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GoBack A new view on phantom views Andreas Rieder Institut f ur Angewandte und Numerische Mathematik Universit at Karlsruhe Fakult at f ur Mathematik (jointly with Arne Schneck, Karlsruhe) c Andreas Rieder, Wien, AIP 09 1

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GoBack

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c Andreas Rieder, Wien, AIP 09 – 1 / 22

A new view on phantom views

Andreas Rieder Institut f¨ ur Angewandte und Numerische Mathematik Universit¨ at Karlsruhe Fakult¨ at f¨ ur Mathematik

(jointly with Arne Schneck, Karlsruhe)

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Overview

Introduction: FBA augmented by phantom views Phantom views reduce streak artifacts Phantom views increase angular convergence rate Bibliographical notes Conclusion c Andreas Rieder, Wien, AIP 09 – 2 / 22

Introduction: FBA augmented by phantom views Phantom views reduce streak artifacts Phantom views increase angular convergence rate Bibliographical notes Conclusion

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Introduction: FBA augmented by phantom views

⊲

Introduction: FBA augmented by phantom views Phantom views reduce streak artifacts Phantom views increase angular convergence rate Bibliographical notes Conclusion c Andreas Rieder, Wien, AIP 09 – 3 / 22

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2D-Radon transform (parallel scanning geometry)

c Andreas Rieder, Wien, AIP 09 – 4 / 22

Rf(s, ϑ) :=

  • l(s,ϑ)∩Ω

f(x) dσ(x)

ϑ

( ) s, s

ϑ

l

tomographic inversion: Rf(s, ϑ) = g(s, ϑ) R: L2(Ω) → L2(Z), Z = [−1, 1] × [0, 2π]

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Inversion formula

c Andreas Rieder, Wien, AIP 09 – 5 / 22

f = 1 4πR∗(Λ ⊗ I)Rf R∗ : L2(Z) → L2(Ω) Backprojection operator R∗g(x) = 2π g(xtω(ϑ), ϑ) dϑ, ω(ϑ) = (cos ϑ, sin ϑ)t Λ: Hα(R) → Hα−1(R) Riesz potential

  • Λu(ξ) = |ξ|

u(ξ).

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Filtered backprojection algorithm (FBA)

c Andreas Rieder, Wien, AIP 09 – 6 / 22

discrete Radon data D = {Rf(kh, jhϑ) : k = −q, . . . , q, j = 0, . . . , 2p − 1}, h = 1/q, hϑ = π/p fFBA(x) := 1 4πR∗

hϑ(IhΛEh ⊗ I)Rf(x)

where Eh, Ih generalized interpolation operators and R∗

hϑg(x) := hϑ 2p−1

  • j=0

g(xtω(ϑj), ϑj), ϑj = jhϑ Remark: The action of IhΛEh can be implemented as a convolution (filtering) followed by an interpolation. The convolution kernel (reconstruction filter) depends on Ih and Eh.

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Angular under-sampling causes artifacts

c Andreas Rieder, Wien, AIP 09 – 7 / 22

FBA works well for standard parallel scanning geometry under optimal sampling, that is, h ≈ hϑ (p ≈ πq). In case of severe angular under-sampling (hϑ ≫ h) the FBA reconstructions are corrupted by heavy streak artifacts: h = hϑ/50

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Introducing phantom views

c Andreas Rieder, Wien, AIP 09 – 8 / 22

Lewitt et al.(1978) suggested to increase the angular sampling by interpolating the Radon data linearly w.r.t. the angular variable:

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Introducing phantom views

c Andreas Rieder, Wien, AIP 09 – 8 / 22

Lewitt et al.(1978) suggested to increase the angular sampling by interpolating the Radon data linearly w.r.t. the angular variable: fFBA(x) := 1 4πR∗

hϑ(IhΛEh ⊗ I)Rf(x)

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Introducing phantom views

c Andreas Rieder, Wien, AIP 09 – 8 / 22

Lewitt et al.(1978) suggested to increase the angular sampling by interpolating the Radon data linearly w.r.t. the angular variable: fFBA(x) := 1 4πR∗

hϑ(IhΛEh ⊗ Thϑ)Rf(x)

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Introducing phantom views

c Andreas Rieder, Wien, AIP 09 – 8 / 22

Lewitt et al.(1978) suggested to increase the angular sampling by interpolating the Radon data linearly w.r.t. the angular variable: fPhanFBA(R)(x) := 1 4πR∗

hϑ/R(IhΛEh ⊗ Thϑ)Rf(x),

R ∈ N

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Introducing phantom views

c Andreas Rieder, Wien, AIP 09 – 8 / 22

Lewitt et al.(1978) suggested to increase the angular sampling by interpolating the Radon data linearly w.r.t. the angular variable: fPhanFBA(R)(x) := 1 4πR∗

hϑ/R(IhΛEh ⊗ I)

  • Step 2

(I ⊗ Thϑ)

  • Step 1

Rf(x), R ∈ N Step 1: linear interpolation of Radon data in angular variable Step 2: standard FBA with angular steps size hϑ/R.

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Introducing phantom views

c Andreas Rieder, Wien, AIP 09 – 8 / 22

Lewitt et al.(1978) suggested to increase the angular sampling by interpolating the Radon data linearly w.r.t. the angular variable: fPhanFBA(R)(x) := 1 4πR∗

hϑ/R(IhΛEh ⊗ I)

  • Step 2

(I ⊗ Thϑ)

  • Step 1

Rf(x), R ∈ N Step 1: linear interpolation of Radon data in angular variable Step 2: standard FBA with angular steps size hϑ/R. FBA PhanFBA(2) PhanFBA(5)

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Phantom views reduce streak artifacts

Introduction: FBA augmented by phantom views

⊲

Phantom views reduce streak artifacts Phantom views increase angular convergence rate Bibliographical notes Conclusion c Andreas Rieder, Wien, AIP 09 – 9 / 22

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A different view on PhanFBA I

c Andreas Rieder, Wien, AIP 09 – 10 / 22

With Φ := (IhΛEh ⊗ Thϑ)Rf and ϑj+ ℓ

R := (j + ℓ

R)hϑ we have that

fPhanFBA(R)(x) = hϑ R

2p−1

  • j=0

R−1

  • ℓ=0

Φ(xtω(ϑj+ℓ/R), ϑj+ℓ/R) = hϑ R

2p−1

  • j=0

R−1

  • ℓ=0
  • 1 − ℓ

R

  • Φ(xtω(ϑj+ℓ/R), ϑj) + ℓ

RΦ(xtω(ϑj+ℓ/R), ϑj+1)

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A different view on PhanFBA I

c Andreas Rieder, Wien, AIP 09 – 10 / 22

With Φ := (IhΛEh ⊗ Thϑ)Rf and ϑj+ ℓ

R := (j + ℓ

R)hϑ we have that

fPhanFBA(R)(x) = hϑ R

2p−1

  • j=0

R−1

  • ℓ=0

Φ(xtω(ϑj+ℓ/R), ϑj+ℓ/R) = 1 R

  • fFBA(x) +

R−1

  • ℓ=1
  • 1 − ℓ

R fFBA(U ℓ

R hϑx) + fFBA(U− ℓ R hϑx)

  • where Uϕ ∈ R2×2 is rotation by angle ϕ.
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A different view on PhanFBA I

c Andreas Rieder, Wien, AIP 09 – 10 / 22

With Φ := (IhΛEh ⊗ Thϑ)Rf and ϑj+ ℓ

R := (j + ℓ

R)hϑ we have that

fPhanFBA(R)(x) = hϑ R

2p−1

  • j=0

R−1

  • ℓ=0

Φ(xtω(ϑj+ℓ/R), ϑj+ℓ/R) = 1 R

  • fFBA(x) +

R−1

  • ℓ=1
  • 1 − ℓ

R fFBA(U ℓ

R hϑx) + fFBA(U− ℓ R hϑx)

  • where Uϕ ∈ R2×2 is rotation by angle ϕ. Hence,

fPhanFBA(R)(0) = fFBA(0)

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A different view on PhanFBA I

c Andreas Rieder, Wien, AIP 09 – 10 / 22

With Φ := (IhΛEh ⊗ Thϑ)Rf and ϑj+ ℓ

R := (j + ℓ

R)hϑ we have that

fPhanFBA(R)(x) = hϑ R

2p−1

  • j=0

R−1

  • ℓ=0

Φ(xtω(ϑj+ℓ/R), ϑj+ℓ/R) = 1 R

  • fFBA(x) +

R−1

  • ℓ=1
  • 1 − ℓ

R fFBA(U ℓ

R hϑx) + fFBA(U− ℓ R hϑx)

  • where Uϕ ∈ R2×2 is rotation by angle ϕ. Hence,

fPhanFBA(R)(0) = fFBA(0) and fPhanFBA(R)(x) is the trapezoidal sum with step size 1

R applied to

1 hϑ hϑ

−hϑ

fFBA(Uϕx) Bhϑ(ϕ) dϕ where Bhϑ is the linear B-Spline w.r.t. [−hϑ, hϑ].

−hϑ hϑ 1

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A different view on PhanFBA II

c Andreas Rieder, Wien, AIP 09 – 11 / 22

As PhanFBA(R) is an angular average of FBA, fPhanFBA(R)(x) ≈ 1 hϑ hϑ

−hϑ

fFBA(Uϕx) Bhϑ(ϕ) dϕ, those edges being tangent to a circle centered about the origin are not blurred. The more transversally an edge intersects such a circle the more it gets blurred. FBA PhanFBA(5)

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Phantom views increase angular convergence rate

Introduction: FBA augmented by phantom views Phantom views reduce streak artifacts

⊲

Phantom views increase angular convergence rate Bibliographical notes Conclusion c Andreas Rieder, Wien, AIP 09 – 12 / 22

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The limit R → ∞

c Andreas Rieder, Wien, AIP 09 – 13 / 22

As R → ∞, fPhanFBA(R)(x) = 1 4πR∗

hϑ/R(IhΛEh ⊗ Thϑ)Rf(x)

converges to fPhanFBA(∞)(x) := 1 4πR∗(IhΛEh ⊗ Thϑ)Rf(x) = 1 hϑ hϑ

−hϑ

fFBA(Uϕx) Bhϑ(ϕ) dϕ. Remark: The evaluation of fPhanFBA(∞)(x) can be organized as standard FBA with an additional multiplication of the filtered data by a sparse matrix.

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PhanFBA(R) vs. PhanFBA(∞)

c Andreas Rieder, Wien, AIP 09 – 14 / 22

PhanFBA(5) PhanFBA(∞)

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PhanFBA(∞) vs. FBA: Convergence rates

c Andreas Rieder, Wien, AIP 09 – 15 / 22

Let f ∈ Hα

0 (Ω). Then,

  • 1

4πR∗

hϑ(IhΛEh⊗I)Rf−f

  • L2
  • hmin{αmax, α}+hα

ϑ+hϑhmin{αmax, α−1}

fα, α ≥ 1

  • 1

4πR∗(IhΛEh⊗Thϑ)Rf−f

  • L2
  • hmin{αmax, α}+hmin{5/2, α}

ϑ

  • fα,

α > 1/2 αmax =      3/2 : Shepp-Logan with piecewise constant interpolation 2 : Shepp-Logan with piecewise linear interpolation 5/2 : mod. Shepp-Logan with piecewise linear interpolation

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PhanFBA(∞) vs. FBA: Convergence rates

c Andreas Rieder, Wien, AIP 09 – 15 / 22

Let f ∈ Hα

0 (Ω). Then,

  • 1

4πR∗

hϑ(IhΛEh⊗I)Rf−f

  • L2
  • hmin{αmax, α}+hα

ϑ+hϑhmin{αmax, α−1}

fα, α ≥ 1

  • 1

4πR∗(IhΛEh⊗Thϑ)Rf−f

  • L2
  • hmin{αmax, α}+hmin{5/2, α}

ϑ

  • fα,

α > 1/2 αmax =      3/2 : Shepp-Logan with piecewise constant interpolation 2 : Shepp-Logan with piecewise linear interpolation 5/2 : mod. Shepp-Logan with piecewise linear interpolation f ∈ Hα

0 (Ω), α ≈ 1

2, and hϑ ≈ √ h = ⇒ errFBA ≈ const., errPhanFBA ≈ h1/4

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PhanFBA(∞) vs. FBA: A numerical comparison

c Andreas Rieder, Wien, AIP 09 – 16 / 22

in Hα

0 (Ω), α < 1/2

25 50 75 100 125 150 175 200 0.1 0.15 0.2 0.25 q

  • rel. L2−errors (rect, p=π*q0.5)

PhanFBA(∞) FBA ~q−1/4

h = 1 q , hϑ = π p where p = ⌈π√q⌉ e(q) :=

x∈Xq

  • frecon(x) − f(x)

2

x∈Xq

f(x)2

1/2

, Xq = Ω ∩ Z2/(800)

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PhanFBA(∞) vs. FBA: Reconstructions

c Andreas Rieder, Wien, AIP 09 – 17 / 22 PhanFBA(∞), q=200, p=45, lin.interpol FBA, q=200, p=45, lin. interpol

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Bibliographical notes

Introduction: FBA augmented by phantom views Phantom views reduce streak artifacts Phantom views increase angular convergence rate

⊲

Bibliographical notes Conclusion c Andreas Rieder, Wien, AIP 09 – 18 / 22

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Bibliographical notes

c Andreas Rieder, Wien, AIP 09 – 19 / 22

  • R. M. Lewitt, R. H. T. Bates, T. M. Peters

Image reconstruction from projections II: Modified backprojection methods, Optik 50 (1978), 85–109.

  • R. R. Galigekere, K. Wiesent, D. W. Holdsworth

Techniques to alleviate the effects of view aliasing artifacts in computed tomography, Med. Phys. 26 (1999), 896–904.

  • A. Rieder, A. Faridani

The semi-discrete filtered backprojection algorithm is optimal for tomographic inversion, SIAM J. Numer. Anal. 41 (2003), 869–892.

  • A. Rieder, A. Schneck

Optimality of the fully discrete filtered backprojection algorithm for tomographic inversion, Numer. Math. 108 (2007), 151–175.

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Conclusion

Introduction: FBA augmented by phantom views Phantom views reduce streak artifacts Phantom views increase angular convergence rate Bibliographical notes

⊲ Conclusion

c Andreas Rieder, Wien, AIP 09 – 20 / 22

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What to remember from this talk

c Andreas Rieder, Wien, AIP 09 – 21 / 22

PhanFBA(∞) is an angular average of standard FBA. Streaks intersecting transversally a circle centered about the origin are

  • diminished. Streaks being tangent to such a circle and a whole

neighborhood of the origin remain unaffected by PhanFBA. Phantom views increase the angular convergence rate.

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What to remember from this talk

c Andreas Rieder, Wien, AIP 09 – 21 / 22

PhanFBA(∞) is an angular average of standard FBA. Streaks intersecting transversally a circle centered about the origin are

  • diminished. Streaks being tangent to such a circle and a whole

neighborhood of the origin remain unaffected by PhanFBA. Phantom views increase the angular convergence rate.

Thank you for your attention!

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SLIDE 33

GAMM 2010

MARCH, 22-26

81st Annual Meeting

  • f the International Association
  • f Applied Mathematics and

Mechanics at the University of

Karlsruhe

Gesellschaft für Angewandte Mathematik und Mechanik Institut für Wissenschaftliches Rechnen und Mathematische Modellbildung