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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 / 22

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

2D-Radon transform (parallel scanning geometry) � s R f ( s, ϑ ) := f ( x ) d σ ( x ) ( ) s, ϑ l ϑ l ( s,ϑ ) ∩ Ω tomographic inversion: R f ( s, ϑ ) = g ( s, ϑ ) R : L 2 (Ω) → L 2 ( Z ) , Z = [ − 1 , 1] × [0 , 2 π ] c � Andreas Rieder, Wien, AIP 09 – 4 / 22

Inversion formula f = 1 4 π R ∗ (Λ ⊗ I ) R f R ∗ : L 2 ( Z ) → L 2 (Ω) Backprojection operator � 2 π R ∗ g ( x ) = g ( x t ω ( ϑ ) , ϑ ) d ϑ, ω ( ϑ ) = (cos ϑ, sin ϑ ) t 0 Λ: H α ( R ) → H α − 1 ( R ) Riesz potential � Λ u ( ξ ) = | ξ | � u ( ξ ) . c � Andreas Rieder, Wien, AIP 09 – 5 / 22

Filtered backprojection algorithm (FBA) discrete Radon data D = { R f ( kh, jh ϑ ) : k = − q, . . . , q, j = 0 , . . . , 2 p − 1 } , h = 1 /q, h ϑ = π/p f FBA ( x ) := 1 4 π R ∗ h ϑ (I h Λ E h ⊗ I ) R f ( x ) where E h , I h generalized interpolation operators and 2 p − 1 � R ∗ g ( x t ω ( ϑ j ) , ϑ j ) , h ϑ g ( x ) := h ϑ ϑ j = jh ϑ j =0 Remark: The action of I h Λ E h can be implemented as a convolution (filtering) followed by an interpolation. The convolution kernel (reconstruction filter) depends on I h and E h . c � Andreas Rieder, Wien, AIP 09 – 6 / 22

Angular under-sampling causes artifacts FBA works well for standard parallel scanning geometry under optimal sampling, that is, h ≈ h ϑ ( p ≈ πq ) . In case of severe angular under-sampling ( h ϑ ≫ h ) the FBA reconstructions are corrupted by heavy streak artifacts: h = h ϑ / 50 c � Andreas Rieder, Wien, AIP 09 – 7 / 22

Introducing phantom views Lewitt et al.(1978) suggested to increase the angular sampling by interpolating the Radon data linearly w.r.t. the angular variable: c � Andreas Rieder, Wien, AIP 09 – 8 / 22

Introducing phantom views Lewitt et al.(1978) suggested to increase the angular sampling by interpolating the Radon data linearly w.r.t. the angular variable: f FBA ( x ) := 1 4 π R ∗ h ϑ (I h Λ E h ⊗ I ) R f ( x ) c � Andreas Rieder, Wien, AIP 09 – 8 / 22

Introducing phantom views Lewitt et al.(1978) suggested to increase the angular sampling by interpolating the Radon data linearly w.r.t. the angular variable: f FBA ( x ) := 1 4 π R ∗ h ϑ (I h Λ E h ⊗ T h ϑ ) R f ( x ) c � Andreas Rieder, Wien, AIP 09 – 8 / 22

Introducing phantom views Lewitt et al.(1978) suggested to increase the angular sampling by interpolating the Radon data linearly w.r.t. the angular variable: f PhanFBA ( R ) ( x ) := 1 4 π R ∗ h ϑ /R (I h Λ E h ⊗ T h ϑ ) R f ( x ) , R ∈ N c � Andreas Rieder, Wien, AIP 09 – 8 / 22

Introducing phantom views Lewitt et al.(1978) suggested to increase the angular sampling by interpolating the Radon data linearly w.r.t. the angular variable: f PhanFBA ( R ) ( x ) := 1 4 π R ∗ h ϑ /R (I h Λ E h ⊗ I ) ( I ⊗ T h ϑ ) R f ( x ) , R ∈ N � �� � � �� � Step 1 Step 2 Step 1: linear interpolation of Radon data in angular variable Step 2: standard FBA with angular steps size h ϑ /R . c � Andreas Rieder, Wien, AIP 09 – 8 / 22

Introducing phantom views Lewitt et al.(1978) suggested to increase the angular sampling by interpolating the Radon data linearly w.r.t. the angular variable: f PhanFBA ( R ) ( x ) := 1 4 π R ∗ h ϑ /R (I h Λ E h ⊗ I ) ( I ⊗ T h ϑ ) R f ( x ) , R ∈ N � �� � � �� � Step 1 Step 2 Step 1: linear interpolation of Radon data in angular variable Step 2: standard FBA with angular steps size h ϑ /R . FBA PhanFBA( 2 ) PhanFBA( 5 ) c � Andreas Rieder, Wien, AIP 09 – 8 / 22

Introduction: FBA augmented by phantom views Phantom views reduce streak ⊲ artifacts Phantom views increase angular convergence rate Bibliographical notes Phantom views reduce streak artifacts Conclusion c � Andreas Rieder, Wien, AIP 09 – 9 / 22

A different view on PhanFBA I R := ( j + ℓ With Φ := (I h Λ E h ⊗ T h ϑ ) R f and ϑ j + ℓ R ) h ϑ we have that 2 p − 1 R − 1 � � f PhanFBA( R ) ( x ) = h ϑ Φ( x t ω ( ϑ j + ℓ/R ) , ϑ j + ℓ/R ) R j =0 ℓ =0 �� � � 2 p − 1 R − 1 � � = h ϑ 1 − ℓ Φ( x t ω ( ϑ j + ℓ/R ) , ϑ j ) + ℓ R Φ( x t ω ( ϑ j + ℓ/R ) , ϑ j +1 ) R R j =0 ℓ =0 c � Andreas Rieder, Wien, AIP 09 – 10 / 22

A different view on PhanFBA I R := ( j + ℓ With Φ := (I h Λ E h ⊗ T h ϑ ) R f and ϑ j + ℓ R ) h ϑ we have that 2 p − 1 R − 1 � � f PhanFBA( R ) ( x ) = h ϑ Φ( x t ω ( ϑ j + ℓ/R ) , ϑ j + ℓ/R ) R j =0 ℓ =0 � �� � � � R − 1 � = 1 1 − ℓ f FBA ( x ) + f FBA ( U ℓ R h ϑ x ) + f FBA ( U − ℓ R h ϑ x ) R R ℓ =1 where U ϕ ∈ R 2 × 2 is rotation by angle ϕ . c � Andreas Rieder, Wien, AIP 09 – 10 / 22

A different view on PhanFBA I R := ( j + ℓ With Φ := (I h Λ E h ⊗ T h ϑ ) R f and ϑ j + ℓ R ) h ϑ we have that 2 p − 1 R − 1 � � f PhanFBA( R ) ( x ) = h ϑ Φ( x t ω ( ϑ j + ℓ/R ) , ϑ j + ℓ/R ) R j =0 ℓ =0 � �� � � � R − 1 � = 1 1 − ℓ f FBA ( x ) + f FBA ( U ℓ R h ϑ x ) + f FBA ( U − ℓ R h ϑ x ) R R ℓ =1 where U ϕ ∈ R 2 × 2 is rotation by angle ϕ . Hence, f PhanFBA( R ) (0) = f FBA (0) c � Andreas Rieder, Wien, AIP 09 – 10 / 22

A different view on PhanFBA I R := ( j + ℓ With Φ := (I h Λ E h ⊗ T h ϑ ) R f and ϑ j + ℓ R ) h ϑ we have that 2 p − 1 R − 1 � � f PhanFBA( R ) ( x ) = h ϑ Φ( x t ω ( ϑ j + ℓ/R ) , ϑ j + ℓ/R ) R j =0 ℓ =0 � �� � � � R − 1 � = 1 1 − ℓ f FBA ( x ) + f FBA ( U ℓ R h ϑ x ) + f FBA ( U − ℓ R h ϑ x ) R R ℓ =1 where U ϕ ∈ R 2 × 2 is rotation by angle ϕ . Hence, f PhanFBA( R ) (0) = f FBA (0) and f PhanFBA( R ) ( x ) is the trapezoidal sum with step size 1 R applied to � h ϑ 1 1 f FBA ( U ϕ x ) B h ϑ ( ϕ ) d ϕ h ϑ − h ϑ where B h ϑ is the linear B-Spline w.r.t. [ − h ϑ , h ϑ ] . − h ϑ h ϑ c � Andreas Rieder, Wien, AIP 09 – 10 / 22

A different view on PhanFBA II As PhanFBA(R) is an angular average of FBA, � h ϑ f PhanFBA( R ) ( x ) ≈ 1 f FBA ( U ϕ x ) B h ϑ ( ϕ ) d ϕ, h ϑ − h ϑ 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 ) c � Andreas Rieder, Wien, AIP 09 – 11 / 22

Introduction: FBA augmented by phantom views Phantom views reduce streak artifacts Phantom views increase angular ⊲ convergence rate Phantom views increase angular convergence Bibliographical notes rate Conclusion c � Andreas Rieder, Wien, AIP 09 – 12 / 22

The limit R → ∞ As R → ∞ , f PhanFBA ( R ) ( x ) = 1 4 π R ∗ h ϑ /R (I h Λ E h ⊗ T h ϑ ) R f ( x ) converges to f PhanFBA ( ∞ ) ( x ) := 1 4 π R ∗ (I h Λ E h ⊗ T h ϑ ) R f ( x ) � h ϑ = 1 f FBA ( U ϕ x ) B h ϑ ( ϕ ) d ϕ. h ϑ − h ϑ Remark: The evaluation of f PhanFBA ( ∞ ) ( x ) can be organized as standard FBA with an additional multiplication of the filtered data by a sparse matrix. c � Andreas Rieder, Wien, AIP 09 – 13 / 22

PhanFBA( R ) vs. PhanFBA( ∞ ) PhanFBA( 5 ) PhanFBA( ∞ ) c � Andreas Rieder, Wien, AIP 09 – 14 / 22

PhanFBA( ∞ ) vs. FBA: Convergence rates Let f ∈ H α 0 (Ω) . Then, � � � ϑ + h ϑ h min { α max , α − 1 } � � 1 � � 4 π R ∗ h min { α max , α } + h α h ϑ (I h Λ E h ⊗ I ) R f − f � f � α , L 2 � � α ≥ 1 � � � � � 1 � � h min { α max , α } + h min { 5 / 2 , α } 4 π R ∗ (I h Λ E h ⊗ T h ϑ ) R f − f � f � α , α > 1 / 2 L 2 � � ϑ  3 / 2 : Shepp-Logan with piecewise constant interpolation   α max = 2 : Shepp-Logan with piecewise linear interpolation   5 / 2 : mod. Shepp-Logan with piecewise linear interpolation c � Andreas Rieder, Wien, AIP 09 – 15 / 22