Abstract

Based on the linearity of the Radon transform and the convolution-backprojection reconstruction algorithm, a new linear-vector space notation is introduced that is of general use in computed tomography (CT) Using this notation a consistency condition for the completion of incomplete projection data is described. This consistency condition leads to singular or ill-conditioned systems of linear equations for the unknown projection data. Using regularization methods, an algorithm for the consistent projection completion is presented that can exploit symmetries of the missing data region The performance of the new algorithm is documented with simulated and actualy measured CT projection data. The algorithm quantitatively improves CT reconstructions with realistic amounts of data and noise and can be used for the completion of arbitrary regions of missing projections.

© 1985 Optical Society of America

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References

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  1. B. E. Oppenheim, Reconstruction tomography from incomplete projections,” in Reconstruction Tomography in Diagnostic Radiology and Nuclear Medicine, M. M. Terr-Pogossian et al., eds. (University Park, Baltimore, Md., 1977) pp. 155–165.
  2. R. M. Lewitt, R. H. T. Bates, “Image reconstruction from projections—part III—projection completion methods,” Optik 50, 189–204 (1978).
  3. A. K. Louis, “Picture reconstruction from projections in restricted range,” Math. Meth. Appl. Sci. 2, 209–220 (1980).
    [CrossRef]
  4. F. Natterer, “Efficient implementation of optimal algorithms in computerized tomography,” Math. Meth. Appl. Sci. 2, 545–555 (1980).
    [CrossRef]
  5. P. Seitz, P. Rüegsegger, “Bone densitometry in the vicinity of metallic implants,” J. Comput. Assist. Tomogr. 6 (1), 198–199 (1982).
    [CrossRef]
  6. M. Nassi, W. R. Brody, B. P. Medoff, A. Macovski, “Iterative reconstruction-reprojection: an algorithm for limited data cardiac-computed tomography,” IEEE Trans. Biomed. Eng. BME-29, 333–340 (1982).
    [CrossRef]
  7. B. P. Medoff, W. R. Brody, M. Nassi, A. Macovski, “Iterative convolution backprojection algorithms for image reconstruction from limited data,” J. Opt. Soc. Am. 73, 1493–1500 (1983).
    [CrossRef]
  8. M. E. DavisonThe ill-conditioned nature of the limited angle tomography problem,” SIAM J. Appl. Math. 43, 428–448 (1983).
    [CrossRef]
  9. A. M. Darling, T. J. Hall, M. A. Fiddy, “Stable, noniterative object reconstruction from incomplete data using a priori knowedge,” J. Opt. Soc. Am. 73, 1466–1469 (1983).
    [CrossRef]
  10. P. Seitz, “Computertomographische Osteodensitometrie beim metallischen Kunstgelenk,” Ph.D. dissertation (Federal Institute of Technology, Zurich, Switzerland, 1984).
  11. L. M. Chen, A. S. Ho, R. E. Burge, “Use of a priori knowledge in image restoration,” J. Opt. Soc. Am. A 1, 386–391 (1984).
    [CrossRef]
  12. A. K. Louis, F. Natterer, “Mathematical problems of computerized tomography,” Proc. IEEE 71, 379–389 (1983).
    [CrossRef]
  13. L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).
    [CrossRef]
  14. S. Helgason, “The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds,” Acta Math. 113, 153–180 (1965).
    [CrossRef]
  15. D. Ludwig, “The Radon transform on Euclidean spaces,” Commun. Pure Appl. Math. 19, 49–81 (1966).
    [CrossRef]
  16. S. Helgason, “The Radon transform,” in Progress in Mathematics, J. Coates, S. Helgason, eds. (Birkhäuser, Cambridge, Mass., 1980), Vol. 5.
  17. A. Tikhonov, V. Arsenin, Solution of Ill-Posed Problems (Wiley, New York, 1977).
  18. G. T. Herman, Image Reconstruction from Projections (Academic, New York, 1980).
  19. R. R. Bitmead, B. D. O. Anderson, “Asymptotically fast solution of Toeplitz and related systems of linear equations,” Lin. Alg. Its Appl. 34, 103–116 (1980).
    [CrossRef]
  20. H. Akaike, “Block Toeplitz matrix inversion,” SIAM (Soc Ind Appl. Math.) Rev. 24, 234–241, 1973.
    [CrossRef]
  21. T. Hinderling, P. Rüegsegger, M. Anliker, C. Dietschi, “Computed tomography from hollow projections: an application to in vivo evaluation of artificial hip joints,” J. Comput. Assist. Tomogr. 3(1) 52–57 (1979).
    [CrossRef] [PubMed]
  22. B. Stebler, P. Rüegsegger, “Special purpose CT-system for quantitative bone evaluation in the appendicular skeleton,” Biomed. Tech. 28, 196–205 (1983).
    [CrossRef]

1984 (1)

1983 (5)

A. M. Darling, T. J. Hall, M. A. Fiddy, “Stable, noniterative object reconstruction from incomplete data using a priori knowedge,” J. Opt. Soc. Am. 73, 1466–1469 (1983).
[CrossRef]

B. P. Medoff, W. R. Brody, M. Nassi, A. Macovski, “Iterative convolution backprojection algorithms for image reconstruction from limited data,” J. Opt. Soc. Am. 73, 1493–1500 (1983).
[CrossRef]

M. E. DavisonThe ill-conditioned nature of the limited angle tomography problem,” SIAM J. Appl. Math. 43, 428–448 (1983).
[CrossRef]

A. K. Louis, F. Natterer, “Mathematical problems of computerized tomography,” Proc. IEEE 71, 379–389 (1983).
[CrossRef]

B. Stebler, P. Rüegsegger, “Special purpose CT-system for quantitative bone evaluation in the appendicular skeleton,” Biomed. Tech. 28, 196–205 (1983).
[CrossRef]

1982 (2)

P. Seitz, P. Rüegsegger, “Bone densitometry in the vicinity of metallic implants,” J. Comput. Assist. Tomogr. 6 (1), 198–199 (1982).
[CrossRef]

M. Nassi, W. R. Brody, B. P. Medoff, A. Macovski, “Iterative reconstruction-reprojection: an algorithm for limited data cardiac-computed tomography,” IEEE Trans. Biomed. Eng. BME-29, 333–340 (1982).
[CrossRef]

1980 (3)

A. K. Louis, “Picture reconstruction from projections in restricted range,” Math. Meth. Appl. Sci. 2, 209–220 (1980).
[CrossRef]

F. Natterer, “Efficient implementation of optimal algorithms in computerized tomography,” Math. Meth. Appl. Sci. 2, 545–555 (1980).
[CrossRef]

R. R. Bitmead, B. D. O. Anderson, “Asymptotically fast solution of Toeplitz and related systems of linear equations,” Lin. Alg. Its Appl. 34, 103–116 (1980).
[CrossRef]

1979 (1)

T. Hinderling, P. Rüegsegger, M. Anliker, C. Dietschi, “Computed tomography from hollow projections: an application to in vivo evaluation of artificial hip joints,” J. Comput. Assist. Tomogr. 3(1) 52–57 (1979).
[CrossRef] [PubMed]

1978 (1)

R. M. Lewitt, R. H. T. Bates, “Image reconstruction from projections—part III—projection completion methods,” Optik 50, 189–204 (1978).

1974 (1)

L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).
[CrossRef]

1973 (1)

H. Akaike, “Block Toeplitz matrix inversion,” SIAM (Soc Ind Appl. Math.) Rev. 24, 234–241, 1973.
[CrossRef]

1966 (1)

D. Ludwig, “The Radon transform on Euclidean spaces,” Commun. Pure Appl. Math. 19, 49–81 (1966).
[CrossRef]

1965 (1)

S. Helgason, “The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds,” Acta Math. 113, 153–180 (1965).
[CrossRef]

Akaike, H.

H. Akaike, “Block Toeplitz matrix inversion,” SIAM (Soc Ind Appl. Math.) Rev. 24, 234–241, 1973.
[CrossRef]

Anderson, B. D. O.

R. R. Bitmead, B. D. O. Anderson, “Asymptotically fast solution of Toeplitz and related systems of linear equations,” Lin. Alg. Its Appl. 34, 103–116 (1980).
[CrossRef]

Anliker, M.

T. Hinderling, P. Rüegsegger, M. Anliker, C. Dietschi, “Computed tomography from hollow projections: an application to in vivo evaluation of artificial hip joints,” J. Comput. Assist. Tomogr. 3(1) 52–57 (1979).
[CrossRef] [PubMed]

Arsenin, V.

A. Tikhonov, V. Arsenin, Solution of Ill-Posed Problems (Wiley, New York, 1977).

Bates, R. H. T.

R. M. Lewitt, R. H. T. Bates, “Image reconstruction from projections—part III—projection completion methods,” Optik 50, 189–204 (1978).

Bitmead, R. R.

R. R. Bitmead, B. D. O. Anderson, “Asymptotically fast solution of Toeplitz and related systems of linear equations,” Lin. Alg. Its Appl. 34, 103–116 (1980).
[CrossRef]

Brody, W. R.

B. P. Medoff, W. R. Brody, M. Nassi, A. Macovski, “Iterative convolution backprojection algorithms for image reconstruction from limited data,” J. Opt. Soc. Am. 73, 1493–1500 (1983).
[CrossRef]

M. Nassi, W. R. Brody, B. P. Medoff, A. Macovski, “Iterative reconstruction-reprojection: an algorithm for limited data cardiac-computed tomography,” IEEE Trans. Biomed. Eng. BME-29, 333–340 (1982).
[CrossRef]

Burge, R. E.

Chen, L. M.

Darling, A. M.

Davison, M. E.

M. E. DavisonThe ill-conditioned nature of the limited angle tomography problem,” SIAM J. Appl. Math. 43, 428–448 (1983).
[CrossRef]

Dietschi, C.

T. Hinderling, P. Rüegsegger, M. Anliker, C. Dietschi, “Computed tomography from hollow projections: an application to in vivo evaluation of artificial hip joints,” J. Comput. Assist. Tomogr. 3(1) 52–57 (1979).
[CrossRef] [PubMed]

Fiddy, M. A.

Hall, T. J.

Helgason, S.

S. Helgason, “The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds,” Acta Math. 113, 153–180 (1965).
[CrossRef]

S. Helgason, “The Radon transform,” in Progress in Mathematics, J. Coates, S. Helgason, eds. (Birkhäuser, Cambridge, Mass., 1980), Vol. 5.

Herman, G. T.

G. T. Herman, Image Reconstruction from Projections (Academic, New York, 1980).

Hinderling, T.

T. Hinderling, P. Rüegsegger, M. Anliker, C. Dietschi, “Computed tomography from hollow projections: an application to in vivo evaluation of artificial hip joints,” J. Comput. Assist. Tomogr. 3(1) 52–57 (1979).
[CrossRef] [PubMed]

Ho, A. S.

Lewitt, R. M.

R. M. Lewitt, R. H. T. Bates, “Image reconstruction from projections—part III—projection completion methods,” Optik 50, 189–204 (1978).

Logan, B. F.

L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).
[CrossRef]

Louis, A. K.

A. K. Louis, F. Natterer, “Mathematical problems of computerized tomography,” Proc. IEEE 71, 379–389 (1983).
[CrossRef]

A. K. Louis, “Picture reconstruction from projections in restricted range,” Math. Meth. Appl. Sci. 2, 209–220 (1980).
[CrossRef]

Ludwig, D.

D. Ludwig, “The Radon transform on Euclidean spaces,” Commun. Pure Appl. Math. 19, 49–81 (1966).
[CrossRef]

Macovski, A.

B. P. Medoff, W. R. Brody, M. Nassi, A. Macovski, “Iterative convolution backprojection algorithms for image reconstruction from limited data,” J. Opt. Soc. Am. 73, 1493–1500 (1983).
[CrossRef]

M. Nassi, W. R. Brody, B. P. Medoff, A. Macovski, “Iterative reconstruction-reprojection: an algorithm for limited data cardiac-computed tomography,” IEEE Trans. Biomed. Eng. BME-29, 333–340 (1982).
[CrossRef]

Medoff, B. P.

B. P. Medoff, W. R. Brody, M. Nassi, A. Macovski, “Iterative convolution backprojection algorithms for image reconstruction from limited data,” J. Opt. Soc. Am. 73, 1493–1500 (1983).
[CrossRef]

M. Nassi, W. R. Brody, B. P. Medoff, A. Macovski, “Iterative reconstruction-reprojection: an algorithm for limited data cardiac-computed tomography,” IEEE Trans. Biomed. Eng. BME-29, 333–340 (1982).
[CrossRef]

Nassi, M.

B. P. Medoff, W. R. Brody, M. Nassi, A. Macovski, “Iterative convolution backprojection algorithms for image reconstruction from limited data,” J. Opt. Soc. Am. 73, 1493–1500 (1983).
[CrossRef]

M. Nassi, W. R. Brody, B. P. Medoff, A. Macovski, “Iterative reconstruction-reprojection: an algorithm for limited data cardiac-computed tomography,” IEEE Trans. Biomed. Eng. BME-29, 333–340 (1982).
[CrossRef]

Natterer, F.

A. K. Louis, F. Natterer, “Mathematical problems of computerized tomography,” Proc. IEEE 71, 379–389 (1983).
[CrossRef]

F. Natterer, “Efficient implementation of optimal algorithms in computerized tomography,” Math. Meth. Appl. Sci. 2, 545–555 (1980).
[CrossRef]

Oppenheim, B. E.

B. E. Oppenheim, Reconstruction tomography from incomplete projections,” in Reconstruction Tomography in Diagnostic Radiology and Nuclear Medicine, M. M. Terr-Pogossian et al., eds. (University Park, Baltimore, Md., 1977) pp. 155–165.

Rüegsegger, P.

B. Stebler, P. Rüegsegger, “Special purpose CT-system for quantitative bone evaluation in the appendicular skeleton,” Biomed. Tech. 28, 196–205 (1983).
[CrossRef]

P. Seitz, P. Rüegsegger, “Bone densitometry in the vicinity of metallic implants,” J. Comput. Assist. Tomogr. 6 (1), 198–199 (1982).
[CrossRef]

T. Hinderling, P. Rüegsegger, M. Anliker, C. Dietschi, “Computed tomography from hollow projections: an application to in vivo evaluation of artificial hip joints,” J. Comput. Assist. Tomogr. 3(1) 52–57 (1979).
[CrossRef] [PubMed]

Seitz, P.

P. Seitz, P. Rüegsegger, “Bone densitometry in the vicinity of metallic implants,” J. Comput. Assist. Tomogr. 6 (1), 198–199 (1982).
[CrossRef]

P. Seitz, “Computertomographische Osteodensitometrie beim metallischen Kunstgelenk,” Ph.D. dissertation (Federal Institute of Technology, Zurich, Switzerland, 1984).

Shepp, L. A.

L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).
[CrossRef]

Stebler, B.

B. Stebler, P. Rüegsegger, “Special purpose CT-system for quantitative bone evaluation in the appendicular skeleton,” Biomed. Tech. 28, 196–205 (1983).
[CrossRef]

Tikhonov, A.

A. Tikhonov, V. Arsenin, Solution of Ill-Posed Problems (Wiley, New York, 1977).

Acta Math. (1)

S. Helgason, “The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds,” Acta Math. 113, 153–180 (1965).
[CrossRef]

Biomed. Tech. (1)

B. Stebler, P. Rüegsegger, “Special purpose CT-system for quantitative bone evaluation in the appendicular skeleton,” Biomed. Tech. 28, 196–205 (1983).
[CrossRef]

Commun. Pure Appl. Math. (1)

D. Ludwig, “The Radon transform on Euclidean spaces,” Commun. Pure Appl. Math. 19, 49–81 (1966).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

M. Nassi, W. R. Brody, B. P. Medoff, A. Macovski, “Iterative reconstruction-reprojection: an algorithm for limited data cardiac-computed tomography,” IEEE Trans. Biomed. Eng. BME-29, 333–340 (1982).
[CrossRef]

IEEE Trans. Nucl. Sci. (1)

L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).
[CrossRef]

J. Comput. Assist. Tomogr. (2)

T. Hinderling, P. Rüegsegger, M. Anliker, C. Dietschi, “Computed tomography from hollow projections: an application to in vivo evaluation of artificial hip joints,” J. Comput. Assist. Tomogr. 3(1) 52–57 (1979).
[CrossRef] [PubMed]

P. Seitz, P. Rüegsegger, “Bone densitometry in the vicinity of metallic implants,” J. Comput. Assist. Tomogr. 6 (1), 198–199 (1982).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Lin. Alg. Its Appl. (1)

R. R. Bitmead, B. D. O. Anderson, “Asymptotically fast solution of Toeplitz and related systems of linear equations,” Lin. Alg. Its Appl. 34, 103–116 (1980).
[CrossRef]

Math. Meth. Appl. Sci. (2)

A. K. Louis, “Picture reconstruction from projections in restricted range,” Math. Meth. Appl. Sci. 2, 209–220 (1980).
[CrossRef]

F. Natterer, “Efficient implementation of optimal algorithms in computerized tomography,” Math. Meth. Appl. Sci. 2, 545–555 (1980).
[CrossRef]

Optik (1)

R. M. Lewitt, R. H. T. Bates, “Image reconstruction from projections—part III—projection completion methods,” Optik 50, 189–204 (1978).

Proc. IEEE (1)

A. K. Louis, F. Natterer, “Mathematical problems of computerized tomography,” Proc. IEEE 71, 379–389 (1983).
[CrossRef]

SIAM (Soc Ind Appl. Math.) Rev. (1)

H. Akaike, “Block Toeplitz matrix inversion,” SIAM (Soc Ind Appl. Math.) Rev. 24, 234–241, 1973.
[CrossRef]

SIAM J. Appl. Math. (1)

M. E. DavisonThe ill-conditioned nature of the limited angle tomography problem,” SIAM J. Appl. Math. 43, 428–448 (1983).
[CrossRef]

Other (5)

P. Seitz, “Computertomographische Osteodensitometrie beim metallischen Kunstgelenk,” Ph.D. dissertation (Federal Institute of Technology, Zurich, Switzerland, 1984).

B. E. Oppenheim, Reconstruction tomography from incomplete projections,” in Reconstruction Tomography in Diagnostic Radiology and Nuclear Medicine, M. M. Terr-Pogossian et al., eds. (University Park, Baltimore, Md., 1977) pp. 155–165.

S. Helgason, “The Radon transform,” in Progress in Mathematics, J. Coates, S. Helgason, eds. (Birkhäuser, Cambridge, Mass., 1980), Vol. 5.

A. Tikhonov, V. Arsenin, Solution of Ill-Posed Problems (Wiley, New York, 1977).

G. T. Herman, Image Reconstruction from Projections (Academic, New York, 1980).

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Figures (9)

Fig. 1
Fig. 1

Graphical representation in the ℝ3 of the consistent completion problem of hollow projections. The inconsistent project vector p is composed of the measurable part p′ ∈ V′ and the inconsistent p″ ∈ V″. We look for a completion vector x ∈ V″ so that x compensates for the inconsistency in p″, i.e., p + x is a consistent vector ∈ W (the space of all consistent vectors in V) and p + x can be reconstructed to an artifact-free image. In this way we have to determine only an x ∈ V″, where for practical purposes dim(V″) ≪ dim(V′).

Fig. 2
Fig. 2

Ai:m × m − matrix. High symmetry of the missing projection region implies high symmetry of the resulting matrix. In the case of hollow projections in which the x-ray-opaque region is a circular area in the center of the reconstruction circle, the resulting matrix for the consistent-projection completion is block symmetric and block Toeplitz, with k being the number of views and m being the number of missing samples per view. In the case of subsequently missing angular views, the resulting matrix is also block symmetric and block Toeplitz, where k is the number of missing angular views and m is the number of samples per view.

Fig. 3
Fig. 3

Reconstructions of inconsistent and consistently restored simulated projection data. (a) Reconstructed CT image of a simulated homogeneous cylinder where we artificially contaminated the central six samples of each of the 128 views with high-amplitude noise. This picture closely resembles actually measured CT images of patients with implanted metallic objects. (b) Reconstructed image of the consistently completed projection vector where 6 × 128 = 768 missing samples had to be restored. The restoration was carried out using the symmetry-preserving algorithm of Eq. (28) with β = 0.9.

Fig. 4
Fig. 4

Realistic phantom of the lower leg for the simulations using different projection-completion methods. The phantom consists of a simulated tibia (left) and a simulated fibula (right) in muscle tissue, where we used the attentuation coefficients of muscle, compact and spongy bone at 42 keV. The white circle in the tibia indicates the place of the x-ray-opaque region that causes the hollow projections.

Fig. 5
Fig. 5

Examples of projection completions by linear interpolation or consistent completion using a regularization method. The hollow projections of the phantom of Fig. 4 (6 missing samples in each of the 128 views) are restored either inconsistently by linear interpolation or by consistent completion using the symmetry-preserving regularization method of Eq. (28) with β = 0.9 (new algorithm). A comparison of (a)–(c) with the original projection data shows much better agreement of the consistent completion algorithm than the inconsistent linear interpolation. (a) View number 47, (b) view number 50, (c) view number 53.

Fig. 6
Fig. 6

Reconstructed images of the hollow-projection data of Fig. 4 completed with different methods. (a) Completion by zero padding: All unknown projection values are set to zero. (b) Completion by (inconsistent) linear interpolation. (c) Consistent completion using the Tikhonov regularization with the optimal regularization parameter α = 0.0007. (d) Symmetry-preserving regularization with β =0.9. Three views of the completed projections are shown in Figs. 5(a)–5(c).

Fig. 7
Fig. 7

Comparison of local mean values in the reconstructed images of the hollow projections of Fig. 4 completed with different methods. (a) Placement of the eight circular areas for which the mean CT values were calculated and compared with the mean values of the original CT image in Fig. 4. (b) Comparison of local mean values with correct values of perfect reconstruction: ●, inconsistent linear interpolation;♢, Tikhonov regularization with optimal regularization parameter α = 0.0007; □, symmetry-preserving regularization, β = 0.5;∇, symmetry-preserving regularization, β = 0.8;△, symmetry-preserving regularization, β = 0.9.

Fig. 8
Fig. 8

Original and restored projections in the case of missing angular views of actually measured CT data. Using the actually measured CT data (128 samples, 128 views), which can be reconstructed to give the image in Fig. 9(a), we considered the first 8 complete angular views to be missing. Restoration of these missing projection samples was carried out by Tikhonov regularization with the optimal regularization parameter α = 0.00005. (a) Originally measured projection values of view number 0. (b) Originally measured projection values of view number 7. (c) Restored projection values of view number 0. (d) Restored projection values of view number 7.

Fig. 9
Fig. 9

Reconstructed CT images using actually measured, incomplete, and restored projection data. (a) Reconstruction of actually measured projection data of which the first and the eighth views are shown in Figs. 8(a) and 8(b). (b) Reconstruction of the same projection data as in (a), where all samples of the first eight angular views were set to zero. (c) Reconstruction of the completed projection data, where we used Tikhonov regularization with α = 0.00005. The first and the last of the eight restored views are shown in Figs. 8(c) and 8(d).

Equations (31)

Equations on this page are rendered with MathJax. Learn more.

p ( t , ϕ ) = P f = f ( x , y ) δ ( x cos ϕ + y sin ϕ t ) d x d y .
p ̂ ( ω , ϕ ) = f ̂ ̂ ( ω , ϕ ) ,
p ̂ ( ω , ϕ ) = p ( t , ϕ ) exp ( i ω t ) d t
f ̂ ̂ ( ω , ϕ ) = f ( x , y ) exp [ i ω ( x × cos ϕ + y × sin ϕ ) ] d x d y .
f ( x , y ) = B q = 1 2 0 π q ( x × cos ϕ + y × sin ϕ , ϕ ) d ϕ ,
q ( s , ϕ ) = C p = p ( t , ϕ ) × c ( s t ) d t .
P : U V ,
p ( t , ξ ) = 0 for all | t | > ρ , ξ = ( cos ϕ sin ϕ ) , p ( t , ξ ) = p ( t , ξ ) , t k p ( t , ξ ) d t is a polynomial in ξ of degree less than or equal to k for each nonnegative integer k .
R : W U , one - to - one
R P f = f .
P R p = p ,
V = V V ,
Q : V V , Q : V V ,
Q + Q = I ( identity ) .
P : = Q P , P : = Q P ,
P + P = P .
p = ( Q + Q ) p = p + p .
P R ( p + x ) = p + x .
P R ( p + x ) + P R ( p + x ) = p + p + x .
P R ( p + x ) = p + x .
x = P R ( p + x ) p
A x = r
A : = P R I , r : = p P R p .
A x r δ ,
x α = ( A t A + α I ) 1 A t r ( t = transose )
W p = p .
P R W p = p .
B x = s
B : = P R W I , s : = p P R W p .
( W x ) i = γ x i 1 + β x i + γ x i + 1 , γ = ( 1 β ) / 2 ,
W ( ω ) = β + ( 1 β ) cos ( ω ) β [ 0.5 , 1.0 ] .

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