Tomographic Image Reconstruction.An Introduction.
Milan ZvolskýLecture on Medical Physics
by Erika Garutti & Florian Grüner
28.11.2014
Outline.
1 Introduction2 PET, SPECT, CT3 Basic Idea: Projections4 Analytic Image Reconstruction
BackprojectionFiltered Backprojection
5 Iterative Image ReconstructionBasic IdeaML-EM Algorithm
6 The EndoTOFPET-US Project7 Summary
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Introduction.
Cutting into slices is a bad idea to perform on humans
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TomographyImaging by sectioning: The word tomography is derived from thegreek tome (cut) and graphein (to write).
Introduction.
Cutting into slices is a bad idea to perform on humans
Milan Zvolský (DESY.) 3 / 48
TomographyImaging by sectioning: The word tomography is derived from thegreek tome (cut) and graphein (to write).
Introduction.
Figure: PET-CT
Figure: PET brain images
Figure: CT: 2D → 3D
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Introduction.Reconstruction quality is a crucial point in medical imagingNo use in building accurate detector when image quality is poorVery active field of research
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Computer Tomography (CT).
Rotating X-ray tube + detectorX-rays propagate through a x-section of patient
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Computer Tomography (CT).
Rotating X-ray tube + detectorX-rays propagate through a x-section of patient
Measure the exit beam intensity integrated alonga line between X-ray source and detector
Id = I0 exp
[−∫ d
0µ(s;E)ds
]with µ linear attenuation coefficient as a function of the
location s and the effective energy E
Basic measurement of CT: Line integral of thelinear attenuation coefficient
Milan Zvolský (DESY.) 6 / 48
Positron Emission Tomography (PET).
Glucose-like 18FDG (β+ emitter) concentrates inthe metabolical active areasWe are interested in the radioactivitydistribution within the bodye+e− → 2γ (back-to-back, 511 keV each)
Milan Zvolský (DESY.) 7 / 48
Positron Emission Tomography (PET).
Glucose-like 18FDG (β+ emitter) concentrates inthe metabolical active areasWe are interested in the radioactivitydistribution within the bodye+e− → 2γ (back-to-back, 511 keV each)
Detect the two γs in coincidence2 detectors fire at the same time → Draw a lineThese are called Lines of Response (LOR)Measure the integrated activity of the LOR
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SPECT.Single Photon Emission Computed Tomographyγ-emitting radionuclide → 1 γ → no LORGamma camera with collimators → reconstructlines perpendicular to the detector
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SPECT.Single Photon Emission Computed Tomographyγ-emitting radionuclide → 1 γ → no LORGamma camera with collimators → reconstructlines perpendicular to the detector
Similar to PET (morphologic images)Cheaper/simpler than PETWorse spacial resolution, lower sensitivity
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Emission & Transmission Tomography.
Transmission Tomography (CT)Radiation source outside the patient: X-ray tubeor long-lived radionuclide rotates around thebodyQuantity to be reconstructed is the photon linearattenuation coefficient of the body
Emission Tomography (PET & SPECT)Radiation source inside the patient: γ- ore+-emitting radionuclideQuantity to be reconstructed is the activityconcentration of the radiopharmacon inside thebody
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1 Introduction
2 PET, SPECT, CT
3 Basic Idea: Projections
4 Analytic Image ReconstructionBackprojectionFiltered Backprojection
5 Iterative Image ReconstructionBasic IdeaML-EM Algorithm
6 The EndoTOFPET-US Project
7 Summary
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Basic Idea.Example: PhotographyTwo trees in a park, make 2 pictures from east and south, try tocreate a map of the park.
A photo is a projection of an object onto a plane
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Basic Idea.Example: Another PhotographyOther configuration: If you see two separate trees on both views, canyou uniquely reconstruct the map of trees?Here you cannot reconstruct the position and height of both trees.
Figure: Two trees seen on both viewsFigure: There are two solutions
If we take another picture at 45◦, we are able to solve the ambiguity.
Milan Zvolský (DESY.) 12 / 48
Basic Idea.Example: Another PhotographyOther configuration: If you see two separate trees on both views, canyou uniquely reconstruct the map of trees?Here you cannot reconstruct the position and height of both trees.
Figure: Two trees seen on both viewsFigure: There are two solutions
If we take another picture at 45◦, we are able to solve the ambiguity.
Milan Zvolský (DESY.) 12 / 48
Basic Idea: Projections.Before: photo, now: Projection is a line integralProjection p(s, φ) at angle φ, s is coordinate on detector
Projection p(s) the same for any φ
p(s,ϕ2)
p(s,ϕ1)
Projection p(s, φ) depends on orientation
Milan Zvolský (DESY.) 13 / 48
Projections: Angle dependency.Example: Point source on the y axisLocation s of the spike on the 1D detector: s = r sinφ.The projection p(s, φ) in the s-φ-coordinate system is a sine function.
x
sp(s,Φ)
yΦ
s'
s'
0
0
Point Source
Detector
Φ
r
SinogramA sinogram is a representation of the projections on the s-φ plane.
Milan Zvolský (DESY.) 14 / 48
Projections: Angle dependency.Example: Point source on the y axisLocation s of the spike on the 1D detector: s = r sinφ.The projection p(s, φ) in the s-φ-coordinate system is a sine function.
SinogramA sinogram is a representation of the projections on the s-φ plane.
Milan Zvolský (DESY.) 14 / 48
Projections: Angle dependency.TomographyFind the image f(x, y) from the measured projections p(s, φ)
We measure a sinogram: We want an image:
Milan Zvolský (DESY.) 15 / 48
1 Introduction
2 PET, SPECT, CT
3 Basic Idea: Projections
4 Analytic Image ReconstructionBackprojectionFiltered Backprojection
5 Iterative Image ReconstructionBasic IdeaML-EM Algorithm
6 The EndoTOFPET-US Project
7 Summary
Milan Zvolský (DESY.) 16 / 48
Backprojection Procedure.Backprojection
Placing a value of p(s, φ) back into the position of theappropriate LORBut the knowledge of where the values came from was lost inthe projection stepThe best we can do is place a constant value into all elementsalong the line
Milan Zvolský (DESY.) 17 / 48
Backprojection Example.1st projection
a b
c d
a+b
c+d
Milan Zvolský (DESY.) 18 / 48
Backprojection Example.2nd projection
a b
c d
a+b
c+d
a+d
b
c
Milan Zvolský (DESY.) 18 / 48
Backprojection Example.3rd projection
a b
c d
b+da+c
a+b
c+d
a+d
b
c
Milan Zvolský (DESY.) 18 / 48
Backprojection Example.4th projection
a b
c d
b+da+c
a
c+bd
a+b
c+d
a+d
b
c
Milan Zvolský (DESY.) 18 / 48
Backprojection Example.Backproject
b+da+c
a
c+bd
a+b
c+d
a+d
b
c
? ?
? ?
Milan Zvolský (DESY.) 18 / 48
Backprojection Example.1st backprojection
b+da+c
a
c+bd
a+b
c+d
a+d
b
c
a+ba+b
c+d c+d
Milan Zvolský (DESY.) 18 / 48
Backprojection Example.2nd backprojection
b+da+c
a
c+bd
a+b
c+d
a+d
b
c
a+b+a+b+
c+d+ c+d+
ba+d
c a+d
Milan Zvolský (DESY.) 18 / 48
Backprojection Example.3rd backprojection
b+da+c
a
c+bd
a+b
c+d
a+d
b
c
a+b+a+b+
c+d+ c+d+
b+a+d+
c+ a+d+
a+c
a+c
b+d
b+d
Milan Zvolský (DESY.) 18 / 48
Backprojection Example.4th backprojection
b+da+c
a
c+bd
a+b
c+d
a+d
b
c
a+b+a+b+
c+d+ c+d+
b+a+d+
c+ a+d+
a+c+
a+c+c+b
ab+d+c+b
b+d+d
Milan Zvolský (DESY.) 18 / 48
Backprojection Example.Subtract projection sum from each entry
b+da+c
a
c+bd
a+b
c+d
a+d
b
c
a+ba+b
c+d c+d
ba+d
c a+d
a+c
a+cc+b
ab+dc+b
b+dd
Milan Zvolský (DESY.) 18 / 48
Backprojection Example.Subtract projection sum from each entry
b+da+c
a
c+bd
a+b
c+d
a+d
b
c
c c+d
ba+d
c a+d
a
c
abc+b
b+d
Milan Zvolský (DESY.) 18 / 48
Backprojection Example.Divide by number of projections −1 = 3
b+da+c
a
c+bd
a+b
c+d
a+d
b
c
c c+d
ba+d
Milan Zvolský (DESY.) 18 / 48
Backprojection Procedure.1 view: spike of intensity 1.This is sum of activity alongprojection pathRe-distribute activity back toits original pathGive equal activity everywherealong the lineMany angles → Tall spike atthe location of the point source(d) Ups...
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Backprojection: Blurring of the image.
Figure: OriginalShepp-Logan phantom
Figure: Backprojected image for(1,3,7,15,31,63) projections
Milan Zvolský (DESY.) 20 / 48
Backprojection.The Radon transformThe Radon transform of a distribution f(x, y) is given by
p(s, φ) =
∫ ∞−∞
∫ ∞−∞
f(x, y) · δ(x cosφ+ y sinφ− s)dxdy
.Delta function: Integrand is zero everywhere except on the line L(s, φ)
Backprojected image:
b(x, y) =
∫ π
0p(s, φ)|s=x cosφ+y sinφdφ
.Integrate over 180◦, the other half doesn’t give extra information
Reconstructed image is blurred:
b(x, y) = f(x, y)× 1√x2 + y2
Milan Zvolský (DESY.) 21 / 48
Interlude: Fourier Transform.There is a close relationship between Radon and the Fourier trafo!
Fourier transform (FT)
F{p(s)} = P (ω) =1√2π
∫ ∞−∞
p(s)e−iωsds
Summation of lines causes duplication in the centerOversampling in the center of the Fourier space
Milan Zvolský (DESY.) 22 / 48
Central Slice Theorem.Central Slice Theorem (CST)
F1{p(s, φ′)} = F2{f(x, y)}|φ=φ′
The following operations are the equivalent:Take a 2D function f(x, y), project it onto a line, and do a FT of that projection.
Do a 2D FT of f(x, y) first, and then take a slice through the origin, parallel to
the projection line.
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Milan Zvolský (DESY.) 23 / 48
Filtered Backprojection: “Proof”.Re-write the image f(x, y) via the inverse FT:
f(x, y) = F−12 {F (vx, vy)}
=
∫ ∞−∞
dvx∫ ∞−∞
dvyF (vx, vy)e2πi(vxx+vyy)
Polar coordinates: vx = ω cosφ, vy = ω sinφ, dvxdvy = ωdωdφ:
f(x, y) =
∫ 2π
0dφ∫ ∞0
dωω F (ω cosφ, ω sinφ)︸ ︷︷ ︸=P (ω) (CST)
e2πiω(
=s︷ ︸︸ ︷x cosφ+ y sinφ)
=
∫ π
0dφ∫ ∞−∞
dω|ω|P (ω)e2πiωs (changing the integration limits)
f(x, y) =
∫ π
0dφ
[∫ ∞−∞
dω|ω|P (ω)e2πiωs]
︸ ︷︷ ︸=F−1{|ω|P (ω)}≡p′(s,φ)
=
∫ π
0dφp′(s, φ)
Milan Zvolský (DESY.) 24 / 48
Filtered Backprojection: “Proof”.Re-write the image f(x, y) via the inverse FT:
f(x, y) = F−12 {F (vx, vy)} =∫ ∞−∞
dvx∫ ∞−∞
dvyF (vx, vy)e2πi(vxx+vyy)
Polar coordinates: vx = ω cosφ, vy = ω sinφ, dvxdvy = ωdωdφ:
f(x, y) =
∫ 2π
0dφ∫ ∞0
dωω F (ω cosφ, ω sinφ)︸ ︷︷ ︸=P (ω) (CST)
e2πiω(
=s︷ ︸︸ ︷x cosφ+ y sinφ)
=
∫ π
0dφ∫ ∞−∞
dω|ω|P (ω)e2πiωs (changing the integration limits)
f(x, y) =
∫ π
0dφ
[∫ ∞−∞
dω|ω|P (ω)e2πiωs]
︸ ︷︷ ︸=F−1{|ω|P (ω)}≡p′(s,φ)
=
∫ π
0dφp′(s, φ)
Milan Zvolský (DESY.) 24 / 48
Filtered Backprojection: “Proof”.Re-write the image f(x, y) via the inverse FT:
f(x, y) = F−12 {F (vx, vy)} =∫ ∞−∞
dvx∫ ∞−∞
dvyF (vx, vy)e2πi(vxx+vyy)
Polar coordinates: vx = ω cosφ, vy = ω sinφ, dvxdvy = ωdωdφ:
f(x, y) =
∫ 2π
0dφ∫ ∞0
dωω F (ω cosφ, ω sinφ)︸ ︷︷ ︸=P (ω) (CST)
e2πiω(
=s︷ ︸︸ ︷x cosφ+ y sinφ)
=
∫ π
0dφ∫ ∞−∞
dω|ω|P (ω)e2πiωs (changing the integration limits)
f(x, y) =
∫ π
0dφ
[∫ ∞−∞
dω|ω|P (ω)e2πiωs]
︸ ︷︷ ︸=F−1{|ω|P (ω)}≡p′(s,φ)
=
∫ π
0dφp′(s, φ)
Milan Zvolský (DESY.) 24 / 48
Filtered Backprojection: “Proof”.Re-write the image f(x, y) via the inverse FT:
f(x, y) = F−12 {F (vx, vy)} =∫ ∞−∞
dvx∫ ∞−∞
dvyF (vx, vy)e2πi(vxx+vyy)
Polar coordinates: vx = ω cosφ, vy = ω sinφ, dvxdvy = ωdωdφ:
f(x, y) =
∫ 2π
0dφ∫ ∞0
dωω F (ω cosφ, ω sinφ)︸ ︷︷ ︸=P (ω) (CST)
e2πiω(
=s︷ ︸︸ ︷x cosφ+ y sinφ)
=
∫ π
0dφ∫ ∞−∞
dω|ω|P (ω)e2πiωs (changing the integration limits)
f(x, y) =
∫ π
0dφ
[∫ ∞−∞
dω|ω|P (ω)e2πiωs]
︸ ︷︷ ︸=F−1{|ω|P (ω)}≡p′(s,φ)
=
∫ π
0dφp′(s, φ)
Milan Zvolský (DESY.) 24 / 48
Filtered Backprojection: “Proof”.Re-write the image f(x, y) via the inverse FT:
f(x, y) = F−12 {F (vx, vy)} =∫ ∞−∞
dvx∫ ∞−∞
dvyF (vx, vy)e2πi(vxx+vyy)
Polar coordinates: vx = ω cosφ, vy = ω sinφ, dvxdvy = ωdωdφ:
f(x, y) =
∫ 2π
0dφ∫ ∞0
dωω F (ω cosφ, ω sinφ)︸ ︷︷ ︸=P (ω) (CST)
e2πiω(
=s︷ ︸︸ ︷x cosφ+ y sinφ)
=
∫ π
0dφ∫ ∞−∞
dω|ω|P (ω)e2πiωs (changing the integration limits)
f(x, y) =
∫ π
0dφ
[∫ ∞−∞
dω|ω|P (ω)e2πiωs]
︸ ︷︷ ︸=F−1{|ω|P (ω)}≡p′(s,φ)
=
∫ π
0dφp′(s, φ)
Milan Zvolský (DESY.) 24 / 48
Filtered Backprojection.
f(x, y) =
∫ π
0dφ[ ∫ ∞−∞
dω|ω|P (ω)e2πiωs]s=x cosφ+y sinφ
FT of projection p(s, φ)Multiply by frequency filter |ω|Inverse-transform this productThis filtered projection is backprojectedThen sum over all filtered projections
h(t)
x
|ω|
ω-ωs ωs
Ramp filter |ω|Blurring (low ω) minimisedContrasts (high ω) accentuated
Milan Zvolský (DESY.) 25 / 48
Filtered Backprojection.
f(x, y) =
∫ π
0dφ[ ∫ ∞−∞
dω|ω|P (ω)e2πiωs]s=x cosφ+y sinφ
FT of projection p(s, φ)
Multiply by frequency filter |ω|Inverse-transform this productThis filtered projection is backprojectedThen sum over all filtered projections
h(t)
x
|ω|
ω-ωs ωs
Ramp filter |ω|Blurring (low ω) minimisedContrasts (high ω) accentuated
Milan Zvolský (DESY.) 25 / 48
Filtered Backprojection.
f(x, y) =
∫ π
0dφ[ ∫ ∞−∞
dω|ω|P (ω)e2πiωs]s=x cosφ+y sinφ
FT of projection p(s, φ)Multiply by frequency filter |ω|
Inverse-transform this productThis filtered projection is backprojectedThen sum over all filtered projections
h(t)
x
|ω|
ω-ωs ωs
Ramp filter |ω|Blurring (low ω) minimisedContrasts (high ω) accentuated
Milan Zvolský (DESY.) 25 / 48
Filtered Backprojection.
f(x, y) =
∫ π
0dφ[ ∫ ∞−∞
dω|ω|P (ω)e2πiωs]s=x cosφ+y sinφ
FT of projection p(s, φ)Multiply by frequency filter |ω|Inverse-transform this product
This filtered projection is backprojectedThen sum over all filtered projections
h(t)
x
|ω|
ω-ωs ωs
Ramp filter |ω|Blurring (low ω) minimisedContrasts (high ω) accentuated
Milan Zvolský (DESY.) 25 / 48
Filtered Backprojection.
f(x, y) =
∫ π
0dφ[ ∫ ∞−∞
dω|ω|P (ω)e2πiωs]s=x cosφ+y sinφ
FT of projection p(s, φ)Multiply by frequency filter |ω|Inverse-transform this productThis filtered projection is backprojected
Then sum over all filtered projectionsh(t)
x
|ω|
ω-ωs ωs
Ramp filter |ω|Blurring (low ω) minimisedContrasts (high ω) accentuated
Milan Zvolský (DESY.) 25 / 48
Filtered Backprojection.
f(x, y) =
∫ π
0dφ[ ∫ ∞−∞
dω|ω|P (ω)e2πiωs]s=x cosφ+y sinφ
FT of projection p(s, φ)Multiply by frequency filter |ω|Inverse-transform this productThis filtered projection is backprojectedThen sum over all filtered projections
h(t)
x
|ω|
ω-ωs ωs
Ramp filter |ω|Blurring (low ω) minimisedContrasts (high ω) accentuated
Milan Zvolský (DESY.) 25 / 48
Filtered Backprojection.f(x, y) =
∫ π
0dφ[ ∫ ∞−∞
dω|ω|P (ω)e2πiωs]s=x cosφ+y sinφ
FT of projection p(s, φ)Multiply by frequency filter |ω|Inverse-transform this productThis filtered projection is backprojectedThen sum over all filtered projections
h(t)
x
|ω|
ω-ωs ωs
Ramp filter |ω|Blurring (low ω) minimisedContrasts (high ω) accentuated
A complete set of 1D projections allows the reconstruction of the original 2D
distribution without loss of informationMilan Zvolský (DESY.) 25 / 48
BP vs. FBP.
One single projection for both methods:
Figure: Backprojection Figure: Filtered Backprojection
Milan Zvolský (DESY.) 26 / 48
Filtered Backprojection.p( )ξ,φ
f(x,y)
p ( )ξ,φ’
BP
ramp filterFigure 8. Flow of the FBP algorithm.
Demonstration of the FBP algorithm:
projection
ramp filter
backprojection
original image
sinogram
FBP image
filtered sinogram
1 view 2 views 10 views
30 views 50 views 80 views
Figure 8. Demonstration of the FBP algorithm.
6
Milan Zvolský (DESY.) 27 / 48
BP vs. FBP.
Figure: Backprojection Figure: Filtered Backprojection
Milan Zvolský (DESY.) 28 / 48
Image Quality vs. Number of Events.
102 103 104
105 106
fFBP(x)
f (x)
107
Number of events follows Poisson distribution→ Signal-to-noise ratio ≈
√N
Analytical Reconstruction works exactly for N =∞Milan Zvolský (DESY.) 29 / 48
Effect of Poissonian Noise.
σ=0 σ=0.05 σ=0.1 σ=0.2 σ=0.5
Typically low noise for CT, high noise for PETHigh signal variability but FBP assumes exact distributionAmplification of high-frequency noise (→ use cut-off)
SignalNoiseMeasure
RampfilterFilteredsignalFilterednoise
=+
νc
FBP bad if data is noisy → Iterative Image Reconstruction
Milan Zvolský (DESY.) 30 / 48
1 Introduction
2 PET, SPECT, CT
3 Basic Idea: Projections
4 Analytic Image ReconstructionBackprojectionFiltered Backprojection
5 Iterative Image ReconstructionBasic IdeaML-EM Algorithm
6 The EndoTOFPET-US Project
7 Summary
Milan Zvolský (DESY.) 31 / 48
FBP vs. Reality.Reality is more complex:
Data is discrete: FBP only precise if all angles are availableData is noisy: Measurements follow a probability distributionDetectors are unprecise: mis-positioning of photonsDetector geometry may not provide complete dataNot all photons travel along straight lines: scatter, absorption
Input prior knowledge of the detector & the physics process
Milan Zvolský (DESY.) 32 / 48
Iterative Methods - Basic Idea.Emission tomography described asy = S(x) + noisey: measurement, S: System operator, x: activity
Model system operator S with systemmatrix A → linear approximationAij : Probability that emission in voxel i(xi) results in a detection in detector j (yj)y = Ax or yj =
∑iAijxi
Inverse problem: Know yi, want xiCould do x = A−1yBut: A is huge & cannot be inverted
LOR i
Voxel j
f(x,y)
LOR i
Modeling of detectionimperfections
Iterative Analytical
Iterative image reconstructionSolve inverse problem iterativelyAssume Poissonian noise on yi
Milan Zvolský (DESY.) 33 / 48
Iterative Methods - Basic Idea.Measurements yi: independent random variables (Poisson)Expectation value: µi = Ai · x =
∑j Aijxj
Probability to measure k given λ: p(k|λ) = e−λλk
k!
Likelihood:
L(x) = p(y|x) =∏i
p(yi|x) =∏i
e−Ai·x(Ai · x)yiyi!
Among all possible images x we choose the one that maximisesthe probability of producing the data (find most likely image)Maximise the log-likelihood, i.e. maximise log(p(y|x))
Maximum Likelihood - Expectation Maximisation (ML-EM)Objective function to maximise: log-Likelihood (ML)Maximisation algorithm: Expectation Maximisation (EM)
Milan Zvolský (DESY.) 34 / 48
ML-EM Algorithm.Initial guess for the image (uniform)x(0)i
Simulate measurements from estimate (forward proj.)ysimuj =
∑k Akjx
(0)k
Compare this with actual measurementsRatio Rj =
yjysimuj
Improve image estimate (backward projection)x(1)i = x
(0)i ·
1∑j Aij
·∑
j AijRj
Repeat until convergence
ML-EM
x(n+1)i = x
(n)i ·
1∑j Aij
·∑j
Aijyj∑
k Akjx(n)k
Milan Zvolský (DESY.) 35 / 48
ML-EM Algorithm.Initial guess for the image (uniform)x(0)i
Simulate measurements from estimate (forward proj.)ysimuj =
∑k Akjx
(0)k
Compare this with actual measurementsRatio Rj =
yjysimuj
Improve image estimate (backward projection)x(1)i = x
(0)i ·
1∑j Aij
·∑
j AijRj
Repeat until convergence
ML-EM
x(n+1)i = x
(n)i ·
1∑j Aij
·∑j
Aijyj∑
k Akjx(n)k
Milan Zvolský (DESY.) 35 / 48
ML-EM Algorithm.Initial guess for the image (uniform)x(0)i
Simulate measurements from estimate (forward proj.)ysimuj =
∑k Akjx
(0)k
Compare this with actual measurementsRatio Rj =
yjysimuj
Improve image estimate (backward projection)x(1)i = x
(0)i ·
1∑j Aij
·∑
j AijRj
Repeat until convergence
ML-EM
x(n+1)i = x
(n)i ·
1∑j Aij
·∑j
Aijyj∑
k Akjx(n)k
Milan Zvolský (DESY.) 35 / 48
ML-EM Algorithm.Initial guess for the image (uniform)x(0)i
Simulate measurements from estimate (forward proj.)ysimuj =
∑k Akjx
(0)k
Compare this with actual measurementsRatio Rj =
yjysimuj
Improve image estimate (backward projection)x(1)i = x
(0)i ·
1∑j Aij
·∑
j AijRj
Repeat until convergence
ML-EM
x(n+1)i = x
(n)i ·
1∑j Aij
·∑j
Aijyj∑
k Akjx(n)k
Milan Zvolský (DESY.) 35 / 48
ML-EM Algorithm.Initial guess for the image (uniform)x(0)i
Simulate measurements from estimate (forward proj.)ysimuj =
∑k Akjx
(0)k
Compare this with actual measurementsRatio Rj =
yjysimuj
Improve image estimate (backward projection)x(1)i = x
(0)i ·
1∑j Aij
·∑
j AijRj
Repeat until convergence
ML-EM
x(n+1)i = x
(n)i ·
1∑j Aij
·∑j
Aijyj∑
k Akjx(n)k
Milan Zvolský (DESY.) 35 / 48
ML-EM Algorithm.ML-EM
x(n+1)i = x
(n)i ·
1∑j Aij
·∑j
Aijyj∑
k Akjx(n)k
Image Domain Projection Domain
ForwardProject
Backproject
Compare with measurement
Initialguess
ith estimate
Correction factorfor each projection
Correction factorfor image estimate
Multiply byimage estimate
Divide by weighting term
Milan Zvolský (DESY.) 36 / 48
ML-EM Algorithm.ML-EM
x(n+1)i = x
(n)i ·
1∑j Aij
·∑j
Aijyj∑
k Akjx(n)k
Image Domain Projection Domain
ForwardProject
Backproject
Compare with measurement
Initialguess
ith estimate
Correction factorfor each projection
Correction factorfor image estimate
Multiply byimage estimate
Divide by weighting term
??
?? 3
7
4 6
2.52.5
2.52.5 5
5
5 5
x=(3+7)/4=10/4=2.5
AssumeAij = 1 ∀i, j∑
j Aij = 2 ∀i
Milan Zvolský (DESY.) 37 / 48
ML-EM Algorithm.ML-EM
x(n+1)i = x
(n)i ·
1∑j Aij
·∑j
Aijyj∑
k Akjx(n)k
Image Domain Projection Domain
ForwardProject
Backproject
Compare with measurement
Initialguess
ith estimate
Correction factorfor each projection
Correction factorfor image estimate
Multiply byimage estimate
Divide by weighting term
??
?? 3
7
4 6
2.52.5
2.52.5 5
5
5 5
c11 = (3/5 + 4/5)/2 = 0.7 x11 = 1.75
AssumeAij = 1 ∀i, j∑
j Aij = 2 ∀i
Milan Zvolský (DESY.) 37 / 48
ML-EM Algorithm.ML-EM
x(n+1)i = x
(n)i ·
1∑j Aij
·∑j
Aijyj∑
k Akjx(n)k
Image Domain Projection Domain
ForwardProject
Backproject
Compare with measurement
Initialguess
ith estimate
Correction factorfor each projection
Correction factorfor image estimate
Multiply byimage estimate
Divide by weighting term
??
?? 3
7
4 6
2.52.5
2.51.75 5
5
5 5
c12 = (3/5 + 6/5)/2 = 0.9x12 = 2.25
AssumeAij = 1 ∀i, j∑
j Aij = 2 ∀i
Milan Zvolský (DESY.) 37 / 48
ML-EM Algorithm.ML-EM
x(n+1)i = x
(n)i ·
1∑j Aij
·∑j
Aijyj∑
k Akjx(n)k
Image Domain Projection Domain
ForwardProject
Backproject
Compare with measurement
Initialguess
ith estimate
Correction factorfor each projection
Correction factorfor image estimate
Multiply byimage estimate
Divide by weighting term
??
?? 3
7
4 6
2.52.5
5
5
5 5
c13 = (7/5 + 4/5)/2 = 1.1x13 = 2.75
1.752.25
AssumeAij = 1 ∀i, j∑
j Aij = 2 ∀i
Milan Zvolský (DESY.) 37 / 48
ML-EM Algorithm.ML-EM
x(n+1)i = x
(n)i ·
1∑j Aij
·∑j
Aijyj∑
k Akjx(n)k
Image Domain Projection Domain
ForwardProject
Backproject
Compare with measurement
Initialguess
ith estimate
Correction factorfor each projection
Correction factorfor image estimate
Multiply byimage estimate
Divide by weighting term
??
?? 3
7
4 6
5
5
5 5
c14 = (7/5 + 6/5)/2 = 1.3x14 = 3.25
1.752.25
2.75 2.5
AssumeAij = 1 ∀i, j∑
j Aij = 2 ∀i
Milan Zvolský (DESY.) 37 / 48
ML-EM Algorithm.ML-EM
x(n+1)i = x
(n)i ·
1∑j Aij
·∑j
Aijyj∑
k Akjx(n)k
Image Domain Projection Domain
ForwardProject
Backproject
Compare with measurement
Initialguess
ith estimate
Correction factorfor each projection
Correction factorfor image estimate
Multiply byimage estimate
Divide by weighting term
3
7
4 6
5
5
5 5
1.752.25
2.753.25
1 2
3 4
AssumeAij = 1 ∀i, j∑
j Aij = 2 ∀i
Milan Zvolský (DESY.) 37 / 48
ML-EM Algorithm.ML-EM
x(n+1)i = x
(n)i ·
1∑j Aij
·∑j
Aijyj∑
k Akjx(n)k
Image Domain Projection Domain
ForwardProject
Backproject
Compare with measurement
Initialguess
ith estimate
Correction factorfor each projection
Correction factorfor image estimate
Multiply byimage estimate
Divide by weighting term
1 1.5 2 2.5 3 3.5 4
1
1.5
2
2.5
3
3.5
4
4 projections
No convergence to true image
1 1.5 2 2.5 3 3.5 4
1
1.5
2
2.5
3
3.5
4
5 projections
Deviation < 1% after 50 it.
Milan Zvolský (DESY.) 38 / 48
ML-EM: Number of Iterations.0 1 2 3 4 6 8 16 32
Figure: Number of iterations
With noise: > 50 iterations needed≈ 100 times slower than FBPLow frequencies are reconstructed first. For higher iterations,the alg. will attempt to recover noise → high spacial variance
Early termination (fast, but non-uniform convergence)Post-smoothing with Gaussian kernel
k = 4 k = 16 k = 64 k = 1600
Milan Zvolský (DESY.) 39 / 48
System Model.Reconstruction can only be as good as our model agrees with reality→ Consider all physical effects governing our system:
Positron rangePhoton acollinearityScattering in the bodyAttenuation in the bodyScattering in the detectorAbsorption in the detectorDeadtime of detectorRandom coincidences comingfrom the background
LOR i
Voxel j
f(x,y)
LOR i
Modeling of detectionimperfections
Iterative Analytical
Milan Zvolský (DESY.) 40 / 48
ML-EM vs. FBP.Iterative methods perform much better than analytical methods ifdata is noisy:
FBP
EM-ML withattenuation pre-correction
EM-ML withattenuation
correction build-in
Milan Zvolský (DESY.) 41 / 48
1 Introduction
2 PET, SPECT, CT
3 Basic Idea: Projections
4 Analytic Image ReconstructionBackprojectionFiltered Backprojection
5 Iterative Image ReconstructionBasic IdeaML-EM Algorithm
6 The EndoTOFPET-US Project
7 Summary
Milan Zvolský (DESY.) 42 / 48
The EndoTOFPET-US Project
Pancreas
Cecum
Commonbile duct
Stomach
Pancreaticduct
Liver
Gallbladder
Duodenum
Transverse colon
Ascending colon
Descending colon
Colon
Milan Zvolský (DESY.) 43 / 48
Positron Emission Tomography (PET)
Radiotracer (β+ emitter) concentrates in the metabolical active arease+e− → 2γ (back-to-back, 511 keV each)Detect the two γs in coincidence
The EndoTOFPET-US Project
Pancreas
Cecum
Commonbile duct
Stomach
Pancreaticduct
Liver
Gallbladder
Duodenum
Transverse colon
Ascending colon
Descending colon
Colon
Milan Zvolský (DESY.) 43 / 48
Objectives
Development of new biomarkersIntra-operative Time-of-Flight (TOF) PET DetectorPrototype for prostate & pancreas cancer
The EndoTOFPET-US Project
Pancreas
Milan Zvolský (DESY.) 43 / 48
Objectives
Development of new biomarkersIntra-operative Time-of-Flight (TOF) PET DetectorPrototype for prostate & pancreas cancer
The EndoTOFPET-US Project
Pancreas
Milan Zvolský (DESY.) 43 / 48
Objectives
Development of new biomarkersIntra-operative Time-of-Flight (TOF) PET DetectorPrototype for prostate & pancreas cancer
The EndoTOFPET-US Project
Pancreas
Milan Zvolský (DESY.) 43 / 48
Objectives
Development of new biomarkersIntra-operative Time-of-Flight (TOF) PET DetectorPrototype for prostate & pancreas cancer
Challenges
Extreme miniaturisationFast crystals & ultra-fast photodetectionAim for our project: Coincidence timeresolution 200 ps FWHM (3 cm)This reduces background noise from otherorgans → better image qualityImage reconstruction for free-handimaging, image resolution of 1 mm
EndoTOFPET Image ReconstructionChallenges
Time of flight (TOF)Limited angle problemFreehand→ undefined volume of interestLow sensitivity (≈ 1%), high noiseReconstruct the image on-line toprovide guidance for the physician
Solution
ML-EM iterative reconstruction: Good performance in case of Poissonian noiseGPU Computation: Solving a massively parallel problem
GPU speedup by factor O(10)
Image reconstruction in O(min.)
Milan Zvolský (DESY.) 44 / 48
Full System Simulation.Simulation of whole detector system → optimise design, test reconstructionGeant4-based toolkit GAMOS with custom extensions (e.g. TOF)Study sensitivity, image resolution etc. on simple phantomsStudy medical procedures on full-body phantoms
Expected resolution in X & Y: ≈ 1 mm(limited by crystal size and tracking precision)Expected resolution in Z: ≈ 5 mm(limited by low angular coverage)
Quantify image qualityHow good can wesuppress backgroundfrom other organs?How much data do weneed to acquire?
Milan Zvolský (DESY.) 45 / 48
Patient Data: PSMA.PET/CT scans performed at TU Munich (Klinikum rechts der Isar)Prostate-specific membrane antigen (PSMA)One specific patient with injected dose of 140 MBqBladder uptake ≈ 20 MBq, prostatic lesion ≈ 0.5 MBqFull-body PET resolution: ≈ 6 mmDownsize datasets: Merge voxels to speed up simulationLoad PET (activity) and CT (scatter) DICOM images into the simulation
Bladder
Prostatic Lesion
BladderProstatic Lesion
Milan Zvolský (DESY.) 46 / 48
Patient Data: PSMA.First reconstructed image of full-body PSMA PET data with the EndoTOFPET system
(a) Transverse (b) Coronal (c) Sagittal
External Plate
Increase lesion uptake ×10:
(a) Transverse (b) Coronal (c) Sagittal
External Plate
Endoscope
Prostatic lesion can be well separated from bladder backgroundMilan Zvolský (DESY.) 47 / 48
Patient Data: PSMA.First reconstructed image of full-body PSMA PET data with the EndoTOFPET system
(a) Transverse (b) Coronal (c) Sagittal
External Plate
Increase lesion uptake ×10 and tilt the PET head:
4.2 cm
(a) Transverse (b) Coronal (c) Sagittal
External Plate
Endoscope
Prostatic lesion can be well separated from bladder backgroundMilan Zvolský (DESY.) 47 / 48
Summary.Tomography
Take projection data,convert it into cross-section imagesA projection is a line integral of an objectBackprojection: sums data from all projection views
Analytic ReconstructionSimple backprojection → BlurringUse the Central Slice Theorem to solve the oversampling(blurring) in the Fourier space → Filtered Backprojection
Iterative ReconstructionStart with guess, compare with measurement, update the guess,iterate until best solution is reachedModel scatter, attenuation and limited detector resolution tobetter reflect reality (e.g. ML-EM)
Milan Zvolský (DESY.) 48 / 48
Thank you for your attention!
Literatur: [1, 2, 3, 4, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12]
Literature I
J.L. Prince and J.M. Links.Medical imaging signals and systems.Pearson Prentice Hall, 2006.
G. Zeng.Medical image reconstruction: A conceptual Tutorial.Springer, 2010.
R. Brinks and T.M. Buzug.Image reconstruction in positron emission tomography (pet): the 90th anniversaryof radon’s solution.Biomedizinische Technik, 52(6):361–364, 2007.
A. Alessio and P. Kinahan.Pet image reconstruction.Nuclear Medicine, 1, 2006.
C. Grupen and I. Buvat.Handbook of particle detection and imaging, volume 1.Springer, 2011.
M. Slaney and A. Kak.Principles of computerized tomographic imaging.SIAM, Philadelphia, 1988.
Milan Zvolský (DESY.) 51 / 48
Literature II
R.M. Leahy and J. Qi.Statistical approaches in quantitative positron emission tomography.Statistics and Computing, 10(2):147–165, 2000.
J.M. Ollinger and J.A. Fessler.Positron-emission tomography.Signal Processing Magazine, IEEE, 14(1):43–55, 1997.
L.A. Shepp and Y. Vardi.Maximum likelihood reconstruction for emission tomography.Medical Imaging, IEEE Transactions on, 1(2):113–122, 1982.
J. Beyerer and F. Puente León.Die Radontransformation in der digitalen Bildverarbeitung.at-Automatisierungstechnik, 50(10/2002):472, 2002.
J. Cui, G. Pratx, S. Prevrhal, and C.S. Levin.Fully 3D list-mode time-of-flight PET image reconstruction on GPUs using CUDA.Medical Physics, 38(12):6775, 2011.
A.J. Reader and H. Zaidi.Advances in PET image reconstruction.PET Clinics, 2(2):173–190, 2007.
Milan Zvolský (DESY.) 52 / 48
Spare Slides
Milan Zvolský (DESY.) 53 / 48
4D NURBS XCAT* Cardiac Torso Phantom.
Realistic model of human anatomy incl.cardiac and respiratory patient motionPlace Tumor in Prostate, injectRadiopharmacon
Milan Zvolský (DESY.) 54 / 48
Frequency cut-off.
Milan Zvolský (DESY.) 55 / 48
Interlude: Fast Fourier Transform (FFT).Discrete FTs are computationally very expensive. Solution: FFTFT: O(N2), FFT: O(N log2N) (2 weeks vs. 30 s CPU time)FT of length N can be written as sum of 2 FTs of length N/2,one for even (e) and one for odd (o) parts:
Fk =
N−1∑j=0
fje− 2πi
Njk =
N/2−1∑j=0
f2je− 2πi
N(2j)k +
N/2−1∑j=0
f2j+1e− 2πi
N(2j+1)k
= F(e)k +W kF
(o)k , W k = exp(2πik/N) (root of unity)
= F(e...e)k + . . .+W ...F
(eoo...o)k + . . .+W pkF
(o...o)k
Divide data all the way down to FTs of length 1.FT of length 1 is function value itselfF
(eoee...o)k = fn for some n→ Each combination (eoeoo...o)
corresponds to an element of the input vector f .Milan Zvolský (DESY.) 56 / 48
Interlude: Fast Fourier Transform (FFT).1 signal of16 points
2 signals of8 points
4 signals of4 points
8 signals of2 points
16 signals of1 point
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 2 4 6 8 10 12 14 1 3 5 7 9 11 13 15
0 4 8 12 2 6 10 14 1 5 9 13 3 7 11 15
8 4 12 2 10 6 14 1 9 5 13 3 11 7 150
8 4 12 2 10 6 14 1 9 5 13 3 11 7 150
Figure: An N point signal is decomposed into N signals à 1 point
Reverse the pattern (eoee...), assign binary number (e = 0,o = 1)E.g.: (eoeo)→ (oeoe) = (1010) = 10
Combine adjacent elements (with the Wnk’s) to get 2-pointtrafo, etc
Milan Zvolský (DESY.) 57 / 48
Interlude: Fast Fourier Transform (FFT).1 signal of16 points
2 signals of8 points
4 signals of4 points
8 signals of2 points
16 signals of1 point
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 2 4 6 8 10 12 14 1 3 5 7 9 11 13 15
0 4 8 12 2 6 10 14 1 5 9 13 3 7 11 15
8 4 12 2 10 6 14 1 9 5 13 3 11 7 150
8 4 12 2 10 6 14 1 9 5 13 3 11 7 150
e
o
e
o
Figure: An N point signal is decomposed into N signals à 1 point
Reverse the pattern (eoee...), assign binary number (e = 0,o = 1)E.g.: (eoeo)→ (oeoe) = (1010) = 10
Combine adjacent elements (with the Wnk’s) to get 2-pointtrafo, etc
Milan Zvolský (DESY.) 57 / 48
Backprojection Procedure.Sinogram represents theprojection of theradiopharmacon distributiononto the detectorFixed point in the objecttraces a sinusoidal path inprojection spaceSinogram: Superposition of allsinusoids corresponding to eachpoint of activity in the objectLORs described by p(s, φ)f(x, y)→ p(s, φ): RadontransformThe task of tomography is tofind f(x, y) from p(s, φ)
Milan Zvolský (DESY.) 58 / 48
Some examples.
102 103 104
105 106
fFBP(x)
f (x)
107
Figure: The effect of the number of detected photons on the quality of thereconstructed image. A synthetic image was projected and Poisson noisewas added to the projections. Reconstructed with the FBP algorithm
Noise has a high impact on the image quality
Milan Zvolský (DESY.) 59 / 48
Backprojection: Blurring of the image.
Assign equal intensity alongeach LORThe projections are formed oneview at a timeSum up the LORsThe final backprojected imageis the summation of thebackprojections from all viewsBackprojected image 6=original imageSolution? For this we needsome math...
Milan Zvolský (DESY.) 60 / 48
Backprojection.
Discrete caseProjection: P = AXBackprojected image: B = ATP
X = (x1, x2, x3, x4)T
P =(p(1, 0◦), p(2, 0◦), p(1, 90◦), p(2, 90◦)
)T= (7, 2, 5, 4)T
B = ATP =
1 0 1 00 1 0 11 1 0 00 0 1 1
T
7254
=
127116
Milan Zvolský (DESY.) 61 / 48
Interlude: GPU Programming.ML-EM is much slower than FBP. But this is not a real problemnowadays...
Parallelise your work384 cores, 32768 registers/blockEach one can perform simple tasks very quickly
Milan Zvolský (DESY.) 62 / 48
Backprojection Numerical Example.a b cdg
e fh i
a+b+c
d+e+f
g+h+i
a+d+gb+e+h
c+f+i
c
g
a
i
d+b
g+e+c
h+f d+h
a+e+i
b+f
e→ 3e+∑
proj⇒ e→ e
a→ a+ b+ c+ a+ e+ i+ a+ d+ g + a = 3a+∑
proj− f − h
⇒ a→ a− 1/3(f + h)
Blurring of the image!
Milan Zvolský (DESY.) 63 / 48
Filtered Backprojection.
Milan Zvolský (DESY.) 64 / 48
Backprojection: Blurring.original image after backprojection
image comparison
1/r blurring of the image!
Milan Zvolský (DESY.) 65 / 48
Central Slice Theorem.Central Slice Theorem (CST)
F1{p(s, φ′)} = F2{f(x, y)}|φ=φ′
The CST relates the Fourier transform (FT) of f(x, y) with theFT of p(s;φ)Summation of lines causes duplication in the centerHigh density of central slices in Fourier spaceOversampling in the center of the Fourier space needs to befiltered in order to have equal sampling throughout the Fourierspace
A complete set of 1D projections allows the reconstruction of theoriginal 2D distribution without loss of information
Milan Zvolský (DESY.) 66 / 48
Iterative Methods - Basic Idea.1 Image Model:.Discretisation of image into N distinct voxels fj
2 System Model:.System matrix Hij : probability that an emission
from voxel j is detected in projection i:
pi =∑Nj=1Hijfj
3 Data Model:.Variation of projection measurements around
expected mean values (Poisson). Derived from our
basic understanding of the acquisition process
4 Governing Principle:.Defines the ’best’ image..e.g. Maximum Likelihood cost function
5 Algorithm:.Optimises the cost function: Maximum
Likelihood - Expectation Maximisation
𝑙1
𝑙𝑓𝑢 𝑙𝑙
𝑙𝑣
𝑙2
Milan Zvolský (DESY.) 67 / 48
FBP Numerical Example.black = 1, white = 0
For projections at 0◦ and 90◦ the spacial resolution is one
pixel (i.e. τ = 1), for the other projection it is√2 pixels.
Convolute filter with projection(Product in Fourier space = convolution in position space)
Discrete convolution: (f ∗ g)(t) =∑
k f(k)g(t− k)
0
* =
Filter
2.0
1QP1
0.25 for tau =10.354 for tau =1/sqrt(2)
-0.1013 for tau =1-0.1433 for tau =1/sqrt(2)
1.0
0.39
-0.20
0.05
-0.10-0.1
0.25
Milan Zvolský (DESY.) 68 / 48
FBP Numerical Example.
0.40
-0.10
0.05
-0.20
-0.20
-0.41
-0.20
0.09
0.80
0.09
0.30
-0.10
0.05
-0.20
0.40
2Q
1Q
Q
4Q
3
Push filtered projections back into image
-0.20
0.40
0.05
-0.100.05-0.20
-0.10
-0.20
0.40
0.05
-0.10
-0.20
0.40
0.05
-0.10
-0.100.05-0.20
-0.20
-0.200.09
-0.20
0.05-0.20
-0.100.05-0.20
-0.100.05
-0.10
0.09
-0.410.800.09
-0.20
-0.20
0.80
0.80
0.09
0.09
-0.410.800.09
-0.41
-0.200.40
0.05
-0.10
-0.20
0.40
0.05
-0.20
-0.20
0.40
0.40
0.40
0.40
0.40
-0.10
0.30
-0.20 0.30 0.09
-0.20 0.30 0.09
-0.20
0.09 0.30
0.30 0.09 -0.200.30
0.30
0.30
1
Q backprojectedQ backprojected
Q backprojectedQ backprojected
43
2
.Usually do filtering in Fourier space..FT computationally expensive → use FFT (2 weeks vs. 1 min. CPU time)Milan Zvolský (DESY.) 69 / 48
Basic Idea.Example: 2× 2 MatrixSum of 1st row is 2, sum of 2nd row is 3, sumof 1st column is 0, sum of 2nd column is 5.⇒ x1 = 0, x2 = 2, x3 = 0, x4 = 3
A B
C D
0 5
2
5
Example: Another 2× 2 MatrixSum of 1st row is 5, sum of 2nd row is 4, sumof 1st column is 7, sum of 2nd column is 2.⇒ No unique solution.
Way out: Take more views?
Hypothesis: With many views frommany angles, we are able toperfectly reconstruct the image
Milan Zvolský (DESY.) 70 / 48
Backprojection.
SummaryTomography is a process of taking projection data andconverting the data into cross-section images. Projection datafrom multiple views are requiredA projection is a line integral of an objectBackprojection is a superposition procedure and it sums thedata from all projection views. Backprojection evenlydistributes the projection domain data back along the same linesfrom which the line-integrals were formed.
Milan Zvolský (DESY.) 71 / 48
Filtered Backprojection.Eliminate blurring by introducing negative “wings” around the spikein the projections before backprojecting
Milan Zvolský (DESY.) 72 / 48
BP vs. FBP.
Figure: Reconstruction of a pointsource with simple backprojection (top)and filtered Backprojection (bottom)
Milan Zvolský (DESY.) 73 / 48