2455-5 Joint ICTP-TWAS Workshop on Portable X-ray Analytical Instruments for Cultural Heritage 29 April - 3 May, 2013 Lecture NoteBasic principles of X-ray Computed Tomography Diego Dreossi Elettra, Trieste Italy
Workshop on Portable X-ray Analytical Instruments for Cultural Heritage ICTP, Trieste, 29 april - 3 may, 2013 Basic principles of X-ray Computed Tomography Diego Dreossi Sincrotrone Trieste, S.S. 14 km 163.5 AREA Science Park 34012 Basovizza (TS), Italy
Acknowledgements Thanks to the big CT community for all the information and material available on the web!
Tomography The word tomography is derived from the Greek ("part" or "section") and ("to write") Object Projected image Sections
4 7 11 11 11 5 3 8 8 8 4 7 5 3 7 5 7 11+7 18 8+5 13 11+7 18 8+7 15
3 5 7 4 3 5 7 4 9 10 9 10 15+10 25 13+9 22 18+10 28 18+9 27 15+10 25 13+9 22 18+10 28 18+9 27 4 7 5 3 3 12 4 25+3 28 22+12 34 28+12 40 27+4 31 25+3 28 22+12 34 28+12 40 27+4 31
31 40 34 28 12 21-19 15 9 4 7 /3 5 3
Basic interactions
Ni No N o N i e x x Ni No N o N i e ( 3 ) x 1 2 x Attenuation
Ni No N o x N e k i x k x k k ln N N i o ( x) dx ln N N i o Line integral
p (t) t A y ds t 1 B x The projection function is also determined by the angle of view.
y A t 1 y 1 x 1 x B The equation of line AB is x cosy sin t
The projection function can be written as p ( t) (, t) line ( x, y) ds p ( t) ( x, y) ( x cos y sin t) dxdy p (t) is known as the Radon Transform of the function x,y). A projection is formed by combining a set of line integrals. The simplest projection is a collection of parallel ray integrals as given by p (t) for a constant. This is known as parallel projection. Laminar beam object detector
Image reconstruction algorithms are derived to construct x,y) from p (t). Classification of Algorithms Backprojection Fourier Domain Approach Filtered Backprojection Iterative Methods Algebraic Reconstruction Technique (ART) Iterative Least Squares Simultaneous Iterative Reconstruction Technique (SIRT)
The above given process can be expressed mathematically. The reconstructed back-projection image b (x,y) at a particular view is : b ( x, y) p ( t) ( x cos y sin t) dt Adding up the images at all angles (0-) f b ( x, y) 0 b ( x, y) d 0 p ( t) ( x cos y sin t) dtd
dudv e v u F v u F F y x f dxdy e y x f y x f F v u F vy ux j vy ux j ) ( 2 1 ) ( 2 ), ( ), ( ), ( ), ( ), ( ), ( f(x,y) F(u,v) Image Space Fourier Space
Thus the Fourier Transform of a projection at angle forms a line in the 2-D Fourier plane at this angle. After filling the entire F(,) plane with the transforms of the projections at all angles, the object can be reconstructed using 2-D Inverse Fourier Transform
Convolution Back-projection Algorithm The back-projected function can rewritten in space domain as follows: ) ( ) ( ) ( 1 1 1 1 1 t c F t p t p F F Thus, instead of filtering in the frequency domain, p (t) can be convolved with a function c (t) and then back-projected.
0.5 0.4 rampa Shepp-Logan gen-hamming 0.3 0.2 RamLak Shepp-Logan gen-hamming ampiezza (u.a.) 0.3 0.2 ampiezza (u.a.) 0.1 0 0.1-0.1 0 0 0.1 0.2 0.3 0.4 0.5 frequenza (1/step) -0.2-10 -5 0 5 10 pixel Ramp filter (RamLak): enhancement of high frequencies noise Gen-Hamming, Shepp-Logan: enhancement of intermediate frequencies Convolution theorem convolution in the direct space as an alternative to multiplication in the Fourier space
Projection Projection t Image t Sinogram
Back Projection Sinogram Filtration Image
Reconstruction in frequency domain
ART x y 12 z w 8 13 11 9 7
First generation EMI Mark I (Hounsfield), pencil beam or parallel-beam scanner 180-240 rotation angle, angular step ~1 Scan time 5 min, reconstruction time 20 min Resolution: 80 x 80 pixels (ea. 3 x 3 mm 2 ), Slice thickness 13 mm
Second generation Hybrid system: Fan beam + linear array (~30 elements) traslation and rotation total scan time ~30 s more complex agorithms ( fan geometry)
Third generation the fan beam is covering all the sample 500-700 elements (ionizating chambers or scintillators) No traslations total scan time ~ seconds reconstruction time ~ seconds
Fourth generation Stationary ring of detectors (600 4800 scintillators) Rotating X-ray source total scan time ~ seconds reconstruction time ~ seconds Slice thickness 1mm 3D volumes constructed as a series of 2D slices Fifth: electron beam scanner
Helical (sixth generation)
Conventional CT Helical CT scanning N 360 scans at positions z 1 to z n One scan of n-360 from positions z 1 to z n Pre-processing corrections corrections intermediate Z-interpolation reconstruction Convolution and backprojection Convolution and backprojection result N images at positions z 1 to z n Images at arbitrary positions from z 1 to z n
Acq. time Data 360 Data helical Matrix Power Slice thick. Spatial res. Contrast res
Status 3D medical CT: Helical trajectory Similar to 3 rd generation CT but with multiple rows of detectors (4, 8, 16, 32, 64, now even up to 640 rows) FDK-like approximate reconstruction 3D lab-based CT: 3D cone-beam micro-ct using circular trajectory 512 2, 1024 2, 2048 2 (flat-panel CCD II detectors- ) FDK approximate reconstruction
Design concepts Source Energy Current Focal spot size Focal spot stability Detector Screen: efficiency, spatial resolution Pixel size: spatial resolution,signal, Fov Dynamic range, daq time Magnification x-ray flux Penumbra field of view
Source Focal spot size Emission angle Material Technology 5-20
Source unsharpness Heat dissipation Stability Phase effects M 1 U FS M FS
Source 100 kvp, = 15, 0.2 mm Be window 500 mm air Internal filtering Characteristic Radiation lines 1.50 mm Al Mean energy 46.2 kev 1 st HVL 2.38 mm Al, 2 nd HVL 4.20 mm Al, HVL1/HVL2 = 0.567 0.25 mm Cu Mean energy 57.3 kev 1 st HVL 6.46 mm Al, 2 nd HVL 8.00 mm Al, HVL1/HVL2 = 0.807
Source
1.E+05 1.E+04 1.E+03 Bone Calcium Carbonate paraffin wax cm 1.E+02 1.E+01 1.E+00 1.E-01 0 50 100 E (kev) 1.0 N o k N e x Lower energy i k Rel. intensity 1 2 Higher energy x 0.2 0.1 0.2 transmission > 5 n
Detector Xray screen Direct (ase, Si) Powders Structured light guide Detector Fov High energy
Detector CCD Area, scmos Flat Panel (large area CMOS) GOS (+FOP) CsI (+FOP) Flat Panel (asi/ase) Photodiode array Single photon counting Direct conversion CsI Ceramic MediPix, Pixirad,.
Detector Dynamic range 72 db ~ 4000:1 -> 12 bit
Detector LAG: not suitable for CT!
Reconstruction Volume 150 x 150 x 150 mm 3 Isotropic voxel size 50 m dataset = 3000 3 x 4 = 108 GB Isotropic voxel size 60 m dataset = 2500 3 x 4 = 62.5 GB DATA I/O / PROCESSING / STORAGE
Reconstruction Pre-processing Cone beam Fan Helical Helical+cone pct HBCT Local Area CT Post-processing Artifacts reduction
Geometry Intensity~1/d 2 FDK filtered back-projection is reasonable up to ~10 With microfocus sources phase contrast imaging possible
Unsharpness Pixel Size 2 U PS M PS Focal spot M 1 U FS M FS ASTM E 2698 Examples U TOT U TOT 1 M U 2 FS ( M U 1) 2 PS FS (1.6PS) 3 3 3 3 FS = 8 m, PS = 25 m, M =5 U TOT = 9.2 m PS eq = 5 m FS = 16 m, PS = 50 m, M =1.25 U TOT = 64 m PS eq = 40 m
Number of projections Spatial and angular sampling on Fourier space K Projections 64 128 256 512 N Samples 512 256 128 64 K /2 N Reconstruction of an ellipse
pn pixel noise (standard deviation evaluated on 2D slices) pn PS Np I 1 t Na k = constant dependend on rec algorithm (smooth kernel reduces noise, but also spatial resolution) PS = pixel size (mm 2 ) Np = number of projections I = source current (ma) t = integration time of the detector (s) Na = image averaging number
Artifacts Sample 1- motion 2- metallic materials 3- dimension > scan field Scanner 1- rings 2- geometry 3- stability Physics 1- Beam Hardening Cupping & Streaks 2- Undersampling 3- Photon starvation 4- Partial Volume Technique 1- helical in transverse plane 2- helical in multislice 3- multiplanar and 3D reconstrucion 4- cone beam effect
Artifacts simulation
Artifacts simulation
Artifacts simulation single pixel traversed by individual rays: I PIXEL will underestimate the attenuation Motion artifacts rod moved during acquisition
More realistic cases Beam hardening: cupping Beam hardening: streaks Undersampling Photon starvation
More realistic cases Metal artifacts Rings
More realistic cases Dimensions > FOV
Imaging Requirements Something to know: 1. The size of the object you want to scan 2. The material the object is made of 3. The level of fine detail or density differences you want to see 4. How fast you want to see the results 5. Where you want to place the instrument 6. Who will use it
New high-speed Computed Tomography system for 3D mass production process control