# Computed Tomography - PowerPoint PPT Presentation

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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

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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

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

## 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

• 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.

## CT

Image Reconstruction

### CT

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

### Tomographic reconstruction

detectors

= 0o

The main idea

detectors

= 20o

The main idea

detectors

= 90o

Reconstruct the image of a non uniform sample using

its x-ray projection at different angles

### The Sinogram

Projection angle

• 

• = 0

• 

• 

Detectors position

### Image reconstruction

• Back projection

• Filtered Back projection

• Iterative methods (CH 22)

### Back-projection

• Given a sample with 4 different spatial absorption properties

A

B

D1= A+B=7

C

D

D2=C+D=7

 =0o

A

B

C

D

 = 90o

D3= A+C=6

D4= B+D=8

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

• 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

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

2D FT of an image at angle

f

### 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