Feature Reconstruction in Tomography Alfred K. Louis Institut fr - - PowerPoint PPT Presentation

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

Feature Reconstruction in Tomography

Images

Feature Reconstruction in Tomography

Typical Procedure

Feature Reconstruction in Tomography

Typical Procedure

Image Reconstruction

Feature Reconstruction in Tomography

Typical Procedure

Image Reconstruction Image Enhancement

Feature Reconstruction in Tomography

Typical Procedure

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

Feature Reconstruction in Tomography

Requirements for Repeated Solution of Same Problem with Different Data

Feature Reconstruction in Tomography

Requirements for Repeated Solution of Same Problem with Different Data

Combination of reconstruction and image analysis in ONE step

Feature Reconstruction in Tomography

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

Feature Reconstruction in Tomography

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 Use properties of the operator for fast algorithms

Feature Reconstruction in Tomography

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

Feature Reconstruction in Tomography 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

Feature Reconstruction in Tomography 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

Feature Reconstruction in Tomography Formulation of Problem

Feature Determination

Image Enhancement Image Registration Classification

Feature Reconstruction in Tomography 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

Feature Reconstruction in Tomography Formulation of Problem

Image Analysis

compute enhanced image Lf

• perate on Lf

Feature Reconstruction in Tomography Formulation of Problem

Image Analysis

compute enhanced image Lf

• perate on Lf

Example for Calculation Lkβf = ∂ ∂xk

• Gβ ∗ f
• Edge Detection

Feature Reconstruction in Tomography Formulation of Problem

Image Analysis

compute enhanced image Lf

• perate on Lf

Example for Calculation Lkβf = ∂ ∂xk

• Gβ ∗ f
• Edge Detection
• L. 96: Compute directly derivative of solution

Schuster, 2000 : Compute divergence of vector fields

Feature Reconstruction in Tomography Formulation of Problem

Applications for X-Ray CT

Detect defects like blowholes In-line inspection 3D Dimensioning ( dimensionelles Messen )

Feature Reconstruction in Tomography Formulation of Problem

X-Ray CT

Feature Reconstruction in Tomography 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 gL = ln(I0/IL) =

• L

f dt

Feature Reconstruction in Tomography 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 gL = ln(I0/IL) =

• L

f dt

Feature Reconstruction in Tomography 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

Feature Reconstruction in Tomography 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 )

Feature Reconstruction in Tomography Inverse Problems

Linear Regularization

Y1

A+

ւ ↑ ˜ Mγ A : X − → Y Mγ ↑

A+

ւ X−1 Sγ = MγA+ = A+ ˜ Mγ

Feature Reconstruction in Tomography Inverse Problems

General Purpose Regularization Methods

Rγg =

• µ

σ−1

µ g, uµYvµ

Feature Reconstruction in Tomography Inverse Problems

General Purpose Regularization Methods

Rγg =

• µ

σ−1

µ Fγ(σµ)g, uµYvµ

Feature Reconstruction in Tomography Inverse Problems

General Purpose Regularization Methods

Rγg =

• µ

σ−1

µ Fγ(σµ)g, uµYvµ

Truncated SVD Fγ(t) = 1 : t ≥ γ : t < γ

Feature Reconstruction in Tomography Inverse Problems

General Purpose Regularization Methods

Rγg =

• µ

σ−1

µ Fγ(σµ)g, uµYvµ

Truncated SVD Fγ(t) = 1 : t ≥ γ : t < γ Tikhonov - Phillips - Regularization ( Levenberg-Marquardt ) Fγ(t) = t2/(t2 + γ2)

Feature Reconstruction in Tomography Inverse Problems

General Purpose Regularization Methods

Rγg =

• µ

σ−1

µ Fγ(σµ)g, uµYvµ

Truncated SVD Fγ(t) = 1 : t ≥ γ : t < γ Tikhonov - Phillips - Regularization ( Levenberg-Marquardt ) Fγ(t) = t2/(t2 + γ2) ⇔ arg min{Af − g2

Y + γ2f2 X : f ∈ X}

Feature Reconstruction in Tomography Inverse Problems

General Purpose Regularization Methods

Rγg =

• µ

σ−1

µ Fγ(σµ)g, uµYvµ

Truncated SVD Fγ(t) = 1 : t ≥ γ : t < γ Tikhonov - Phillips - Regularization ( Levenberg-Marquardt ) Fγ(t) = t2/(t2 + γ2) ⇔ arg min{Af − g2

Y + γ2f2 X : f ∈ X}

Iterative Methods (Landweber, CG, etc.)

Feature Reconstruction in Tomography Inverse Problems

Regularization by Linear Functionals

Likht, 1967 Backus-Gilbert, 1967 L.-Maass, 1990 L, 1996

Feature Reconstruction in Tomography Inverse Problems

Regularization by Linear Functionals

Likht, 1967 Backus-Gilbert, 1967 L.-Maass, 1990 L, 1996 Approximate Inverse: choose mollifier eγ(x, ·) ≈ δx and solve A∗ψγ(x, ·) = eγ(x, ·) put fγ(x) := Eγf(x) := f, eγ(x, ·) = g, ψγ(x, ·)

Feature Reconstruction in Tomography 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

Feature Reconstruction in Tomography 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, eγ with mollifier eγ

Feature Reconstruction in Tomography 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, eγ with mollifier eγ Assume a solution exists of A∗ψγ(x, ·) = L∗eγ(x, ·)

Feature Reconstruction in Tomography 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, eγ with mollifier eγ Assume a solution exists of A∗ψγ(x, ·) = L∗eγ(x, ·) Then (Lf)γ(x) = g, ψγ(x, ·)

Feature Reconstruction in Tomography Approximate Inverse for Combining Reconstruction and Analysis

Proof

(Lf)γ(x) = Lf, eγ(x, ·) = f, L∗eγ(x, ·) = f, A∗ψγ(x, ·) = g, ψγ(x, ·)

Feature Reconstruction in Tomography Approximate Inverse for Combining Reconstruction and Analysis

Proof

(Lf)γ(x) = Lf, eγ(x, ·) = f, L∗eγ(x, ·) = f, A∗ψγ(x, ·) = g, ψγ(x, ·) Conditions on eγ such that Sγ 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 : L2(0, 1) → L2(0, 1) with Af(x) = x f(y)dy Then f = g′ Mollifier eγ(x, y) = 1 2γ χ[x−γ,x+γ](y)

Feature Reconstruction in Tomography Approximate Inverse for Combining Reconstruction and Analysis

Example 1 continued

Auxiliary Equation: A∗ψγ(x, y) = 1

y

ψγ(x, t)dt = eγ(x, y) leading to ψγ(x, y) = 1 2γ (δx+γ − δx−γ)(y) and Sγg(x) = g(x + γ) − g(x − γ) 2γ

Feature Reconstruction in Tomography Approximate Inverse for Combining Reconstruction and Analysis

Example 2: L = d

dx

Consider A : L2(0, 1) → L2(0, 1) with Af(x) = x f(y)dy Then Lf = g′′ Mollifier eγ(x, y) = y−(x−γ)

γ

for x − γ ≤ y ≤ x

x+γ−y γ

for x ≤ y ≤ x + γ

Feature Reconstruction in Tomography Approximate Inverse for Combining Reconstruction and Analysis

Example 2 continued

Auxiliary Equation A∗ψγ(x, y) = 1

y

ψγ(x, t)dt = L∗eγ(x, y) = − ∂ ∂y eγ(x, y) leading to ψγ(x, y) = 1 γ2 (δx+γ − 2δx + δx−γ)(y) and Sγg(x) = g(x + γ) − 2g(x) + g(x − γ) γ2

Feature Reconstruction in Tomography Approximate Inverse for Combining Reconstruction and Analysis

Existence

Consider A : Hs → Hs+α with AfHs+α ≥ c1fHs and L : D(L) ⊂ L2 → L2 with LfHs−t ≤ c2fHs Then formally LA† : L2 → Ht−α and regularization via Eγ : Ht−α → L2

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 T1, T2, T3 be such that L∗T1 = T2L∗ T2A∗ = A∗T3 and solve for a reference mollifier Eγ the equation R∗Ψγ = L∗Eγ Then the general reconstruction kernel for the general mollifier eγ = T1Eγ is ψγ = T3Ψγ and fγ = g, T3Ψγ

Feature Reconstruction in Tomography Approximate Inverse for Combining Reconstruction and Analysis

Possible Calculation of Reconstruction Kernel

If the reconstruction of f is achieved as Eγf(x) =

• ψγ(x, y)g(y)dy

then the reconstruction kernel for calculating Lβf can be computed as ¯ ψβγ(·, y) = Lβψγ(·, y)

Feature Reconstruction in Tomography 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

Feature Reconstruction in Tomography Radon Transform and Edge Detection

Radon Transform

Rf(θ, s) =

• R2 f(x)δ(s − x⊤θ)dx

θ ∈ S1 and s ∈ R.

• Rf(θ, σ) = (2π)1/2ˆ

f(σθ) R−1 = 1 4πR∗I−1 where R∗ is the adjoint operator from L2 to L2 known as backprojection R∗g(x) =

• S1 g(θ, x⊤θ)dθ

and the Riesz potential I−1 is defined with the Fourier transform

• I−1g(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 ∂ ∂xk f(θ, s) = θk ∂ ∂sRf(θ, s)

Feature Reconstruction in Tomography Radon Transform and Edge Detection

Invariances

Let for x ∈ R2 the shift operators T x

2 f(y) = f(y − x) and

T x⊤θ

3

g(θ, s) = g(θ, s − x⊤θ) then RθT x

2 = T x⊤θ 3

Rθ Let U be a unitary 2 × 2 matrix and DU

2 f(y) = f(Uy). then

RDU

2 = DU 3 R

where DU

3 g(θ, s) = g(Uθ, s).

(TR)∗ = R∗T ∗

Feature Reconstruction in Tomography Radon Transform and Edge Detection

Reconstruction Kernel

Theorem Let the mollifier be given as eγ(x, y) = Eγ(x − y). Then the reconstruction kernel for Lk = ∂/∂xk is ψk,γ = θkΨγ

• s − x⊤θ
• with

Ψγ = − 1 4π ∂ ∂sI−1REγ

Feature Reconstruction in Tomography Radon Transform and Edge Detection

Reconstruction Kernel

Theorem Let the mollifier be given as eγ(x, y) = Eγ(x − y). Then the reconstruction kernel for Lk = ∂/∂xk is ψk,γ = θkΨγ

• s − x⊤θ
• with

Ψγ = − 1 4π ∂ ∂sI−1REγ Same costs for calculating

∂f ∂xk as for calculating f.

Feature Reconstruction in Tomography Radon Transform and Edge Detection

Proof

R∗ψkγ = L∗

keγ

= R−1RL∗

keγ

= 1 4πR∗I−1RL∗

keγ

Hence ψkγ = 1 4πI−1RL∗

keγ

L∗

k = −Lk

and differentiation rule completes the proof.

Feature Reconstruction in Tomography Radon Transform and Edge Detection

Example

Let Eβγ be given as

• Eβγ(ξ) = (2π)−1sincξπ

β sincξπ 2γ χ[−γ,γ](ξ)

• Shepp−Logan kernel

Then the reconstruction kernel for β = γ is ψπ/h(ℓh) = 1 π2h3 8ℓ

• 3 + 4ℓ22 − 64ℓ2 , ℓ ∈ Z

Feature Reconstruction in Tomography Radon Transform and Edge Detection

Kernel with β = γ and β = 2γ

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∗TMΓT where T is a differential and MΓ a trajectory dependent multiplication operator

Feature Reconstruction in Tomography 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

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 ut = ∆ u |∇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 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 LTu(0, ·) = u(T, ·) = G(T, ·) ∗ f

Feature Reconstruction in Tomography Radon Transform and Diffusion

Combining Reconstruction and Diffusion

Let eγ(x, ·) = δx and L = LT Then the reconstruction kernel for time t = T has to fulfil R∗ψT(x, y) = L∗

Tδx(y) = G(T, x − y)

and is of convolution type.

Feature Reconstruction in Tomography Radon Transform and Diffusion

Reconstruction Kernel

Let Eγ(x) = δx Then ψT(s) = 1 2π ∞ σ exp(−Tσ2) cos(sσ)dσ and ψT(s) = 1 4π2T

• 1 −

s √ T D

• s

2 √ T

• with the Dawson integral

D(s) = exp(−s2) s exp(t2)dt

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 xk ψT,k(s) = − θk 4π2T 3/2

• s

2 √ T +

• 1 − s2

2T

• D(

s 2 √ 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

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

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.,MAASS,RIEDER 94 Represent the solution of Rf = g as f =

• k

cM

k ϕMk +

• m
• ℓ

dm

ℓ ψmℓ

Feature Reconstruction in Tomography Radon Transform and Wavelet Decomposition

Inversion with Wavelets

L.,MAASS,RIEDER 94 Represent the solution of Rf = g as f =

• k

cM

k ϕMk +

• m
• ℓ

dm

ℓ ψmℓ

Precompute vMk, wmℓ as R∗vMk = ϕMk R∗wmℓ = ψmℓ

Feature Reconstruction in Tomography Radon Transform and Wavelet Decomposition

Inversion with Wavelets

L.,MAASS,RIEDER 94 Represent the solution of Rf = g as f =

• k

cM

k ϕMk +

• m
• ℓ

dm

ℓ ψmℓ

Precompute vMk, wmℓ as R∗vMk = ϕMk R∗wmℓ = ψmℓ Then cM

k = g, vMk

dm

ℓ = g, wmℓ

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

f, ψjk = 1 4π

• S1
• I

R

I−1g(θ, s)Rθψjk(s)dsdθ = 1 4π

• S1
• I−1g(θ, ·) ∗ R−θψj0
• (Djk, θ)dθ

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

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

Feature Reconstruction in Tomography Nonlinear Problems

Considered Nonlinearity

Af =

∞

• ℓ=1

Aℓf where A1f(x) =

• k1(x, y)f(y)dy

A2f(x) =

• k2(x, y1, y2)f(y1)f(y2)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 Sγg = S1g + S2g with S1g(x) = g, ψ1(x, ·) and S2g(x) =

• g, Ψ2(x, ·, ·), g
• Replace g by Af and omit higher order terms. Then

SγAf = S1A1f

≈LEγf

+ S1A2f + S2A1f

• ≈0

Feature Reconstruction in Tomography Nonlinear Problems

Determination of the Square Term

Consider A : L2(Ω) → RN Minimizing the defect leads to the equation A1A∗

1Ψγ(x)A1A∗ 1 = − N

• n=1

ψγ,n(x)Bn where Bn =

• Ω×Ω

k1(y1)k2n(y1, y2)k1(y2)dy1dy2 Note: Bn is independent of x !

Feature Reconstruction in Tomography Nonlinear Problems

Example: A : L2 → RN

Vibrating String u′′(x) + ρ(x)ω2u(x) = 0, u(0) = u(1) = 0 ρ(x) = 1 + f(x) Data gn = ω2

n − (ω0 n)2

(ω0

n)2

, , n = 1, . . . , N k1,n(y) = −2 sin2(nπy) k2,n(y1, y2) = 4 sin2(nπy1) sin2(nπy2) +4

• n=m

n2 n2 − m2 sin(nπy1) sin(mπy1) sin(nπy2) sin(mπy2)

Feature Reconstruction in Tomography Nonlinear Problems

Numerical Test, L = d

dx

Linear and quadratic approximation of function (left) and derivative (right).

Feature Reconstruction in Tomography 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

Feature Reconstruction in Tomography 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

Feature Reconstruction in Tomography Future

Open Problems

Nonlinear Problems, esp. nonlinear in L Time dependent problems

• ther scanning geometries
• ther operators L