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


Computed tomography

The main idea

Reconstruct the image of a non uniform sample using

its x-ray projection at different angles


Computed tomography

The main idea

Reconstruct the image of a non uniform sample using

its x-ray projection at different angles


Computed tomography

The main idea

Reconstruct the image of a non uniform sample using

its x-ray projection at different angles


Computed tomography

The main idea

Reconstruct the image of a non uniform sample using

its x-ray projection at different angles


Computed tomography

The main idea

Reconstruct the image of a non uniform sample using

its x-ray projection at different angles


Projection

Projection

Radon transform


Inverse back projection is used to reconstruct the original image from the projected image

Inverse back-projection is used to reconstruct the original image from the projected image


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.


Break

BREAK


Computed tomography

CT

Image Reconstruction


Computed tomography

CT

  • Please read Ch 13.

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


Tomographic reconstruction

Tomographic reconstruction

detectors

= 0o


Computed tomography

The main idea

detectors

= 20o


Computed tomography

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


Computed tomography

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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