Feature Reconstruction in Tomography Approximate Inverse for Combining Reconstruction and Analysis Proof ( Lf ) γ ( x ) = � Lf , e γ ( x , · ) � � f , L ∗ e γ ( x , · ) � = � f , A ∗ ψ γ ( x , · ) � = = � g , ψ γ ( x , · ) � Conditions on e γ such that S γ is a regularization for determining Lf are given in the above mentioned paper.
Feature Reconstruction in Tomography Approximate Inverse for Combining Reconstruction and Analysis Example 1: L = I Consider A : L 2 ( 0 , 1 ) → L 2 ( 0 , 1 ) with � x Af ( x ) = f ( y ) dy 0 Then f = g ′ Mollifier e γ ( x , y ) = 1 2 γ χ [ x − γ, x + γ ] ( y )
Feature Reconstruction in Tomography Approximate Inverse for Combining Reconstruction and Analysis Example 1 continued Auxiliary Equation: � 1 A ∗ ψ γ ( x , y ) = ψ γ ( x , t ) dt = e γ ( x , y ) y leading to ψ γ ( x , y ) = 1 2 γ ( δ x + γ − δ x − γ )( y ) and S γ g ( x ) = g ( x + γ ) − g ( x − γ ) 2 γ
Feature Reconstruction in Tomography Approximate Inverse for Combining Reconstruction and Analysis Example 2: L = d dx Consider A : L 2 ( 0 , 1 ) → L 2 ( 0 , 1 ) with � x Af ( x ) = f ( y ) dy 0 Then Lf = g ′′ Mollifier � y − ( x − γ ) for x − γ ≤ y ≤ x γ e γ ( x , y ) = x + γ − y for x ≤ y ≤ x + γ γ
Feature Reconstruction in Tomography Approximate Inverse for Combining Reconstruction and Analysis Example 2 continued Auxiliary Equation � 1 ψ γ ( x , t ) dt = L ∗ e γ ( x , y ) = − ∂ A ∗ ψ γ ( x , y ) = ∂ y e γ ( x , y ) y leading to ψ γ ( x , y ) = 1 γ 2 ( δ x + γ − 2 δ x + δ x − γ )( y ) and S γ g ( x ) = g ( x + γ ) − 2 g ( x ) + g ( x − γ ) γ 2
Feature Reconstruction in Tomography Approximate Inverse for Combining Reconstruction and Analysis Existence Consider A : H s → H s + α with � Af � H s + α ≥ c 1 � f � H s and L : D ( L ) ⊂ L 2 → L 2 with � Lf � H s − t ≤ c 2 � f � H s Then formally LA † : L 2 → H t − α and regularization via E γ : H t − α → L 2
Feature Reconstruction in Tomography Approximate Inverse for Combining Reconstruction and Analysis Possibilities
Feature Reconstruction in Tomography Approximate Inverse for Combining Reconstruction and Analysis Possibilities t < 0: addtional regularization needed ( Ex.: L differentiation )
Feature Reconstruction in Tomography Approximate Inverse for Combining Reconstruction and Analysis Possibilities t < 0: addtional regularization needed ( Ex.: L differentiation ) t = 0: only regularization of A needed ( Ex.: Wavelet decomposition )
Feature Reconstruction in Tomography Approximate Inverse for Combining Reconstruction and Analysis Possibilities t < 0: addtional regularization needed ( Ex.: L differentiation ) t = 0: only regularization of A needed ( Ex.: Wavelet decomposition ) t > α : no regularization at all ( Ex.: diffusion )
Feature Reconstruction in Tomography Approximate Inverse for Combining Reconstruction and Analysis Invariances Theorem (L, 1997/2007) Let the operator T 1 , T 2 , T 3 be such that L ∗ T 1 T 2 L ∗ = T 2 A ∗ A ∗ T 3 = and solve for a reference mollifier E γ the equation R ∗ Ψ γ = L ∗ E γ Then the general reconstruction kernel for the general mollifier e γ = T 1 E γ is ψ γ = T 3 Ψ γ and f γ = � g , T 3 Ψ γ �
Feature Reconstruction in Tomography Approximate Inverse for Combining Reconstruction and Analysis Possible Calculation of Reconstruction Kernel If the reconstruction of f is achieved as � E γ f ( x ) = ψ γ ( x , y ) g ( y ) dy then the reconstruction kernel for calculating L β f can be computed as ¯ ψ βγ ( · , y ) = L β ψ γ ( · , y )
Feature Reconstruction in Tomography Radon Transform and Edge Detection Content Formulation of Problem 1 Inverse Problems and Regularization 2 Approximate Inverse for Combining Reconstruction and 3 Analysis Radon Transform and Edge Detection 4 Radon Transform and Diffusion 5 Radon Transform and Wavelet Decomposition 6 Nonlinear Problems 7 Future 8
Feature Reconstruction in Tomography Radon Transform and Edge Detection Radon Transform � R 2 f ( x ) δ ( s − x ⊤ θ ) dx R f ( θ, s ) = θ ∈ S 1 and s ∈ R . R f ( θ, σ ) = ( 2 π ) 1 / 2 ˆ � f ( σθ ) R − 1 = 1 4 π R ∗ I − 1 where R ∗ is the adjoint operator from L 2 to L 2 known as backprojection � R ∗ g ( x ) = S 1 g ( θ, x ⊤ θ ) d θ and the Riesz potential I − 1 is defined with the Fourier transform � I − 1 g ( θ, σ ) = | σ | ˆ g ( θ, σ )
Feature Reconstruction in Tomography Radon Transform and Edge Detection Radon Transform and Derivatives The Radon transform of a derivative is R ∂ ∂ f ( θ, s ) = θ k ∂ s R f ( θ, s ) ∂ x k
Feature Reconstruction in Tomography Radon Transform and Edge Detection Invariances Let for x ∈ R 2 the shift operators T x 2 f ( y ) = f ( y − x ) and T x ⊤ θ g ( θ, s ) = g ( θ, s − x ⊤ θ ) then 3 2 = T x ⊤ θ R θ T x R θ 3 Let U be a unitary 2 × 2 matrix and D U 2 f ( y ) = f ( Uy ) . then R D U 2 = D U 3 R where D U 3 g ( θ, s ) = g ( U θ, s ) . ( T R ) ∗ = R ∗ T ∗
Feature Reconstruction in Tomography Radon Transform and Edge Detection Reconstruction Kernel Theorem Let the mollifier be given as e γ ( x , y ) = E γ ( � x − y � ) . Then the reconstruction kernel for L k = ∂/∂ x k is � � s − x ⊤ θ ψ k ,γ = θ k Ψ γ with Ψ γ = − 1 ∂ ∂ s I − 1 R E γ 4 π
Feature Reconstruction in Tomography Radon Transform and Edge Detection Reconstruction Kernel Theorem Let the mollifier be given as e γ ( x , y ) = E γ ( � x − y � ) . Then the reconstruction kernel for L k = ∂/∂ x k is � � s − x ⊤ θ ψ k ,γ = θ k Ψ γ with Ψ γ = − 1 ∂ ∂ s I − 1 R E γ 4 π ∂ f Same costs for calculating ∂ x k as for calculating f .
Feature Reconstruction in Tomography Radon Transform and Edge Detection Proof R ∗ ψ k γ L ∗ = k e γ R − 1 R L ∗ = k e γ 1 4 π R ∗ I − 1 R L ∗ = k e γ Hence ψ k γ = 1 4 π I − 1 R L ∗ k e γ L ∗ k = − L k and differentiation rule completes the proof.
Feature Reconstruction in Tomography Radon Transform and Edge Detection Example Let E βγ be given as E β γ ( ξ ) = ( 2 π ) − 1 sinc � ξ � π sinc � ξ � π � 2 γ χ [ − γ,γ ] ( � ξ � ) β � �� � Shepp − Logan kernel Then the reconstruction kernel for β = γ is 1 8 ℓ ψ π/ h ( ℓ h ) = 3 + 4 ℓ 2 � 2 − 64 ℓ 2 , ℓ ∈ Z � π 2 h 3
Feature Reconstruction in Tomography Radon Transform and Edge Detection Kernel with β = γ and β = 2 γ
Feature Reconstruction in Tomography Radon Transform and Edge Detection Exact Data Shepp - Logan phantom with original densities
Feature Reconstruction in Tomography Radon Transform and Edge Detection Noisy Data
Feature Reconstruction in Tomography Radon Transform and Edge Detection Differentiation of Image vs. New Method
Feature Reconstruction in Tomography Radon Transform and Edge Detection New Differentiation with respect to x and y .
Feature Reconstruction in Tomography Radon Transform and Edge Detection Lambda and wrong Parameter β
Feature Reconstruction in Tomography Radon Transform and Edge Detection Differentiation of Image vs. New Method Sum of absolute values of the two derivatives Apply support theorem of Boman and Quinto.
Feature Reconstruction in Tomography Radon Transform and Edge Detection Reconstruction from Measured Data, Fan Beam Geometry
Feature Reconstruction in Tomography Radon Transform and Edge Detection Applications 3D CT ( obvious with Feldkamp algorithm ) Phase Contrast Phase Contrast with Eikonal ( High Energy ) Approximation Electron Microscopy Adapt for cone beam tomography and inversion formula D − 1 = cD ∗ TM Γ T where T is a differential and M Γ a trajectory dependent multiplication operator
Feature Reconstruction in Tomography Radon Transform and Diffusion Content Formulation of Problem 1 Inverse Problems and Regularization 2 Approximate Inverse for Combining Reconstruction and 3 Analysis Radon Transform and Edge Detection 4 Radon Transform and Diffusion 5 Radon Transform and Wavelet Decomposition 6 Nonlinear Problems 7 Future 8
Feature Reconstruction in Tomography Radon Transform and Diffusion Diffusion Processes Smoothing of noisy images f by diffusion ∂ ∂ t u = ∆ u u ( 0 , x ) = f ( x ) Problem Edges are smeared out Perona-Malik: Nonlinear Diffusion u u t = ∆ |∇ u |
Feature Reconstruction in Tomography Radon Transform and Diffusion Example Shepp-Logan phantom, 12% noise
Feature Reconstruction in Tomography Radon Transform and Diffusion Example Shepp-Logan phantom, 12% noise Normal reconstruction (left) and smoothing by linear diffusion, T = 0 . 1
Feature Reconstruction in Tomography Radon Transform and Diffusion Example Shepp-Logan phantom, 12% noise Normal reconstruction (left) and smoothing by linear diffusion, T = 0 . 1
Feature Reconstruction in Tomography Radon Transform and Diffusion Fundamental Solution L.: Inverse Problems and Imaging, to appear Fundamental Solution of the heat equation 1 ( 4 π t ) n / 2 exp ( −| x | 2 / 4 t ) G ( t , x ) = The solution of the heat equation for initial condition u ( 0 , x ) = f ( x ) can be calculated as L T u ( 0 , · ) = u ( T , · ) = G ( T , · ) ∗ f
Feature Reconstruction in Tomography Radon Transform and Diffusion Combining Reconstruction and Diffusion Let e γ ( x , · ) = δ x and L = L T Then the reconstruction kernel for time t = T has to fulfil R ∗ ψ T ( x , y ) = L ∗ T δ x ( y ) = G ( T , x − y ) and is of convolution type.
Feature Reconstruction in Tomography Radon Transform and Diffusion Reconstruction Kernel Let E γ ( x ) = δ x Then � ∞ ψ T ( s ) = 1 σ exp ( − T σ 2 ) cos ( s σ ) d σ 2 π 0 and � �� 1 � s s 1 − ψ T ( s ) = √ D √ 4 π 2 T 2 T T with the Dawson integral � s D ( s ) = exp ( − s 2 ) exp ( t 2 ) dt 0
Feature Reconstruction in Tomography Radon Transform and Diffusion Numerical Tests with noisy Data, 12 % noise
Feature Reconstruction in Tomography Radon Transform and Diffusion Numerical Tests with noisy Data, 12 % noise Smoothing by linear diffusion ( left ) and combined reconstruction (right).
Feature Reconstruction in Tomography Radon Transform and Diffusion Reconstruction of Derivatives for Noisy Data Reconstruction Kernel for Derivative in Direction x k � � 1 − s 2 � � θ k s s ψ T , k ( s ) = − √ + D ( √ ) 4 π 2 T 3 / 2 2 T 2 2 T T
Feature Reconstruction in Tomography Radon Transform and Diffusion Noise and Differentiation, 6 % Noise
Feature Reconstruction in Tomography Radon Transform and Diffusion Noise and Differentiation, 6 % Noise Smoothing by linear diffusion ( left ) and combined reconstruction (right).
Feature Reconstruction in Tomography Radon Transform and Diffusion Noise and Differentiation, 6 % and 12 % Noise Combined reconstruction of sum of absolute values of derivatives for 6 % and 12 % noise .
Feature Reconstruction in Tomography Radon Transform and Diffusion Noise and Differentiation, 6 % and 12 % Noise Combined reconstruction of sum of absolute values of derivatives for 6 % and 12 % noise . No useful results with 12 % noise with separate calculation of reconstruction and smoothed derivative.
Feature Reconstruction in Tomography Radon Transform and Wavelet Decomposition Content Formulation of Problem 1 Inverse Problems and Regularization 2 Approximate Inverse for Combining Reconstruction and 3 Analysis Radon Transform and Edge Detection 4 Radon Transform and Diffusion 5 Radon Transform and Wavelet Decomposition 6 Nonlinear Problems 7 Future 8
Feature Reconstruction in Tomography Radon Transform and Wavelet Decomposition Wavelet Components Combination of reconstruction and wavelet decomposition: L, Maass, Rieder, 1994 Image reconstruction and wavelets : Holschneider, 1991 Berenstein, Walnut, 1996 Bhatia, Karl, Willsky, 1996 Bonnet, Peyrin, Turjman, 2002 Wang et al, 2004
Feature Reconstruction in Tomography Radon Transform and Wavelet Decomposition Inversion with Wavelets L.,M AASS ,R IEDER 94 Represent the solution of Rf = g as � � � c M d m f = k ϕ Mk + ℓ ψ m ℓ m k ℓ
Feature Reconstruction in Tomography Radon Transform and Wavelet Decomposition Inversion with Wavelets L.,M AASS ,R IEDER 94 Represent the solution of Rf = g as � � � c M d m f = k ϕ Mk + ℓ ψ m ℓ m k ℓ Precompute v Mk , w m ℓ as R ∗ v Mk = ϕ Mk R ∗ w m ℓ = ψ m ℓ
Feature Reconstruction in Tomography Radon Transform and Wavelet Decomposition Inversion with Wavelets L.,M AASS ,R IEDER 94 Represent the solution of Rf = g as � � � c M d m f = k ϕ Mk + ℓ ψ m ℓ m k ℓ Precompute v Mk , w m ℓ as R ∗ v Mk = ϕ Mk R ∗ w m ℓ = ψ m ℓ Then c M k = � g , v Mk � d m ℓ = � g , w m ℓ �
Feature Reconstruction in Tomography Radon Transform and Wavelet Decomposition This is joint work with Steven Oeckl Department Process Integrated Inspection Systems Fraunhofer IIS
Feature Reconstruction in Tomography Radon Transform and Wavelet Decomposition New Application, first in NDT In-line inspection in production process Reconstruction result: Three-dimensional registration of the object Main inspection task Dimensional measurement Detection of blow holes and porosity
Feature Reconstruction in Tomography Radon Transform and Wavelet Decomposition Typical Scanning Geometry in NDT
Feature Reconstruction in Tomography Radon Transform and Wavelet Decomposition Wavelet Coefficient for 2D Parallel Geoemtry � � 1 I − 1 g ( θ, s ) R θ ψ jk ( s ) dsd θ � f , ψ jk � = 4 π S 1 R I � 1 � � I − 1 g ( θ, · ) ∗ R − θ ψ j 0 ( � D j k , θ � ) d θ = 4 π S 1
Feature Reconstruction in Tomography Radon Transform and Wavelet Decomposition 2D Shepp-Logan Phantom, Fan Beam
Feature Reconstruction in Tomography Radon Transform and Wavelet Decomposition Cone Beam: decentralized slice
Feature Reconstruction in Tomography Radon Transform and Wavelet Decomposition Cone Beam ’Local Reconstruction’
Feature Reconstruction in Tomography Nonlinear Problems Content Formulation of Problem 1 Inverse Problems and Regularization 2 Approximate Inverse for Combining Reconstruction and 3 Analysis Radon Transform and Edge Detection 4 Radon Transform and Diffusion 5 Radon Transform and Wavelet Decomposition 6 Nonlinear Problems 7 Future 8
Feature Reconstruction in Tomography Nonlinear Problems Considered Nonlinearity � ∞ Af = A ℓ f ℓ = 1 where � A 1 f ( x ) = k 1 ( x , y ) f ( y ) dy � A 2 f ( x ) = k 2 ( x , y 1 , y 2 ) f ( y 1 ) f ( y 2 ) dy Considered by : Snieder for Backus-Gilbert Variants, 1991 L: Approximate Inverse, 1995
Feature Reconstruction in Tomography Nonlinear Problems Ansatz for Inversion, Presented for 2 Terms Put S γ g = S 1 g + S 2 g with S 1 g ( x ) = � g , ψ 1 ( x , · ) � and � � S 2 g ( x ) = g , � Ψ 2 ( x , · , · ) , g � Replace g by Af and omit higher order terms. Then S γ Af = S 1 A 1 f + S 1 A 2 f + S 2 A 1 f � �� � � �� � ≈ 0 ≈ LE γ f
Feature Reconstruction in Tomography Nonlinear Problems Determination of the Square Term Consider A : L 2 (Ω) → R N Minimizing the defect leads to the equation N � A 1 A ∗ 1 Ψ γ ( x ) A 1 A ∗ 1 = − ψ γ, n ( x ) B n n = 1 where � B n = k 1 ( y 1 ) k 2 n ( y 1 , y 2 ) k 1 ( y 2 ) dy 1 dy 2 Ω × Ω Note: B n is independent of x !
Feature Reconstruction in Tomography Nonlinear Problems Example: A : L 2 → R N Vibrating String u ′′ ( x ) + ρ ( x ) ω 2 u ( x ) = 0 , u ( 0 ) = u ( 1 ) = 0 ρ ( x ) = 1 + f ( x ) Data g n = ω 2 n − ( ω 0 n ) 2 , n = 1 , . . . , N , ( ω 0 n ) 2 k 1 , n ( y ) = − 2 sin 2 ( n π y ) k 2 , n ( y 1 , y 2 ) = 4 sin 2 ( n π y 1 ) sin 2 ( n π y 2 ) � n 2 + 4 n 2 − m 2 sin ( n π y 1 ) sin ( m π y 1 ) sin ( n π y 2 ) sin ( m π y 2 ) n � = m
Feature Reconstruction in Tomography Nonlinear Problems Numerical Test, L = d dx Linear and quadratic approximation of function (left) and derivative (right).
Recommend
More recommend
