Transcript Mammography

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
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
detectors
= 0o
The main idea
detectors
= 90o
The main idea
Reconstruct the image of a non uniform sample using
its x-ray projection at different angles
The Sinogram

 


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
Back-projection
A
B
C
D
 = 90o
D3= A+C=6
D4= B+D=8
Back-projection
9
A
B
C
D
6
8
7
A+B=7
A+C=6
A+D=5
B+C=9
B+D=8
C+D=7
7
5
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

    f x, y x cos  y sin  x'dxdy   f 
p x ' 

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

    f x, y x cos  y sin  x'dxdy   f 
p x ' 

  
 p x '    f (x, y)

p ( )  F  cos    sin  
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