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Introduction to tomographic reconstruction
Jakob Sauer Jørgensen
Postdoc, Technical University of Denmark (DTU)
02946 Scientific Computing for X-Ray Computed TomographyDTU Training School, Week 1, Days 1+2
January 2–6, 2017
SC for CT, week 1, Mon+Tues: Overview
Monday
09.00 – 09.30 Welcome
09.30 – 12.00 Forward problem: Radon transform & Lambert-Beer
12.00 – 13.00 - - - Lunch break - - -
13:00 – 15.30 Reconstruction: Filtered Back-Projection (FBP)
15.30 – 16.30 Intro to micro-CT scan and micro project
Tuesday
09.00 – 12.00 Micro-CT scan at DTU Physics
12.00 – 13.00 - - - Lunch break - - -
13:00 – 15.30 Reconstruction real data set
15.30 – 16.30 Time for micro project
Outline
1 Data model: Lambert-Beer law and Radon transform
2 Reconstruction: Filtered back-projection (FBP) algorithm
3 Introduction to hands-on CT scan and micro project
4 Practical aspects for reconstruction from real data
3 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
Tomography
What is tomography?
I From Greek: Tomos a section or slice, Graphos: to describe.
I Imaging of slices of an object – without actually slicing it!
I These days not just slices but 3D maps can be obtained.
I To see the inside, need information obtained from the outside.
I Some applications of tomography...
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Medical imaging
5 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
Materials scienceDevelopment of advanced materials requires understanding theirproperties at the micro and nano scale.
Example: Maximize strength of glass fibre for wind turbine blades.
Laboratory micro-CT scanners and large-scale synchrotron facilities
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Non-destructive testing
Production, security, metrology, etc.
Example: Airport luggage scanner for threat detection.
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Tomography: Imaging from projectionsI Projections are measured around an object using X-rays.I Goal is to reconstruct the object from the projections.I Simplest is 2D parallel beam geometry, which we focus on.I Used in early scanners and in large scale synchrotron facilities.
θ
θ
1
2
8 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
Cone-beam geometry
I Cone-beam (medical CT scanners, lab-based micro-CT, etc)
I Cone-beam restricted to central slice: Fan-beam
I Move source far away: Parallel-beam (synchrotron)
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Contrast mechanism: X-ray attenuation
Heavier matter attenuate X-rays more: Air – tissue – bone – metal.Quantified by so-called linear attenuation coefficient µ.
Wilhelm Conrad Rontgen and the first X-ray image ever takenshowing his wife’s hand (1895).
10 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
Lambert-Beer law of attenuationHomogeneous material:
I = I0 exp{− µ0D
}Non-homogeneous (more interesting) material:
I = I0 exp
{−∫Lµ(x)dx
}Rearrange to line integral form:
− logI
I0=
∫Lµ(x)dx
A tomographic scan:I Measure I along many lines to get many line integral values
through the object from which to determine µ(x).I The intensity I is called the transmission, while the
corresponding − log(I/I0) is called absorption or projection.
I0 Iµ0-
D
I0 Iµ(x)
X-ray L
��
-
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Transmission vs absorption
I0 = 10000 I I/I0 − log (I/I0)
∫L µ(x)dx =
-
-
-
-
-
10000
5000
2500
1250
625
1.0000
0.5000
0.2500
0.1250
0.0625
0.0
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1.4
2.1
2.8
12 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
The origin of tomographic reconstruction
Original reference:
Uber die Bestimmung von Funktionen durchihre Integralwerte langs gewisserMannigfaltigkeiten. (Johann Radon, 1917):
An object can be reconstructed perfectlyfrom a full set of line integrals.
What does a “full set” mean?
13 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
Parametrizing lines in the planeHow to describe a line?One way is by slope and y intercept. Vertical lines excluded.Alternative: “normal form”:
Lφ,ρ = {(x, y) | x cosφ+ y sinφ = ρ}ρ is signed orthogonal distance of line to origin.φ is angle between positive x-axis and unit normal vector to Lφ,ρ
φ x
y
(cosφsinφ
) ρ
Lφ,ρ
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The Radon transform for parallel-beam geometryObject f(x, y):
I contained in disk of radius R
Line of integration Lφ,ρ given by:
I φ: angle of line to be projected onto
I ρ: position on line
Projection: All line integrals at one φ:
pφ(ρ) =
∫Lφ,ρ
f(x, y)ds for ρ ∈ [−R,R].
The Radon transform is:
[Rf ](φ, ρ) = pφ(ρ) =
∫Lφ,ρ
f(x, y)ds
for φ ∈ [0◦, 180◦[ and ρ ∈ [−R,R].
φ x
y
ρ
ρ
pφ(ρ)
15 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
Pen and paper exercise: Radon transform of a disk
Given image of a small centered disk of radius r:
f(x, y) =
{1 for x2 + y2 ≤ r20 otherwise
Derive that the Radon transform is:
[Rf ](φ, ρ) ={
2√r2 − ρ2 for |ρ| ≤ r
0 otherwise
Image Radon transformed image
-�
RadontransformR
R−1 ?
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Radon transform of an ellipse (Kak & Slaney)
φ x
y
ρ
pφ(ρ)
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Radon transform of an ellipse (Kak & Slaney)
Given ellipse
f(x, y) =
{c for x2
A2 + y2
B2 ≤ 1 (inside ellipse)0 otherwise (outside the ellipse)
If centered (x1 = y1 = 0) and not rotated (α = 0) then
pφ(ρ) =
{2cABa2(φ)
√a2(φ)− ρ2 for |ρ| ≤ a(φ)
0 otherwise
where a2(φ) = A2 cos2 φ+B2 sin2 φ.
If centered at (x1, y1), rotated by α then given from pφ(ρ) above as
pφ(ρ) = pφ−α(ρ− s cos(γ − φ)),
where s =√x21 + y21 and γ = tan−1(y1/x1).
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Parametrized form of Radon transformThe Radon transform
pφ(ρ) =
∫Lφ,ρ
f(x, y)ds, (1)
can be written explicitly using a parametrization of the line:
pφ(ρ) =
∫ ∞−∞
f(x(s), y(s))ds, (2)
where at fixed φ and ρ:
x(s) = ρ cosφ− s sinφy(s) = ρ sinφ+ s cosφ
Line is traced as s runsfrom −∞ to ∞.
φ x
y
(cosφsinφ
)(− sinφ
cosφ
)ρ
Lφ,ρ
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A convenient explicit form of Radon transform
Useful alternative expression for the Radon transform:
pφ(ρ) =
∫ ∞−∞
∫ ∞−∞
f(x, y)δ(x cosφ+ y cosφ− ρ)dxdy
Interpretation:The line Lφ,ρ consists of exactly those (x, y) at whichx cosφ+ y cosφ− ρ = 0, which is the argument of the Dirac deltafunction δ. Thus, the integrand is restricted to function values off(x, y) on Lφ,ρ, which amounts to the corresponding line integral.
20 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
Dirac delta function
Generalized function with heuristic definition:
δ(t) =
{+∞ for t = 00 for t 6= 0
and ∫ ∞−∞
δ(t)dt = 1
Important property of Dirac delta function:∫ ∞−∞
f(t)δ(t− T )dt = f(T )
This is called the sifting property, because the Dirac delta functionacts as a sieve and “sifts out” the value at t = T .
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Connection Radon Transform and Lambert-Beer
The Radon transform
pφ(ρ) =
∫Lφ,ρ
f(x, y)ds
and the Lambert-Beer law along the same line Lφ,ρ
Iφ,ρ = I0 exp
(−∫Lφ,ρ
µ(x, y)ds
)
are connected through the identifications
f(x, y) = µ(x, y)
pφ(ρ) = − log
(Iφ,ρI0
)
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Projections all around the object
The Radon transform describes the forward problem of how (ideal)X-ray projection data arises in a parallel-beam scan geometry.
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Image and sinogram
The output of the Radon transform is called a sinogram:
Example image
20 40 60 80 100
10
20
30
40
50
60
70
80
90
100 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Sinogram
φ50 100 150
ρ
20
40
60
80
100
120
1400
0.1
0.2
0.3
0.4
0.5
Angle φ
Positionρ
Note that 180◦ captures all necessary projections of the object.The other 180◦ are mirror images of the opposite projections.
24 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
A bit about noise in CTLambert-Beer law normally expressed in intensity I, i.e., number ofphotons per time. Equivalently if during an interval the sourceemits N photons, the average transmitted number of photons is
N = N0 · p
where the transmission probability is
p = exp
(−∫Lµ(x, y)ds
)The number of transmitted photons in interval is a Poisson randomvariable, i.e., the probability that k photons are transmitted is
P (N = k) =λk
k!exp(−λ)
where the parameter
λ = N0 · p
is also the expected value of the number of transmitted photons.25 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
Gaussian approximation and effect on projection
For large number of photons a Poisson distribution can be closelyapproximated by a Gaussian distribution. In particular theapproximating Gaussian distribution has both mean and standarddeviation of N0p, i.e., approximately the number of photonstransmitted
N ∼ N (N0p,N0p)
How does this affect the reconstruction? It turns out that for theprojection value b = − log(N/N0) to be used in the reconstructionapproximately
b ∼ N (− log p,1
N0p)
Thus, with increasing intensity (larger N0) the expected projectionvalue is constant but the standard deviation decays; in other wordsnoise is reduced.
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Some other error sources
In practice, in addition to Poisson noise data can be affected bynumerous other issues:
I Detector noise
I Scatter (some X-rays do not follow straight line)
I X-rays not monochromatic, but full spectrum. Attenuationcoefficient depends on energy, e.g., beam hardening
I Bad detectors, e.g., void measurements
I Too dense features, e.g., metal blocking rays completely
I Object changing during acquisition, e.g., motion
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Exercises: Radon transform
Available from:http://www2.compute.dtu.dk/~pcha/HDtomo/SCforCT.html
File:ExWeek1Days1and2.pdf
28 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
Outline
1 Data model: Lambert-Beer law and Radon transform
2 Reconstruction: Filtered back-projection (FBP) algorithm
3 Introduction to hands-on CT scan and micro project
4 Practical aspects for reconstruction from real data
29 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
Reconstruction
Overview:
I Now we have seen how the Radon transform describes theforward problem of how projections arise from an object.
I Question: How can we do the inverse operation, namelyreconstruct an image of the object from the projections?
I Class of analytical reconstruction methods construct aninverse transform. For parallel beam case, inverse Radontransform.
I This gives rise to the most common reconstruction method:Filtered Back-Projection (FBP).
I To derive the FBP method, we need an important resultknown as the Fourier slice theorem, and to set some notation.
30 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
Back-projection
Mathematically, back-projection is written as integration over all φ:
B[pφ(ρ)](x, y) =∫ π
0bφ(x, y)dφ
of the back-projection images at each angle φ:
bφ(x, y) = pφ(x cosφ+ y sinφ)
Interpretation: “Smearing and summation”Each point (x, y) and each angle φ define a unique linear positionρ = x cosφ+ y sinφ. In the back-projection image the point (x, y)is assigned the value at ρ from the projection pφ(ρ). This is“smearing”. Back-projection then adds up all contributions to each(x, y) by integrating over φ. This is “summation”.
31 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
Back-projection: Does it invert projection? (No)
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The Fourier transformFourier transform (“Frequency representation”):
−10 −5 0 5 10−0.5
0
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2
2.5
−10 −5 0 5 10−0.5
0
0.5
1
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-�
1D Fouriertransform F1
1D InverseFouriertransform F−11
ρ ω
Also possible for functions of more than 1 variable:
-�
2D Fouriertransform F2
2D InverseFouriertransform F−12
33 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
The Fourier transform pair definitions
1D Fourier transform and inverse:
f(ω) = F1[f(t)](ω) =
∫ ∞−∞
∫ ∞−∞
f(t)e−j2πωtdt (3)
f(t) = F−11 [f(ω)](t) =
∫ ∞−∞
∫ ∞−∞
f(ω)e+j2πωtdω (4)
2D Fourier transform and inverse:
F (u, v) = F2[f(x, y)](u, v) =
∫ ∞−∞
∫ ∞−∞
f(x, y)e−j2π(ux+yv)dxdy (5)
f(x, y) = F−12 [F (u, v)](x, y) =
∫ ∞−∞
∫ ∞−∞
F (u, v)e+j2π(ux+yv)dudv (6)
34 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
Fourier Slice Theorem: Key to reconstruction
Projection at angle φ 1D Fourier transform At angle φ in Fourier space
All 1D Fouriertransformed projections
=
2D Fourier transform
All angles φ ∈ [0◦, 180◦[ needed to build complete 2D Fourier transform
35 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
Derivation of Fourier Slice Theorem (1)
Strategy: Manipulate 1D Fourier-transformed projection into slicethrough 2D Fourier-transformed image.
pφ(ω) =
∫ ∞−∞
pφ(ρ)e−j2πωρdρ
=
∫ ∞−∞
∫ ∞−∞
∫ ∞−∞
f(x, y)δ(x cosφ+ y sinφ− ρ)e−j2πωρdxdydρ
=
∫ ∞−∞
∫ ∞−∞
f(x, y)
∫ ∞−∞
δ((x cosφ+ y sinφ)− ρ)e−j2πωρdρdxdy
=
∫ ∞−∞
∫ ∞−∞
f(x, y)
∫ ∞−∞
δ(ρ− (x cosφ+ y sinφ))e−j2πωρdρdxdy
=
∫ ∞−∞
∫ ∞−∞
f(x, y)e−j2πω(x cosφ+y sinφ)dxdy
using definition of 1D Fourier transform, Dirac delta expression of pφ(ρ),
reordering, that δ(−t) = δ(t), and sifting property.
36 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
Derivation of Fourier Slice Theorem (2)Continuing by reordering and recognizing as 2D Fouirer transform
pφ(ω) =
∫ ∞−∞
∫ ∞−∞
f(x, y)e−j2π(xω cosφ+yω sinφ)dxdy
=
∫ ∞−∞
∫ ∞−∞
f(x, y)e−j2π(xu+yv)dxdy
∣∣∣∣u=ω cosφ,v=ω sinφ
= F (u, v)
∣∣∣∣u=ω cosφ,v=ω sinφ
yields finally the Fourier slice theorem:
pφ(ω) = F (ω cosφ, ω sinφ). (7)
Interpretation: (u, v) = (ω cosφ, ω sinφ) for φ ∈ [0, π) and
ω ∈ (−∞,∞) specifies a line in 2D Fourier space rotated by φ relative to
the positive u axis. This corresponds to the ρ axis in real space. Thus,
the 1D Fourer transform of a projection is equivalent to the
corresponding slice/line through the 2D Fourier transform.
37 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
Fourier reconstruction method
−10 −5 0 5 10−0.5
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Radontransform R
�
Piece 1Dstogetherto form 2D
?
1D FouriertransformsF1
62D inverseFouriertransformF−12
38 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
Problems for Fourier reconstruction methodI In practice we do not have all φ and ρ since we record a finite
number of projections with a detector of fixed size.I Interpolation from polar to Cartesian grid known as
“regridding” required for 2D inverse Fourier transform:
I Accurate interpolation in Fourier domain difficult.I 2D inverse Fourier transform relatively expensive and requires
all data simultaneously.I Result: Fourier method rarely used in practice.I Alternative: Filtered Back-Projection
39 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
Derive filtered back-projection, 1
Strategy: Rewrite inverse 2D Fourier transform:
f(x, y) = F−12 [F2f ]
[2D Fourier transform definitions]
=
∫ ∞−∞
∫ ∞−∞
F (u, v)ej2π(ux+vy)dudv
[Change to polar coordinates, including Jacobian]
=
∫ 2π
0
∫ ∞0
F (ω cosφ, ω sinφ)ej2π(ω cosφx+ω sinφy)ωdωdφ
[Split integral over 2π in two:]
=
∫ π
0
∫ ∞0
F (ω cosφ, ω sinφ)ej2π(ω cosφx+ω sinφy)ωdωdφ
+
∫ π
0
∫ ∞0
F (ω cos(φ+ π), ω sin(φ+ π))ej2π(ω cos(φ+π)x+ω sin(φ+π)y)ωdωdφ
40 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
Derive filtered back-projection, 2
[Using sin(φ+ π) = − sinφ and cos(φ+ π) = − cosφ :]
=
∫ π
0
∫ ∞0
F (ω cosφ, ω sinφ)ej2π(ω cosφx+ω sinφy)ωdωdφ
+
∫ π
0
∫ ∞0
F (−ω cosφ,−ω sinφ)ej2π(−ω cosφx−ω sinφy)ωdωdφ
[Change sign/bounds in second integral:]
=
∫ π
0
∫ ∞0
F (ω cosφ, ω sinφ)ej2π(ω cosφx+ω sinφy)ωdωdφ
+
∫ π
0
∫ 0
−∞F (ω cosφ, ω sinφ)ej2π(ω cosφx+ω sinφy)(−ω)dωdφ
Recollect using absolute value:
=
∫ π
0
∫ ∞−∞
F (ω cosφ, ω sinφ)ej2πω(x cosφ+y sinφ)|ω|dωdφ
41 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
Derive filtered back-projection, 3
[Apply Fourier Slice Theorem:]
f(x, y) =
∫ π
0
∫ ∞−∞
pφ(ω)ej2πω(x cosφ+y sinφ)|ω|dωdφ
[Use that x cosφ+ y sinφ is constant wrt ω-integration, say ρ]
=
∫ π
0
[∫ ∞−∞
pφ(ω)ej2πωρ|ω|dω
]ρ=x cosφ+y sinφ
dφ.
Recognizing inner integral as 1D inverse Fourier transform wedefine the filtered projection:
qφ(ρ) =
∫ ∞−∞
pφ(ω)ej2πωρ|ω|dω
= F−11 [pφ(ω) · |ω|](ρ) = F−11 [F1[pφ](ω) · |ω|] (ρ)
Projections filtered by multiplying ramp filter |ω| in Fourier domain.42 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
Derive filtered back-projection, 4
Finally we can write
f(x, y) =
∫ π
0qφ(x cosφ+ y sinφ)dφ = B[qφ](x, y)
by recognizing the back-projection operation.
This is the Filtered Back-Projection inversion formula for theRadon transform.
Interpretation:Given each point (x, y) of the image f to be reconstructed eachangle φ defines a position ρ = x cosφ+ y sinφ. Throughback-projection, the point (x, y) is assigned the value at ρ from thefiltered projection, and contributions at all angles are summed up.
43 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
Back-projection: Does it invert projection?
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44 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
Filter projections by “ramp” before back-projection
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45 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
Back-projection vs. filtered back-projection
Projections must be filteredwith a “ramp” filter beforeback-projection.
In Fourier domain: |ω|
ω
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
|ω|
0
0.2
0.4
0.6
0.8
1Ramp filter
[Buzug 2008: Computedtomography, Springer.]
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Additional filtering needed in practice
Ramp filter is part of the inversion formula but is a high-pass filterwhich is problematic in practice when noise is present.
In practice additional filters can be used:
ω
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
|ω|
0
0.2
0.4
0.6
0.8
1Ramp filter
[Buzug 2008: Computed tomography, Springer.]
(Ram-Lak)
47 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
Filtered back-projection (FBP) step by step
To compute FBP reconstructed image from projections:
1. Fourier-transform projections.
2. Apply ramp filter by multiplication of |ω|.3. Optionally apply additional (low-pass) filters to handle noise.
4. Inverse Fourier-transform to obtain filtered projections.
5. Backproject filtered projections and sum up.
In practice:
I Often done automatically for you by scanner/instrument
I FBP implementations available.
I Easy to use: Main user input is choice of low-pass filter.
I In MATLAB: iradon.
48 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
FBP for fan-beam and cone-beam
Parallel-beam FBP:
Projections → Filter → Back-project → Reconstruction
Fan-beam – 2 strategies:
I Rebinning to parallel-beam
I Dedicated filtered back-projection algorithm
Cone-beam:
I Dedicated filtered back-projection algorithm
I Feldkamp-Davis-Kress (FDK) is standard
Projections → Weighting → Filter → Weighting → Back-project → Reconstruction
49 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
Strengths and weaknesses of FBP
Strengths:
I Fast: Based on FFT and a single back-projection
I Few parameters to adjust
I Conceptually easy to understand and implement
I Reconstruction behavior well understood
I Typically works very well (for complete/good data)
Weaknesses:
I Large number of projections required.
I Full angular range required (limited angle problem)
I Only modest amount of noise in data can be tolerated.
I Fixed scan geometries – others require own inversion formulas
I Cannot make use of prior knowledge such as non-negativity.
50 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
Exercises
Exercises on reconstruction using FBP (exercise 2).
51 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
Outline
1 Data model: Lambert-Beer law and Radon transform
2 Reconstruction: Filtered back-projection (FBP) algorithm
3 Introduction to hands-on CT scan and micro project
4 Practical aspects for reconstruction from real data
52 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
SC for CT, week 1, Mon+Tues: Overview
Monday
09.00 – 09.30 Welcome
09.30 – 12.00 Forward problem: Radon transform & Lambert-Beer
12.00 – 13.00 - - - Lunch break - - -
13:00 – 15.30 Reconstruction: Filtered Back-Projection (FBP)
15.30 – 16.30 Intro to micro-CT scan and micro project
Tuesday
09.00 – 12.00 Micro-CT scan at DTU Physics
12.00 – 13.00 - - - Lunch break - - -
13:00 – 15.30 Reconstruction real data set
15.30 – 16.30 Time for micro project
Hands-on CT scan session – Meeting time and place
I Micro CT scans take place in building 309, ground floor, from9:00 to 12.00 tomorrow (Tuesday) morning.
I We will meet in the cafe area of building 306, ground floor at8.50 and walk to 309 as a group.
54 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
How to find it
55 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
Hands-on CT scan session
I To get started there will be a joint introduction given byCarsten Gundlach, DTU Physics.
I There will be 2 groups doing separate scans, one on a Zeissinstrument, one on a Nikon instrument.
I Once the data has been acquired a 3D reconstruction will becomputed automatically.
I After the scans, there will be time to experiment with 3Dvisualization using in-house software until lunch break.
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Zeiss micro-CT instrument
I Rotate sample, keep X-ray source and detector fixed.
I Cone-beam scan geometry.
I Reconstruction by FDK.
I Similar Nikon micro-CT instrument.
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Micro project – exterior tomography
I In deriving the FBP reconstruction formula, we assume fulldata is available, i.e., 180◦ and sample fully contained withinfield of view.
I In many cases full data is not available, e.g., limited angularrange, few projections or the region-of-interest/interiorproblem all encountered in the exercises.
I In the ROI/interior problem only rays passing through centralpart of the object are available, because object is too large orwe have zoomed in on a small region.
I In the exterior problem, only rays through outer annulus ofobject are measured, not through the center. This can happenfor example in non-destructive testing of rockets, which arelarge and dense so that X-rays do not penetrate sufficiently.
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Shepp Logan, full data 180◦
Original phantom and generated sinogram.
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Shepp Logan, reconstruction from full data 180◦
Full sinogram and FBP reconstruction.
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Shepp Logan, reconstruction region-of-interest data
Region-of-interest/interior sinogram and reconstruction.
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Shepp Logan, reconstruction exterior data
Exterior sinogram.
Reconstruction?
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Goals of project
Investigate the exterior problem, in particular try to establishwhat can and cannot be reconstructed.
The micro project is quite open-ended. In the lectures andexercises you are introduced to theory, techniques and tools fortomographic reconstruction. In the project it is up to you tochoose from this material and apply to the exterior problem inorder to understand the problem and the limitations ofreconstruction from exterior data.
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Some ideas
I Design phantoms with features to illustrate howreconstruction quality depends on size of features, closeness toboundary, radius of missing interior region, etc. (Mon+Tues)
I Apply FBP including practical corrections such as padding asused for ROI data (Mon+Tues)
I Apply SVD analysis and compare singular values and vectorswith full data case (Wed)
I Apply ideas from micro-local analysis to assess which featurescan be reconstructed (Thurs)
You are free to pick from this list or pursue your own ideas withinthe course material and exterior tomography – please do nothesitate to discuss ideas with us.
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Time for micro project
I Form groups of 2–3 persons.
I Start experimenting and familiarizing yourselves with theexterior problem.
I For example,I create code to simulate missing data in the sinogram caused
by the exterior problem.I create interesting phantoms and do FBP reconstruction from
simulated exterior data.I ...
I Suggestions:I Take notes/keep a log of the experiments you do and what you
learn from each, etc.I Make separate scripts for each study you do to make it easy
for yourself to remember and reproduce later as well as tocreate figures for the oral presentation on Friday.
65 / 73 Introduction to tomographic reconstruction Jakob Sauer Jørgensen, DTU
Outline
1 Data model: Lambert-Beer law and Radon transform
2 Reconstruction: Filtered back-projection (FBP) algorithm
3 Introduction to hands-on CT scan and micro project
4 Practical aspects for reconstruction from real data
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Flat and dark field correction
Typical data acquired:
I I: Transmission images (sample in, source on)
I I0: Flat (or white) field (sample out, source on)
I ID: Dark field (sample out, source off)
FBP needs line integrals (Radon transform):
I Convert using Lambert-Beer:∫Lf(x, y)ds = − log
I
I0
To obtain projections:
Z = − log Y, Y =I − IDI0 − ID
← pixelwise division
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Center-of-rotation (COR) correction
I Standard FBP implementationssuch as MATLAB’s iradon assumea perfectly centered object.
I In other words the center ofrotation should be mapped to thecentral detector pixel.
I In practice only approximatecentering physically possible.
I Naively reconstructing yieldsartifacts. Need to docenter-of-rotation correction.
I Can be done by “shifting”projections by padding sinogramwith sufficiently many artificaldetector pixel values.
&%'$s s
---------Centered
&%'$s s
---------
Non-centered
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Region-of-interest (ROI) correction
I In some cases the object to be scanned is too large to fit inthe field of view.
I Or we want to focus on some small region-of-interest (ROI).
I Projections are truncated (not covering entire object).
I Can we still reconstruct the object? Or just the ROI?
Image: [Wang & Yu 2009, Med. Phys. 36, 3575–3581]
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The region-of-interest problem
I Mathematically, the answer is NO.
I Interior Radon data contaminated by exterior Radon data.
I Interior reconstruction will also be contaminated.
Fig. from [Bilgot et al. 2011, IEEE Nuc. Sci. Symp. Conf. Rec. , 4080–4085]
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ROI correction in practice
I Boundaries correct.
I Large artifact apparent.
I Trick of padding of sinogramyields large improvement,but values still off.
I The exercise data set is ROIdata!
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Many other possible artifacts and corrections
I Ring artifacts
I Beam hardening
I Zingers
I Misalignment
I Metal artifacts
I Motion
I ...
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Exercises
Reconstruction of a real data set (2D parallel-beam). Exercise 3.
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