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