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## PowerPoint Slideshow about ' Computed Tomography' - mackenzie-payne

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### Computed Tomography

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

### CT

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

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

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

Projection

Radon transform

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.

Image Reconstruction

CT

- Please read Ch 13.
- Homework is due 1 week from today at 12 pm.

detectors

= 90o

Reconstruct the image of a non uniform sample using

its x-ray projection at different angles

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

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

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

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