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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