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)