Uncertainty of Non-Destructive Interior Imaging...

Uncertainty ofNon-

DestructiveInteriorImaging

Techniques

Laszlo Varga

Introduction:ImagingTechniques

TransmissionTomography

Problemformulation

Uncertainties intransmissiontomography

Noise

Lowinformationcontent

Advancedreconstructiontechniques

Datauncertainty inMRI

Uncertainty of Non-Destructive InteriorImaging Techniques

Laszlo Varga

University of Szeged, HungaryDepartment of Image Processing and Computer Graphics

19 July 2017

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Interior imaging techniques

Some imaging techniques

• Computer Tomography

• Magnetic Resonance Imaging (MRI)

• Positron Emission Tomography,

• Ultrasound imaging

• Electric Impedance Tomography.

Common properties

• Gathers secondary information,

• Uses mathematical tools forreconstruction,

• Data gathering has some cost.

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Common properties




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Common properties




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Common properties




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Common properties




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Common properties




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Common properties




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Transmission tomography

• We are interested in theinner structure of somegiven object.

• We can measure theprojections of the object ofstudy (the densities of theobject along the path ofsome projection beams).

• The goal is to reconstructthe original structure from agiven set of projections.

• Usually done slice-by-slice.

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Object ofstudy

Projection

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• The object of study is represented by a function f (u, v).

f : R2 → R

• We take the line integrals of the image(Radon-Transform).

[Rf ](α, t) =

∫ ∞−∞

f (t cos(α)−q sin(α), t sin(α)+q cos(α)) dq

• We are looking for an f ′(u, v) function that has the sameprojections as the original f (u, v).

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Formulation of the reconstructionproblem

• We assume a discrete representation of the object of study(i.e., it is represented on an n × n sized discrete image).

• The projections are given by the integrals of the imagealong a set of straight lines.

x1 x2 x3 x4

x5 x6 x7 x8

x9 x10 x11 x12

x13 x14 x15 x16 Source

Detector

xjbi

bi+1

ai,j

ai+1,j

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Projections and Sinogram

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Problem with reconstruction

This was the ideal case, which is not so common.

Taking many projection of good quality has high costs.

• High radiation dosage.

• High acquisition time.

• Simply costs much money.

Consequences of the limitations

• Noise in the projections.

• Limited amount of projections.

• Leading to uncertainty of the data.

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Noise in the projections.

Basic background:

• We emit a given number ofX-ray photons.

• Some of them are absorbed bythe material.

In formulation:

• I0 emitted number of photons.

• If measured number ofphotons.

• If = I0e−

∫f (x)dx

Projection:

• ∫f (x)dx = − If

I0

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Basic background:



In formulation:



• If = I0e−

∫f (x)dx

Projection:

• ∫f (x)dx = − If

I0

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Basic background:



In formulation:



• If = I0e−

∫f (x)dx

Projection:

• ∫f (x)dx = − If

I0

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Source of the noise:If follows Poisson distribution:

Causing:

• Less photons lead to morenoise.

• More photons mean lessnoise.

• But also moreradiation.

• i.e.: more harm to thepatient, more cost, etc.

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Causing:





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Causing:





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Causing:





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Causing:





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Causing:





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Noise in practice

100000 photons 1000 photons 100 photons/ pixel / pixel / pixel

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Noise in practice

100000 photons

1000 photons 100 photons

/ pixel

/ pixel / pixel

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Noise in practice


100 photons

/ pixel / pixel

/ pixel

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Noise in practice


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Noise in practice


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Noise in practice

100000 photons


/ pixel

/ pixel / pixel

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Noise in practice


100 photons

/ pixel / pixel

/ pixel

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Noise in practice


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Handling noise

Easy ways to handle noise

• Use high photon counts.

• Increases radiation dosage and cost.• Makes better measurements.

• Use many projections.

• Might also increases radiation dosage and cost,• Projections average out each other, and suppress noise.

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Handling noise


• Use high photon counts.

• Increases radiation dosage and cost.• Makes better measurements.



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Handling noise


• Use high photon counts.• Increases radiation dosage and cost.• Makes better measurements.



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Handling noise



• Use many projections.• Might also increases radiation dosage and cost,• Projections average out each other, and suppress noise.

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Noise and many projections

45 projs., 45 projs., 180 projs.,100000 photons 10000 photons 10000 photons

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Handling noise




Algorithmic ways to handle noise

• Use more advanced reconstruction techniques.

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Handling noise






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Handling noise






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Reconstruction from fewprojections

Sometimes we only have only few projections

Possible causes:

• We want to reduce radiation dosage,

• One projection needs long exposure time (e.g., whenimaging dense objects),

• Exposure damages the object (e.g., crystallography.)

New problems arise

The data is sparse:

• We have less measurements then pixels.

• There are many possible reconstruction, all possibleaccording to projections.

• Algorithms start to ’guess’ and find the wrong result.

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Possible causes:




New problems arise

The data is sparse:




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Possible causes:




New problems arise

The data is sparse:




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Possible causes:




New problems arise

The data is sparse:




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Possible causes:




New problems arise

The data is sparse:




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Possible causes:




New problems arise

The data is sparse:




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Possible causes:




New problems arise

The data is sparse:




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Possible causes:




New problems arise

The data is sparse:




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Low projection Count in practice

180 projs., 30 projs., 6 projs.,FBP FBP FBP

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180 projs.,

30 projs., 6 projs.,

FBP

FBP FBP

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6 projs.,

FBP FBP

FBP

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180 projs., 30 projs., 6 projs.,FBP FBP FBP

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Handling effects of low projectioncount

Easy ways:

• Take more projections.

• Not always possible.• should be considered...

• Take more projections with lower photon counts.

• Sometimes possible (e.g.: half the exposure time perprojection, and double projection count),

• Leads to more but more noisy projections.



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Easy ways:

• Take more projections.

• Not always possible.• should be considered...






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Easy ways:

• Take more projections.• Not always possible.• should be considered...






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Easy ways:


• Take more projections with lower photon counts.• Sometimes possible (e.g.: half the exposure time per

projection, and double projection count),• Leads to more but more noisy projections.



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Easy ways:


• Take more projections with lower photon counts.• Sometimes possible (e.g.: half the exposure time per

projection, and double projection count),• Leads to more but more noisy projections.



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Reconstruction algorithms

Common basic techniques

• Filtered Back-Projection

• Fast method based on mathematical concept. (Filteringand back-projection)

• Needs many projections for good results.

Continuous algebraic reconstruction

• ART, SART, CGLS, etc.

• Iterative equation system solvers,• Slightly better then FBP, but need more time (many

filtering + back-projection cycles).

Discrete algebraic reconstruction

• DART, Energy minimization techniques, etc.

• Iterative equation system solvers, with priors,• Good results, but huge time requirement.

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• Filtered Back-Projection• Fast method based on mathematical concept. (Filtering

and back-projection)• Needs many projections for good results.








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• ART, SART, CGLS, etc.• Iterative equation system solvers,

• Slightly better then FBP, but need more time (manyfiltering + back-projection cycles).




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• ART, SART, CGLS, etc.• Iterative equation system solvers,• Slightly better then FBP, but need more time (many





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• DART, Energy minimization techniques, etc.• Iterative equation system solvers, with priors,

• Good results, but huge time requirement.

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• DART, Energy minimization techniques, etc.• Iterative equation system solvers, with priors,• Good results, but huge time requirement.

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Continuous algebraicreconstruction algorithms

Iterative approximations of the solution

• Usually more accurate then FBP, because of the iterativeimprovement of the result.

• Can incorporate basic prior information

• L1, L2 norm.• bounds on intensities.

• Has higher computational time.

• Each iteration takes as much time as FBP itself.

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• Can incorporate basic prior information• L1, L2 norm.• bounds on intensities.



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• Can incorporate basic prior information• L1, L2 norm.• bounds on intensities.

• Has higher computational time.• Each iteration takes as much time as FBP itself.

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• With this the reconstruction problem can be reformulated as asystem of equations Ax = b, where:

• b, is the vector of m projection values,• x, represents the vector of the image pixel values,• A, describes the connection between the image pixels, and

the projection values, with all aij giving the length linesegment of the i-th projection line in the j pixel.

x1 x2 x3 x4

x5 x6 x7 x8

x9 x10 x11 x12


Detector

xjbi

bi+1

ai,j

ai+1,j

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• With this the reconstruction problem can be reformulated as asystem of equations Ax = b, where:

• b, is the vector of m projection values,• x, represents the vector of the image pixel values,• A, describes the connection between the image pixels, and

the projection values, with all aij giving the length linesegment of the i-th projection line in the j pixel.

x1 x2 x3 x4

x5 x6 x7 x8

x9 x10 x11 x12


Detector

xjbi

bi+1

ai,j

ai+1,j

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Algebraic reconstruction withsimple prior

Algebraic reconstruction with lower and upper bounds

• Pixel values can be in a well defined range (which can bedetermined by previous measurements)

Solve

Ax = b

Subject to

xi ∈ [0, 1]

10

1

x1

x2 Convex set ofsolutions

Ax = b

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Solve

Ax = b

Subject to

xi ∈ [0, 1]

10

1

x1


Ax = b

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Solve

Ax = b

Subject to

xi ∈ [0, 1]

10

1

x1


Ax = b

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Results of Iterative algorithms

180 projs., 30 projs., 30 projs.,FBP FBP SIRT

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180 projs.,


FBP

FBP SIRT

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30 projs.,

FBP FBP

SIRT

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180 projs., 30 projs., 30 projs.,FBP FBP SIRT

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E.g., binary tomography. The pixel values can be either 0 or 1.

Solve

Ax = b

Subject to

xi ∈ {0, 1}

10

1

x1

x2

Ax = b

Binary values

Can also be formulated with energy function.

E =1

2‖Ax + b‖2

2 +α

2

n∑i=1

∑j∈N4i

(xi − xj)2 +

µ

2〈x, 1− x〉

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Solve

Ax = b

Subject to

xi ∈ {0, 1}

10

1

x1

x2

Ax = b

Binary values


E =1

2‖Ax + b‖2

2 +α

2

n∑i=1

∑j∈N4i

(xi − xj)2 +

µ

2〈x, 1− x〉

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Solve

Ax = b

Subject to

xi ∈ {0, 1}

10

1

x1

x2

Ax = b

Binary values


E =1

2‖Ax + b‖2

2 +α

2

n∑i=1

∑j∈N4i

(xi − xj)2 +

µ

2〈x, 1− x〉

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Solve

Ax = b

Subject to

xi ∈ {0, 1}

10

1

x1

x2

Ax = b

Binary values


E =1

2‖Ax + b‖2

2 +α

2

n∑i=1

∑j∈N4i

(xi − xj)2 +

µ

2〈x, 1− x〉

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Solve

Ax = b

Subject to

xi ∈ {0, 1}

10

1

x1

x2

Ax = b

Binary values


E =1

2‖Ax + b‖2

2 +α

2

n∑i=1

∑j∈N4i

(xi − xj)2 +

µ

2〈x, 1− x〉

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Results of discrete algorithms

10 projs., 10 projs., 10 projs.,FBP SIRT Discrete

0.001725 sec. 0.408 sec. 19.878 sec.

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10 projs.,


FBP

SIRT Discrete0.001725 sec. 0.408 sec. 19.878 sec.

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10 projs.,

FBP SIRT

Discrete0.001725 sec. 0.408 sec. 19.878 sec.

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0.001725 sec. 0.408 sec. 19.878 sec.

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0.001725 sec.

0.408 sec. 19.878 sec.

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0.001725 sec. 0.408 sec.

19.878 sec.

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0.001725 sec. 0.408 sec. 19.878 sec.

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What about other imaging likeMRI?

Magnetic Resonance imaging

• Nuclei in our atoms are made of protons and electrons.

• Each particle has two attributes• Spin,• optionally charge.

• If the munber of spins is even, then they cancell eachotherout, but atoms with an odd number of spins has aaccumuted spin.

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What about other imaging likeMRI?

Magnetic Resonance imaging

• Nuclei in our atoms are made of protons and electrons.

• Each particle has two attributes• Spin,• optionally charge.

• If the munber of spins is even, then they cancell eachotherout, but atoms with an odd number of spins has aaccumuted spin.

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Magnetic Resonance Imaging

In a strong magnetic field, spin of the atoms get aligned withthe direction of the field.

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• Periodic radio signals can change

direction of the nuclei.

• Excitation frequencycorresponds to the atomand magnetic field energy.

• After stopping the radio signalthe nuclei start to return to theiroriginal direction.

• In the process they emitelectromagnetic signals.

• Measuring the attenuation of thesignal gives information on theatoms along a plane in space.

• Having data on enough planesthe task is similar to transmissiontomography.

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Noise and lack of data in MRI

Cost of imaging is time

• In MRI imaging every one measurement has distortions(noise).

• It has to be repeated many times.

• The measurement has to be repeated on many planes.

• Each measurement takes time, while we wait for the nucleito get back in order.

• Measurement can be highly time-consuming.

• High resolution imaging can take up to half an our or more.

Problems and possibilities

• Less measurements with more noise?

• More imaging time?

• Advanced algorithms?

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• Measurement can be highly time-consuming.• High resolution imaging can take up to half an our or more.





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Summary

There are many ways to take accurate images of the interior ofobjects.

• The simple way is to take many data (measurement) ofgood quality.

• If the data quality is not good it can be balanced by hugeamount.

• If the amount of data is not sufficient?

• It might be handled by advanced methods.• These methods use extra resources for improved imaging.

• Prior knowledge on the data (takes time to find out goodpriors.)

• Extra computational time to incorporate prior int theimaging.

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Summary








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• If the amount of data is not sufficient?• It might be handled by advanced methods.

• These methods use extra resources for improved imaging.



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Summary




• If the amount of data is not sufficient?• It might be handled by advanced methods.• These methods use extra resources for improved imaging.



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Techniques

Laszlo Varga



Problemformulation


Noise




Summary







Uncertainty ofNon-


Techniques

Laszlo Varga



Problemformulation


Noise




Summary







Date post:	10-Mar-2020
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Uncertainty of Non-Destructive Interior Imaging...

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