limited-data ct for underwater pipeline...

Limited-Data CT for Underwater Pipeline Inspection

Nicolai André Brogaard Riis ([email protected])

Tuesday 24th September, 2019

IDA Matematik 2019, Copenhagen, Denmark.

Industrial collaborator:

Joint work with:

Per Christian Hansen, Yiqiu Dong – Technical University of Denmark

Jacob Frøsig, Rasmus D. Kongskov – 3Shape

Torben Klit Pedersen – FORCE Technology

Arvid P. L. Böttiger – Exensor Technology

Jürgen Frikel – OTH Regensburg

Todd Quinto – Tu�s University

Subsea CT-Scanner by FORCE Technology, Denmark

Alternative to ultrasound: use X-ray scanning to compute cross-sectional images ofoil pipes lying on the seabed, to detect defects, cracks, etc. in the pipe.

Illustration courtesy of FORCE Technology.

2 DTU Compute Nicolai A. B. Riis: Limited-Data CT for Underwater Pipeline Inspection 24.9.2019


https://forcetechnology.com/da/innovation/afsluttede-projekter/subsea-inspektion-konstruktion-levetid-reparation-planlaegning



Comparing Geometries

Limitations in the scanner deviceData: Narrow high intensity X-Ray beam→

available from a limited view.Goal: Design set-up to capture as much detail

of the pipe as possible.

Full illuminationNot possible

CenteredPossible set-up

O�-centeredPossible set-up


The Computed Tomograhy Forward Model

Continuous formulation, limited data

Themeasured projections g for the object f are described by

g(θ, s) = (R`f)(θ, s) + noise

where (R`f)(θ, s) = (Rf)(θ, s) for those pairs (θ, s) corresponding to the `imitedillumination, andR is the Radon transform.

Corresponding algebraic model

Themeasured data b for a discretized objectx is described by

b = A` x+ e, b ∈ Rm, x ∈ Rn, A` ∈ Rm×n,

where e ∈ Rm with ei ∼ N(0, σ2) andA` is the discretion ofR`.


Full and Two Di�erent Limited Illuminations


Characterising What We Can Measure – Centered Beam

Microlocal analysis: a singularity at position χwith direction ξ is visible if and onlyif data from the line through χ perpendicular to ξ is present.

Measured data from oneview/projection.

Visible from oneview/projection.

Visible from allviews/projections.


Characterising What We Can Measure – O�-Center Beam

Microlocal analysis: a singularity at position χwith direction ξ is visible if and onlyif data from the line through χ perpendicular to ξ is present.

Measured data from oneview/projection.

Visible from oneview/projection.

Visible from allviews/projections.


Reconstruction: Incorporating Sparsity

We need a robust algorithm that can utilize all information in the data.

Use the algebraic system

b = A` x+ e, b ∈ Rm, x ∈ Rn, A` ∈ Rm×n.

and compute a sparse representation of the solution x in a “basis” that is wellsuited for the problem.If we can represent xwith few non-zero basis functions, we have fewer unknownsto determine→ same data, fewer unknowns, “easier” problem.


Representing pipe in a sparse “basis”

Frame decomposition Given an object, x, we decompose it into building blocks

x =∑µ

cµϕµ,

whereϕµ are the building blocks and cµ = 〈x,ϕµ〉. are coe�icients.*If the object is fully represented by few building blocks, we have a sparse representation.

Example:

= c1 + c2 + c3

+c4 + c5

Why is this useful?


Why enforce sparsity of frame coe�icients?

Noise/artefact reduction:

−−−−−−→Decompose

c1 + c2 + c3

+c4 + c5

=





c1 + c2 + c3

+c4 + c5

=

Too strong prior!





c1 + c2 + c3

+c4 + c5

=

Too strong prior!Not all images arefully represented bythe decompositon.


Shearlets: A tight frame

Tight frames,Φ ={ϕµ}, generalises ONB, i.e., for all images xwe have

x =∑µ

〈x,ϕµ〉ϕµ.

Examples of 2D Shearlets:

Shearlets yield a sparse representation of defects, contours, etc.


Shearlet-based optimization problem

Recall discrete model of scanning process: b = A` x+ e.

Reconstruction with a weighted shearlet-based sparsity penalty

Optimization problem:

argminx≥0

12‖A`x− b‖22 + α‖W c‖1,

s.t. c = Φx

with α > 0 regularization parameter,W = diag(wi) ∈ Rp×p with weightswi > 0,Φ ∈ Rp×n is the shearlet analysis transform and cµ = 〈x,ϕµ〉 are coe�icients.

Shearlet transform contains the basis functions:ΦT =[ϕ1, . . . , ϕp

]. hey






argminx≥0

12‖A`x− b‖22 + α‖W c‖1,

s.t. c = Φx



].

Data-fitting






argminx≥0

12‖A`x− b‖22 + α‖W c‖1,

s.t. c = Φx



].

Weighted sparsity penalty on shearlet coe�icients


Final algorithmRecall optimization problem:

arg minx≥0

12‖A

`x− b‖22 + α‖W c‖1,

s.t. c = Φx

ADMM-based algorithm

We solve it using the ADMMmethod [Boyd et. al. 2011]. Auxiliary variable c = Φx. Iterativeupdates:

xk+1 := minx≥0

12‖A` x−b‖22+ ρ2‖Φx−c

k+uk‖22,

ck+1 := minc

α‖W c‖1 + ρ2‖Φxk+1−c+uk‖22,

uk+1 := uk+Φxk+1−ck+1,

where u are the scaled Lagrange multipliers and ρ > 0 the penalty parameter.

The updates are calculated using:xk+1: CGLS + non-negativity projection.ck+1: Element-wise so� thresholding.15 DTU Compute Nicolai A. B. Riis: Limited-Data CT for Underwater Pipeline Inspection 24.9.2019

Synthetic andmeasured data from both geometries

Synthetic centered

Projection angle

Detectorpixel

0

0.5

1

1.5

2

2.5

3

3.5

4Measured centered

Projection angle

Detectorpixel

0

0.5

1

1.5

2

2.5

3

3.5

4Synthetic off-centered

Projection angle

Detectorpixel

0

0.5

1

1.5

2

2.5

3

3.5

4Measured off-centered

Projection angle

Detectorpixel

0

0.5

1

1.5

2

2.5

3

3.5

4


Reconstructions from Real Data – Both GeometriesCentered beam: there are many artifacts.

Prev. alg.

Kaczmarz, 5 iterations

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2Shearlet, α = 0.010

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Our alg.

O�-center beam: singularities are easy to detect; artifacts are reduced.

Prev. alg.

Kaczmarz, 5 iterations

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2Ground truth

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Our alg.


Reconstructions from Real Data – Zoom


Prev. alg. Our alg.

0

0.05

0.1

0.15

0.2

0

0.05

0.1

0.15

0.2


Reconstructions from Real Data – Zoom


arg minx≥0‖Ax− b‖22 + λ‖∇x‖2,1 Our alg. hey

0

0.05

0.1

0.15

0.2

0

0.05

0.1

0.15

0.2


Centered Versus O�-Center Beam

Centered beam O�-center beamPros Good reconstruction in the

center domain.Captures singularities out-side the center domain.

Cons Terrible reconstruction outsidethe center domain.

Less good reconstruction inthe center domain.

Comments Requires less projections be-cause the center domain iswellcovered by rays.

Requiresmore projections togive good reconstruction ev-erywhere.Better suited for this applica-tion.


Conclusions

• For technical reasons the X-ray beam cannot cover the whole pipe.• An o�-centered beam can give a satisfactory reconstruction.• A weighted shearlets-based sparsity penalty gives better reconstructions – especially withfew projections.• It is important to include weights in the sparsity penalty.

Future work:• Optimize the algorithm for performance and robustness.• Design heuristics for choosing the weights and the reg. parameter.• Derive more theory for the continuous model with limited data.• Quantify the uncertainties in the model and the solution.


IntroductionFormulation of problem

Uncertain view angle CT

b = A(θ)x + e, θ ∼ πθ(·), e ∼ πe(·). (1)

Measured / fixed:b ∈ Rm: measured noisy sinogram.A ∈ Rm×n: discretized Radon transform.

Known but with uncertainty:θ ∈ Rq : view angles.e ∈ Rm: measurement noise.

Unknown:x ∈ Rn: attenuation coe�icients.

πθ1 (θ

1 )

πθ2(θ2)

b1

=A

(θ1 )x

+e1

b2 =A(θ2

)x +e2

x

ModelMachine




b = A(θ)x + e, θ ∼ πθ(·), e ∼ πe(·). (1)

Measured / fixed:b ∈ Rm: measured noisy sinogram.A ∈ Rm×n: discretized Radon transform.

Known but with uncertainty:θ ∈ Rq : view angles.e ∈ Rm: measurement noise.

Unknown:x ∈ Rn: attenuation coe�icients.

πθ1 (θ

1 )

πθ2(θ2)

b1

=A

(θ1 )x

+e1

b2 =A(θ2

)x +e2

x

ModelMachine

Actual data comes from b := A(θ̄) x + enoise.




b = A(θ)x + e, θ ∼ πθ(·), e ∼ πe(·). (1)

Goal:Reconstruct x from bwith uncertainty in θ and e!

Applications:Inaccuracies in rotation, patient motion etc.


IntroductionEarly results


IntroductionEarly results 2


IntroductionOngoing Research - Uncertain View Angles

• Large scale problems (2D→ 3D)• Apply to pipe scanner (with shearlet regularizer)


Thank you!

Contact:Nicolai André Brogaard [email protected] ComputeBuilding 303B Room 118


Introduction

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