Download CT Lecture 1 - Lorentz Center

Fan-beam and cone-beam tomography:
principles, artefacts, and examples from
the geosciences
Richard Ketcham
Jackson School of Geosciences
The University of Texas at Austin
Origin: 3 Cool CATs
Decca Studios
Jan 1, 1962
EMI 2001
Colour TV Camera
Phonograph (circa 1930)
Electric and Musical Industries Ltd.
June 6, 1962
H2S Radar (WW II)
Origin: 3 Cool CATs
Decca Studios
Jan 1, 1962
EMIDEC 1100
Logic unit
Godfrey Newbold
Hounsfield
$£
Electric and Musical Industries Ltd.
June 6, 1962
Abbey Road Studio
EMI Prototype
EMI-Scanner
Basic elements of CT
(3)
(2)
1) Make X-rays
2) X-rays hit sample,
some get stopped
3) Remaining X-rays are
detected
4) Cleverly collect data,
and a computer does
the rest…
(1)
(1)
Make X-rays I
• X-ray tube components
– Source of electrons: filament
– Large potential difference
through which electrons are
accelerated: cathode + anode
– Target
• Tungsten, copper,
molybdenum; high-Z
– Vacuum
– Window to let X-rays escape
• Beryllium, aluminum; low-Z
The basics: directional tube
+
-
Making X-rays II
• What you get
Low mA
High kV
High mA
Low kV
– X-ray tube settings
• filament in mA (milliamperes)
– How many electrons you get
• potential in kV (kilovolts)
– How energetic you make them (how fast they’re going)
– Input energy is in watts (W = mA x kV)
– X-ray production efficiency = 0.9-9 ZV
• example: efficiency with tungsten target and 100 kV potential is
0.6%; rest of energy dissipated as heat
Making X-rays III
• Let’s take a look at the photons
– Bremsstrahlung
• Electron passes through
Coulomb field of nucleus, slows
down, excess energy given off as
photons
• Maximum energy (keV) =
source voltage
Photons per keV/(mA s mm²) @ 750 mm
Pantak 420 kV spectrum
10000000
1000000
100000
10000
1000
0
25
50
– Characteristic X-rays
– Bremsstrahlung continuous,
characteristic discrete
– Most photons low-energy
• Mean energy usually less than half of
maximum
Photon Energy (keV)
Same spectrum, Non-log scale
Photons per keV/(mA s mm²) @ 750 mm
• Incoming electron knocks out
inner shell electron, outer-shell
electron cascades to replace it,
gives off energy equal to energy
loss of transition
75 100 125 150 175 200 225 250 275 300 325 350 375 400 425
2000000
1500000
1000000
500000
0
0
25
50
75 100 125 150 175 200 225 250 275 300 325 350 375 400 425
Photon Energy (keV)
Making X-rays IV
Pantak 420 kV spectrum
Filtering
– X-rays lose energy whenever
they pass through something
– Higher Z (atomic number), r
(density), thickness = more loss
– Some filtering inherent in tube
housing
•
Intentional filtering
– Objective: only use higherenergy X-rays
– Procedure: Send X-rays through
additional material (brass, Al)
before they hit sample
– Result: Lower-energy X-rays
are preferentially diminished
– Drawback: All energies
diminished somewhat
Original spectrum
Spectrum after filtering by 3 mm copper
1000000
100000
10000
1000
0
25
50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425
Photon Energy (keV)
Pantak 420 kV spectrum
2000000
Photons per keV/(mA s mm²) @
750 mm
•
Photons per keV/(mA s mm²) @
750 mm
10000000
Original spectrum
Spectrum after filtering by 3 mm copper
1500000
1000000
500000
0
0
25
50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425
Photon Energy (keV)
Making X-rays V
• “What’s good” -L. Reed
– High X-ray intensity
• Reduces image noise
• More photons = better counting statistics
– Small X-ray focal spot
• Improves image clarity, resolution
• Reduces number of paths through any one point in object being
scanned
– Appropriate energy (kV) for your object
• High kV = better penetration, less noise
• Low kV = better discrimination, more contrast
Making X-rays VI
Transmission source
Filament grid
Anode
Xylon.com
• Transmission vs. Directional Sources
– Transmission: X-rays emanate “through”
thin target
• Smaller focal spot (< 1 um)
• Lower intensity, energy
– Directional: X-rays from “thick” target
• Other sources
– “Liquid Metal” target
– Synchrotron, of course…
Magnetic
focusing
Target
X-rays
Directional source
X-rays hit sample I
• Some get through
– Basic equation: P = µ I
• P = rate of removal of photons (num/cm)
• µ = linear attenuation coefficient (cm-1)
– Characteristic of material: density, composition
– Also X-ray energy
• I = number of photons
• Some don’t
– Photoelectric absorption
• Photon hits inner-shell electron, causing it to eject
– All energy transferred to electron
• Attenuation varies with Z3 - very atom-dependent!
• Dominant at low E, high Z
– Compton scattering
• Photon hits one or more outer electrons, losing energy with each
collision
• Attenuation varies with electron density
– Depends more on overall density than atoms
– Less discrimination ability
• Dominant at higher energy (above 30-100 keV, depending on Z)
X-rays hit sample II
• Integrating the equation: I = I0 exp(-µ x)
–
–
–
–
x
I0
I0 = number of photons going in
µ = linear attenuation coefficient (1/cm)
x = distance (cm)
I = number of photons getting through
• Multiple materials: I = I0 S exp(-µ(xi) xi)
– This is what CT reconstruction algorithm
solves for
• Complication: I = SI0,jS exp(-µ(xi,Ej) xi)
– µ also a function of X-ray energy
• Low-energy X-rays easier to stop
• High-energy more likely to get through
I
µ
x1
x2
I
I0
µ1
µ2
x1
x2
µ1
µ2
– CT does not take this into account
• Leads to artifacts
X-rays hit sample III
Low-energy
High-energy
25 mm core of graphic granite
(quartz + orthoclase)
Remaining X-rays detected I
• General idea
– Detector array counts the number of
photons that get through sample along
certain ray paths
• Detection mechanism: scintillation
– Incoming photon causes flash of light;
flashes counted
• Efficiency
– Low-energy X-ray photons easier to
count
• Think about it: higher energy X-rays
might not be stopped by detector, either
– Different detectors can be optimal for
different energies
Remaining X-rays detected II
• What’s good
– High efficiency
– Uniform, consistent signal across all energies
– Fast
• Get reading quickly, clear for next one
– Small size
• Minimize number of paths through subject averaged together
• Only applies to surface area facing X-ray source
– Minimal crosstalk, shadowing
• Communication between adjacent detectors
• “Memory” of previous image
– High bit depth
• Typical range 12-16 bits; more provides better contrast
– Durability
• To stand up to years of radiation exposure…
Clever Data Collection
Volume CT
Various vendors
•
Phoenix/GE
X-Tek/Metris/Nikon
SkyScan/Bruker
Xradia/Zeiss
Advantage: Time
–
•
1024 or 2048 slices per rotation; each view can take a
long time
Disadvantage: Scattering
–
Scanning dense materials at high energies leads to
blurring, hard-to-correct artifacts.
How We Magnify
Detector
(10242, 20482; 100’s µm)
Detectors
(20482; 10’s-0.1’s µm)
Source (1’s-0.1’s µm)
Source (1’s µm)
Most lab scanners
Source-dominated
Xradia (similar to synchrotron)
Detector-dominated
Reconstruction:
Fourier slice theorem
Filtered back-projection
The idea: projections of the normalized intensity data…
𝐼(𝜃, 𝑡)
𝑃𝜃 𝑡 = 𝑓 𝑥, 𝑦 𝑑𝑠 = −ln
𝐼0 (𝑡)
ℒ
𝑡 = 𝑥 cos 𝜃 + 𝑦 sin(𝜃)
are filtered in the Fourier domain by a ramp filter…
∞
𝑆𝜃 𝑤 =
∞
𝑃𝜃 (𝑡)𝑒 −2𝜋𝑗𝑤𝑡 𝑑𝑤
𝑄𝜃 𝑡 =
−∞
𝑆𝜃 (𝑤) 𝑤 𝑒 2𝜋𝑗𝑤𝑡 𝑑𝑤
−∞
after which the function value is obtained by summing over all angles:
𝜋
𝑓 𝑥, 𝑦 =
0
𝑄𝜃 𝑡 𝑑𝜃
Cone-beam acquisition
Feldkamp algorithm
2𝜋
𝑓 𝑥, 𝑦, 𝑧 =
0
𝑅2
′ , 𝑟 ′ 𝑑𝜃
𝑄
𝑡
𝜃
𝑈2
R = source-sample distance
𝑈 = 𝑅 + 𝑥 cos 𝜃 + 𝑦 sin 𝜃
(i.e. source-point distance)
t′, r′ project detector to origin
𝑄𝜃
𝑆𝜃
𝑤, 𝑟 ′
𝑡′, 𝑟′
∞
=
−∞
∞
=
−∞
𝑆𝜃
(𝑤, 𝑟 ′ )
𝑅
𝑅2
+ 𝑡 ′2
+ 𝑟 ′2
𝑤
𝑃𝜃
′
2𝜋𝑗𝑤𝑡
𝑒
𝑑𝑤
′
′
′
−2𝜋𝑗𝑤𝑡
(𝑡 , 𝑟 )𝑒
𝑑𝑤
Newer trends in reconstruction
• Iterative (examples: SIRT, SART)
– Forward and back projection to converge on data
– Relies on good description of beam, attenuation
– Generally much more computationally expensive
• Ready for prime time yet?
Incomplete cone-beam data
Samples Radon space from conebeam acquisition
Defrise phantom
One antidote: spiral CT
• Rotate continuously, while
moving vertically
• Eliminates missing data
• Good scans for “tall”
objects
– But recon more demanding
– New artefacts?
UTCT instrumentation
• Sources
– 450 kV GE Titan; dual spots: 0.4 mm (700W), 1.0mm (1500W)
– 225 kV FeinFocus; 8W for <5 µm spot, usually run defocused
• Detectors
– 2048x2048 16” Perkin Elmer flat panel
– 3096-channel 24” LDA
UTCT instrumentation
• XRadia microCT
–
–
–
–
150 kV Hamamatsu closed source; spot size to 4-7 µm
2048x2048 CCD video camera
Cone beam data collection (up to 2048 slices)
Voxel size down to 0.2 µm, line pair detection to 1.5 µm
Artefacts
• What’s an “artefact?”
– An image feature or characteristic that reflects
(imperfections in) the scanning process rather
than the physical object.
– Errors due to:
• Geometric imprecision
• Detector (or source) behavior
• Math oversimplifies the physics
– The earlier they’re dealt with in the scanning
process, the better.
Beam hardening
• Typical manifestation: edges of image appear brighter than center
• Cause: X-ray spectrum “hardens” as it passes through object
• Effect:
– µ changes as f(position)
• As mean E rises, µ falls
– And as f(beam path)!
Photons per keV/(mA s mm²)
X-ray spectrum
Initial Initial
and Hardened
X-ray spectrum
10000000
10000000
1000000
1000000
100000
100000
Lower intensity, but
higher mean energy
10000
10000
1000
1000
00
50
50
100
100
150
150
200
200
250
250
300
300
Photon
Photon Energy
Energy (keV)
(keV)
350
350
400
400
Beam hardening
• Another way of thinking about it
– Preferential to total attenuation of low-energy x-rays
– Makes short x-ray paths equivalent to long ones
– Thus they appear denser, ergo brighter
High keV
Low keV
Longer path attenuates more
x
Both paths attenuate same
x
More attenuation per length = more dense?
No, but CT thinks so…
Recognizing beam hardening
1 cm
5 cm
Prosauropod
Rooneyia
It’s also in the air…
• Dark and light
streaks in air are
“reflections” of
beam hardening
• Some hide this
by setting air
gray value well
below zero, so
it’s all just black
– This loses
information!
Fixing beam hardening
• Solutions
–
–
–
–
Use higher-energy x-rays
Pre-harden the x-rays
Take wedge calibration through similar material
Software correction during reconstruction
5 cm
Saurosuchus
Beam Hardening: Software correction
•
Most CT systems allow a form of linearization
•
•
•
Attempts to transform polychromatic to monochromatic data
Recent work: Iterative algorithm finds a linearization function to minimize artifacts in marked
locations where artifact is dominant feature (Ketcham and Hanna, 2014)
Upcoming (any news at this workshop?)
•
•
Iterative forward/backward projection incorporating X-ray energy spectrum, material properties
Generally assume few, known materials; computationally expensive
Ring Artefacts
• Manifestation: rings in the images
• Correspond to non-ideal detector behavior
– One or more channels different from neighbors
• Caused by beam change in scanning conditions
(intensity, hardness) vs. calibration conditions
Archaeopteryx, the first known bird
5 mm
Signal in sinograms
Basalt (24 mm FOV)
Sinogram
Signal in sinograms
Basalt (24 mm FOV)
Sinogram
Signal in sinograms
Basalt (24 mm FOV)
Sinogram
One algorithm
• Assumption 1: error form 𝐷 𝑐, 𝜙 = 𝐹 𝑐, 𝜙 𝐸(𝑐)
– Could also be additive
• Assumption 2: F averaged over f is smooth
• So, define: 𝐺 𝑐 = 𝜙 𝐷(𝑐, 𝜙)
• To estimate error as: 𝐸 ≅ 𝐺/𝐺𝑆
– GS = Smooth(G); use moving window mean, median
• Then, recover F using: 𝐹 𝑐, 𝜙 = 𝐷(𝑐, 𝜙)/𝐸 (𝑐)
• Can also apply to images by converting to polar
coordinates (interpolated radial sampling)
Rings: Archaeopteryx
Uncorrected
Software corrections
OK
(image-based)
Other solutions
• Wedge calibration
• Dithering
• Move detector during
acquisition (SkyScan)
• Multi-energy calibration
• North-Star (only?)
Better
(sinogram-based)
Ring artefact resliced
Ring artefact: Confuciusornis
Artifacts
• Beam starvation
– Manifestation: linear streaks along long axis of objects
with high aspect ratio
• often combines with rings to make them more severe along
long axis
– Cause: very low signal-to-noise ratio along long paths
– Solutions
• Longer acquisition
• Smoothing sinograms in low-signal regions (“De-streak”)
Long-axis lines: Prosauropod 1
Long-axis lines: Prosauropod 2
Long-axis lines: Prosauropod 3
Lines are noise in one direction;
Salt-and-pepper is in all directions.
Artefacts
• Starbursts
– Result from large attenuation contrasts
combined with low-energy X-rays
– Sources:
• metal pins
• oxides/sulfides
Startbursts: Sipalocyon
X-ray reflection
in single crystals
Quartz
Scattering
Quartz with fluorite and gold
10 cm
Fan beam (collimated)
Cone beam
Barber-pole (or candy cane)
Outcome of spiral acquisition; cause not yet ascertained
Artefact correction:
final, cautionary notes
• Be wary – inappropriate processing can be the
most nefarious artefact of all!
– Reason: An experienced analyst can recognize a CT
artifact, but not necessarily a (botched) correction.
– “First, do no harm”
• Avoid saturating your images
– i.e. have no voxels at minimum or maximum gray
value
– Reason A: air contains evidence of artefacts
– Reason B: it throws away information