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An Overview on Mathematical Methods in Tomography
Andreas Rieder
UNIVERSITAT KARLSRUHE (TH)
Institut fur Wissenschaftliches Rechnenund Mathematische Modellbildung
und
Institut fur Praktische Mathematik
Straßburg, 25.05.2004 – p.1/21
Preface
Focus: 2D X-ray computerized tomography
CT variants not addressed here:3D CT
SPECT
Doppler CT
Diffusive (optical) CT
MR imaging
Impedance CT
Ultrasound CT
F. Natterer, F. Wübbeling: Mathematical Methods in Image Reconstruction,SIAM 2001
Straßburg, 25.05.2004 – p.2/21
Contents
Mathematical model for CT: the Radon transform
Inversion formula: global and local tomography
Non-uniqueness for discrete data
Approximate inversion
Filtered backprojection algorithm
Computational examples
Straßburg, 25.05.2004 – p.3/21
Principle of CT scanning device
Straßburg, 25.05.2004 – p.4/21
The mathematical model
I0 I1L
x + ∆xX-ray detector
f
sourcex
physical assump.: I(x + ∆x) − I(x) = −f(x) ‖∆x‖ I(x)
I(x + ∆x) − I(x)
‖∆x‖= −f(x) I(x)
∆x → 0 =⇒ ∂L ln I(x) = −f(x)
∫
L
f(x) dσ(x) = ln(I0/I1
)
Straßburg, 25.05.2004 – p.5/21
CT scanning geometries
Straßburg, 25.05.2004 – p.6/21
2D-Radon-Transform (parallel scanning geometry)
Rf(s, ϑ) :=
∫
l(s,ϑ)∩Ω
f(x) dσ(x)ϑ
s ϑ( )s ,l
tomographic inversion: Rf(s, ϑ) = g(s, ϑ)
R : L2(Ω) → L2(Z), Z = [−1, 1] × [0, π]
Johann Radon 1917, A. M. Cormack 1963, G. N. Houndsfield 1967
Straßburg, 25.05.2004 – p.7/21
Inversion formula
Riesz potential Λα : Ht(Rd) → Ht−α(Rd), α > −d
Λαf(ξ) := ‖ξ‖α f(ξ), Λα = (−∆)α/2
backprojection R∗ : L2(Z) → L2(Ω)
R∗g(x) =
∫ π
0
g(xt ω(ϑ), ϑ) dϑ
Λαf =1
2πR
∗Λ1+αs Rf, f ∈ L2(Ω)
α = 0: Radon 1917, general result: Smith, Solomon and Wagner 1977
Straßburg, 25.05.2004 – p.8/21
Global and local tomography
Λαf =1
2πR
∗Λ1+αs Rf
α = 0: Λs = Hd
ds, H Hilbert transform
inversion formula for f is global
α = 1: Λ2s = −
d2
ds2, sing supp Λf ⊂ sing supp f
inversion formula for Λf is local
Straßburg, 25.05.2004 – p.9/21
Local tomography
f Λf
Straßburg, 25.05.2004 – p.10/21
Non-Uniqueness for discrete data
Smith et al. 1977: si, i = 1, . . . , q, ϑj , j = 1, . . . , p
∃f 6= 0 : Rf(si, ϑj) = 0 ∀ i, j
Natterer 1980: (si, ϑj) rectangular grid with h = 2/q = π/p
f ghost =⇒ ‖f‖L2 . hβ ‖f‖Hβ0
, β > 1/2
Louis 1984: 0 < τ < 1
f ghost =⇒
∫
|ξ|≤τ(p−1)
|f(ξ)| dξ . e−λ(τ) p ‖f‖L1
Further analytical aspects: stability, sampling and resolution
Straßburg, 25.05.2004 – p.11/21
Non-Uniqueness for discrete data
Smith et al. 1977: si, i = 1, . . . , q, ϑj , j = 1, . . . , p
∃f 6= 0 : Rf(si, ϑj) = 0 ∀ i, j
Natterer 1980: (si, ϑj) rectangular grid with h = 2/q = π/p
f ghost =⇒ ‖f‖L2 . hβ ‖f‖Hβ0
, β > 1/2
Louis 1984: 0 < τ < 1
f ghost =⇒
∫
|ξ|≤τ(p−1)
|f(ξ)| dξ . e−λ(τ) p ‖f‖L1
Further analytical aspects: stability, sampling and resolution
Straßburg, 25.05.2004 – p.11/21
Non-Uniqueness for discrete data
Smith et al. 1977: si, i = 1, . . . , q, ϑj , j = 1, . . . , p
∃f 6= 0 : Rf(si, ϑj) = 0 ∀ i, j
Natterer 1980: (si, ϑj) rectangular grid with h = 2/q = π/p
f ghost =⇒ ‖f‖L2 . hβ ‖f‖Hβ0
, β > 1/2
Louis 1984: 0 < τ < 1
f ghost =⇒
∫
|ξ|≤τ(p−1)
|f(ξ)| dξ . e−λ(τ) p ‖f‖L1
Further analytical aspects: stability, sampling and resolution
Straßburg, 25.05.2004 – p.11/21
Non-Uniqueness for discrete data
Smith et al. 1977: si, i = 1, . . . , q, ϑj , j = 1, . . . , p
∃f 6= 0 : Rf(si, ϑj) = 0 ∀ i, j
Natterer 1980: (si, ϑj) rectangular grid with h = 2/q = π/p
f ghost =⇒ ‖f‖L2 . hβ ‖f‖Hβ0
, β > 1/2
Louis 1984: 0 < τ < 1
f ghost =⇒
∫
|ξ|≤τ(p−1)
|f(ξ)| dξ . e−λ(τ) p ‖f‖L1
Further analytical aspects: stability, sampling and resolution
Straßburg, 25.05.2004 – p.11/21
CT ghost
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
−1 −0.5 0 0.5 1
−1
−0.5
0
0.5
1
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
s
ϑ
−1 −0.5 0 0.5 1
0
1
2
3
4
5
6
Straßburg, 25.05.2004 – p.12/21
Approximate inversion I
inversion formula: f =1
2πR
∗ Λs g g = Rf
approx. inversion: f ? e = R∗ (υ ?s Rf), e = R
∗υ
e mollifier (e ≈ δ, centered about 0 with mean value 1)υ reconstruction filter/kernel
υ = (2π)−1 ΛsRe =⇒ e = R∗υ
eγ(x) = γ−2 e(x/γ), υγ(s) = γ−2 υ(s/γ), γ > 0
f ? eγ → f as γ → 0
Straßburg, 25.05.2004 – p.13/21
Approximate inversion II
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
Mollifier
e3
e6
0 0.5 1 1.5
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7Rekonstruktionskerne zur Radon−Transformation
υ3
υ6
Straßburg, 25.05.2004 – p.14/21
Reconstruction algorithm I
approx. inversion: f ? eγ = R∗ (υγ ?s g)
discrete data: g`,j = Rf(`/q, j π/p), ` = −q, . . . , q, j = 0, . . . , p − 1
filtered backprojection: fR(x) = R∗p
(υγ ?h g
)(x)
(υγ ?h g)k,j = h
q∑
` =−q
υγ
(h(k − `)
)g`,j , h = 1/q
How to choose γ in relation to h?
Straßburg, 25.05.2004 – p.15/21
Reconstruction algorithm II
R. 2000: f essentially b-band-limited and h ≤ π/b
fR = f ? eγ + m(γ, h) Λ−1f + discr. error
Strategy: Determine γ = γh as a zero of m(·, h), that is,m(γh, h) = 0
Straßburg, 25.05.2004 – p.16/21
Reconstruction algorithm III
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
Mollifier
e3
e6
h = 0.01
0.018 0.022 0.026 0.03
-10
-5
5
10
0.028 0.032 0.036
-3
-2
-1
1
2
Straßburg, 25.05.2004 – p.17/21
Shepp-Logan head phantom
1.005
1.01
1.015
1.02
1.025
1.03
1.035
1.04
1.045
1.05
Tomographic Data1.8
0.2
1.0
−1.0 1.0
π
0
Straßburg, 25.05.2004 – p.18/21
Reconstructions I
e(x) =
(1 − ‖x‖2)6 : ‖x‖ ≤ 1
0 : otherwise, h = 0.01
original γ = 0.01765299..
(sm. zero of m),rel. `2-error: 0.0816
Straßburg, 25.05.2004 – p.19/21
Reconstructions II
original γ = 0.02357177..
(2nd sm. zero of m),rel. `2-error: 0.1001
Straßburg, 25.05.2004 – p.20/21
Violating the zero condition
γ = 0.01765299..
rel. `2-error: 0.0816
0.8
0.9
0.85
γ = 0.0177..
rel. `2-error: 0.1730
0.028 0.032 0.036
-3
-2
-1
1
2
fR = f ? eγ + m(γ, h) Λ−1f + discr. error
Straßburg, 25.05.2004 – p.21/21
Violating the zero condition
γ = 0.01765299..
rel. `2-error: 0.0816
0.8
0.9
0.85
γ = 0.0177..
rel. `2-error: 0.1730
0.028 0.032 0.036
-3
-2
-1
1
2
fR = f ? eγ + m(γ, h) Λ−1f + discr. error
Straßburg, 25.05.2004 – p.21/21
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