Feature Reconstruction in Tomography - Ricam · Feature Reconstruction in Tomography Typical Procedure Image Reconstruction Image Enhancement Disadvantage: The two operations are

Feature Reconstruction in Tomography


Alfred K. Louis

Institut für Angewandte MathematikUniversität des Saarlandes

66041 Saarbrückenhttp://www.num.uni-sb.de

[email protected]

Wien, July 20, 2009


Images


Typical Procedure

Image ReconstructionImage EnhancementDisadvantage: The two operations are non adjusted andpossibly too time-consuming


Typical Procedure

Image Reconstruction

Image EnhancementDisadvantage: The two operations are non adjusted andpossibly too time-consuming


Typical Procedure

Image ReconstructionImage Enhancement

Disadvantage: The two operations are non adjusted andpossibly too time-consuming


Typical Procedure

Image ReconstructionImage EnhancementDisadvantage: The two operations are non adjusted andpossibly too time-consuming


Requirements for Repeated Solution of SameProblem with Different Data

Combination of reconstruction and image analysis in ONEstepPrecompute solution operator for efficient evaluation of thesame problem with different data setsUse properties of the operator for fast algorithms



Combination of reconstruction and image analysis in ONEstep

Precompute solution operator for efficient evaluation of thesame problem with different data setsUse properties of the operator for fast algorithms



Combination of reconstruction and image analysis in ONEstepPrecompute solution operator for efficient evaluation of thesame problem with different data sets

Use properties of the operator for fast algorithms



Combination of reconstruction and image analysis in ONEstepPrecompute solution operator for efficient evaluation of thesame problem with different data setsUse properties of the operator for fast algorithms


Content

1 Formulation of Problem

2 Inverse Problems and Regularization

3 Approximate Inverse for Combining Reconstruction andAnalysis

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

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8 Future



Target

Combine the two steps in ONE algorithmInstead of

Solve reconstruction part Af = gEvaluate the result Lf

Compute directlyLA†g



Feature Determination

Image EnhancementImage RegistrationClassification



Examples of Existing Methods

A = R, L = I−1

Lambda CT 2D : Smith, VainbergLambda 3D: L-Maass, KatsevichLambda with SPECT, ET: Quinto, ÖktemLambda with SONAR : L., Quinto



Image Analysis

compute enhanced image Lfoperate on Lf

Example for Calculation

Lkβf =∂

∂xk

(Gβ ∗ f

)Edge Detection

L. 96: Compute directly derivative of solutionSchuster, 2000 : Compute divergence of vector fields



Image Analysis



Lkβf =∂

∂xk

(Gβ ∗ f

)Edge Detection




Image Analysis



Lkβf =∂

∂xk

(Gβ ∗ f

)Edge Detection




Applications for X-Ray CT

Detect defects like blowholesIn-line inspection3D Dimensioning ( dimensionelles Messen )



X-Ray CT



X-Ray Tomography

Assumptions:X-Rays travel on straight linesAttenuation proportional to path lengthAttenuation proportional to number of photons

Radon Transform

4I = −I4t f

gL = ln(I0/IL) =

∫L

f dt



X-Ray Tomography

Assumptions:X-Rays travel on straight linesAttenuation proportional to path lengthAttenuation proportional to number of photons

Radon Transform

4I = −I4t f

gL = ln(I0/IL) =

∫L

f dt


Inverse Problems

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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 existSolution is not uniqueSolution does not depend continuously on g

⇒ Problem ill - posed ( mal posé )


Inverse Problems

Linear Regularization

Y1

A+

↑ Mγ

A : X −→ Y

Mγ ↑A+

X−1

Sγ = MγA+ = A+Mγ


Inverse Problems

General Purpose Regularization Methods

Rγg =∑µ

σ−1µ 〈g,uµ〉Yvµ

Truncated SVD

Fγ(t) =

1 : t ≥ γ0 : t < γ

Tikhonov - Phillips - Regularization( Levenberg-Marquardt )

Fγ(t) = t2/(t2 + γ2)

⇔ arg min‖Af − g‖2Y + γ2‖f‖2X : f ∈ X

Iterative Methods (Landweber, CG, etc.)


Inverse Problems


Rγg =∑µ

σ−1µ Fγ(σµ)〈g,uµ〉Yvµ

Truncated SVD

Fγ(t) =

1 : t ≥ γ0 : t < γ


Fγ(t) = t2/(t2 + γ2)




Inverse Problems


Rγg =∑µ


Truncated SVD

Fγ(t) =

1 : t ≥ γ0 : t < γ


Fγ(t) = t2/(t2 + γ2)




Inverse Problems


Rγg =∑µ


Truncated SVD

Fγ(t) =

1 : t ≥ γ0 : t < γ


Fγ(t) = t2/(t2 + γ2)




Inverse Problems


Rγg =∑µ


Truncated SVD

Fγ(t) =

1 : t ≥ γ0 : t < γ


Fγ(t) = t2/(t2 + γ2)




Inverse Problems


Rγg =∑µ


Truncated SVD

Fγ(t) =

1 : t ≥ γ0 : t < γ


Fγ(t) = t2/(t2 + γ2)




Inverse Problems

Regularization by Linear Functionals

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

Approximate Inverse: choose mollifier eγ(x , ·) ≈ δx and solve

A∗ψγ(x , ·) = eγ(x , ·)

putfγ(x) := Eγ f (x) := 〈f ,eγ(x , ·)〉 = 〈g, ψγ(x , ·)〉


Inverse Problems

Regularization by Linear Functionals

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

Approximate Inverse: choose mollifier eγ(x , ·) ≈ δx and solve

A∗ψγ(x , ·) = eγ(x , ·)

putfγ(x) := Eγ f (x) := 〈f ,eγ(x , ·)〉 = 〈g, ψγ(x , ·)〉


Approximate Inverse for Combining Reconstruction and Analysis

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8 Future



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 , ·)〉



Approximate Inverse





A∗ψγ(x , ·) = L∗eγ(x , ·)




Approximate Inverse





A∗ψγ(x , ·) = L∗eγ(x , ·)




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 fordetermining Lf are given in the above mentioned paper.



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 fordetermining Lf are given in the above mentioned paper.



Example 1: L = I

ConsiderA : L2(0,1)→ L2(0,1)

with

Af (x) =

∫ x

0f (y)dy

Thenf = g′

Mollifiereγ(x , y) =

12γχ[x−γ,x+γ](y)



Example 1 continued

Auxiliary Equation:

A∗ψγ(x , y) =

∫ 1

yψγ(x , t)dt = eγ(x , y)

leading to

ψγ(x , y) =1

2γ(δx+γ − δx−γ)(y)

andSγg(x) =

g(x + γ)− g(x − γ)

2γ



Example 2: L = ddx

ConsiderA : L2(0,1)→ L2(0,1)

with

Af (x) =

∫ x

0f (y)dy

ThenLf = g′′

Mollifier

eγ(x , y) =

y−(x−γ)

γ for x − γ ≤ y ≤ xx+γ−y

γ for x ≤ y ≤ x + γ



Example 2 continued

Auxiliary Equation

A∗ψγ(x , y) =

∫ 1

yψγ(x , t)dt = L∗eγ(x , y) = − ∂

∂yeγ(x , y)

leading to

ψγ(x , y) =1γ2 (δx+γ − 2δx + δx−γ)(y)

andSγg(x) =

g(x + γ)− 2g(x) + g(x − γ)

γ2



Existence

ConsiderA : Hs → Hs+α

with‖Af‖Hs+α ≥ c1‖f‖Hs

andL : D(L) ⊂ L2 → L2

with‖Lf‖Hs−t ≤ c2‖f‖Hs

Then formally

LA† : L2 → H t−α

and regularization via

Eγ : H t−α → L2



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 )



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 )



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 )



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 )



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 mollifiereγ = T1Eγ is

ψγ = T3Ψγ

andfγ = 〈g,T3Ψγ〉



Possible Calculation of Reconstruction Kernel

If the reconstruction of f is achieved as

Eγ f (x) =

∫ψγ(x , y)g(y)dy

then the reconstruction kernel for calculating Lβf can becomputed as

ψβγ(·, y) = Lβψγ(·, y)


Radon Transform and Edge Detection

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8 Future



Radon Transform

Rf (θ, s) =

∫R2

f (x)δ(s − x>θ)dx

θ ∈ S1 and s ∈ R.

Rf (θ, σ) = (2π)1/2 f (σθ)

R−1 =1

4πR∗I−1

where R∗ is the adjoint operator from L2 to L2 known asbackprojection

R∗g(x) =

∫S1

g(θ, x>θ)dθ

and the Riesz potential I−1 is defined with the Fourier transformI−1g(θ, σ) = |σ|g(θ, σ)



Radon Transform and Derivatives

The Radon transform of a derivative is

R∂

∂xkf (θ, s) = θk

∂

∂sRf (θ, s)



Invariances

Let for x ∈ R2 the shift operators T x2 f (y) = f (y − x) and

T x>θ3 g(θ, s) = g(θ, s − x>θ) then

RθT x2 = T x>θ

3 Rθ

Let U be a unitary 2× 2 matrix and DU2 f (y) = f (Uy). then

RDU2 = DU

3 R

where DU3 g(θ, s) = g(Uθ, s).

(T R)∗ = R∗T ∗



Reconstruction Kernel

TheoremLet the mollifier be given as eγ(x , y) = Eγ(‖x − y‖).Then the reconstruction kernel for Lk = ∂/∂xk is

ψk ,γ = θk Ψγ

(s − x>θ

)with

Ψγ = − 14π

∂

∂sI−1REγ

Same costs for calculating ∂f∂xk

as for calculating f .




TheoremLet the mollifier be given as eγ(x , y) = Eγ(‖x − y‖).Then the reconstruction kernel for Lk = ∂/∂xk is

ψk ,γ = θk Ψγ

(s − x>θ

)with

Ψγ = − 14π

∂

∂sI−1REγ

Same costs for calculating ∂f∂xk

as for calculating f .



Proof

R∗ψkγ = L∗keγ= R−1RL∗keγ

=1

4πR∗I−1RL∗keγ

Henceψkγ =

14π

I−1RL∗keγ

L∗k = −Lk

and differentiation rule completes the proof.



Example

Let Eβγ be given as

Eβγ(ξ) = (2π)−1sinc‖ξ‖πβ

sinc‖ξ‖π2γ

χ[−γ,γ](‖ξ‖)︸︷︷︸Shepp−Logan kernel

Then the reconstruction kernel for β = γ is

ψπ/h(`h) =1

π2h38`(

3 + 4`2)2 − 64`2

, ` ∈ Z



Kernel with β = γ and β = 2γ



Exact Data

Shepp - Logan phantom with original densities



Noisy Data



Differentiation of Image vs. New Method



New

Differentiation with respect to x and y .



Lambda and wrong Parameter β



Differentiation of Image vs. New Method

Sum of absolute values of the two derivativesApply support theorem of Boman and Quinto.



Reconstruction from Measured Data, Fan BeamGeometry



Applications

3D CT ( obvious with Feldkamp algorithm )Phase ContrastPhase Contrast with Eikonal ( High Energy ) ApproximationElectron Microscopy

Adapt for cone beam tomography and inversion formula

D−1 = cD∗TMΓT

where T is a differential and MΓ a trajectory dependentmultiplication operator


Radon Transform and Diffusion

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8 Future



Diffusion Processes

Smoothing of noisy images f by diffusion

∂

∂tu = ∆u

u(0, x) = f (x)

Problem Edges are smeared outPerona-Malik: Nonlinear Diffusion

ut = ∆u|∇u|



Example

Shepp-Logan phantom, 12% noise

Normal reconstruction (left) and smoothing by linear diffusion,T = 0.1



Example





Example





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 conditionu(0, x) = f (x) can be calculated as

LT u(0, ·) = u(T , ·) = G(T , ·) ∗ f



Combining Reconstruction and Diffusion

Let eγ(x , ·) = δx and L = LTThen the reconstruction kernel for time t = T has to fulfil

R∗ψT (x , y) = L∗T δx (y) = G(T , x − y)

and is of convolution type.




LetEγ(x) = δx

ThenψT (s) =

12π

∫ ∞0

σ exp(−Tσ2) cos(sσ)dσ

andψT (s) =

14π2T

(1− s√

TD( s

2√

T

))with the Dawson integral

D(s) = exp(−s2)

∫ s

0exp(t2)dt



Numerical Tests with noisy Data, 12 % noise

Smoothing by linear diffusion ( left ) and combinedreconstruction (right).



Numerical Tests with noisy Data, 12 % noise




Reconstruction of Derivatives for Noisy Data

Reconstruction Kernel for Derivative in Direction xk

ψT ,k (s) = − θk

4π2T 3/2

( s2√

T+(1− s2

2T)D(

s2√

T))



Noise and Differentiation, 6% Noise




Noise and Differentiation, 6% Noise




Noise and Differentiation, 6% and 12% Noise

Combined reconstruction of sum of absolute values ofderivatives for 6% and 12%noise .

No useful results with 12% noise with separate calculation ofreconstruction and smoothed derivative.



Noise and Differentiation, 6% and 12% Noise

Combined reconstruction of sum of absolute values ofderivatives for 6% and 12%noise .

No useful results with 12% noise with separate calculation ofreconstruction and smoothed derivative.


Radon Transform and Wavelet Decomposition

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8 Future



Wavelet Components

Combination of reconstruction and wavelet decomposition:L, Maass, Rieder, 1994Image reconstruction and wavelets :

Holschneider, 1991Berenstein, Walnut, 1996Bhatia, Karl, Willsky, 1996Bonnet, Peyrin, Turjman, 2002Wang et al, 2004



Inversion with Wavelets

L.,MAASS,RIEDER 94

Represent the solution of Rf = g as

f =∑

k

cMk ϕMk +

∑m

∑`

dm` ψm`

Precompute vMk , wm` as

R∗vMk = ϕMk

R∗wm` = ψm`

ThencM

k = 〈g, vMk 〉

dm` = 〈g,wm`〉




L.,MAASS,RIEDER 94


f =∑

k

cMk ϕMk +

∑m

∑`

dm` ψm`


R∗vMk = ϕMk

R∗wm` = ψm`

ThencM

k = 〈g, vMk 〉

dm` = 〈g,wm`〉




L.,MAASS,RIEDER 94


f =∑

k

cMk ϕMk +

∑m

∑`

dm` ψm`


R∗vMk = ϕMk

R∗wm` = ψm`

ThencM

k = 〈g, vMk 〉

dm` = 〈g,wm`〉



This is joint work with Steven OecklDepartment Process Integrated Inspection SystemsFraunhofer IIS



New Application, first in NDT

In-line inspection in production process

Reconstruction result:Three-dimensional registration of the object

Main inspection taskDimensional measurementDetection of blow holes and porosity



Typical Scanning Geometry in NDT



Wavelet Coefficient for 2D Parallel Geoemtry

〈f , ψjk 〉 =1

4π

∫S1

∫IR

I−1g(θ, s)Rθψjk (s)dsdθ

=1

4π

∫S1

(I−1g(θ, ·) ∗ R−θψj0

)(〈Djk , θ〉)dθ



2D Shepp-Logan Phantom, Fan Beam



Cone Beam: decentralized slice



Cone Beam ’Local Reconstruction’


Nonlinear Problems

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8 Future


Nonlinear Problems

Considered Nonlinearity

Af =∞∑`=1

A`f

whereA1f (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, 1991L: Approximate Inverse, 1995


Nonlinear Problems

Ansatz for Inversion, Presented for 2 Terms

PutSγg = S1g + S2g

withS1g(x) = 〈g, ψ1(x , ·)〉

andS2g(x) =

⟨g, 〈Ψ2(x , ·, ·),g〉

⟩Replace g by Af and omit higher order terms. Then

SγAf = S1A1f︸︷︷︸≈LEγ f

+ S1A2f + S2A1f︸︷︷︸≈0


Nonlinear Problems

Determination of the Square Term

ConsiderA : L2(Ω)→ RN

Minimizing the defect leads to the equation

A1A∗1Ψγ(x)A1A∗1 = −N∑

n=1

ψγ,n(x)Bn

whereBn =

∫Ω×Ω

k1(y1)k2n(y1, y2)k1(y2)dy1dy2

Note: Bn is independent of x !


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 − (ω0n)2

(ω0n)2

, ,n = 1, . . . ,N

k1,n(y) = −2 sin2(nπy)

k2,n(y1, y2) = 4 sin2(nπy1) sin2(nπy2)

+4∑n 6=m

n2

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


Nonlinear Problems

Numerical Test, L = ddx

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


Future

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8 Future


Future

Work in Progress

Weakly nonlinear operators AElectron 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 LTime dependent problemsother scanning geometriesother operators L

Feature Reconstruction in Tomography - Ricam · Feature Reconstruction in Tomography Typical Procedure Image Reconstruction Image Enhancement Disadvantage: The two operations are

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