Computed tomography
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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


The main idea

Reconstruct the image of a non uniform sample using

its x-ray projection at different angles


The main idea

Reconstruct the image of a non uniform sample using

its x-ray projection at different angles


The main idea

Reconstruct the image of a non uniform sample using

its x-ray projection at different angles


The main idea

Reconstruct the image of a non uniform sample using

its x-ray projection at different angles


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 image from the projected image

  • 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 image from the projected image

  • 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 image from the projected image

  • 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 image from the projected image

  • 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 image from the projected image

  • 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.


Break
BREAK image from the projected image


CT image from the projected image

Image Reconstruction


CT image from the projected image

  • Please read Ch 13.

  • Homework is due 1 week from today at 12 pm.


Tomographic reconstruction
Tomographic reconstruction image from the projected image

detectors

= 0o


The main idea image from the projected image

detectors

= 20o


The main idea image from the projected image

detectors

= 90o

Reconstruct the image of a non uniform sample using

its x-ray projection at different angles


The sinogram
The Sinogram image from the projected image

Projection angle

  • 

  • = 0

  • 

  • 

Detectors position


Image reconstruction
Image reconstruction image from the projected image

  • Back projection

  • Filtered Back projection

  • Iterative methods (CH 22)


Back projection
Back-projection image from the projected image

  • 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 image from the projected image

A

B

C

D

 = 90o

D3= A+C=6

D4= B+D=8


A image from the projected image

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 image from the projected image

  • 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 image from the projected image

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


Projections
Projections image from the projected image

  • Are periodic in +/- 

  • The radon transform of an image produces a sinogram


Central slice theorem
Central Slice Theorem image from the projected image

  • 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 image from the projected image

2D FT of an image at angle

f


Why is it important
Why is it important? image from the projected image

  • 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 image from the projected image 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 image from the projected image

  • Prove the center slice theorem.

  • Use imrotate


Imaging in matlab
Imaging in Matlab image from the projected image

  • An image is a 2D matrix of numbers

  • imread - reads an image file

  • imwrite - writes an image to file


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