Abstract

Radon data interpolation is a necessary procedure in computed tomography (CT), especially for reconstruction from divergent beam scanning. In a polar-grid representation, the Radon data of a fanbeam projection are populated on an arc, rather on a radial line. Collectively, the Radon data generated from a fanbeam CT system are unevenly populated: The population becomes sparser as the polar distance increases. In CT reconstruction, the Fourier central slice theorem requires a radial scanline full of Radon data. Therefore the vacant entries of a scanline must be filled by interpolation. In addition, interpolation is also required in polar-to-Cartesian conversion. In this paper we propose a practical interpolation technique for filling the vacant entries by local convex combination. It is a linear interpolant that generates a value for a grid point from the available data lying in its neighborhood, by a weighted average, with the weights corresponding to the inverse distances. In fact, the linear convex combination serves as a general flat-smoothing operation in filling a vacancy. Specifically, this technique realizes a variety of linear interpolations, including nearest-neighbor replication, two-point collinear, three-point triangulation, and four-point quadrilateral, and local extrapolation, in a unified framework. Algorithms and a simulation demonstration are provided.

© 2003 Optical Society of America

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References

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  1. G. Besson, “CT image reconstruction from fan-parallel data,” Med. Phys. 26, 415–426 (1999).
    [CrossRef]
  2. A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1987).
  3. I. Svalbe, D. Spek, “Reconstruction of tomographic images using analog projections and the digital Radon transform,” Linear Algebr. Appl. 339, 125–145 (2001).
    [CrossRef]
  4. X. Pan, “Optimal noise control in and fast reconstruction of fanbeam computed tomography imaging,” Med. Phys. 26, 289–297 (1999).
    [CrossRef]
  5. I. Amidror, “Scattered data interpolation methods for electronic imaging systems: a survey,” J. Electron. Imaging 11, 157–176 (2002).
    [CrossRef]
  6. S. E. Parker, “Nearest-grid-point interpolation in gyrokinetic particle-in-cell simulation,” J. Comput. Phys. 178, 520–532 (2002).
    [CrossRef]
  7. D. Rajan, S. Chaudhuri, “Generalized interpolation and its application in super-resolution imaging,” Image Vision Comput. 19, 957–969 (2001).
    [CrossRef]
  8. W. K. Carey, D. B. Chuang, S. S. Hemami, “Regularity-preserving image interpolation,” IEEE Trans. Image Proc. 8, 1293–1297 (1999).
    [CrossRef]
  9. J. A. Parker, R. V. Kenyon, D. E. Troxel, “Comparison of interpolating methods for image resampling,” IEEE Trans. Med. Imaging 2, 31–39 (1983).
    [CrossRef] [PubMed]
  10. G. R. Davis, “Faster tomographic fanbeam back-projection using Cartesian axes pre-projection,” Nucl. Instrum. Methods Phys. Res. A 410, 329–334 (1998).
    [CrossRef]
  11. F. Noo, M. Defrise, R. Clackdoyle, H. Kudo, “Image reconstruction from fanbeam projections on less than a short scan,” Phys. Med. Biol. 47, 2525–2546 (2002).
    [CrossRef] [PubMed]
  12. H. W. Guggenheimer, Applicable Geometry: Global and Local Convexity, (Krieger, New York, 1977).
  13. S. Bonnet, F. Peyrin, F. Turjman, R. Prost, “Multiresolution reconstruction in fanbeam tomography,” IEEE Trans. Image Proc. 11, 169–176 (2002).
    [CrossRef]
  14. Z. Chen, R. Ning, “Why should breast tumour detection go three dimensional?” Phys. Med. Biol. 48, 2217–2228 (2003).
    [CrossRef] [PubMed]

2003 (1)

Z. Chen, R. Ning, “Why should breast tumour detection go three dimensional?” Phys. Med. Biol. 48, 2217–2228 (2003).
[CrossRef] [PubMed]

2002 (4)

F. Noo, M. Defrise, R. Clackdoyle, H. Kudo, “Image reconstruction from fanbeam projections on less than a short scan,” Phys. Med. Biol. 47, 2525–2546 (2002).
[CrossRef] [PubMed]

S. Bonnet, F. Peyrin, F. Turjman, R. Prost, “Multiresolution reconstruction in fanbeam tomography,” IEEE Trans. Image Proc. 11, 169–176 (2002).
[CrossRef]

I. Amidror, “Scattered data interpolation methods for electronic imaging systems: a survey,” J. Electron. Imaging 11, 157–176 (2002).
[CrossRef]

S. E. Parker, “Nearest-grid-point interpolation in gyrokinetic particle-in-cell simulation,” J. Comput. Phys. 178, 520–532 (2002).
[CrossRef]

2001 (2)

D. Rajan, S. Chaudhuri, “Generalized interpolation and its application in super-resolution imaging,” Image Vision Comput. 19, 957–969 (2001).
[CrossRef]

I. Svalbe, D. Spek, “Reconstruction of tomographic images using analog projections and the digital Radon transform,” Linear Algebr. Appl. 339, 125–145 (2001).
[CrossRef]

1999 (3)

X. Pan, “Optimal noise control in and fast reconstruction of fanbeam computed tomography imaging,” Med. Phys. 26, 289–297 (1999).
[CrossRef]

G. Besson, “CT image reconstruction from fan-parallel data,” Med. Phys. 26, 415–426 (1999).
[CrossRef]

W. K. Carey, D. B. Chuang, S. S. Hemami, “Regularity-preserving image interpolation,” IEEE Trans. Image Proc. 8, 1293–1297 (1999).
[CrossRef]

1998 (1)

G. R. Davis, “Faster tomographic fanbeam back-projection using Cartesian axes pre-projection,” Nucl. Instrum. Methods Phys. Res. A 410, 329–334 (1998).
[CrossRef]

1983 (1)

J. A. Parker, R. V. Kenyon, D. E. Troxel, “Comparison of interpolating methods for image resampling,” IEEE Trans. Med. Imaging 2, 31–39 (1983).
[CrossRef] [PubMed]

Amidror, I.

I. Amidror, “Scattered data interpolation methods for electronic imaging systems: a survey,” J. Electron. Imaging 11, 157–176 (2002).
[CrossRef]

Besson, G.

G. Besson, “CT image reconstruction from fan-parallel data,” Med. Phys. 26, 415–426 (1999).
[CrossRef]

Bonnet, S.

S. Bonnet, F. Peyrin, F. Turjman, R. Prost, “Multiresolution reconstruction in fanbeam tomography,” IEEE Trans. Image Proc. 11, 169–176 (2002).
[CrossRef]

Carey, W. K.

W. K. Carey, D. B. Chuang, S. S. Hemami, “Regularity-preserving image interpolation,” IEEE Trans. Image Proc. 8, 1293–1297 (1999).
[CrossRef]

Chaudhuri, S.

D. Rajan, S. Chaudhuri, “Generalized interpolation and its application in super-resolution imaging,” Image Vision Comput. 19, 957–969 (2001).
[CrossRef]

Chen, Z.

Z. Chen, R. Ning, “Why should breast tumour detection go three dimensional?” Phys. Med. Biol. 48, 2217–2228 (2003).
[CrossRef] [PubMed]

Chuang, D. B.

W. K. Carey, D. B. Chuang, S. S. Hemami, “Regularity-preserving image interpolation,” IEEE Trans. Image Proc. 8, 1293–1297 (1999).
[CrossRef]

Clackdoyle, R.

F. Noo, M. Defrise, R. Clackdoyle, H. Kudo, “Image reconstruction from fanbeam projections on less than a short scan,” Phys. Med. Biol. 47, 2525–2546 (2002).
[CrossRef] [PubMed]

Davis, G. R.

G. R. Davis, “Faster tomographic fanbeam back-projection using Cartesian axes pre-projection,” Nucl. Instrum. Methods Phys. Res. A 410, 329–334 (1998).
[CrossRef]

Defrise, M.

F. Noo, M. Defrise, R. Clackdoyle, H. Kudo, “Image reconstruction from fanbeam projections on less than a short scan,” Phys. Med. Biol. 47, 2525–2546 (2002).
[CrossRef] [PubMed]

Guggenheimer, H. W.

H. W. Guggenheimer, Applicable Geometry: Global and Local Convexity, (Krieger, New York, 1977).

Hemami, S. S.

W. K. Carey, D. B. Chuang, S. S. Hemami, “Regularity-preserving image interpolation,” IEEE Trans. Image Proc. 8, 1293–1297 (1999).
[CrossRef]

Kak, A. C.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1987).

Kenyon, R. V.

J. A. Parker, R. V. Kenyon, D. E. Troxel, “Comparison of interpolating methods for image resampling,” IEEE Trans. Med. Imaging 2, 31–39 (1983).
[CrossRef] [PubMed]

Kudo, H.

F. Noo, M. Defrise, R. Clackdoyle, H. Kudo, “Image reconstruction from fanbeam projections on less than a short scan,” Phys. Med. Biol. 47, 2525–2546 (2002).
[CrossRef] [PubMed]

Ning, R.

Z. Chen, R. Ning, “Why should breast tumour detection go three dimensional?” Phys. Med. Biol. 48, 2217–2228 (2003).
[CrossRef] [PubMed]

Noo, F.

F. Noo, M. Defrise, R. Clackdoyle, H. Kudo, “Image reconstruction from fanbeam projections on less than a short scan,” Phys. Med. Biol. 47, 2525–2546 (2002).
[CrossRef] [PubMed]

Pan, X.

X. Pan, “Optimal noise control in and fast reconstruction of fanbeam computed tomography imaging,” Med. Phys. 26, 289–297 (1999).
[CrossRef]

Parker, J. A.

J. A. Parker, R. V. Kenyon, D. E. Troxel, “Comparison of interpolating methods for image resampling,” IEEE Trans. Med. Imaging 2, 31–39 (1983).
[CrossRef] [PubMed]

Parker, S. E.

S. E. Parker, “Nearest-grid-point interpolation in gyrokinetic particle-in-cell simulation,” J. Comput. Phys. 178, 520–532 (2002).
[CrossRef]

Peyrin, F.

S. Bonnet, F. Peyrin, F. Turjman, R. Prost, “Multiresolution reconstruction in fanbeam tomography,” IEEE Trans. Image Proc. 11, 169–176 (2002).
[CrossRef]

Prost, R.

S. Bonnet, F. Peyrin, F. Turjman, R. Prost, “Multiresolution reconstruction in fanbeam tomography,” IEEE Trans. Image Proc. 11, 169–176 (2002).
[CrossRef]

Rajan, D.

D. Rajan, S. Chaudhuri, “Generalized interpolation and its application in super-resolution imaging,” Image Vision Comput. 19, 957–969 (2001).
[CrossRef]

Slaney, M.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1987).

Spek, D.

I. Svalbe, D. Spek, “Reconstruction of tomographic images using analog projections and the digital Radon transform,” Linear Algebr. Appl. 339, 125–145 (2001).
[CrossRef]

Svalbe, I.

I. Svalbe, D. Spek, “Reconstruction of tomographic images using analog projections and the digital Radon transform,” Linear Algebr. Appl. 339, 125–145 (2001).
[CrossRef]

Troxel, D. E.

J. A. Parker, R. V. Kenyon, D. E. Troxel, “Comparison of interpolating methods for image resampling,” IEEE Trans. Med. Imaging 2, 31–39 (1983).
[CrossRef] [PubMed]

Turjman, F.

S. Bonnet, F. Peyrin, F. Turjman, R. Prost, “Multiresolution reconstruction in fanbeam tomography,” IEEE Trans. Image Proc. 11, 169–176 (2002).
[CrossRef]

IEEE Trans. Image Proc. (2)

W. K. Carey, D. B. Chuang, S. S. Hemami, “Regularity-preserving image interpolation,” IEEE Trans. Image Proc. 8, 1293–1297 (1999).
[CrossRef]

S. Bonnet, F. Peyrin, F. Turjman, R. Prost, “Multiresolution reconstruction in fanbeam tomography,” IEEE Trans. Image Proc. 11, 169–176 (2002).
[CrossRef]

IEEE Trans. Med. Imaging (1)

J. A. Parker, R. V. Kenyon, D. E. Troxel, “Comparison of interpolating methods for image resampling,” IEEE Trans. Med. Imaging 2, 31–39 (1983).
[CrossRef] [PubMed]

Image Vision Comput. (1)

D. Rajan, S. Chaudhuri, “Generalized interpolation and its application in super-resolution imaging,” Image Vision Comput. 19, 957–969 (2001).
[CrossRef]

J. Comput. Phys. (1)

S. E. Parker, “Nearest-grid-point interpolation in gyrokinetic particle-in-cell simulation,” J. Comput. Phys. 178, 520–532 (2002).
[CrossRef]

J. Electron. Imaging (1)

I. Amidror, “Scattered data interpolation methods for electronic imaging systems: a survey,” J. Electron. Imaging 11, 157–176 (2002).
[CrossRef]

Linear Algebr. Appl. (1)

I. Svalbe, D. Spek, “Reconstruction of tomographic images using analog projections and the digital Radon transform,” Linear Algebr. Appl. 339, 125–145 (2001).
[CrossRef]

Med. Phys. (2)

X. Pan, “Optimal noise control in and fast reconstruction of fanbeam computed tomography imaging,” Med. Phys. 26, 289–297 (1999).
[CrossRef]

G. Besson, “CT image reconstruction from fan-parallel data,” Med. Phys. 26, 415–426 (1999).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. A (1)

G. R. Davis, “Faster tomographic fanbeam back-projection using Cartesian axes pre-projection,” Nucl. Instrum. Methods Phys. Res. A 410, 329–334 (1998).
[CrossRef]

Phys. Med. Biol. (2)

F. Noo, M. Defrise, R. Clackdoyle, H. Kudo, “Image reconstruction from fanbeam projections on less than a short scan,” Phys. Med. Biol. 47, 2525–2546 (2002).
[CrossRef] [PubMed]

Z. Chen, R. Ning, “Why should breast tumour detection go three dimensional?” Phys. Med. Biol. 48, 2217–2228 (2003).
[CrossRef] [PubMed]

Other (2)

H. W. Guggenheimer, Applicable Geometry: Global and Local Convexity, (Krieger, New York, 1977).

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1987).

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Figures (7)

Fig. 1
Fig. 1

Fanbeam scanning geometry.

Fig. 2
Fig. 2

Parameters for fanbeam projection.

Fig. 3
Fig. 3

Radon circle, Radon arc, Radon domain, and object support circle in fanbeam geometry.

Fig. 4
Fig. 4

Local convex combination for interpolation (filling open circle) and extrapolation (filling open diamond).

Fig. 5
Fig. 5

Manifestations of data filling using local convex combination. (a) Nearest-neighbor replication, (b) collinear interpolation, (c) triangulation interpolation, (d) quadrilateral interpolation, (e) local extrapolation (case 1), (f) local extrapolation (case 2).

Fig. 6
Fig. 6

Simulation results of Radon domain filling. (a) The original Radon data generated from 80 fanbeam projections, and its interpolations by (b) nearest-neighbor replication, (c) 3 × 3 neighborhood interpolation, and (d) 5 × 5 neighborhood interpolation.

Fig. 7
Fig. 7

Reconstruction results. (a) Shepp-Logan phantom superimposed by a scanline (horizontal white line) and (b) the scanline profile; (c) reconstruction from nearest-neighbor replication and (d) the scanline profile; (e) reconstruction from 3 × 3 neighborhood filling and (f) the scanline profile; (g) reconstruction from 5 × 5 neighborhood filling and (h) the scanline profile.

Equations (23)

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β=tan-1s/SDD,
ρ=Rorbit sinβ,
θ=α-β+π/2β0α-β-π/2β<0,
ρmax=maxρ<, βmax=maxβ<,
gρ, θ= fx, yδx cos θ+y sin θ-ρdxdy.
Gω; θ= gρ; θexp-i2πρωdρ,
Gω; θ=Fω; θ, θ[0, 360°),
fx, y= dθ  Gω, θexpi2πx2+y2ωωdω.
fx, y=IFFTFu, v,
u=ω cos θ, v=ω sin θ, Fu, v=Gu, v,
gfanρ, θ=cos β  fx, yδx cos θ+y sin θ-ρdxdy.
gfanρ, θ=gfanρ, θ,
g¯fanρ, θ=meanρ,θgfanρ, θ|ρ=ρ, θ=θ.
Rρ, θ=fillg¯fanρ, θ.
fx, y=fx, yMx, y
Mx, y=1x2+y2ρmax0otherwise
Mρ=1ρ<ρmax0otherwise,
gfanρ, θ=gfanρ, θMρ.
Nm×mp=p|maxp-pm/2,
Nm×map=p|pNm×mp and g¯p was available.
R=piNm×map λig¯pi, p is a vacancy.
i λi=1, λi0.
λi=1/dii 1/di=1pi-p i 1/pi-p,

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