique value of the magnetic field strength at each point within the instrument (see illustration).Some nuclei in the atoms of a patient's tissues have a spin, which makes them behave as tiny nuclear magnets.
The purpose of the large magnetic field is to align these nuclear magnets. Having achieved this alignment, the area under examination is subjected to pulses of radio-frequency (RF) radiation. At a resonant frequency of theRF pulses the nuclei under examination undergo Larmor precession. This phenomenon may be thought of as a ‘tipping’ of the nuclear magnets away from the strong field alignment. The nuclear magnets then precess, or ‘wobble’, about the axis of the main field as the nuclei regain their alignment with that field.
The speed at which the nuclei return to the steady state gives rise to two parameters, known as relaxation times. Because these relaxation times for nuclei depend on their atomic environment, they may be used to identify nuclei. Small changes in the magnetic field produced as the nuclei precess induce currents in a receiving coil. These signals are digitized before being stored in a computer.
MRI has produced spectacular results in studies of the brain and central nervous system, providing excellent images of delicate structures without the risk of the damage associated with ionizing radiation. Systems using very strong fields, in the region of 2 tesla or above, produce images of extremely high quality.
MRI: the way unique field strengths are produced at differentpoints in a specimen.
"nuclear magnetic resonance" A Dictionary of Physics.Ed. Alan Isaacs. Oxford University Press, 2000.
Oxford Reference Online. Oxford University Press.
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Industrial Tomography Slide 16
Transducer
Incident Wave
Reflected Wave
Time delay proportional to distance between
source and reflector
Sonics Principles – Active Sonar(Courtesy J&S Marine Ltd)
Dense reflecting object
Mismatch in acoustic impedance
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Industrial Tomography Slide 17
Object
Transmitter
Receiver
Sonics Principles – Types of Scan
A – Scan:
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Industrial Tomography Slide 18
Object
Z modulation
Time baseX Axis
Scan mechanism
Y Axis
Receiver
Transmitter
Sonics Principles – Types of Scan
B – Scan:
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Industrial Tomography Slide 19
Object
Z modulation
TimebaseX Axis
Y Axis
Beam scanned over object
Scan control
Sonics Principles – Types of Scan
Medicalscan:
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Industrial Tomography Slide 20
Ө
Sonics Principles – Types of Scan
Phasedarray:
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Industrial Tomography Slide 21
Two mechanisms result in need for Time Varied Gain (TVG)
1) Spherical spreading 2) AbsorptionLoss (dB) = 40 log (R) Loss (dB) = 2 α R
R = distance from transducer to reflector
Sonics Principles – Time Varied Gain (TVG)
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Industrial Tomography Slide 22
Fixed gain
preamps
TVG amplifiers
Quadraturedetectors
Analog to Digital
Converters
FIFO memory
DSP
System Timing
generator
TVG generattion
DDS signal generators
Fixed transmit power
amplifier
Steered transmit power
amplifiers
Phased array signal
generators
RS485 serial link
To AUV control PC
Sonar System Overview(Courtesy J&S Marine Ltd)
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Industrial Tomography Slide 23
Method Sensor elements
Typical arrangement
Measure values
Typical material properties ??
Typical material
ECT
Capacitive plates
CapacitanceC
εr 1 – 100 σ < 10-1 S/m (low)
Oil, water, non-metallic
powders, polymers, burning gasses
ERT (EIT)
Electrodes
Resistance (Impedance)
R / Z
σ 10-1 - 107 S/m (wide) εr 100 - 102
Water / saline, biological tissue, rock /geological
materials, semi-conductors e.g. silicon
EMT (MIT)
Coils
Self/ mutualInductance
L / M
σ 102 - 107 S/m (high) μr 1 to 10,000
Metals, some minerals, magnetic materials and
ionised water ?
Sensing techniques:Electrical techniques
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Industrial Tomography Slide 24
1
2
34 5
6
7
8
91011
12
Measure:
C1-2C1-3etc...C1-12
C2-3C2-4etc...C2-12
then
( )2
1−nn. independent measurements
Sensing techniquesOperation of ECT
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Industrial Tomography Slide 25
Excitation coils
Detection coils
Sensor array Conditioning electronics
Host computer
D1D2D3....DM
I1I2I3...IN
CM1CM2CM3...CMN
C21C22C23...C2N
C11C12C13...C1N
.
.
.
.
.
.
.
.
.
.
.
.
.
.
=
Reconstruction algorithm
AC magneticfield
Field control&
Measured signals
Data&
Control
Sensing techniquesHardware
“Typical” electrical tomography system:
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Industrial Tomography Slide 26
Image of 3 copper bars.(15 mm dia, 10% of object space)
Image of 2 copper bars & 1 ferrite.(15 mm dia, 10% of object space)
Sample images(SIRT & ART)
Designed and built experimental systems
Sensing techniquesExample of an EMT system
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Industrial Tomography Slide 27
Example of Hardware Design: ECT
Typical ECT sensor
• 11 times excitation• 66 measurements• Circular or square Measurement
electrode
Insulating pipe
1
234
5
6
7
8
Earthed screen
Radial screen
9 1011
12
Imaging area
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Industrial Tomography Slide 28
Capacitance Values
Requires:
• Highly sensitive circuit • Large measurement range (>100 times)• Stray-immune (150 pF stray C)
0
0.02
0.04
0.06
0.08
0.1
2 3 4 5 6 7 8 9 10 11 12
Detection electrode number
Cap
acita
nce
(pF)
Change in C (<0.1 pF)
0
0.2
0.4
0.6
2 3 4 5 6 7 8 9 10 11 12
Detection electrode number
Cap
acita
nce
(pF)
Standing C (<0.5 pF)
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Industrial Tomography Slide 29
Switched Capacitor Input Circuit
V1 V2
SW, frequency fSW
C
ICharge transferred each cycle ΔQ = C.(V1 – V2)
Current, I = ΔQ.fSW = C. fSW.(V1 – V2)
Equivalent resistance,SW
EQ fCR
.1
=
Simple schematic of a switched capacitor C to V converter:
VREFVOUT
fSW
C
-
+
RF
REFSWOUT VfCV ..−=
A major practical difficulty is the effects of charge injection
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Industrial Tomography Slide 30
C
R
V
C C
VCx
f
foi
s1 s2
1C Rf f
ω
VV
o
i
-90
-45
0
0.01 0.1 1 10 100
Vj C R
j C RVo
x f
f fi= −
+ω
ω 1
Charge amplifier:
AC-based Input Circuit
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Industrial Tomography Slide 31
221)(
ωω++
−=sRsC
RsCsV
ff
fxo
Output Laplace transform with a sine wave input, frequency, ω
( )ωα ff RC1cot−=( )
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡+⎟
⎟⎠
⎞⎜⎜⎝
⎛−−
+= αω
ω
ωt
RCt
RC
RCtV
ffff
fxo sinexp
1)(
2
Time domain response:
ff RC=τTime constant
Transient Analysis
fRC<<
ω1 V
C
CVo
x
fi= −
Capacitive feedback,
• Independent on frequency, good for spectroscopy
• Large τ = RfCf >> 1/ω→ long transient process
ff C
Rω
1<< ifxo VRCjV ω−=
• Stable frequency required• Small τ = RfCf << 1/ω→ short transient process
Resistive feedback,
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Industrial Tomography Slide 32
• DDS signal generators (AD7008) – A, f, φ programmable• AC-PGA necessary (SNR of multiplier)• Multiplier-based demodulator – no odd harmonics• 4th order Butterworth low-pass filter -- 80 dB/decade• C+R• Spectroscopy
Block Diagram of one Channel
ACPGA
Analoguemultiplier
Low-passfilter
DDS signalgenerator
DDS signalgeneratorClock
digital controlsignal
capacitancemeasurement
CxCf
Cs1 Cs2
Rf
latch
latch
Vi
VoVd
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Industrial Tomography Slide 33
System Block Diagram
• Standing capacitance compensation• DC PGA for large measurement range• PCI data acquisition card
M
Electrode 1
Capacitancetransducer
Capacitancetransducer
DDS signalgenerators
Differentialamplifier PGA ADC
DAC PC
Digitaloutputport
Data acquisitioncard
DC
offset voltage
digital control signals
Electrode NUX
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Industrial Tomography Slide 34
Circuit Details - Demodulation
What is the output for a purely resistive object?How would you measure R?
( ) ( )[ ]αωααωω +−=+= tS
ABttBAS
Vd 2coscos2
sinsin1
( )[ ]tS
ABVd ω2cos12
−=
SABVd 2
=
In phase component,
After low-pass
Multiplier-based demodulator
Electrical Tomography:How could you modify the system for ERT or EMT?
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Industrial Tomography Slide 35
C
VoVi
RR
2C
⎟⎟⎠
⎞⎜⎜⎝
⎛+
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−
=++
=
nn
nn
n
i
o
jjjjV
jV
ωωξ
ωωωωξωω
ωωω
21
12)()(
)(222
2
RCfo 22
1π
=ξ = 0.707 Derive f0 →
Feature → maximally flat in the pass band
Circuit Details -Butterworth low-pass filter
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Industrial Tomography Slide 36
Circuit Details –Excite / Detect Switching
DDS signalgenerator
1
2Electrode
Switch couplingcapacitance
Principle
Minimises problems due to parasitic “off” capacitance
T-configuration switch:
DDS signalgenerator
1 2
3
4Electrode Switch coupling
capacitances
Practical
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Industrial Tomography Slide 37
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
++
=+4096
5.040961 DV
KK
FFE
KCC ref
dgcpx
System Model
K K+ + +-
AC-basedcircuit with PGA amp
4096
FE
ADC
Offset signal
Diff.amp
DC
c d
D
4096
Reference voltage, Vref
Cx
Cp
AC PGA
K
DC
generator
g+
0.5F
+
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Industrial Tomography Slide 38
Calibration:
> 0max.↨ (0-4096)0Parasitic< 4095↨ (1-16) keep samemax.Full pipe> 0max.↨ (0-4096)max.Empty pipe
ADC, D3PGA, D2Offset, D1Excitation
DAC
PGA ADC
D1
D2
D3Vx
Simplified System Model
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Industrial Tomography Slide 39
Image reconstruction techniques
• Basic concepts of image reconstruction• Difficulties• Sensitivity maps• reconstruction algorithms• Sample images
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Industrial Tomography Slide 40
Image reconstructionSome basic conceptsBasic concepts
Permittivity
distribution
Capacitance
measurements
Conductivity
distribution
Resistance
measurements
Permeability
distribution
Inductance
measurements
Forward problemInverse problem
)),(( yxfC ε= )),(( yxfR σ= )),(( yxfI μ=
)(),( 1 Cfyx −=ε )(),( 1 Rfyx −=σ )(),( 1 Ifyx −=μ
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Industrial Tomography Slide 41
Image reconstructionSome basic concepts
Projections
X
Y
Objectdistributions
Reconstructedimages
X
Y
ProjectionPoint
distribution
Radial projections Point spread function
X
Y
Clearly we need sufficient projections to obtain a unique solution:
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Industrial Tomography Slide 42
Image reconstructionDifficulties
Several difficulties associated:
“Soft field” effect
Ill-condition of the problem (ill posed)
Limited number of independent measurement
Non-linearity
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Industrial Tomography Slide 43
Image reconstruction:The “soft field” effect
Aluminium targetThe magnetic field cannot penetrate the target due to eddy current effects
(d)
Ferrite targetFlux lines drawn into the target
(c)(b)
Air targetObject does not affect the lines of magnetic flux
aluminium
Model
Ferrite μR = 1000
insulator
target
(a)
Object space diameter, 150 mm excitation frequency 100kHz
The distribution of the excitation field lines is determined by the object material
Simple model of an EMT sensor:
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Industrial Tomography Slide 44
Image reconstruction:The “soft field” effect – ERT example
2
1
2
1
)tan()tan(
σσ
αα
=2
1
2
1
)tan()tan(
εε
αα
=2
1
2
1
)tan()tan(
μμ
αα
=
α 1 and α 2 denotes the angles between field lines and the direction normal to interface
In electromagnetic theory, at the interface between two media
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Industrial Tomography Slide 45
• There are only a limited number of independent measurements per frame, i.e. ( )
21nn +
• Cannot expect high resolution images,No. of independent ~ No. of independent
pixels measurements
Image reconstructionLimited independent measurements
• Smoothing used to improve appearance of the image.
( )2
1nn − ( )2
3−nn
EMT ECT ERT
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Industrial Tomography Slide 46
( )2
1nn +
Image reconstructionnumber of independent measurements
( )2
1nn − ( )2
3−nnEMT ECT ERT
Detector channel
Exci
tatio
n so
urceE1
E2
E3
E4
E5
E6
E7
E8
D1 D2 D3 D4 D5 D6 D7D8
( ) 362
88=
+1Ex
cita
tion
sour
ceE1
E2
E3
E4
E5
E6
E7
D2 D3 D4 D5 D6 D7D8
Detector channel
( ) 282
88=
−1 ( ) 202
388=
−
Exci
tatio
n so
urce
E1-2
E2-3
E3-4
E4-5
E5-6
E6-7
D34 D45 D56 D67 D78 D81
8-coils example 8-electrodes example 8-electrodes example
Detector channel
Increase the number of independent measurements?
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Industrial Tomography Slide 47
Image reconstructionIll-conditioning (ill-posed)
0100000200000300000400000500000600000
1 2 3 4 5 6 7 8 9 10 11 12 13 14
Elements Across Diameter
Sens
itivi
ty
• Expect blurring images near the center
The spatial sensitivity distribution is highly non-uniform, i.e. the sensitivity near the wall is very high and the sensitivity near the centre is very low, which is linked with an ill-conditioned sensitivity matrix. The very large condition number of the sensitivity matrix can result in the magnification of both measurement error and numerical error in the reconstructed image.
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Industrial Tomography Slide 48
Image reconstructionNon-linearity
≠
[ ]TyxfC ..............)),(( 11 == ε
[ ]TyxfC ..............)),(( 22 == ε
[ ]TyxfC ..............)),(( 33 == ε
+
target
(a)
(b)
(c)
+
=
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Industrial Tomography Slide 49
Image reconstructionAlgorithms
Lowercomputationrequirements
Highercomputationrequirements
Rulebased
algorithms
Weightedback-projection
Sensitivitycoefficient
NOSER
ART SIRTNeuralnetworks
Parametricalgorithms
QuantitativeFE based
algorithms
Several approaches:
Non-iterative Iterative
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Industrial Tomography Slide 50
Image reconstructionBack-projection (along field lines)
One of the simplest methods involves projecting back along the field lines:
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Industrial Tomography Slide 51
The main approaches:
1. Measure them.- Tedious unless automated- Only useful for the simpler algorithms- Effectively calibrates offset and gain errors at the same time
2. Sweep a perturbation over the model.- Slow- Subject to FE quantisation error
3. Calculate from field values extracted from the model.• The sensitivity maps are strongly affected by boundaries.• So static sensitivity maps are very poor for looking inside
conductive objects.• Need to know where the “main” boundaries are and
dynamically update the maps.
Determining the sensitivity maps
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Industrial Tomography Slide 52
Sensitivity maps are commonly used.These quantify the response of a particular excite / detector pair to each pixel location
Image reconstructionExamples of sensitivity maps (ECT)
5
4
32
1
8
76
How many sensitivity maps for a ECT sensor with 8 electrodes?
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Industrial Tomography Slide 53
Sensitivity maps are commonly used.These quantify the response of a particular excite / detector pair to each pixel location
Image reconstructionExamples of sensitivity maps (EMT)
Coil 1
Coil 6
Coil 3 Coil
5
Coil 2
Coil 4
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Industrial Tomography Slide 54
Image reconstructionAlgebraic techniques
Algebraic techniques are widely used to in image reconstructionAs a first step both measurement and image values can be re-arranged into a vector format, i.e.
Measured data
Exci
tatio
n so
urce
Detector channel
= D
Shown with common excitation / detection elements, i.e. triangular array
. etc ..
Image(Pixel positions)
= P
. etc ..
M×1 N×1
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Industrial Tomography Slide 55
Image reconstructionBack-projection
P = AT.D
etc.
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
=
TM
T
T
a
a
a
A.
2
1
m
M
mm dP ⋅= ∑
=1α
A linear combination of sensitivity maps⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
=
Md
d
d
D.
2
1
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Industrial Tomography Slide 56
Image reconstructionLinear forward model
For small changes in the pixel values or for a first order approximation, we can make a linear approximation:
δD = A.δP
The matrix, A, is know as the Jacobian and represents a linear model of the system. It has M rows and N columns, where M is the number of measurements and N is the number of pixels.For this presentation, we will drop the δ, so
D = A.PThe values in the rectangular matrix A are obtained by re-organizing the M sensitivity matrices (maps) on a row by row basis. The values are re-arranged to be consistent with the organization of the vectors Dand P.
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Industrial Tomography Slide 57
For image reconstruction, we need to determine P from measured data D. Unfortunately, the matrix A cannot be directly inverted.A natural solution would be to choose the Moore-Penrose generalisedinverse, i.e.
A† = (AT.A)-1.AT
P = A†.D is the least squares solutions to D = A.P
Unfortunately the problem is extremely ill-posed and the calculation of (A.AT )-1 or (AT.A)-1 will be swamped by numerical error.
Image reconstructionFormulating an inverse solution
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Industrial Tomography Slide 58
Image reconstructionFormulating an inverse solution
1 2
1 3=
3
4
x
y
x=1
y=1
1 2
1 3=
3.3
4
x
y
x=1.9
y=0.7
0.01 2
0.01 3=
2.01
3.01
x
y
x=1
y=1
0.01 2
0.01 3=
2.211
3.01
x
y
x=61.3
y=0.799 Condition number =1300Condition number =1300
Condition number =14.9Condition number =14.9
Condition number is normally used to describe inevitable loss in solution of linear equations.
the largest singular value the smallest singular valueCondition number =
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Industrial Tomography Slide 59
Image reconstructionRegularising the inverse solution (Tikhonov)
The previous solution, i.e., (AT.A)-1.AT provide a solution of min || D - A.P ||2
This is irrespective of the magnitude of vector P. A better solution would be to seek the minimum of
|| D - A.P ||2 + α2 || P ||2
The coefficient α controls a compromise between fitting the data and controlling the size of the solution. Note, ∑=
kkxx 2
A better solution, called the Tikhonov regularized solution, is given by,P =(AT.A + α.I)-1.AT.D
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Industrial Tomography Slide 60
Image reconstructionRegularising the inverse solution(TSVD)
SVD – singular value decompositionA = U S VT
Where U is an M by M orthogonal matrix, V is am N by N orthogonal matrix and S is M by N matrix with all elements zero except diagonal components (δ1, δ2, .. δp).
P = V S-1 UT . D
A better solution, called the Truncated singular value decompositionP = V S-1 UT
T . D
δ1
δ2
δ3
δp
S =
δ1
δ2
δ3
δr
ST =
rp δδ
δδ 11 >
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Industrial Tomography Slide 61
Some of the most effective algorithms employ iterative schemes:
• Linear model• Finite element model (FEM)• Parametric model• Analytical (rare)
Measurements from the sensor array, D
Latest estimate of the image, P
Σ+
-
APPROXIMATE INVERSE SOLVER
FORWARD SOLUTION
Update / constrain / programme flow
λ
Iterative Image Reconstruction
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Industrial Tomography Slide 62
Linear model,i.e. D = A.P
Measurements from the sensor array, D
Latest estimate of the image, P
Σ+
-
APPROXIMATE INVERSE
SOLVER ≈ A-1
FORWARD SOLUTION
Update / constrain / programme flow
λ
Relaxation,often adaptive
Regularisedpseudo-inverse.
Some varietye.g. ART vs. SIRTAdaptable flow
Iterative Linear Schemes
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Industrial Tomography Slide 63
( ) TT s
ssgsgg k
kk
kkkkk ⋅
−−= −
−λ1
1ˆˆˆ
ART (Algebraic reconstruction technique)
Image is updated after each pixel calculation.Converges more quickly.But, more sensitive to noise
ART and SIRT
( )TT
SSSS
diagˆˆˆ 1
dppp kkk
−−=+ λ
A new image is computed before updating.A type of descent gradient method
SIRT (Simultaneous iterative reconstruction technique)
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Industrial Tomography Slide 64
Measurements from the sensor array, D
Latest estimate of the image, P
Σ+
-
APPROXIMATE INVERSE
SOLVER ≈ A-1
FORWARD SOLUTION
Update / constrain / programme flow
λ
Parameterised model.Pixels are a very basic form of parameterisation
Based on a priori knowledge
Prior knowledge can be used to
dictate the constraining or regularisation.
Iterative parametric algorithms
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Industrial Tomography Slide 65
Parametric algorithms - Examples
• Linear image reconstruction algorithm• Change threshold to match area
Implicit model
Explicit model
Determine θ, d Determine x, y, r
Requires prior knowledge and accurate forward model
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Industrial Tomography Slide 66
Measurements from the sensor array, D
Latest estimate of the image, P
Σ+
-
APPROXIMATE INVERSE
SOLVER ≈ A-1
FORWARD SOLUTION
Update / constrain / programme flow
λ
Full FE (or analytical) modelMesh adapted to pixel geometries
Regularisedpseudo-inverse.as earlier slides
Update the sensitivity maps on each iteration
Iterative FE based algorithms
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Industrial Tomography Slide 67
Image reconstructionFEM – 2D
Simple mesh used for previous examples
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Industrial Tomography Slide 68
Image reconstructionFEM – 3D
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Industrial Tomography Slide 69
Image reconstruction Comparison of algorithms (EMT)
Back-projectionNo Constraining
ART20 iterations
No Constraining
SIRT500 iterations
No Constraining
NOSERNo Constraining
Object space 150 mm diameter, 16 pole system (separate excite and detect coils) 100kHz.Target 15 mm copper tube, at radius 75 mm.
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Industrial Tomography Slide 70
Image reconstruction Illustration of spatial resolution (EMT)
20 mm diameter
ART, 10 it.s, λ = 0.9min. = -0.075max. = 0.3mean = -0.022
ART, 10 it.s, λ = 0.9min. = -0.16max. = 0.46mean = -0.038
25 mm diameter
ART, 10 it.s, λ = 0.9min. = -0.21max. = 0.67mean = -0.046
30 mm diameter
23d
2d
d
Object space 150 mm diameter, 16 pole system (separate excite and detect coils) 100kHz, Aluminium rods
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Industrial Tomography Slide 71
Image reconstruction Comparison of algorithms (ECT)
Back-projection TSVD Tikhonov Iterative
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Industrial Tomography Slide 72
Single object Stratified Annular Two objects
Simulated test object
LBP
SVD
Tikhonov
Iterative
Tikhonov
Projected
Landweber
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Industrial Tomography Slide 73
Two rods
SIRT
Tikhonovregularization
SVD
Single rod Three rods Four rods
EMT Images - Rods
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Industrial Tomography Slide 74
Square Rectangle Quarter cylinder U-shape
Coaxial tube and rod Tube alone Tube with rod Difference image
EMT Images – 8 coil array
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Industrial Tomography Slide 75
Example Applications
•Biomedical experiment •Body composition•Metal production processes•Hydraulic conveying•Hydraulic conveying •Flow monitoring•Bubble Column
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Industrial Tomography Slide 76
LadleLadle shroud
Tundish
Water Cooled Mould
Submerged Entry Nozzle (S.E.N)
Spray Banks
Rollers
Tundish
Submerged Entry Nozzle
EM Imaging of metal production processes
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Industrial Tomography Slide 77
Example of Predicted of Flow Regimes
Full Half-full Annular
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Industrial Tomography Slide 78
Pilot Plant Experiments
Photograph of a pilot plant experiment:
Transparent quartz tube:
Example of flow:
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Industrial Tomography Slide 79
Small bar(19 mm dia.)at the centreof the SEN
Medium bar(25 mm dia.)at the centreof the SEN
Large bar(38 mm dia.)at the centreof the SEN
Sample Images
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Industrial Tomography Slide 80
Tomographic Imaging of Hot Steel
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Industrial Tomography Slide 81
Images of molten steel flow profiles
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Industrial Tomography Slide 82
Opening a taphole: Closing:
Taphole Monitoring
Wear mechanisms:• Aggressive nature of the hot materials• Opening and closing methods• Thermal cycling
Maintenance:Outer Insert Change - furnace on line,
2-3 hr jobInner Insert Change - furnace shut down,
labour intensive(2 outer changes for every inner changed)
Risks• Unable to plug hole, leading to a run out• Structural integrity of tapping assembly
may be compromised• Contact between molten materials and
cooling water channels
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Industrial Tomography Slide 83
Screened cubical
Weight measurement
Electromagnetic array
Camera system
Embedded PC
Body Composition
Dave
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Industrial Tomography Slide 84
• Mixingliquid-liquid
gas-liquid
solid-liquid
gas-solid-liquid
• Separationhydrocyclone
filtration
• Transportationhydraulic
powder conveying
• On-line monitoringproduct consistency
diffusion in foodstuffs
• Material characterisationmicro-structure
Applications (ITS Ltd)Applications (ITS Ltd)
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Industrial Tomography Slide 85
Liquid mixing example
Outputs Sensor
10mm
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Industrial Tomography Slide 86
Liquid mixing example
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Industrial Tomography Slide 87
Visualization of swirling flow in a hydraulic conveyor
0
0.5
1
1.5
2
2.5
0 5 10 15 20 25
dis tance (L/D)
flow
vel
ocity
(m/s
)Hydraulic conveying example
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Industrial Tomography Slide 88
Hydraulic conveying: Tomographs and Photographs
Visualization of swirling flow in a hydraulic conveyor
Side view
Tomograms
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Industrial Tomography Slide 89
Air-water flows in a horizontal pipeline
Reconstructed 2D images in respect to typical air cavity formation in the flow loop
Photograph of a slug flow
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Industrial Tomography Slide 90
(from Korjenevsky's web site)
Circular MIT sensor
Image of brain
Experimental Biomedical System
Human head cross-section: one of the first in-vivoimages. Two bright spots in the central part may be identified as ventricles of the brain filled with CSF.
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Industrial Tomography Slide 91
Conclusions
• Overview of electrical tomography as applied to industrial applications.
• Summarised- Sensing modalities- Applications- Image reconstruction