Abstract

It is shown that the absorption field inside an inhomogeneous, rotationally symmetric medium with a spatially variable refractive index can be reconstructed by means of a tomographic technique. The classic Abel transform is extended to non-Euclidean optical media. The optical behavior of such a medium is described and, provided that the product of the refractive index with the radial distance is a monotonic function, an exact inverse formula is found. Both a numerical and an analytical test on a phantom function is carried out to prove the exactness of this formula. In contrast, when the assumption of a monotonic function is not true, it is shown that the reconstruction problem becomes subdeterminate because of the presence of annular regions, known as blind areas, inside of which no curved path reaches an extremum. The spatial localization and the size of these regions are related to the extrema of the index of refraction times the radial distance.

© 2000 Optical Society of America

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References

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  1. R. N. Bracewell, A. C. Riddle, “Inversion of fan beam scans in radio astronomy,” Astrophys. J. 150, 427–434 (1967).
    [CrossRef]
  2. C. J. Dasch, “One-dimensional tomography: a comparison of Abel, onion-peeling, and filtered backprojection methods,” Appl. Opt. 31, 1146–1152 (1992).
    [CrossRef] [PubMed]
  3. G. N. Ramachandran, A. V. Lakshminarayanan, “Three dimensional reconstruction from radiographs and electron micrographs: application of convolutions instead of Fourier transforms,” Proc. Natl. Acad. Sci. USA 68, 2236–2240 (1971).
    [CrossRef]
  4. H. H. Barret, W. Swindel, “Analog reconstruction methods for transaxial tomography,” Proc. IEEE 65, 89–107 (1977).
    [CrossRef]
  5. A. M. Cormack, “Representation of a function by its line integrals, with some radiological applications,” J. Appl. Phys. 34, 2722–2727 (1963).
    [CrossRef]
  6. P. Ben-Abdallah and V. Le Dez, “Sur la mesure locale du champ d’absorption d’un milieu axisymétrique à indice de réfraction variable,” C. R. Acad. Sci. IIb: Mec. Phys. Chim. Astron. 327, 471–474 (1999).
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1964).
  8. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
  9. R. Courant, Differential and Integral Calculus (Interscience, New York, 1968), Vol. 1.
  10. F. G. Tricomi, Integral Equations (Dover, New York, 1985), pp. 39–40.
  11. A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).
  12. A. Bakushinsky, A. Goncharsky, Ill-Posed Problems: Theory and Applications (Kluwer Academic, Dordrecht, The Netherlands, 1994).
    [CrossRef]

1999 (1)

P. Ben-Abdallah and V. Le Dez, “Sur la mesure locale du champ d’absorption d’un milieu axisymétrique à indice de réfraction variable,” C. R. Acad. Sci. IIb: Mec. Phys. Chim. Astron. 327, 471–474 (1999).

1992 (1)

1977 (1)

H. H. Barret, W. Swindel, “Analog reconstruction methods for transaxial tomography,” Proc. IEEE 65, 89–107 (1977).
[CrossRef]

1971 (1)

G. N. Ramachandran, A. V. Lakshminarayanan, “Three dimensional reconstruction from radiographs and electron micrographs: application of convolutions instead of Fourier transforms,” Proc. Natl. Acad. Sci. USA 68, 2236–2240 (1971).
[CrossRef]

1967 (1)

R. N. Bracewell, A. C. Riddle, “Inversion of fan beam scans in radio astronomy,” Astrophys. J. 150, 427–434 (1967).
[CrossRef]

1963 (1)

A. M. Cormack, “Representation of a function by its line integrals, with some radiological applications,” J. Appl. Phys. 34, 2722–2727 (1963).
[CrossRef]

Arsenin, V. Y.

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

Bakushinsky, A.

A. Bakushinsky, A. Goncharsky, Ill-Posed Problems: Theory and Applications (Kluwer Academic, Dordrecht, The Netherlands, 1994).
[CrossRef]

Barret, H. H.

H. H. Barret, W. Swindel, “Analog reconstruction methods for transaxial tomography,” Proc. IEEE 65, 89–107 (1977).
[CrossRef]

Ben-Abdallah and V. Le Dez, P.

P. Ben-Abdallah and V. Le Dez, “Sur la mesure locale du champ d’absorption d’un milieu axisymétrique à indice de réfraction variable,” C. R. Acad. Sci. IIb: Mec. Phys. Chim. Astron. 327, 471–474 (1999).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1964).

Bracewell, R. N.

R. N. Bracewell, A. C. Riddle, “Inversion of fan beam scans in radio astronomy,” Astrophys. J. 150, 427–434 (1967).
[CrossRef]

Cormack, A. M.

A. M. Cormack, “Representation of a function by its line integrals, with some radiological applications,” J. Appl. Phys. 34, 2722–2727 (1963).
[CrossRef]

Courant, R.

R. Courant, Differential and Integral Calculus (Interscience, New York, 1968), Vol. 1.

Dasch, C. J.

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Goncharsky, A.

A. Bakushinsky, A. Goncharsky, Ill-Posed Problems: Theory and Applications (Kluwer Academic, Dordrecht, The Netherlands, 1994).
[CrossRef]

Lakshminarayanan, A. V.

G. N. Ramachandran, A. V. Lakshminarayanan, “Three dimensional reconstruction from radiographs and electron micrographs: application of convolutions instead of Fourier transforms,” Proc. Natl. Acad. Sci. USA 68, 2236–2240 (1971).
[CrossRef]

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Ramachandran, G. N.

G. N. Ramachandran, A. V. Lakshminarayanan, “Three dimensional reconstruction from radiographs and electron micrographs: application of convolutions instead of Fourier transforms,” Proc. Natl. Acad. Sci. USA 68, 2236–2240 (1971).
[CrossRef]

Riddle, A. C.

R. N. Bracewell, A. C. Riddle, “Inversion of fan beam scans in radio astronomy,” Astrophys. J. 150, 427–434 (1967).
[CrossRef]

Swindel, W.

H. H. Barret, W. Swindel, “Analog reconstruction methods for transaxial tomography,” Proc. IEEE 65, 89–107 (1977).
[CrossRef]

Tikhonov, A. N.

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

Tricomi, F. G.

F. G. Tricomi, Integral Equations (Dover, New York, 1985), pp. 39–40.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1964).

Appl. Opt. (1)

Astrophys. J. (1)

R. N. Bracewell, A. C. Riddle, “Inversion of fan beam scans in radio astronomy,” Astrophys. J. 150, 427–434 (1967).
[CrossRef]

C. R. Acad. Sci. IIb: Mec. Phys. Chim. Astron. (1)

P. Ben-Abdallah and V. Le Dez, “Sur la mesure locale du champ d’absorption d’un milieu axisymétrique à indice de réfraction variable,” C. R. Acad. Sci. IIb: Mec. Phys. Chim. Astron. 327, 471–474 (1999).

J. Appl. Phys. (1)

A. M. Cormack, “Representation of a function by its line integrals, with some radiological applications,” J. Appl. Phys. 34, 2722–2727 (1963).
[CrossRef]

Proc. IEEE (1)

H. H. Barret, W. Swindel, “Analog reconstruction methods for transaxial tomography,” Proc. IEEE 65, 89–107 (1977).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

G. N. Ramachandran, A. V. Lakshminarayanan, “Three dimensional reconstruction from radiographs and electron micrographs: application of convolutions instead of Fourier transforms,” Proc. Natl. Acad. Sci. USA 68, 2236–2240 (1971).
[CrossRef]

Other (6)

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1964).

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

R. Courant, Differential and Integral Calculus (Interscience, New York, 1968), Vol. 1.

F. G. Tricomi, Integral Equations (Dover, New York, 1985), pp. 39–40.

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

A. Bakushinsky, A. Goncharsky, Ill-Posed Problems: Theory and Applications (Kluwer Academic, Dordrecht, The Netherlands, 1994).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Geometry of the generalized Abel transform.

Fig. 2
Fig. 2

According to Bouguer’s law, the extremum location ρ of the light path is generally a multivalued function of incoming angle φ0.

Fig. 3
Fig. 3

Different rn(r) profiles inside an inhomogeneous axisymmetric medium: (a) homogeneous case; (b) monotonic case, ρ is a single-value function of φ0; (c) single extremum case; (d) general case, rn(r) possesses a finite set of extrema.

Fig. 4
Fig. 4

Optical behavior of an inhomogeneous medium with a monotonic n(r)r profile.

Fig. 5
Fig. 5

Transmission optical behavior of an inhomogeneous medium having an rn(r) profile with a unique extremum. The extrema are confined inside the disk of radius ρmax.

Fig. 6
Fig. 6

General transmission optical behavior of an inhomogeneous medium. The extrema are confined inside areas (1) and (2).

Fig. 7
Fig. 7

Comparison of Abel transform parameters for both homogeneous and inhomogeneous media.

Fig. 8
Fig. 8

Reconstruction of an absorption field with 20 noise projections inside an inhomogeneous medium.

Tables (1)

Tables Icon

Table 1 Summary of Abel Transforms and Their Inversions

Equations (31)

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δL=δsAsB nxds0=0,
L=sAsB nxds0=0Λ ng0ν, ν1/2ds,
L=0Λgν, ν1/2ds.
ds2=n2rdr2+r2dθ2.
Dν=0,  ν=dx/ds.
nrr sin φ=β
sin φ=rr2+drdθ2-1/2.
nrr2r2+drdθ2-1/2=β,
drdθ2=rβ2n2rr2-β2.
nρρ=β=nRR sin φ0,
θ-π=±β Rrdssn2ss2-β21/2.
θ-π=β Rrdssn2ss2-β21/2
θ=θ1-β ρ1Rdrrn2rr2-β21/2,
θ1=π+β Rρ1drrn2rr2-β21/2.
tλ=lnI0I=0 κλds,
dψ=nλrdθ.
ds2=nλ2rdr2+r2dψ2.
ζ=r2dψds,
ds=nλrrdrr2-ζ21/2.
1=nλ2rdrds2+ζr2.
tλρκλ=0+ κλds=2 ρRnλrκλrrdrr2-ρ21/2.
κλr=-1πnλrrRdtλ/dρρ2-r21/2dρ.
κλr=anλr,r10,elsewhere,
tλρκλ=2a ρ1rr2-ρ21/2dr=2a1-ρ21/2.
κλr=2aπnλrr1ρρ2-r21/21-ρ21/2dρ,
ρ2=1-r2z+r2
κλr=aπnλr01dz1-z1/2z1/2.
υ=1-z1/2,
κλr=2aπnλr01dυ1-υ21/2=anλr,
ti=j=iN-1 2njκjrjrj+1rdrr2-ρi21/2=j=iN-1 Aijκj.
Fκ=Aκ-tδ2+ακ2,

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