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 Reconstruction HST.S14, February 19, 2013 1 / 36

Transcript of Image Reconstruction 2 Cone Beam Reconstruction · 2 Reconstruction from fan projections Resorting...

  • 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

  • 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

  • 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

  • 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

  • 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

  • Backprojection

    Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 6 / 36

  • Central Slice Theorem

    Thomas Bortfeld (MGH, HMS, Rad. Onc.) Image Reconstruction 2 – Cone Beam ReconstructionHST.S14, February 19, 2013 7 / 36

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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)

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

    2