Feature Reconstruction in Tomography

Feature Reconstruction in Tomography Alfred K. Louis Institut für Angewandte Mathematik Universität des Saarlandes 66041 Saarbrücken http://www.num.uni-sb.de louis@num.uni-sb.de Wien, July 20, 2009

Images

Typical Procedure

Typical Procedure Image Reconstruction

Typical Procedure Image Reconstruction Image Enhancement

Typical Procedure Image Reconstruction Image Enhancement Disadvantage: The two operations are non adjusted and possibly too time-consuming

Requirements for Repeated Solution of Same Problem with Different Data

Requirements for Repeated Solution of Same Problem with Different Data Combination of reconstruction and image analysis in ONE step

Requirements for Repeated Solution of Same Problem with Different Data Combination of reconstruction and image analysis in ONE step Precompute solution operator for efficient evaluation of the same problem with different data sets

Content 1 Formulation of Problem 2 Inverse Problems and Regularization 3 Approximate Inverse for Combining Reconstruction and Analysis 4 Radon Transform and Edge Detection 5 Radon Transform and Diffusion 6 Radon Transform and Wavelet Decomposition 7 Nonlinear Problems 8 Future

Formulation of Problem Content 1 Formulation of Problem 2 Inverse Problems and Regularization 3 Approximate Inverse for Combining Reconstruction and Analysis 4 Radon Transform and Edge Detection 5 Radon Transform and Diffusion 6 Radon Transform and Wavelet Decomposition 7 Nonlinear Problems 8 Future

Formulation of Problem Target Combine the two steps in ONE algorithm Instead of Solve reconstruction part Af = g Evaluate the result Lf Compute directly LA g

Formulation of Problem Feature Determination Image Enhancement Image Registration Classification

Formulation of Problem Examples of Existing Methods A = R, L = I 1 Lambda CT 2D : Smith, Vainberg Lambda 3D: L-Maass, Katsevich Lambda with SPECT, ET: Quinto, Öktem Lambda with SONAR : L., Quinto

Formulation of Problem Image Analysis compute enhanced image Lf operate on Lf

Formulation of Problem Image Analysis compute enhanced image Lf operate on Lf Example for Calculation Edge Detection L kβ f = ( ) G β f x k

Formulation of Problem Image Analysis compute enhanced image Lf operate on Lf Example for Calculation Edge Detection L kβ f = ( ) G β f x k L. 96: Compute directly derivative of solution Schuster, 2000 : Compute divergence of vector fields

Formulation of Problem Applications for X-Ray CT Detect defects like blowholes In-line inspection 3D Dimensioning ( dimensionelles Messen )

Formulation of Problem X-Ray CT

Formulation of Problem X-Ray Tomography Assumptions: X-Rays travel on straight lines Attenuation proportional to path length Attenuation proportional to number of photons Radon Transform I = I t f g L = ln(i 0 /I L ) = f dt L

Inverse Problems Content 1 Formulation of Problem 2 Inverse Problems and Regularization 3 Approximate Inverse for Combining Reconstruction and Analysis 4 Radon Transform and Edge Detection 5 Radon Transform and Diffusion 6 Radon Transform and Wavelet Decomposition 7 Nonlinear Problems 8 Future

Inverse Problems Inverse Problems OBSERVATION g SEARCHED-FOR DISTRIBUTION f Af = g A L(X, Y) Integral or Differential - Operator Difficulties: Solution does not exist Solution is not unique Solution does not depend continuously on g Problem ill - posed ( mal posé )

Inverse Problems Linear Regularization Y 1 A + Mγ A : X Y M γ X 1 A + S γ = M γ A + = A + Mγ

Inverse Problems General Purpose Regularization Methods R γ g = µ σ 1 µ g, u µ Y v µ

Inverse Problems General Purpose Regularization Methods R γ g = µ σ 1 µ F γ (σ µ ) g, u µ Y v µ

Inverse Problems General Purpose Regularization Methods R γ g = µ σ 1 µ F γ (σ µ ) g, u µ Y v µ Truncated SVD F γ (t) = { 1 : t γ 0 : t < γ

Inverse Problems General Purpose Regularization Methods R γ g = µ σ 1 µ F γ (σ µ ) g, u µ Y v µ Truncated SVD F γ (t) = { 1 : t γ 0 : t < γ Tikhonov - Phillips - Regularization ( Levenberg-Marquardt ) F γ (t) = t 2 /(t 2 + γ 2 )

Inverse Problems Regularization by Linear Functionals Likht, 1967 Backus-Gilbert, 1967 L.-Maass, 1990 L, 1996

Inverse Problems Regularization by Linear Functionals Likht, 1967 Backus-Gilbert, 1967 L.-Maass, 1990 L, 1996 Approximate Inverse: choose mollifier e γ (x, ) δ x and solve A ψ γ (x, ) = e γ (x, ) put f γ (x) := E γ f (x) := f, e γ (x, ) = g, ψ γ (x, )

Approximate Inverse for Combining Reconstruction and Analysis Content 1 Formulation of Problem 2 Inverse Problems and Regularization 3 Approximate Inverse for Combining Reconstruction and Analysis 4 Radon Transform and Edge Detection 5 Radon Transform and Diffusion 6 Radon Transform and Wavelet Decomposition 7 Nonlinear Problems 8 Future

Approximate Inverse for Combining Reconstruction and Analysis Approximate Inverse L.: SIAM J. Imaging Sciences 1 188-208, 2008 Aim Solve Af = g and compute Lf in one step Smoothed version (Lf ) γ = Lf, e γ with mollifier e γ

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, )

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.

Approximate Inverse for Combining Reconstruction and Analysis Example 1: L = I Consider with Then Mollifier A : L 2 (0, 1) L 2 (0, 1) Af (x) = x 0 f = g f (y)dy e γ (x, y) = 1 2γ χ [x γ,x+γ](y)

Approximate Inverse for Combining Reconstruction and Analysis Example 1 continued Auxiliary Equation: A ψ γ (x, y) = 1 y ψ γ (x, t)dt = e γ (x, y) leading to and ψ γ (x, y) = 1 2γ (δ x+γ δ x γ )(y) S γ g(x) = g(x + γ) g(x γ) 2γ

Approximate Inverse for Combining Reconstruction and Analysis Example 2: L = d dx Consider with Then Mollifier A : L 2 (0, 1) L 2 (0, 1) Af (x) = x 0 Lf = g f (y)dy e γ (x, y) = { y (x γ) γ x+γ y γ for x γ y x for x y x + γ

Approximate Inverse for Combining Reconstruction and Analysis Example 2 continued Auxiliary Equation A ψ γ (x, y) = leading to 1 y ψ γ (x, t)dt = L e γ (x, y) = y e γ(x, y) ψ γ (x, y) = 1 γ 2 (δ x+γ 2δ x + δ x γ )(y) and S γ g(x) = g(x + γ) 2g(x) + g(x γ) γ 2

Approximate Inverse for Combining Reconstruction and Analysis Existence Consider with and with Then formally and regularization via A : H s H s+α Af H s+α c 1 f H s L : D(L) L 2 L 2 Lf H s t c 2 f H s LA : L 2 H t α E γ : H t α L 2

Approximate Inverse for Combining Reconstruction and Analysis Possibilities

Approximate Inverse for Combining Reconstruction and Analysis Possibilities t < 0: addtional regularization needed ( Ex.: L differentiation )

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 )

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

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)

Radon Transform and Edge Detection Content 1 Formulation of Problem 2 Inverse Problems and Regularization 3 Approximate Inverse for Combining Reconstruction and Analysis 4 Radon Transform and Edge Detection 5 Radon Transform and Diffusion 6 Radon Transform and Wavelet Decomposition 7 Nonlinear Problems 8 Future

Radon Transform and Edge Detection Radon Transform Rf (θ, s) = f (x)δ(s x θ)dx R 2 θ S 1 and s R. Rf (θ, σ) = (2π) 1/2ˆf (σθ) R 1 = 1 4π R I 1 where R is the adjoint operator from L 2 to L 2 known as backprojection R g(x) = g(θ, x θ)dθ S 1 and the Riesz potential I 1 is defined with the Fourier transform I 1 g(θ, σ) = σ ĝ(θ, σ)

Radon Transform and Edge Detection Radon Transform and Derivatives The Radon transform of a derivative is R f (θ, s) = θ k Rf (θ, s) x k s

Radon Transform and Edge Detection Invariances Let for x R 2 the shift operators T2 x f (y) = f (y x) and T x θ 3 g(θ, s) = g(θ, s x θ) then R θ T x 2 = T x θ 3 R θ Let U be a unitary 2 2 matrix and D2 U f (y) = f (Uy). then where D3 U g(θ, s) = g(uθ, s). RD U 2 = DU 3 R (T R) = R T

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 ψ k,γ = θ k Ψ γ ( s x θ ) with Ψ γ = 1 4π s I 1 RE γ

Radon Transform and Edge Detection Proof R ψ kγ = L k e γ = R 1 RL k e γ = 1 4π R I 1 RL k e γ Hence ψ kγ = 1 4π I 1 RL k e γ L k = L k and differentiation rule completes the proof.

Radon Transform and Edge Detection Example Let E βγ be given as Ê β γ(ξ) = (2π) 1 sinc ξ π β Then the reconstruction kernel for β = γ is sinc ξ π 2γ χ [ γ,γ]( ξ ) }{{} Shepp Logan kernel ψ π/h (lh) = 1 8l π 2 h 3 ( ) 3 + 4l 2 2, l Z 64l 2

Radon Transform and Edge Detection Kernel with β = γ and β = 2γ

Radon Transform and Edge Detection Exact Data Shepp - Logan phantom with original densities

Radon Transform and Edge Detection Noisy Data

Radon Transform and Edge Detection Differentiation of Image vs. New Method

Radon Transform and Edge Detection New Differentiation with respect to x and y.

Radon Transform and Edge Detection Lambda and wrong Parameter β

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.

Radon Transform and Edge Detection Reconstruction from Measured Data, Fan Beam Geometry

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

Radon Transform and Diffusion Content 1 Formulation of Problem 2 Inverse Problems and Regularization 3 Approximate Inverse for Combining Reconstruction and Analysis 4 Radon Transform and Edge Detection 5 Radon Transform and Diffusion 6 Radon Transform and Wavelet Decomposition 7 Nonlinear Problems 8 Future

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 t = u u

Radon Transform and Diffusion Example Shepp-Logan phantom, 12% noise

Radon Transform and Diffusion Example Shepp-Logan phantom, 12% noise Normal reconstruction (left) and smoothing by linear diffusion, T = 0.1

Radon Transform and Diffusion Fundamental Solution L.: Inverse Problems and Imaging, to appear Fundamental Solution of the heat equation G(t, x) = 1 (4πt) n/2 exp( x 2 /4t) 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

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 and is of convolution type. R ψ T (x, y) = L T δ x(y) = G(T, x y)

Radon Transform and Diffusion Reconstruction Kernel Let Then and ψ T (s) = 1 2π 0 E γ (x) = δ x σ exp( T σ 2 ) cos(sσ)dσ ψ T (s) = 1 ( 4π 2 1 s D ( s T T 2 ) ) T with the Dawson integral s D(s) = exp( s 2 ) exp(t 2 )dt 0

Radon Transform and Diffusion Numerical Tests with noisy Data, 12 % noise

Radon Transform and Diffusion Numerical Tests with noisy Data, 12 % noise Smoothing by linear diffusion ( left ) and combined reconstruction (right).

Radon Transform and Diffusion Reconstruction of Derivatives for Noisy Data Reconstruction Kernel for Derivative in Direction x k ψ T,k (s) = θ ( k s 4π 2 T 3/2 2 T + ( 1 s2 2T ) D( s 2 T ) )

Radon Transform and Diffusion Noise and Differentiation, 6% Noise

Radon Transform and Diffusion Noise and Differentiation, 6% Noise Smoothing by linear diffusion ( left ) and combined reconstruction (right).

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.

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.

Radon Transform and Wavelet Decomposition Content 1 Formulation of Problem 2 Inverse Problems and Regularization 3 Approximate Inverse for Combining Reconstruction and Analysis 4 Radon Transform and Edge Detection 5 Radon Transform and Diffusion 6 Radon Transform and Wavelet Decomposition 7 Nonlinear Problems 8 Future

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

Radon Transform and Wavelet Decomposition Inversion with Wavelets L.,MAASS,RIEDER 94 Represent the solution of Rf = g as f = k c M k ϕ Mk + m dl m ψ ml l

Radon Transform and Wavelet Decomposition Inversion with Wavelets L.,MAASS,RIEDER 94 Represent the solution of Rf = g as f = k c M k ϕ Mk + m dl m ψ ml l Precompute v Mk, w ml as R v Mk = ϕ Mk R w ml = ψ ml

Radon Transform and Wavelet Decomposition Inversion with Wavelets L.,MAASS,RIEDER 94 Represent the solution of Rf = g as f = k c M k ϕ Mk + m dl m ψ ml l Precompute v Mk, w ml as R v Mk = ϕ Mk Then R w ml = ψ ml c M k = g, v Mk d m l = g, w ml

Radon Transform and Wavelet Decomposition This is joint work with Steven Oeckl Department Process Integrated Inspection Systems Fraunhofer IIS

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 New Application, first in NDT In-line inspection in production process

Radon Transform and Wavelet Decomposition Typical Scanning Geometry in NDT

Radon Transform and Wavelet Decomposition Wavelet Coefficient for 2D Parallel Geoemtry f, ψ jk = 1 I 1 g(θ, s)r θ ψ jk (s)dsdθ 4π S 1 IR = 1 ( I 1 ) g(θ, ) R θ ψ j0 ( D j k, θ )dθ 4π S 1

Radon Transform and Wavelet Decomposition 2D Shepp-Logan Phantom, Fan Beam

Radon Transform and Wavelet Decomposition Cone Beam: decentralized slice

Radon Transform and Wavelet Decomposition Cone Beam Local Reconstruction

Nonlinear Problems Content 1 Formulation of Problem 2 Inverse Problems and Regularization 3 Approximate Inverse for Combining Reconstruction and Analysis 4 Radon Transform and Edge Detection 5 Radon Transform and Diffusion 6 Radon Transform and Wavelet Decomposition 7 Nonlinear Problems 8 Future

Nonlinear Problems Considered Nonlinearity where Af = A l f l=1 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

Nonlinear Problems Ansatz for Inversion, Presented for 2 Terms Put with and S γ g = S 1 g + S 2 g S 1 g(x) = g, ψ 1 (x, ) 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 }{{} LE γf 0

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 where B n = Ω Ω Note: B n is independent of x! n=1 k 1 (y 1 )k 2n (y 1, y 2 )k 1 (y 2 )dy 1 dy 2

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 = ω2 n (ω 0 n) 2 (ω 0 n) 2,, n = 1,..., N k 1,n (y) = 2 sin 2 (nπy) k 2,n (y 1, y 2 ) = 4 sin 2 (nπy 1 ) sin 2 (nπy 2 ) +4 n m n 2 n 2 m 2 sin(nπy 1) sin(mπy 1 ) sin(nπy 2 ) sin(mπy 2 )

Nonlinear Problems Numerical Test, L = d dx Linear and quadratic approximation of function (left) and derivative (right).

Future Content 1 Formulation of Problem 2 Inverse Problems and Regularization 3 Approximate Inverse for Combining Reconstruction and Analysis 4 Radon Transform and Edge Detection 5 Radon Transform and Diffusion 6 Radon Transform and Wavelet Decomposition 7 Nonlinear Problems 8 Future

Future Work in Progress Weakly nonlinear operators A Electron microscopy ( Kohr ) Inverse problems for Maxwell s equation ( A. Lakhal) Spherical Radon Transform ( Riplinger ) System Biology ( Groh ) Gait Analysis ( Bechtel, Johann ) Mathematical Finance ( M. Lakhal ) Semi - Discrete Problems ( Krebs ) 3D X-ray CT

Future Open Problems Nonlinear Problems, esp. nonlinear in L Time dependent problems other scanning geometries other operators L