Image Reconstruction 2 Cone Beam Reconstruction · 2 Reconstruction from fan projections Resorting...
Transcript of Image Reconstruction 2 Cone Beam Reconstruction · 2 Reconstruction from fan projections Resorting...
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Image Reconstruction 2 – Cone Beam Reconstruction
Thomas Bortfeld
Massachusetts General Hospital, Radiation Oncology, HMS
HST.S14, February 19, 2013
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 1 / 36
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Projection
Mathematically, the projection λis the integral of f(x) along a(parallel) set of projection lines:
λφ(p) =
∫A
f(x) δ(p−x·n̂φ) d2x
1 Note: A projection line isdescribed in the Hessian normalform by the equation p = x · n̂φ.
2 Note also: The δ-function“picks” those points x from theplane A that lie on theprojection line.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 2 / 36
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Radon Transform
We will consider all projections of f as a two-dimensional functionwith the arguments p and φ, and write it as λ(p, φ). The transformf(x1, x2)→ λ(p, φ) is called a Radon transform1
In symbols:λ(p, φ) = R {f(x)} .
The problem of reconstructing f(x) from the (known) projections λ(p, φ)is basically the determination of the inverse Radon transform, R−1.
1After the mathematician Johann Radon, who described the first mathematicalmethod for a reconstruction from projections as early as in 1917
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 3 / 36
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The problem: inverting the Radon transform
Object p φ
Sinogram
p
φ
?
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 4 / 36
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Backprojection
By backprojection we mean “smearing out” of the values of λφ(p) alongthe projection lines, over the plane A, which results in a streak image.Mathematically, backprojection under an angle φ is simply given by:
fφ(x) = λφ(x · n̂φ).
If we perform backprojections for all angles within the interval [0, π) andintegrate the results, we get
fb(x) =
π∫0
λφ(x · n̂φ) dφ.
fb(x) = B {λ(p, φ)} = BR {f(x)} .
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 5 / 36
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Backprojection
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 6 / 36
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Central Slice Theorem
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 7 / 36
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Filtered Backprojection: algorithm
The function f(x) can be reconstructed from the projection profiles λφ(p)using the following steps:
1 Fourier transform of λφ(p) → Λφ(ν);2 multiplication of Λφ(ν) with |ν| → Λ∗φ(ν);3 inverse Fourier transform of Λ∗φ(ν) → λ∗φ(p′);4 backprojection of λ∗φ(p
′) and integration over φ → f(x).
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 8 / 36
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Outline
1 Practical implementationDiscrete dataFiltering in the spatial domainRam-Lak filterShepp-Logan filter
2 Reconstruction from fan projectionsResorting fan to parallel projectionsWeighted filtered back-projection for fan beams
3 Cone-beam reconstructionThe Feldkamp algorithmLimitations of the Feldkamp algorithm
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 9 / 36
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Practical implementation Discrete data
Discrete data
Discrete projection data (sinogram):
We know λm·∆φ(n ·∆p) for n = −N, . . . , N , and m = 1, . . . ,M withM = π/∆φ.
Assume that the sampling interval, ∆p, satisfies the Nyquist samplingcondition. This means, we assume that projection profiles in theFourier domain, Λφ(ν), are bandlimited within − 12∆p < ν <
12∆p .
Then the inverse transfer function H−1(ν) = |ν| can be restricted tothe same interval,
[− 12∆p ,
12∆p
].
The modified function
H−1r (ν) =
{|ν| for |ν| ≤ 12∆p0 otherwise
is called “ramp filter”.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 10 / 36
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Practical implementation Discrete data
Discrete data
Discrete projection data (sinogram):
We know λm·∆φ(n ·∆p) for n = −N, . . . , N , and m = 1, . . . ,M withM = π/∆φ.
Assume that the sampling interval, ∆p, satisfies the Nyquist samplingcondition. This means, we assume that projection profiles in theFourier domain, Λφ(ν), are bandlimited within − 12∆p < ν <
12∆p .
Then the inverse transfer function H−1(ν) = |ν| can be restricted tothe same interval,
[− 12∆p ,
12∆p
].
The modified function
H−1r (ν) =
{|ν| for |ν| ≤ 12∆p0 otherwise
is called “ramp filter”.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 10 / 36
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Practical implementation Discrete data
Discrete data
Discrete projection data (sinogram):
We know λm·∆φ(n ·∆p) for n = −N, . . . , N , and m = 1, . . . ,M withM = π/∆φ.
Assume that the sampling interval, ∆p, satisfies the Nyquist samplingcondition. This means, we assume that projection profiles in theFourier domain, Λφ(ν), are bandlimited within − 12∆p < ν <
12∆p .
Then the inverse transfer function H−1(ν) = |ν| can be restricted tothe same interval,
[− 12∆p ,
12∆p
].
The modified function
H−1r (ν) =
{|ν| for |ν| ≤ 12∆p0 otherwise
is called “ramp filter”.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 10 / 36
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Practical implementation Discrete data
Discrete data
Discrete projection data (sinogram):
We know λm·∆φ(n ·∆p) for n = −N, . . . , N , and m = 1, . . . ,M withM = π/∆φ.
Assume that the sampling interval, ∆p, satisfies the Nyquist samplingcondition. This means, we assume that projection profiles in theFourier domain, Λφ(ν), are bandlimited within − 12∆p < ν <
12∆p .
Then the inverse transfer function H−1(ν) = |ν| can be restricted tothe same interval,
[− 12∆p ,
12∆p
].
The modified function
H−1r (ν) =
{|ν| for |ν| ≤ 12∆p0 otherwise
is called “ramp filter”.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 10 / 36
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Practical implementation Filtering in the spatial domain
Filtering in the spatial domain
Reconstruction formula:
f(x) =
π∫0
∞∫−∞
|ν|Λφ(ν) exp(2πi νx · n̂φ) dν dφ
The first three steps are a filtering (convolution) of the projectionprofiles with the filter h−1(p), which is the inverse FT of H−1(ν) = |ν|:
f(x) =
π∫0
∞∫−∞
λφ(p)h−1(x · n̂φ − p) dp dφ
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 11 / 36
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Practical implementation Filtering in the spatial domain
To determine the filter h−1r (p) in the spatial domain we have to do aninverse Fourier transform of H−1r (ν), which yields:
h−1r (p) = F−11
{H−1r (ν)
}=
1
4∆p2
(2 sinc
(p
∆p
)− sinc2
(p
2∆p
)),
where sinc(x) stands for sin(πx)/(πx).
A sampling at discrete positions p = n∆p yields the discrete version:
h−1r (n∆p) =
1
4∆p2for n = 0
0 for n even, 6= 0− 1n2π2∆p2
for n odd .
This filter goes back to Ramachandran and Lakshminarayanan.It is known as “Ram-Lak” filter.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 12 / 36
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Practical implementation Filtering in the spatial domain
To determine the filter h−1r (p) in the spatial domain we have to do aninverse Fourier transform of H−1r (ν), which yields:
h−1r (p) = F−11
{H−1r (ν)
}=
1
4∆p2
(2 sinc
(p
∆p
)− sinc2
(p
2∆p
)),
where sinc(x) stands for sin(πx)/(πx).A sampling at discrete positions p = n∆p yields the discrete version:
h−1r (n∆p) =
1
4∆p2for n = 0
0 for n even, 6= 0− 1n2π2∆p2
for n odd .
This filter goes back to Ramachandran and Lakshminarayanan.It is known as “Ram-Lak” filter.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 12 / 36
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Practical implementation Ram-Lak filter
Ram-Lak filter
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 13 / 36
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Practical implementation Shepp-Logan filter
Another commonly used filter is the so-called “Shepp and Logan” filter,which results from averaging (smoothing) of the Ram-Lak filter overintervals of the width ∆p (or in the frequency domain from the ramp filter|ν| by multiplication with sinc(ν∆p)):
h−1s (p) = −2
π2∆p21− 2(p/∆p) sin(πp/∆p)
4(p/∆p)2 − 1.
The discrete version of this filter is very simple:
h−1s (n∆p) = −2
π2∆p2(4n2 − 1).
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 14 / 36
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Practical implementation Shepp-Logan filter
Shepp-Logan filter
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 15 / 36
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Practical implementation Shepp-Logan filter
Simple backprojection Filtered backprojection
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 16 / 36
simple_bp_copy.mpgMedia File (video/mpeg)
filtered_bp.mpgMedia File (video/mpeg)
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Practical implementation Shepp-Logan filter
Homework 1
Homework 1: reconstruct yourself!
a) Take a picture of yourself and convert it to a grayscale 100 x 100pixel square image.
b) Create your sinogram space for 100 projection angles.
c) Reconstruct your image by filtered backprojection using (i) theRam-Lak filter, and (ii) the Shepp-Logan filter. Do the filtering in thespatial domain using filters h−1r (Ram-Lak) and h
−1s (Shepp-Logan).
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 17 / 36
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Reconstruction from fan projections
Outline
1 Practical implementationDiscrete dataFiltering in the spatial domainRam-Lak filterShepp-Logan filter
2 Reconstruction from fan projectionsResorting fan to parallel projectionsWeighted filtered back-projection for fan beams
3 Cone-beam reconstructionThe Feldkamp algorithmLimitations of the Feldkamp algorithm
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 18 / 36
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Reconstruction from fan projections Resorting fan to parallel projections
Resorting fan parallel
k: counter of source positions
l: counter of projection lines within each fan-projection
k + l = 7 = const. yields parallel projections (not equally spaced)
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 19 / 36
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Reconstruction from fan projections Weighted filtered back-projection for fan beams
Weighted filtered back-projection for fan beams
We start with the reconstruction formula (note φ integration from 0 to2π):
f(x) =1
2
2π∫0
∞∫−∞
λφ(p)h−1(x · n̂φ − p) dp dφ
Now introduce fan beam coordinates γ, θ such that:
p = R sin(γ)
φ = θ + γ
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 20 / 36
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Reconstruction from fan projections Weighted filtered back-projection for fan beams
Weighted filtered back-projection for fan beams
We start with the reconstruction formula (note φ integration from 0 to2π):
f(x) =1
2
2π∫0
∞∫−∞
λφ(p)h−1(x · n̂φ − p) dp dφ
Now introduce fan beam coordinates γ, θ such that:
p = R sin(γ)
φ = θ + γ
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 20 / 36
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Reconstruction from fan projections Weighted filtered back-projection for fan beams
p = R sin(γ)
φ = θ + γ
Note: the redprojection line is thesame line onceexpressed as λφ(p)
and then as D̆θ(γ).
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 21 / 36
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Reconstruction from fan projections Weighted filtered back-projection for fan beams
Weighted filtered back-projection for fan beams, cont’d
With the Jacobian determinant dp dφ = R cos γ dγ dθ we obtain:
f(x) =1
2
2π∫0
π/2∫−π/2
D̆θ(γ)R cos γ h−1(x · n̂θ+γ −R sin γ) dγ dθ.
Now defineγθ,x := angle between line (source — x) and central rayLθ,x := distance between source and point x
f(x) =1
2
2π∫0
π/2∫−π/2
D̆θ(γ)R cos γ h−1 (Lθ,x sin(γθ,x − γ)) dγ dθ.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 22 / 36
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Reconstruction from fan projections Weighted filtered back-projection for fan beams
Weighted filtered back-projection for fan beams, cont’d
With the Jacobian determinant dp dφ = R cos γ dγ dθ we obtain:
f(x) =1
2
2π∫0
π/2∫−π/2
D̆θ(γ)R cos γ h−1(x · n̂θ+γ −R sin γ) dγ dθ.
Now defineγθ,x := angle between line (source — x) and central rayLθ,x := distance between source and point x
f(x) =1
2
2π∫0
π/2∫−π/2
D̆θ(γ)R cos γ h−1 (Lθ,x sin(γθ,x − γ)) dγ dθ.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 22 / 36
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Reconstruction from fan projections Weighted filtered back-projection for fan beams
Straight detector
The blue linerepresents the straightdetector, virtuallypositioned at theisocenter (origin)
γ = arctan( uR
)
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 23 / 36
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Reconstruction from fan projections Weighted filtered back-projection for fan beams
Weighted filtered back-projection for fan beams, cont’d
Now: straight detector (projected into the origin) measuring Dθ(u) suchthat γ = arctan
(uR
)and dγ = R
R2+u2du.
Using the angle addition formula for the sine we obtain:
f(x) =1
2
2π∫0
∞∫−∞
Dθ(u)R3
(R2 + u2)3/2h−1
Lθ,xR(uθ,x − u)√R2 + u2θ,x
√R2 + u2
du dθ.Now define Wθ,x := Lθ,x/
√R2 + u2θ,x. Note that Wθ,x equals the ratio of
the projection of point x onto the central ray to R.
f(x) =1
2
2π∫0
∞∫−∞
Dθ(u)R3
(R2 + u2)3/2h−1
(Wθ,xR√R2 + u2
(uθ,x − u))
du dθ.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 24 / 36
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Reconstruction from fan projections Weighted filtered back-projection for fan beams
Weighted filtered back-projection for fan beams, cont’d
Now: straight detector (projected into the origin) measuring Dθ(u) suchthat γ = arctan
(uR
)and dγ = R
R2+u2du.
Using the angle addition formula for the sine we obtain:
f(x) =1
2
2π∫0
∞∫−∞
Dθ(u)R3
(R2 + u2)3/2h−1
Lθ,xR(uθ,x − u)√R2 + u2θ,x
√R2 + u2
du dθ.
Now define Wθ,x := Lθ,x/√R2 + u2θ,x. Note that Wθ,x equals the ratio of
the projection of point x onto the central ray to R.
f(x) =1
2
2π∫0
∞∫−∞
Dθ(u)R3
(R2 + u2)3/2h−1
(Wθ,xR√R2 + u2
(uθ,x − u))
du dθ.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 24 / 36
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Reconstruction from fan projections Weighted filtered back-projection for fan beams
Weighted filtered back-projection for fan beams, cont’d
Now: straight detector (projected into the origin) measuring Dθ(u) suchthat γ = arctan
(uR
)and dγ = R
R2+u2du.
Using the angle addition formula for the sine we obtain:
f(x) =1
2
2π∫0
∞∫−∞
Dθ(u)R3
(R2 + u2)3/2h−1
Lθ,xR(uθ,x − u)√R2 + u2θ,x
√R2 + u2
du dθ.Now define Wθ,x := Lθ,x/
√R2 + u2θ,x. Note that Wθ,x equals the ratio of
the projection of point x onto the central ray to R.
f(x) =1
2
2π∫0
∞∫−∞
Dθ(u)R3
(R2 + u2)3/2h−1
(Wθ,xR√R2 + u2
(uθ,x − u))
du dθ.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 24 / 36
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Reconstruction from fan projections Weighted filtered back-projection for fan beams
Weighted filtered back-projection for fan beams, cont’d
Finally, we make use of the relationship h−1(αx) = 1α2h−1(x). This
follows from the representation of h−1 as the inverse Fourier transform ofH−1 = |ν|:
f(x) =1
2
2π∫0
1
W 2θ,x
∞∫−∞
Dθ(u)R√
R2 + u2h−1 (uθ,x − u) du dθ.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 25 / 36
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Reconstruction from fan projections Weighted filtered back-projection for fan beams
Weighted filtered back-projection for fan beams, algorithm
1 Calculate modified projections from the Detector signal Dθ(u) bymultiplying with R√
R2+u2.
2 Convolve modified fan-beam projections with h−1. The Ram-Lak orShepp-Logan filter derived above for parallel data can be used here aswell.
3 Perform weighted filtered back-projections along the fan using1/W 2θ,x as the weight factor. Remember: Wθ,x equals the ratio of theprojection of point x onto the central ray to R.
4 Integrate (sum up) back-projections from all fans.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 26 / 36
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Reconstruction from fan projections Weighted filtered back-projection for fan beams
Weighted filtered back-projection for fan beams, algorithm
1 Calculate modified projections from the Detector signal Dθ(u) bymultiplying with R√
R2+u2.
2 Convolve modified fan-beam projections with h−1. The Ram-Lak orShepp-Logan filter derived above for parallel data can be used here aswell.
3 Perform weighted filtered back-projections along the fan using1/W 2θ,x as the weight factor. Remember: Wθ,x equals the ratio of theprojection of point x onto the central ray to R.
4 Integrate (sum up) back-projections from all fans.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 26 / 36
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Reconstruction from fan projections Weighted filtered back-projection for fan beams
Weighted filtered back-projection for fan beams, algorithm
1 Calculate modified projections from the Detector signal Dθ(u) bymultiplying with R√
R2+u2.
2 Convolve modified fan-beam projections with h−1. The Ram-Lak orShepp-Logan filter derived above for parallel data can be used here aswell.
3 Perform weighted filtered back-projections along the fan using1/W 2θ,x as the weight factor. Remember: Wθ,x equals the ratio of theprojection of point x onto the central ray to R.
4 Integrate (sum up) back-projections from all fans.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 26 / 36
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Reconstruction from fan projections Weighted filtered back-projection for fan beams
Weighted filtered back-projection for fan beams, algorithm
1 Calculate modified projections from the Detector signal Dθ(u) bymultiplying with R√
R2+u2.
2 Convolve modified fan-beam projections with h−1. The Ram-Lak orShepp-Logan filter derived above for parallel data can be used here aswell.
3 Perform weighted filtered back-projections along the fan using1/W 2θ,x as the weight factor. Remember: Wθ,x equals the ratio of theprojection of point x onto the central ray to R.
4 Integrate (sum up) back-projections from all fans.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 26 / 36
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Cone-beam reconstruction
Outline
1 Practical implementationDiscrete dataFiltering in the spatial domainRam-Lak filterShepp-Logan filter
2 Reconstruction from fan projectionsResorting fan to parallel projectionsWeighted filtered back-projection for fan beams
3 Cone-beam reconstructionThe Feldkamp algorithmLimitations of the Feldkamp algorithm
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 27 / 36
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Cone-beam reconstruction The Feldkamp algorithm
Feldkamp algorithm
Idea: Reconstruct a 3D object from tilted fans. Treat tilted fans in thesame way as in 2D geometry.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 28 / 36
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Cone-beam reconstruction The Feldkamp algorithm
Feldkamp algorithm
Two important differences (due to fan tilting):
The source distance is slightly enlarged for the tilted fans:R̃ =
√R2 + v2.
The angular increment dθ̃ is slightly decreased: dθ̃R̃ = dθR.
Then the ”reconstruction” formula becomes:
f(x) =1
2
2π∫0
1
W 2θ,x
∞∫−∞
Dθ(u, v)R√
R2 + u2 + v2h−1 (uθ,x − u) du dθ.
Note that this is a heuristic, not an exact reconstruction formula!
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 29 / 36
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Cone-beam reconstruction The Feldkamp algorithm
Feldkamp algorithm
Two important differences (due to fan tilting):
The source distance is slightly enlarged for the tilted fans:R̃ =
√R2 + v2.
The angular increment dθ̃ is slightly decreased: dθ̃R̃ = dθR.
Then the ”reconstruction” formula becomes:
f(x) =1
2
2π∫0
1
W 2θ,x
∞∫−∞
Dθ(u, v)R√
R2 + u2 + v2h−1 (uθ,x − u) du dθ.
Note that this is a heuristic, not an exact reconstruction formula!
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 29 / 36
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Cone-beam reconstruction The Feldkamp algorithm
Feldkamp algorithm
1 Calculate modified projections from the Detector signal Dθ(u, v) bymultiplying with R√
R2+u2+v2.
2 Convolve modified projections with h−1 in the u direction. TheRam-Lak or Shepp-Logan filter derived above for parallel data can beused here as well.
3 Perform weighted filtered back-projections along the tilted fans using1/W 2θ,x as the weight factor. Note: Wθ,x is the ratio of the projectionof point x onto the central ray (at u = v = 0) of the cone, to R.
4 Integrate (sum up) back-projections from all cones.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 30 / 36
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Cone-beam reconstruction The Feldkamp algorithm
Feldkamp algorithm
1 Calculate modified projections from the Detector signal Dθ(u, v) bymultiplying with R√
R2+u2+v2.
2 Convolve modified projections with h−1 in the u direction. TheRam-Lak or Shepp-Logan filter derived above for parallel data can beused here as well.
3 Perform weighted filtered back-projections along the tilted fans using1/W 2θ,x as the weight factor. Note: Wθ,x is the ratio of the projectionof point x onto the central ray (at u = v = 0) of the cone, to R.
4 Integrate (sum up) back-projections from all cones.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 30 / 36
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Cone-beam reconstruction The Feldkamp algorithm
Feldkamp algorithm
1 Calculate modified projections from the Detector signal Dθ(u, v) bymultiplying with R√
R2+u2+v2.
2 Convolve modified projections with h−1 in the u direction. TheRam-Lak or Shepp-Logan filter derived above for parallel data can beused here as well.
3 Perform weighted filtered back-projections along the tilted fans using1/W 2θ,x as the weight factor. Note: Wθ,x is the ratio of the projectionof point x onto the central ray (at u = v = 0) of the cone, to R.
4 Integrate (sum up) back-projections from all cones.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 30 / 36
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Cone-beam reconstruction The Feldkamp algorithm
Feldkamp algorithm
1 Calculate modified projections from the Detector signal Dθ(u, v) bymultiplying with R√
R2+u2+v2.
2 Convolve modified projections with h−1 in the u direction. TheRam-Lak or Shepp-Logan filter derived above for parallel data can beused here as well.
3 Perform weighted filtered back-projections along the tilted fans using1/W 2θ,x as the weight factor. Note: Wθ,x is the ratio of the projectionof point x onto the central ray (at u = v = 0) of the cone, to R.
4 Integrate (sum up) back-projections from all cones.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 30 / 36
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Cone-beam reconstruction Limitations of the Feldkamp algorithm
Limitations of the Feldkamp algorithm: Defrise phantom
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 31 / 36
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Cone-beam reconstruction Limitations of the Feldkamp algorithm
Limitations of the Feldkamp algorithm: Defrise phantom
Artifacts occur at cone angles above 10◦:
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 32 / 36
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Cone-beam reconstruction Limitations of the Feldkamp algorithm
Features of the Feldkamp algorithm
The Feldkamp algorithm is exact in the following sense:
1 For an object that has no contrast (no variation of density) in the z(= x3) direction, the reconstruction will be exact.
2 The Feldkamp algorithm produces the correct integral of the imageintensity in the z (= x3) direction.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 33 / 36
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Cone-beam reconstruction Limitations of the Feldkamp algorithm
Homework 2
Homework 2:
a) Prove that the Feldkamp algorithm yields the correct result if theobject has no density variation in the z (= x3) direction. You may dothis either analytically or numerically.
Note: Assume that the planar fan beam reconstruction for the sameobject yields the exact result.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 34 / 36
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Further Reading
L.A. Feldkamp, L.C. Davis, J.W. Kress: Practical cone-beamalgorithm. J. Opt. Soc. Am. A 1(6):612-619, 1984
A.C. Kak, M. Slaney: Principles of Computerized TomographicImaging. Reprint: SIAM Classics in Applied Mathematics, 2001.PDF available: http://www.slaney.org/pct/pct-toc.html
F. Natterer: The Mathematics of Computerized Tomography.Reprint: SIAM Classics in Applied Mathematics, 2001.
A.M. Cormack: Early Two-Dimensional Reconstruction and RecentTopics Stemming from it. Nobel lecture, 1979.http://www.nobelprize.org/nobel_prizes/medicine/
laureates/1979/cormack-lecture.pdf
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 35 / 36
http://www.slaney.org/pct/pct-toc.htmlhttp://www.nobelprize.org/nobel_prizes/medicine/laureates/1979/cormack-lecture.pdfhttp://www.nobelprize.org/nobel_prizes/medicine/laureates/1979/cormack-lecture.pdf
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Further Reading
R.N. Bracewell: The Fourier Transform and its Applications.McGraw-Hill, New York, 3rd edition, revised, 1999.
T. Bortfeld: Röntgencomputertomographie: MathematischeGrundlagen. In: Schlegel W, Bille J, eds. Medizinische Physik 2(Medizinische Strahlenphysik). Heidelberg: Springer; 2002: 229-245.English translation available from author.
J. Radon: Über die Bestimmung von Funktionen durch ihreIntegralwerte längs gewisser Mannigfaltigkeiten.Berichte der Sächsischen Akademie der Wissenschaften – Math.-Phys.Klasse, 69:262–277, 1917.
Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 36 / 36
1Practical implementationDiscrete dataFiltering in the spatial domainRam-Lak filterShepp-Logan filter
Reconstruction from fan projectionsResorting fan to parallel projectionsWeighted filtered back-projection for fan beams
Cone-beam reconstructionThe Feldkamp algorithmLimitations of the Feldkamp algorithm
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