Download Computed Tomography Principles

Computed Tomography Principles
Ge Wang, Ph.D.
Department of Radiology
University of Iowa
Iowa 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 prizes
Roentgen (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
(From Aracor)
Photoelectric effect results in total absorption of
the X-ray photon and the emission of a bound electron
X-ray Interactions - Compton Scatter
(From Aracor)
Compton Scatter results in a free electron &
a scattered (less energetic) photon
Source and Detectors
Source
- Rotating anode disk
- Small focal spot
down to 0.6 mm
- Polychromatic beam
Detectors
- Xenon (50-60%)
- Scintillation (>90%)
(From Siemens)
Exponential Attenuation of X-ray
Ni

No  Nie
No
Ni: input intensity of X-ray
No: output intensity of X-ray
: linear X-ray attenuation
x
Ni


 x

No
N o  N i e  ( 1   2  3 ) x
x
X-rays
Attenuated
more
Ray-Sum of X-ray Attenuation
Ni
k
x
Ray-sum
No
No  Ni e
Ni
  k x  ln
k
No

 k x
k
Line integral

Ni
 ( x)dx  ln N o
Projection & Sinogram
Sinogram:
All projections
Projection:
All ray-sums in a direction
P(t)

y
t

p
x
f(x,y)
X-rays
t
Sinogram
Completeness Condition
There exists at least a source on any line
intersecting a cross-section
Computed Tomography Principles
1. Projection measurement
2. Scanning modes
3. Scanner systems
4. Image reconstruction
First Generation
One detector
Translation-rotation
Parallel-beam
Second Generation
Multiple detectors
Translation-rotation
Small fan-beam
Third Generation
Multiple detectors
Translation-rotation
Large fan-beam
Fourth Generation
Detector ring
Source-rotation
Large fan-beam
Third & Fourth Generations
(From Siemens)
(From Picker)
Spiral/Helical Scanning
Simultaneous
•Source rotation
•Table translation
•Data acquisition
Cone-Beam Geometry
Z
Y
X
Scanning modes
• First generation
One detector, translation-rotation
Parallel-beam
• Second generation
Multiple detectors, translation-rotation
Small fan-beam
• Third generation
Multiple detectors, rotation-rotation
Large fan-beam
Scanning modes
• Fourth generation
Detector ring, source-rotation
Large 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
Network
Gantry
Source
Computer
Table
Parallel
processor
Display
Control
console
Recording
Detectors
Data acquisition
system
Storage units:
Tapes, disks
Data Acquisition System (DAS)
Pre-Collimator
Post-Collimator
Scattering
Source
Detector
Filter
Patient
Data Acquisition System (DAS)
X-ray Tube
Source
Filter
Detectors
CT Gantry
(From Siemens)
Detector
Spiral CT Scanner
• Gantry
Data acquisition system
• Table
• Computer
Parallel processors
• Control console
• Storage units
Tapes, disks
(From Elscint)
• Recording device
• Network interface
X-ray generator
Heat exchanger
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
y
Computed tomography (CT):
Image reconstruction from
projections
P(t)
f(x,y)
P(t)
t

x
f(x,y)
X-rays
Reconstruction Idea
4


4
1   2  7
    3
 3
4





6
1
3

 2   4  4
Algebraic Reconstruction Technique
(ART)
6
4
0
0
4



0
0




4



Guess 0
Guess 2


Error
Error
Guess 1
Update a guess
based on
data differences
Fourier Transformation
Fourier
Transform
F (u, v)  F  f ( x, y ) 
 

f(x,y)
F(u,v)
Image
Space
Fourier
Space
f ( x, y )e  j 2p (ux vy ) dxdy
  
f ( x, y )  F
1
 
F (u, v)    F (u, v)e j 2p (uxvy ) dudv
  
Fourier Slice Theorem
P(t)
y
v
t

F[P(t)]
x

F(u,v)
f(x,y)
X-rays
u
From Projections to Image
y
-1
F [F(u,v)]
v
x
f(x,y)
P(t)
u
F(u,v)
Filtered Backprojection
P’(t)
P(t)
f(x,y)
f(x,y)
1) Convolve projections with a filter
2) Backproject filtered projections
Example: Projection
Projection
Projection
Ideal Image
Sinogram
Example: Backprojection
Projection
Example: Backprojection
Sinogram
Backprojected Image
Example: Filtering
Sinogram
Filtered Sinogram
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)