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Computed Tomography. Tomos = slice. CT scan. Mathematical idea developed by Radon in 1917 Cormack did the instrumentation research 1963 published it A practical x-ray CT scanner was built by Hounsfield. When was the first computer introduced in laboratories?. The main idea.

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computed tomography

Computed Tomography

Tomos = slice

ct scan
CT scan
  • Mathematical idea developed by Radon in 1917
  • Cormack did the instrumentation research 1963 published it
  • A practical x-ray CT scanner was built by Hounsfield.

When was the first computer introduced in laboratories?

the main idea
The main idea

Reconstruct the image of a non uniform sample using

its x-ray projection at different angles

slide6

The main idea

Reconstruct the image of a non uniform sample using

its x-ray projection at different angles

slide7

The main idea

Reconstruct the image of a non uniform sample using

its x-ray projection at different angles

slide8

The main idea

Reconstruct the image of a non uniform sample using

its x-ray projection at different angles

slide9

The main idea

Reconstruct the image of a non uniform sample using

its x-ray projection at different angles

slide10

The main idea

Reconstruct the image of a non uniform sample using

its x-ray projection at different angles

projection
Projection

Radon transform

ct images
CT images
  • Maps of relative linear attenuation of tissue
  • µ relative attenuation coefficient is expressed in Hounsfield units (HU) also known as CT numbers
  • HUx = 1000.(µx - µwater)/µwater
  • HUwater = 0
  • HU depends on photon energy
ct images1
CT images
  • FOV (field of view) Diameter of the region being imaged (head 25 cm)
  • Voxel Volume element in the patient
    • Pixel area x slice thickness
ct scan generations
CT scan generations
  • 1st generation
    • Translate rotate, pencil beam
  • 2nd generation
    • Translate rotate, fan beam
  • 3rd generation
    • Rotate rotate, fan beam
  • 4th generation
    • Rotate, wide fan
  • 5th generation
    • Fixed array of detectors
x ray tube
X-ray tube
  • High voltage xray tubes
  • For large focal spots (1mm) ->high power (60kW), smaller spots (0.5 mm) low power rating (below 25kW)
  • Copper and aluminum filters used for beam hardening effect
  • Collimators both in x ray tube and detector
detectors
Detectors
  • Measure radiation through patient
  • High xray efficiency
  • Scintillation
    • Crystals produce visible range photons coupled with PMT
  • Xenon gas ionization detector
    • Gas chamber anode and cathode at potential. Used in 3rd gen., stable.
slide33

CT

Image Reconstruction

slide34
CT
  • Please read Ch 13.
  • Homework is due 1 week from today at 12 pm.
tomographic reconstruction
Tomographic reconstruction

detectors

= 0o

slide36

The main idea

detectors

= 20o

slide37

The main idea

detectors

= 90o

Reconstruct the image of a non uniform sample using

its x-ray projection at different angles

the sinogram
The Sinogram

Projection angle

  • 
  • = 0
  • 
  • 

Detectors position

image reconstruction
Image reconstruction
  • Back projection
  • Filtered Back projection
  • Iterative methods (CH 22)
back projection
Back-projection
  • Given a sample with 4 different spatial absorption properties

A

B

D1= A+B=7

C

D

D2=C+D=7

 =0o

back projection1
Back-projection

A

B

C

D

 = 90o

D3= A+C=6

D4= B+D=8

slide42

A

B

7

C

D

7

9

5

6

8

Back-projection

A+B=7

A+C=6

A+D=5

B+C=9

B+D=8

C+D=7

2

5

4

3

real back projection
Real back-projection
  • In a real CT we have at least 512 x 512 values to reconstruct
  • We don’t know where one absorber ends where the next begins
  • ~ 800,000 projections
back projection2
Back projection

The projection of a function is the radon transform of that function

projections
Projections
  • Are periodic in +/- 
  • The radon transform of an image produces a sinogram
central slice theorem
Central Slice Theorem
  • Relates the 1 D Fourier transform of a projection of an object
    • F(p(x’)) at a given angle 
  • To a line through the center of the 2D Fourier transform of the object at a given angle 
central slice theorem1
Central Slice Theorem

2D FT of an image at angle

f

why is it important
Why is it important?
  • If you compute the 1D Fourier transform of all the projection (at all angles f) you can “fill” the 2 D Fourier transform of the object.
  • The object can then be reconstructed by a simple 2D Fourier transform.
filtered back projection
FILTERED back-projection
  • If only the 2D inverse Fourier transform is computed you will obtain a “blurry” image. (it is intrinsic in inverse Radon)
  • The blur is eliminated by deconvolution
  • In filtered back projection a RAMP filter is used to filter the data
homework
Homework
  • Prove the center slice theorem.
  • Use imrotate
imaging in matlab
Imaging in Matlab
  • An image is a 2D matrix of numbers
  • imread - reads an image file
  • imwrite - writes an image to file
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