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Computed Tomography: Introduction and Instrumentation
Primary Source: Medical Imaging Signals and Systems
By Jerry Prince and Jonathan Links
CT: Physics
• Same as x-ray radiography: – Image contrast from photoelectric effect
– Image blurring from compton scatter
Particulate ionizing radiation: Electromagnetic ionizing radiation:
CT: Physics – radiation sidenote • http://www.dotmed.com/news/story/11025
• http://emf.mercola.com/sites/emf/archive/2010/09/25/high-ct-scan-radiation-is-deadly.aspx
CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each exposure • Shield all of x-ray beam but slice • Obtain1D projections of 2D axial cross section • A Radon transform takes 1D projections of a 2D object over many angles
and it has an inverse
“raw data” (indirect and tomographic modality)
• The Radon Transform: the integral transform consisting of the integral of a function over straight lines
Projections – general definition/concepts
x
y
θ
l l
l
θ=45o
θ=90o
θ=135o
l θ=0o
θ=179o
l
dxdyyxyxfg )sincos(),(),(
where δ(xcosθ+ysinθ-l) is a line impulse that exists along a line normal to
θ a distance l from the origin.
δ(xcosθ+ysinθ-l)
f(x,y)
g(l,θ), the Radon Transform:
EW
1
EW
2
EW
3
EW
4
EW
5
EW
6
EW
7
EW
8
EW
9
NS1
NS2
NS3
NS4
NS5
NS6
NS7
NS8
NS9
BATTLESHIP
East - West Data
N
o
r
t
h
/
S
o
u
t
h
D
a
t
a
Battleship:
New Rules
Old way
EW 2/ NS 3 Hit!
EW 5/ NS 6 Miss!
EW
1
EW
2
EW
3
EW
4
EW
5
EW
6
EW
7
EW
8
EW
9
NS1
NS2
NS3
NS4
NS5
NS6
NS7
NS8
NS9
1 0 4
BATTLESHIP
East - West Data
N
o
r
t
h
/
S
o
u
t
h
D
a
t
a
Battleship:
New Rules
Tell the other
player how
many times you
see a battleship in
a square along
each column.
EW
1
EW
2
EW
3
EW
4
EW
5
EW
6
EW
7
EW
8
EW
9
NS1
NS2
NS3
NS4
NS5
NS6
NS7
NS8
NS9
1 1 1 0 1 1 1 4 0
BATTLESHIP
East - West Data
N
o
r
t
h
/
S
o
u
t
h
D
a
t
a
Battleship:
New Rules
Here it is finished.
EW
1
EW
2
EW
3
EW
4
EW
5
EW
6
EW
7
EW
8
EW
9
NS1
NS2
NS3
NS4
NS5
NS6
NS7
NS8
NS9
1 1 1 0 1 1 1 4 0
BATTLESHIP
East - West Data
N
o
r
t
h
/
S
o
u
t
h
D
a
t
a
If this is all your
opponent told you,
could you find
where the
battleships were?
Need more info.
Repeat the
procedure for
rows.
EW
1
EW
2
EW
3
EW
4
EW
5
EW
6
EW
7
EW
8
EW
9
NS1 1
NS2
1
NS3 4
NS4 1
NS5 0
NS6 0
NS7 3
NS8 0
NS9 0
1 1 1 0 1 1 1 4 0
BATTLESHIP
East - West Data
N
o
r
t
h
/
S
o
u
t
h
D
a
t
a
Here are all the
answers.
EW
1
EW
2
EW
3
EW
4
EW
5
EW
6
EW
7
EW
8
EW
9
NS1 1 1 1 0 1 1 1 4 0 1
NS2
1 1 1 0 1 1 1 4 0 1
NS3 1 1 1 0 1 1 1 4 0 4
NS4 1 1 1 0 1 1 1 4 0 1
NS5 1 1 1 0 1 1 1 4 0 0
NS6 1 1 1 0 1 1 1 4 0 0
NS7 1 1 1 0 1 1 1 4 0 3
NS8 1 1 1 0 1 1 1 4 0 0
NS9 1 1 1 0 1 1 1 4 0 0
1 1 1 0 1 1 1 4 0
BATTLESHIP
East - West Data
N
o
r
t
h
/
S
o
u
t
h
D
a
t
a
Can you find the
ships now if you just
knew the gray squares?
One idea:
Smear the East West
numbers all the way
up the columns.
This tells us that we
should not spend much
time looking along
EW4 or EW9.
But there is something
probably up in EW8
EW
1
EW
2
EW
3
EW
4
EW
5
EW
6
EW
7
EW
8
EW
9
NS1 1+1
=2
4+1
=5 1
NS2
1
NS3 4
NS4 1
NS5 0
NS6 0
NS7 3+1
=4 3
NS8 0
NS9 0
1 1 1 0 1 1 1 4 0
BATTLESHIP
East - West Data
N
o
r
t
h
/
S
o
u
t
h
D
a
t
a
Next, smear the
North/ South
numbers to the left
and add them
to what was in the
grid before.
Where do you
think the ships are?
By the biggest
numbers?
Is this always true?
EW
1
EW
2
EW
3
EW
4
EW
5
EW
6
EW
7
EW
8
EW
9
NS1 2 2 2 1 2 2 2 5 1 1
NS2
2 2 2 1 2 2 2 5 1 1
NS3 5 5 5 4 5 5 5 8 4 4
NS4 2 2 2 1 2 2 2 5 1 1
NS5 1 1 1 0 1 1 1 4 0 0
NS6 1 1 1 0 1 1 1 4 0 0
NS7 4 4 4 3 4 4 4 7 3 3
NS8 1 1 1 0 1 1 1 4 0 0
NS9 1 1 1 0 1 1 1 4 0 0
1 1 1 0 1 1 1 4 0
BATTLESHIP
East - West Data
N
o
r
t
h
/
S
o
u
t
h
D
a
t
a
Here I finished
smearing the
north/south
numbers to the left
and adding them to
the east/west
numbers.
Where do you
think the ships are?
By the biggest
numbers?
Is this always true?
EW
1
EW
2
EW
3
EW
4
EW
5
EW
6
EW
7
EW
8
EW
9
NS1 2 2 2 1 2 2 2 5 1 1
NS2
2 2 2 1 2 2 2 5 1 1
NS3 5 5 5 4 5 5 5 8 4 4
NS4 2 2 2 1 2 2 2 5 1 1
NS5 1 1 1 0 1 1 1 4 0 0
NS6 1 1 1 0 1 1 1 4 0 0
NS7 4 4 4 3 4 4 4 7 3 3
NS8 1 1 1 0 1 1 1 4 0 0
NS9 1 1 1 0 1 1 1 4 0 0
1 1 1 0 1 1 1 4 0
BATTLESHIP
East - West Data
N
o
r
t
h
/
S
o
u
t
h
D
a
t
a
Next, smear the
North/ South
numbers to the left
and add them
to what was in the
grid before.
Are the battleships
where the biggest
numbers are?
All of the time?
Some of the time?
1 2 3 4 5 6 7 8 9
A
B
C
D
E
F
G
H
I
BATTLESHIP
0
0
1
1
1
0
0
0 1 1 2 1 1
1
0 0 0
What if we can measure along the diagonals?
1 2 3 4 5 6 7 8 9
A 2 2 3 2 3 2 2 6 2 1
B
2 3 3 2 2 2 3 6 2 1
C 6 6 6 4 5 6 6 9 6 4
D 3 3 2 1 3 3 3 7 2 1
E 2 1 1 1 2 2 3 5 1 0
F 1 1 2 1 2 3 2 5 0 0
G 4 5 5 4 6 5 5 7 3 3
H 2 2 2 2 2 2 1 4 0 0
I 2 2 3 1 2 1 1 4 0 0
1 1 1 0 1 1 1 4 0
BATTLESHIP
Now add the
diagonal
information to our
totals.
Are we doing any
better?
Are the battleships
where the biggest
numbers are more
often?
How is “BMEN 420 Battleship” similar to what we do in CT?
http://www.colorado.edu/physics/2000/index.pl
≈
• Step 1: Backprojection – form 2D images (in (x,y) Cartesian space) from your 1D projections (collected in (ℓ,θ) polar space)
Examples of backprojecting: “smearing” your 1-D projection data collected at
an angle back into x-y space
Image Reconstruction from Projections Backprojection Summation
),sincos(),( yxgyxb
b0 (x,y)
b90 (x,y)
b35 (x,y) x
y
θ
l
• Step 1: Backprojection – form 2D images (in (x,y) Cartesian space) from your 1D projections (collected in (ℓ,θ) polar space)
Examples of backprojecting: “smearing” your 1-D projection data collected at
an angle back into x-y space
Image Reconstruction from Projections Backprojection Summation
),sincos(),( yxgyxb
b0 (x,y)
b90 (x,y)
b35 (x,y)
• Step 1: Backprojection – form 2D images (in (x,y) Cartesian space) from your 1D projections (collected in (ℓ,θ) polar space)
Examples of backprojecting: “smearing” your 1-D projection data collected at
an angle back into x-y space
Image Reconstruction from Projections Backprojection Summation
),sincos(),( yxgyxb
b0 (x,y)
b90 (x,y)
b35 (x,y)
• Step 2: Summation – sum the backprojections from step 1 to obtain the reconstructed object fbs(x,y).
Image Reconstruction from Projections Backprojection Summation
tionbackprojec theis ),sincos(),( yxgyxbwhere
0
sincos0
0
),(),sincos(),(),( dlgdyxgdyxbyxf yxlbs
*how many projections you “need” depends on what you want to see
• Step 2: Summation – sum the backprojections from step 1 to obtain the reconstructed object fbs(x,y).
Image Reconstruction from Projections Backprojection Summation
tionbackprojec theis ),sincos(),( yxgyxbwhere
0
sincos0
0
),(),sincos(),(),( dlgdyxgdyxbyxf yxlbs
*how many projections you “need” depends on what you want to see
CT & General Radiography: Broad Comparisons
Advantages Disadvantages
General Radiography - Inexpensive -“fast” (motion not as big an issue) - no computational power necessary -Broad coverage -Very good resolution
-overlaying structures (projection) -lower contrast
CT -tomography (no overlaying structures) -higher contrast
-more radiation -motion artifacts -expensive -computationally challenging -narrow coverage -big, cumbersome -no long-axis images
Disappearing with computation power, 3D data blocks (allows for reformatting) from spiral, helical, and multislice
CT: Instrumentation – 1st Generation
From Webb, Physics of Medical Imaging
0-D projection of 1-D object (pencil beam) Translate then rotate
CT: Instrumentation – 2nd Generation
From Webb, Physics of Medical Imaging
Eliminate some of translation steps… Still have to translate + rotate
1G and 2G – how large is the time savings?
Consider a 1G or 2G scanner whose source detector apparatus can move linearly at a speed of 1.0 m/sec and the field-of-view has a diameter of 0.5m. Suppose that 360 projections over 180o are required and that it takes 0.5 sec for the source-detector apparatus to rotate one angular increment, regardless of the angle.
-What is the scan time for a 1G scanner?
-What is the scan time for a 2G scanner having 9 detectors
spaced 0.5o apart?
1G and 2G – how large is the time savings?
Consider a 1G or 2G scanner whose source detector apparatus can move linearly at a speed of 1.0 m/sec and the field-of-view has a diameter of 0.5m. Suppose that 360 projections over 180o are required and that it takes 0.5 sec for the source-detector apparatus to rotate one angular increment, regardless of the angle.
-What is the scan time for a 1G scanner?
-What is the scan time for a 2G scanner having 9 detectors
spaced 0.5o apart?
Example 6.1 in MISS - ans: 6 mins versus 40 secs
CT: Instrumentation – 3rd Generation
Eliminate translation Just rotate
CT: Instrumentation – 4th Generation
Eliminate rotation of detectors. Now just source rotates. NO DETECTOR COLLIMATION => More efficient, but more blurring
• Generations 1-4 – Single x-ray tube (bulky, expensive, difficult to calibrate)
– Moving the source limits scan speed
• Generation 5: electron beam computed tomography (EBCT) – Source: Flying electron beam, steered electromagnetically,
to hit one of four tungsten anode strips. X-rays generated are collimated into a fan beam and detection is as with 4G
– Extremely fast (stop action cardiac without gating)
CT: Instrumentation – 5th ,6th , and 7th Generations
• Generation 6: helical CT – 3G/4G continuously rotating with moving patient
for 3D volumetric acquisition • 60cm torso scan, 30 secs
• 24cm lung scan, 12 secs
• 15cm angio scan, 30 secs
• Generation 7: multislice CT – “thick” fan beam, parallel rows of detectors
– Collects multiple (up to 64, 256) 1-D projections at one time
• Multislicehelical = larger pitch on helix:
CT: Instrumentation – 5th ,6th , and 7th Generations
CT: Instrumentation
• Instrumentation developments (engineering) have paved the way for modern CT: – Fan beam collimation: 2 pieces of lead with a slit 1-10mm between them,
as close as possible to patient. Controls slice thickness. Done at console. – Copper followed by aluminum filtration to “harden” the beam (make it
more monoenergetic) – Solid state detectors, xenon gas detectors (3G), multiple solid state
detector array – Gantry: holds the x-ray tube, detectors, so can rotate around patient
rapidly and repeatedly : movie of Philips Brilliance 16 slice http://www.youtube.com/watch?v=YAqK-huXQoI http://www.youtube.com/watch?v=2CWpZKuy-NE&feature=related – Slip ring: brush system to deliver power to x-ray tube – Patient table: must account for patient loading and continuous controlled
movement (if helical)