computed tomography - university of california, san diego · 2010. 10. 5. · 1! tt liu, be280a,...
TRANSCRIPT
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TT Liu, BE280A, UCSD Fall 2010
Bioengineering 280A ���Principles of Biomedical Imaging���
Fall Quarter 2010���CT Lecture 1���
TT Liu, BE280A, UCSD Fall 2010
Computed Tomography
Suetens 2002
TT Liu, BE280A, UCSD Fall 2010
Computed Tomography
Suetens 2002
Parallel
Beam
Fan
Beam
TT Liu, BE280A, UCSD Fall 2010
Scanner Generations
Suetens 2002
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TT Liu, BE280A, UCSD Fall 2010
From http://www.sprawls.org/resources/CTIMG/classroom.htm
TT Liu, BE280A, UCSD Fall 2010
Single vs. Multi-slice
Suetens 2002
TT Liu, BE280A, UCSD Fall 2010
Scanner Generations
Prince and Links 2005
TT Liu, BE280A, UCSD Fall 2010
1G vs. 2G scanner
€
Example 6.1 from Prince and Links. Compare 1G vs. 2G scanner whose source - detector apparatus can move linearlyat speed of 1 m/sec; FOV 0.5m; 360 projections over 180 degrees; 0.5 s for apparatusto rotate one angular increment, regardless of angle.Question : Scan time for 1 G scanner? Scan time for 2G scanner with 9 detectors space 0.5 degrees apart?
Answer : 1G scanner : 0.5m/(1m/s) = 0.5s per projection. 360 * 0.5 = 180s scan time 360 * 0.5 =180s for rotation of apparatus. Total time = 360 s or 6 minutes.
2G scanner : Required angular resolution is 180/360 = 0.5 degrees - - agrees with spacing. 360/9 = 40 rotations required. 40*0.5 = 20s for scanning 40*0.5 = 20s for rotations. Total time = 40s.
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TT Liu, BE280A, UCSD Fall 2010
3G, 6G, and 7G scanners
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3G scanner : Typical scanner acquires 1000 projections with fanbeam angleof 30 to 60 degrees; 500 to 700 detectors; 1 to 20 seconds.
6G : Spiral/Helical CT 60 cm torso scan : 30s. 24 cm lung scan : 12s 15 cm angio : 30s
7G : Multislice CT 64 or more parallel 1D projections.
TT Liu, BE280A, UCSD Fall 2010
Suetens 2002
TT Liu, BE280A, UCSD Fall 2010
Detectors
Prince and Links 2005
TT Liu, BE280A, UCSD Fall 2010
CT Line Integral
€
Id = S0 E( )E0Emax∫ exp − µ s; ʹ′ E ( )0
d∫ ds( )dE
Monoenergetic Approximation
Id = I0 exp − µ s;E ( )0d∫ ds( )
gd = −logIdI0
⎛
⎝ ⎜
⎞
⎠ ⎟
= µ s;E ( )0d∫ ds
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TT Liu, BE280A, UCSD Fall 2010
CT Number
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CT_number = µ −µwaterµwater
×1000
Measured in Hounsfield Units (HU)
Air: -1000 HU
Soft Tissue: -100 to 60 HU
Cortical Bones: 250 to 1000 HU
Metal and Contrast Agents: > 2000 HU
TT Liu, BE280A, UCSD Fall 2010
CT Display
Suetens 2002
TT Liu, BE280A, UCSD Fall 2010
Direct Inverse Approach
µ1
µ2
µ3
µ4
p1
p2
p3
p4
p1= µ1+ µ2 p2= µ3+ µ4 p3= µ1+ µ3 p4= µ2+ µ4
4 equations, 4 unknowns.
Are these the correct equations to use?
€
p1p2p3p4
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
=
1 1 0 00 0 1 11 0 1 00 1 0 1
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
µ1µ2µ3µ4
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
No, equations are not linearly independent.
p4= p1+ p2- p3
Matrix is not full rank.
TT Liu, BE280A, UCSD Fall 2010
Direct Inverse Approach
µ1
µ2
µ3
µ4
p1
p2
p3
p4
p1= µ1+ µ2 p2= µ3+ µ4 p3= µ1+ µ3 p4= µ2+ µ4
4 equations, 4 unknowns.
Are these the correct equations to use?
€
p1p2p3p4
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
=
1 1 0 00 0 1 11 0 1 00 1 0 1
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
µ1µ2µ3µ4
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
No, equations are not linearly independent.
p4= p1+ p2- p3
Matrix is not full rank.
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TT Liu, BE280A, UCSD Fall 2010
Direct Inverse Approach
µ1
µ2
µ3
µ4
p1
p2
p3
p4
p1= µ1+ µ2 p2= µ3+ µ4 p3= µ1+ µ3 p5= µ1+ µ4
4 equations, 4 unknowns. These are linearly independent now.
In general for a NxN image, N2 unknowns, N2 equations.
This requires the inversion of a N2xN2 matrix
For a high-resolution 512x512 image, N2=262144 equations.
Requires inversion of a 262144x262144 matrix!
Inversion process sensitive to measurement errors.
€
p1p2p3p5
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
=
1 1 0 00 0 1 11 0 1 01 0 0 1
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
µ1µ2µ3µ4
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
p5
TT Liu, BE280A, UCSD Fall 2010
Iterative Inverse Approach ���Algebraic Reconstruction Technique (ART)
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2.5
2.5
2.5
2.5
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1.5
1.5
3.5
3.5
3
7
5
5
1
2
3
4
3
7
5
5
TT Liu, BE280A, UCSD Fall 2010
Backprojection
Suetens 2002
0
0
0
0
3
0
0
0
0
0
3
0
3
0
3
0
3
0
0
0
1
1
1
0
0
0
1
0
0
1
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1
0
0
1
1
1
0
1
3
1
0
1
1
1
1
1
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4
1
1
1
1
TT Liu, BE280A, UCSD Fall 2010
In-Class Exercise
Suetens 2002
µ1
µ2
µ3
µ4
5.7
11.3
8.2
8.8
10.1
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TT Liu, BE280A, UCSD Fall 2010
Projections
Suetens 2002
€
rs⎡
⎣ ⎢ ⎤
⎦ ⎥ =
cosθ sinθ−sinθ cosθ⎡
⎣ ⎢
⎤
⎦ ⎥ xy⎡
⎣ ⎢ ⎤
⎦ ⎥
xy⎡
⎣ ⎢ ⎤
⎦ ⎥ =
cosθ −sinθsinθ cosθ⎡
⎣ ⎢
⎤
⎦ ⎥ rs⎡
⎣ ⎢ ⎤
⎦ ⎥
TT Liu, BE280A, UCSD Fall 2010
Projections
Suetens 2002
€
I(r,θ) = I0 exp − µ(x,y)dsLr ,θ∫⎛ ⎝ ⎜ ⎞
⎠ ⎟
= I0 exp − µ(rcosθ − ssinθ,rsinθ + scosθ)dsLr ,θ∫⎛ ⎝ ⎜ ⎞
⎠ ⎟
TT Liu, BE280A, UCSD Fall 2010
Projections
Suetens 2002
€
I(r,θ) = I0 exp − µ(rcosθ − ssinθ,rsinθ + scosθ)dsLr ,θ∫⎛ ⎝ ⎜ ⎞
⎠ ⎟
€
p(r,θ) = −ln Iθ (r)I0
= µ(rcosθ − ssinθ,rsinθ + scosθ)dsLr ,θ∫
TT Liu, BE280A, UCSD Fall 2010
Radon Transform
€
g(r,θ) = µ(x(s),y(s))ds−∞
∞
∫= µ(rcosθ − ssinθ,rsinθ + scosθ)ds
−∞
∞
∫= µ(x,y)δ x cosθ + y sinθ − r( )
−∞
∞
∫−∞
∞
∫ dxdy
€
z • r r
= r
xˆ x + yˆ y ( ) • cosθˆ x + sinθˆ y ( ) = rx cosθ + y sinθ = r€
θ
€
z
€
r
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TT Liu, BE280A, UCSD Fall 2010
Example
€
f (x,y) =1 x 2 + y 2 ≤10 otherwise
⎧ ⎨ ⎩
€
g(l,θ = 0) = f (l,y)dy−∞
∞
∫
= dy− 1− l 21− l 2
∫
=2 1− l2 l ≤10 otherwise
⎧ ⎨ ⎪
⎩ ⎪
TT Liu, BE280A, UCSD Fall 2010
Sinogram
Suetens 2002
TT Liu, BE280A, UCSD Fall 2010
Backprojection
Suetens 2002
€
b(x,y) = B g l,θ( ){ }
= g(x cosθ + y sinθ,θ)dθ0
π
∫
x
y
€
b(x0,y) = p l,θ = 0( )Δθ= p(x0,0)Δθ
x0
l
€
bθ (x,y) = g(x cosθ + y sinθ,θ)Δθ
TT Liu, BE280A, UCSD Fall 2010
Backprojection
Suetens 2002
€
b(x,y) = B p l,θ( ){ }
= p(x cosθ + y sinθ,θ)dθ0
π
∫
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TT Liu, BE280A, UCSD Fall 2010
Backprojection
Suetens 2002
€
b(x,y) = B p l,θ( ){ } = p(x cosθ + y sinθ,θ)dθ0
π
∫
TT Liu, BE280A, UCSD Fall 2010
Example
Prince & Links 2006