2D fan-beam CT with independent source and detector...
Transcript of 2D fan-beam CT with independent source and detector...
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2D fan-beam CT with independent source anddetector rotation
Simon Rit1, Rolf Clackdoyle2
1CREATIS / CLB / ESRF, University of Lyon, France
2LHC, Universite Jean Monnet Saint Etienne, France
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Image Guided Radiotherapy (IGRT)
Imaging in the radiotherapy roomFluoroscopyPortal imagingCT on railUltrasound probe...
Cone-beam CT since 10 yearsTreatment guidanceRetrospective studiesAdaptive Radiotherapy
[Jaffray et al., IJROBP, 2002]
2D fan-beam CT with independent source and detector rotation Simon Rit 2
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New PAIR device (Salzburg)
Patient Alignment system with an integrated x-ray Imaging Ring
Ceiling mounted robotic armIndependent rotation of thesource and the flat panelCouch translationSource collimation with 4motorized jawsFast switching between energies41× 41 cm2 flat panel medPhoton G.m.b.H
(courtesy of P. Steininger)
Installation: 1 prototype in Salzburg, 4 planned at MedAustron
2D fan-beam CT with independent source and detector rotation Simon Rit 3
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Geometry in this presentation
[Gullberg et al., IEEE TMI, 1986]
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Geometry in this presentation
[Gullberg et al., IEEE TMI, 1986]
2D fan-beam CT with independent source and detector rotation Simon Rit 4
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Effect of tilt on sampling
S
D
β
u
s
2D fan-beam CT with independent source and detector rotation Simon Rit 5
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Effect of tilt on sampling
Relationship
s =uD
D cosβ + u sinβ(1)
Limits u∗ =
−Dtanβ
s∗ =D
sinβ
(2)
D = 100, β = 30
−300 −200 −100 0 100 200 300−300
−200
−100
0
100
200
300
u
s
u*
s*
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Effect of tilt on sampling
Derivative
dsdu
=D2 cosβ
(D cosβ + u sinβ)2 (3)
At origin,
dsdu
(0) =1
cosβ(4)
which would also be theconstant sampling ratio in theparallel situation.
D = 100, β = 30
−200 −150 −100 −50 0 50 100 150 200 250 3000
50
100
150
200
250
300
350
400
450
500
u
du
/ds
u*
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Potential use to increase spatial resolution
[Muller and Arce,J Opt Soc Am A, 1994]
2D fan-beam CT with independent source and detector rotation Simon Rit 8
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Inversion of the Radon transform
From parallel projections pp(θ, t), the Fourier slice theoremleads to the inversion formula
f (r , φ) =
∫ 2π
0
∫ R
−Rpp(θ, t)h[r cos(θ − φ)− t ] dt dθ (5)
withh(t) =
∫R
|µ|2
exp2iπµt dµ. (6)
Implementation: filtered backprojection algorithm.
2D fan-beam CT with independent source and detector rotation Simon Rit 9
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Change of variable [Gullberg et al., TMI, 1986]
Assuming a flat detector at theorigin (D=D’), we have
pp(θ, t) = pf (α, s) (7)
fort = (s + τ)Z
θ = α + tan−1( s
D
) (8)
with
Z =D√
s2 + D2(9)
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Change of variable [Gullberg et al., TMI, 1986]
Assuming that D and τ are constant, the Jacobian matrix is
dtds
= (D2 − τs)Z 3
D2 (10)
dθds
=Z 2
D(11)
dtdα
= 0 (12)
dθdα
= 1 (13)
so its determinant is
J =
∣∣∣∣(D2 − τs) Z 3
D2
∣∣∣∣ (14)
2D fan-beam CT with independent source and detector rotation Simon Rit 11
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Change of variable [Gullberg et al., TMI, 1986]
f (r , φ) =∫ 2π
0
∫ W
−Wpf (α, s)h
[r cos(α+ tan−1(
sD)− φ)− (s + τ)Z
](D2 − τs)
Z 3
D2ds dα
=
∫ 2π
0
∫ W
−Wpf (α, s)h
[UZ (s′ − s)
](D2 − τs)
Z 3
D2ds dα (15)
with
U =r sin(α− φ) + D
D(16)
s′ =rD cos(α− φ)− τD
r sin(α− φ) + D(17)
2D fan-beam CT with independent source and detector rotation Simon Rit 12
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Change of variable [Gullberg et al., TMI, 1986]
We can then use an essential property of the filter:
h(at) =1a2 h(t) (18)
to obtain
f (r , φ) =
∫ 2π
0
1U2
∫ W
−Wpf (α, s)
D − τsD√
s2 + D2h(s′ − s) ds dα (19)
2D fan-beam CT with independent source and detector rotation Simon Rit 13
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Experiments [Gullberg et al., TMI, 1986]
D = 630 mm, D′ = 1100 mm, τ = 1 mmProjections
1000768 samples0.2 mm spacing
Reconstruction512× 512 pixels0.125× 0.125 mm2 spacing
2D fan-beam CT with independent source and detector rotation Simon Rit 14
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Experiments [Gullberg et al., TMI, 1986]
Gray level window: [-582, -482] HU.
τ = 0 mm τ = 1 mm, no correction
2D fan-beam CT with independent source and detector rotation Simon Rit 15
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Experiments [Gullberg et al., TMI, 1986]
Gray level window: [-582, -482] HU.
τ = 0 mm τ = 1 mm, new algorithm
2D fan-beam CT with independent source and detector rotation Simon Rit 15
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Experiments [Gullberg et al., TMI, 1986]
Gray level window: [-582, -482] HU.
τ = 0 mm τ = 1 mm, uncorrected weights
2D fan-beam CT with independent source and detector rotation Simon Rit 15
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Experiments
4 6 8 10 12 14 16−500
−400
−300
−200
−100
0
100
200
300
400
x (mm)
Inte
nsity (
HU
)
Reference
Uncorrected
New algorithm
Uncorrected weights
2D fan-beam CT with independent source and detector rotation Simon Rit 16
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Experiments
Gray level window: [-582, -482] HU.
β = 30˚, new algorithm β = 30˚, uncorrected weights
2D fan-beam CT with independent source and detector rotation Simon Rit 17
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Experiments
4 6 8 10 12 14 16−800
−600
−400
−200
0
200
400
x (mm)
Inte
nsity (
HU
)
Reference
Uncorrected
New algorithm
2D fan-beam CT with independent source and detector rotation Simon Rit 18
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Point Spread Function
D = τ = 1000 mm, β = 45◦
Ball at (0, 0), radius0.01 mm, density 1000 HUProjections
1000 projections10 mm spacing1000 rays per pixel
ReconstructionCentered on (0, 0)64× 64 pixels1× 1µm2 spacing
2D fan-beam CT with independent source and detector rotation Simon Rit 19
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Point Spread Function
D = τ = 1000 mm, β = 45◦
Ball at (800, 0), radius0.01 mm, density 1000 HUProjections
1000 projections10 mm spacing1000 rays per pixel
ReconstructionCentered on (800, 0)64× 64 pixels1× 1µm2 spacing
2D fan-beam CT with independent source and detector rotation Simon Rit 19
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Point Spread Function
Short scan [Parker, Med Phys, 1982]
Arc [50, 310]◦ Arc [-130, 130]◦
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Independent source and detector rotation
S
FR
x
y
2D fan-beam CT with independent source and detector rotation Simon Rit 21
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Independent source and detector rotation
S
FR
x
y
2D fan-beam CT with independent source and detector rotation Simon Rit 21
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Independent source and detector rotation
S
FR
x
y
2D fan-beam CT with independent source and detector rotation Simon Rit 21
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Independent source and detector rotation
S
FR
x
y
2D fan-beam CT with independent source and detector rotation Simon Rit 21
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Independent source and detector rotation
S
FR
x
y
2D fan-beam CT with independent source and detector rotation Simon Rit 21
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Independent source and detector rotation
S
FR
x
y
2D fan-beam CT with independent source and detector rotation Simon Rit 21
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Independent source and detector rotation
S
FR
x
y
2D fan-beam CT with independent source and detector rotation Simon Rit 21
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Independent source and detector rotation
S
FR
x
y
2D fan-beam CT with independent source and detector rotation Simon Rit 21
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Independent source and detector rotation
S
FR
x
y
2D fan-beam CT with independent source and detector rotation Simon Rit 21
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Independent source and detector rotation
S
FR
x
y
2D fan-beam CT with independent source and detector rotation Simon Rit 21
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Independent source and detector rotation
S
FR
x
y
2D fan-beam CT with independent source and detector rotation Simon Rit 21
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Independent source and detector rotation
SFR
x
y
2D fan-beam CT with independent source and detector rotation Simon Rit 21
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Independent source and detector rotation
0 50 100 150 200 250 300 350−200
−100
0
100
200
300
400
α (°)
Dis
tan
ce (
mm
)
D
τ
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[Crawford et al., Med Phys, 1988]
τ depends on the gantry angle
⇒ τ : α→ τ(α) (20)
Note that the exact samederivation has been performedby [Concepcion et al., IEEETMI, 1992]
2D fan-beam CT with independent source and detector rotation Simon Rit 23
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Change of variable [Crawford et al., Med Phys, 1988]
Assuming D constant and τ constant, the Jacobian matrix is
dtds
= (D2 − τs)Z 3
D2 (21)
dθds
=Z 2
D(22)
dtdα
= Zdτdα
(23)
dθdα
= 1 (24)
so its determinant is
J =
∣∣∣∣(D2 − τs − Ddτdα
)Z 3
D2
∣∣∣∣ (25)
2D fan-beam CT with independent source and detector rotation Simon Rit 24
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Change of variable [Crawford et al., Med Phys, 1988]
Inversion formula
f (r , φ) =
∫ 2π
0
1U2
∫ W
−Wpf (α, s)
D − τsD −
dτdα√
s2 + D2h(s′ − s) ds dα (26)
In [Concepcion et al., IEEE TMI, 1992], opposite sign for thenew term. Mathematically wrong but experimentally correct(work in progress)...
2D fan-beam CT with independent source and detector rotation Simon Rit 25
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Experiments [Crawford et al., Med Phys, 1988]
Gray level window: [-582, -482] HU.
Assuming τ = 0 mm Actual τ(α) = 3 sin (2α) mm
2D fan-beam CT with independent source and detector rotation Simon Rit 26
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Experiments [Crawford et al., Med Phys, 1988]
Gray level window: [-582, -482] HU.
Assuming τ = 0 mm Assuming dτdα = 0 mm
2D fan-beam CT with independent source and detector rotation Simon Rit 26
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Experiments
Gray level window: [-582, -482] HU.
Actual τ(α) =630× tan(30˚) sin (2α) mm
Assuming dτdα = 0 mm
2D fan-beam CT with independent source and detector rotation Simon Rit 27
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Going further
D also depends on the gantryangle
⇒ D : α→ D(α) (27)
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Change of variable
Assuming D constant and τ constant, the Jacobian matrix is
dtds
= (D2 − τs)Z 3
D2 (28)
dθds
=Z 2
D(29)
dtdα
= Zdτdα
+ (s + τ)Z 3s2
D3dDdα
(30)
dθdα
= 1− Z 2sD3
dDdα
(31)
so its determinant is
J =
∣∣∣∣(D2 − τs − Ddτdα− s
dDdα
)Z 3
D2
∣∣∣∣ (32)
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Change of variable
Inversion formula
f (r , φ) =
∫ 2π
0
1U2
∫ W
−Wpf (α, s)
D − τsD −
dτdα −
sD
dDdα√
s2 + D2h(s′−s) ds dα
(33)
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Experiments
S
FR
x
y
R = 200 mm, F = 100 mmProjections
20001000 samples0.25 mm spacing
Reconstruction512× 512 pixels0.125 mm2 spacingCentered around (F,0)
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Experiments
Gray level window: [-582, -482] HU.
New formula Assuming dDdα = 0 mm
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Conclusions
Tilting an ideal detector improves CT resolution
Inversion formula for any 2D derivable trajectory
Requires the derivative of the parameters with respect tothe gantry angle
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Open questions
What is the sensitivity to noise in geometric parameters?
What is the optimal angular sampling?
Will it really improve resolution when hardwareconsiderations come into the picture?
2D fan-beam CT with independent source and detector rotation Simon Rit 34