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Computed Tomography Principles
Ge Wang, Ph.D.Department of Radiology
University of IowaIowa City, Iowa 52242, USA
Learning Objectives
• CT terms• Data acquisition• Basic elements of CT scanner• Scanning modes• Image reconstruction
• Spiral/helical CT• Image resolution and artifacts• Interaction among imaging parameters• Quality assurance• Radiation exposure
A Little Bit History
Nobel prizesRoentgen (1901): Discovery of X-rays Hounsfield & Cormack (1979): Computed tomography
Computed Tomography Principles
1. Projection measurement
2. Scanning modes
3. Scanner systems
4. Image reconstruction
X-ray Interactions - Photoelectric Effect
Photoelectric effect results in total absorption ofthe X-ray photon and the emission of a bound electron
(From Aracor)
X-ray Interactions - Compton Scatter
Compton Scatter results in a free electron &a scattered (less energetic) photon
(From Aracor)
Source- Rotating anode disk- Small focal spot
down to 0.6 mm- Polychromatic beam
Detectors- Xenon (50-60%)- Scintillation (>90%)
Source and Detectors
(From Siemens)
Exponential Attenuation of X-ray
x
No
xio eNN
xio eNN )( 321 Ni
x
X-rays
Attenuatedmore
NoNi
Ni: input intensity of X-rayNo: output intensity of X-ray: linear X-ray attenuation
Ray-Sum of X-ray Attenuation
o
i
kk N
Nx ln
o
i
N
Ndxx ln)(
x
iok
k
eNN
x
NoNi
Ray-sum Line integral
Projection & Sinogram
Sinogramt
Sinogram:All projections
P(t)
f(x,y)
t
y
x
X-rays
Projection:All ray-sums in a direction
Completeness Condition
There exists at least a source on any lineintersecting a cross-section
Computed Tomography Principles
1. Projection measurement
2. Scanning modes
3. Scanner systems
4. Image reconstruction
First Generation
One detectorTranslation-rotationParallel-beam
Second Generation
Multiple detectorsTranslation-rotationSmall fan-beam
Third Generation
Multiple detectorsTranslation-rotationLarge fan-beam
Fourth Generation
Detector ringSource-rotationLarge fan-beam
Third & Fourth Generations
(From Picker)
(From Siemens)
Simultaneous•Source rotation•Table translation•Data acquisition
Spiral/Helical Scanning
Cone-Beam Geometry
X
Z
Y
Scanning modes
• First generationOne detector, translation-rotationParallel-beam
• Second generationMultiple detectors, translation-rotationSmall fan-beam
• Third generationMultiple detectors, rotation-rotationLarge fan-beam
Scanning modes
• Fourth generationDetector ring, source-rotationLarge fan-beam
• Spiral/Helical scanning, cone-beam geometry
Computed Tomography Principles
1. Projection measurement
2. Scanning modes
3. Scanner systems
4. Image reconstruction
Spiral CT Scanner
Gantry
Data acquisitionsystem
Detectors
Storage units:Tapes, disks
Display
Controlconsole
Computer
Parallelprocessor
Table Recording
Source
Network
Filter
Data Acquisition System (DAS)
Source Detector
Pre-Collimator Post-Collimator
Patient
Scattering
Data Acquisition System (DAS)
X-ray Tube
Detectors
CT Gantry(From Siemens)
Filter
Source
Detector
Spiral CT Scanner
• GantryData acquisition system
• Table• Computer
Parallel processors• Control console• Storage units
Tapes, disks• Recording device• Network interfaceX-ray generatorHeat exchanger
(From Elscint)
E-Beam CT Scanner
• Speed: 50, 100 ms• Thickness: 1.5, 3, 6, 10 mm• ECG trigger cardiac images
(From Imatron)
Computed Tomography Principles
1. Projection measurement
2. Scanning modes
3. Scanner systems
4. Image reconstruction
Computed Tomography
P(t) f(x,y)P(t)
f(x,y)
t
y
x
X-rays
Computed tomography (CT):Image reconstruction fromprojections
Reconstruction Idea
4
6
3
7
42
31
43
21
Algebraic Reconstruction Technique(ART)
Update a guessbased on
data differences
Guess 1
Guess 0
Guess 2
Error
Error
Fourier Transformation
dudvevuFvuFFyxf
dxdyeyxfyxfFvuF
vyuxj
vyuxj
)(21
)(2
),(),(),(
),(),(),(
FourierTransform
f(x,y) F(u,v)
ImageSpace
FourierSpace
Fourier Slice Theorem
v
u
F(u,v)
P(t)
f(x,y)
t
y
x
X-rays
F[P(t)]
From Projections to Image
y
x
v
u
F-1[F(u,v)]
f(x,y) P(t) F(u,v)
Filtered Backprojection
f(x,y) f(x,y)
P(t) P’(t)
1) Convolve projections with a filter2) Backproject filtered projections
Example: Projection
SinogramIdeal Image
Projection
Projection
Example: Backprojection
Projection
Example: Backprojection
Sinogram Backprojected Image
Example: Filtering
Filtered SinogramSinogram
Example: Filtered Backprojection
Filtered Sinogram Reconstructed Image
References
T. S. Curry III, J. E. Dowdey, R. C. Murry Jr. Christensen’s physics of diagnostic Radiology (4th edition), Lea & Febiger (for residents)
G. Wang, M. W. Vannier: Computerized tomography. Encyclopedia of Electrical and Electronics Engineering, edited by Webster JG, to be published by John Wiley & Sons (for engineers)
http://dolphin.radiology.uiowa.edu/ge (on-line slides & handouts in the Teaching section)