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

Reconstruct the image of a non uniform sample using its x-ray projection at different angles

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

Projection

Radon transform

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

projected image

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 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

• 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

• 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

• 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

CT

Image Reconstruction

CT

• Please read Ch 13.

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

Tomographic reconstruction

= 0o

detectors

The main idea

=

20o

dete

ctors

Reconstruct the image of a non uniform sample using its x-ray projection at different angles

The main idea

=

90o

dete

ctor

s

The Sinogram

Detectors position

Pro

ject

ion

angl

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Image reconstruction

• Back projection

• Filtered Back projection

• Iterative methods (CH 22)

Back-projection

• Given a sample with 4 different spatial absorption properties

A B

C D

D1= A+B=7

D2=C+D=7

=0o

A B

C D

D3= A+C=6

= 90o

D4= B+D=8

Back-projection

A B

C D

7

7

5869

A+B=7A+C=6A+D=5B+C=9B+D=8C+D=7

Back-projection

2 5

4 3

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 projection

€

pφ x'

( ) = f x, y( )δ x cosφ + y sinφ − x'( )dxdy−∞

∞

∫∫ =ℜ f( )

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

Projections

• Are periodic in +/- • The radon transform of an image

produces a sinogram

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 Theorem

€

pφ x'

( ) = f x, y( )δ x cosφ + y sinφ − x'( )dxdy−∞

∞

∫∫ =ℜ f( )

ℑ pφ x'

( )( ) = ℑ f (x, y)( )φ

pφ (ω) = F ω cosφ +ω sinφ( )

2D FT of an image at angle

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

• 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

• Prove the center slice theorem.

• Use imrotate

Imaging in Matlab

• An image is a 2D matrix of numbers

• imread - reads an image file

• imwrite - writes an image to file