Abstract

We investigate how prior knowledge of the object in the form of linear equality constraints influences the inverse problem of incoherently illuminated object reconstruction by using an elimination method in the context of least squares by regularized singular-value decomposition. Some representative numerical calculations that use noisy images were carried out to illustrate the analysis. When compared with the corresponding unconstrained inversion it appears that the linear constrained inverse does not seem to be any better when viewed in the global sense.

© 1999 Optical Society of America

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References

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  1. R. Mammone, “Image resolution using linear programming,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), p. 128.
  2. C. Matgson, “Error reduction in images using high-quality prior knowledge,” Opt. Eng. 33, 3222–3232 (1994).
  3. A. Lannes, M. Casanova, S. Roques, “Stabilized reconstruction in signal and image processing. II. Iterative reconstruction with and without constraints-interactive implementation,” J. Mod. Opt. 34, 321–330 (1987).
    [Crossref]
  4. M. Roggemann, D. Tyler, “Model-based image reconstruc- tion by means of a constrained least-squares solution,” Appl. Opt. 36, 2360–2369 (1997).
    [Crossref] [PubMed]
  5. R. Fletcher, Practical Methods of Optimization: Constrained Optimization (Wiley, New York, 1987), Vol. 2.
  6. L. Scales, Introduction to Non-Linear Optimization (Springer-Verlag, New York, 1985), Chaps. 5–7.
  7. C. Lawson, R. Hanson, Solving Least-Squares Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974).
  8. L. Crone, “The singular value decomposition of matrices and cheap numerical filtering of systems of linear equations,” J. Franklin Inst. 294, 133–136 (1972).
    [Crossref]
  9. G. Strang, Linear Algebra and its Applications, 2nd ed. (Academic, New York, 1980), p. 321.
  10. W. Press, B. Flannery, S. Teukolsy, W. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1990), p. 52.
  11. G. Golub, C. van Loan, Matrix Computations, 2nd ed. (Johns Hopkins U. Press, Baltimore, Md., 1989).
  12. J. Nash, Compact Numerical Methods for Computers, 2nd ed. (Hilger, Bristol, UK, 1990).
  13. R. Barakat, “Dilute aperture diffraction imagery and object reconstruction,” Opt. Eng. 29, 131–139 (1990).
    [Crossref]
  14. R. Barakat, B. Sandler, “Determination of the wave-front aberration function from measured values of the point-spread function: a two-dimensional phase retrieval problem,” J. Opt. Soc. Am. A 9, 1715–1723 (1992).
    [Crossref]
  15. R. Barakat, G. Newsam, “Remote sensing of the refractive index structure parameter via inversion of Tatarski’s integral equation for both spherical and plane wave situations,” Radio Sci. 19, 1041–1056 (1984).
    [Crossref]

1997 (1)

1994 (1)

C. Matgson, “Error reduction in images using high-quality prior knowledge,” Opt. Eng. 33, 3222–3232 (1994).

1992 (1)

1990 (1)

R. Barakat, “Dilute aperture diffraction imagery and object reconstruction,” Opt. Eng. 29, 131–139 (1990).
[Crossref]

1987 (1)

A. Lannes, M. Casanova, S. Roques, “Stabilized reconstruction in signal and image processing. II. Iterative reconstruction with and without constraints-interactive implementation,” J. Mod. Opt. 34, 321–330 (1987).
[Crossref]

1984 (1)

R. Barakat, G. Newsam, “Remote sensing of the refractive index structure parameter via inversion of Tatarski’s integral equation for both spherical and plane wave situations,” Radio Sci. 19, 1041–1056 (1984).
[Crossref]

1972 (1)

L. Crone, “The singular value decomposition of matrices and cheap numerical filtering of systems of linear equations,” J. Franklin Inst. 294, 133–136 (1972).
[Crossref]

Barakat, R.

R. Barakat, B. Sandler, “Determination of the wave-front aberration function from measured values of the point-spread function: a two-dimensional phase retrieval problem,” J. Opt. Soc. Am. A 9, 1715–1723 (1992).
[Crossref]

R. Barakat, “Dilute aperture diffraction imagery and object reconstruction,” Opt. Eng. 29, 131–139 (1990).
[Crossref]

R. Barakat, G. Newsam, “Remote sensing of the refractive index structure parameter via inversion of Tatarski’s integral equation for both spherical and plane wave situations,” Radio Sci. 19, 1041–1056 (1984).
[Crossref]

Casanova, M.

A. Lannes, M. Casanova, S. Roques, “Stabilized reconstruction in signal and image processing. II. Iterative reconstruction with and without constraints-interactive implementation,” J. Mod. Opt. 34, 321–330 (1987).
[Crossref]

Crone, L.

L. Crone, “The singular value decomposition of matrices and cheap numerical filtering of systems of linear equations,” J. Franklin Inst. 294, 133–136 (1972).
[Crossref]

Flannery, B.

W. Press, B. Flannery, S. Teukolsy, W. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1990), p. 52.

Fletcher, R.

R. Fletcher, Practical Methods of Optimization: Constrained Optimization (Wiley, New York, 1987), Vol. 2.

Golub, G.

G. Golub, C. van Loan, Matrix Computations, 2nd ed. (Johns Hopkins U. Press, Baltimore, Md., 1989).

Hanson, R.

C. Lawson, R. Hanson, Solving Least-Squares Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Lannes, A.

A. Lannes, M. Casanova, S. Roques, “Stabilized reconstruction in signal and image processing. II. Iterative reconstruction with and without constraints-interactive implementation,” J. Mod. Opt. 34, 321–330 (1987).
[Crossref]

Lawson, C.

C. Lawson, R. Hanson, Solving Least-Squares Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Mammone, R.

R. Mammone, “Image resolution using linear programming,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), p. 128.

Matgson, C.

C. Matgson, “Error reduction in images using high-quality prior knowledge,” Opt. Eng. 33, 3222–3232 (1994).

Nash, J.

J. Nash, Compact Numerical Methods for Computers, 2nd ed. (Hilger, Bristol, UK, 1990).

Newsam, G.

R. Barakat, G. Newsam, “Remote sensing of the refractive index structure parameter via inversion of Tatarski’s integral equation for both spherical and plane wave situations,” Radio Sci. 19, 1041–1056 (1984).
[Crossref]

Press, W.

W. Press, B. Flannery, S. Teukolsy, W. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1990), p. 52.

Roggemann, M.

Roques, S.

A. Lannes, M. Casanova, S. Roques, “Stabilized reconstruction in signal and image processing. II. Iterative reconstruction with and without constraints-interactive implementation,” J. Mod. Opt. 34, 321–330 (1987).
[Crossref]

Sandler, B.

Scales, L.

L. Scales, Introduction to Non-Linear Optimization (Springer-Verlag, New York, 1985), Chaps. 5–7.

Strang, G.

G. Strang, Linear Algebra and its Applications, 2nd ed. (Academic, New York, 1980), p. 321.

Teukolsy, S.

W. Press, B. Flannery, S. Teukolsy, W. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1990), p. 52.

Tyler, D.

van Loan, C.

G. Golub, C. van Loan, Matrix Computations, 2nd ed. (Johns Hopkins U. Press, Baltimore, Md., 1989).

Vetterling, W.

W. Press, B. Flannery, S. Teukolsy, W. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1990), p. 52.

Appl. Opt. (1)

J. Franklin Inst. (1)

L. Crone, “The singular value decomposition of matrices and cheap numerical filtering of systems of linear equations,” J. Franklin Inst. 294, 133–136 (1972).
[Crossref]

J. Mod. Opt. (1)

A. Lannes, M. Casanova, S. Roques, “Stabilized reconstruction in signal and image processing. II. Iterative reconstruction with and without constraints-interactive implementation,” J. Mod. Opt. 34, 321–330 (1987).
[Crossref]

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

Opt. Eng. (2)

R. Barakat, “Dilute aperture diffraction imagery and object reconstruction,” Opt. Eng. 29, 131–139 (1990).
[Crossref]

C. Matgson, “Error reduction in images using high-quality prior knowledge,” Opt. Eng. 33, 3222–3232 (1994).

Radio Sci. (1)

R. Barakat, G. Newsam, “Remote sensing of the refractive index structure parameter via inversion of Tatarski’s integral equation for both spherical and plane wave situations,” Radio Sci. 19, 1041–1056 (1984).
[Crossref]

Other (8)

R. Mammone, “Image resolution using linear programming,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), p. 128.

G. Strang, Linear Algebra and its Applications, 2nd ed. (Academic, New York, 1980), p. 321.

W. Press, B. Flannery, S. Teukolsy, W. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1990), p. 52.

G. Golub, C. van Loan, Matrix Computations, 2nd ed. (Johns Hopkins U. Press, Baltimore, Md., 1989).

J. Nash, Compact Numerical Methods for Computers, 2nd ed. (Hilger, Bristol, UK, 1990).

R. Fletcher, Practical Methods of Optimization: Constrained Optimization (Wiley, New York, 1987), Vol. 2.

L. Scales, Introduction to Non-Linear Optimization (Springer-Verlag, New York, 1985), Chaps. 5–7.

C. Lawson, R. Hanson, Solving Least-Squares Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974).

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Figures (5)

Fig. 1
Fig. 1

Behavior of the singular values of T and T as a function of n. T has 103 singular values and T has 57 singular values.

Fig. 2
Fig. 2

Sample realization of unconstrained reconstruction in the presence of 5% measurement noise with q = 0.01.

Fig. 3
Fig. 3

Sample realization of unconstrained reconstruction in the presence of 5% measurement noise with q = 0.05.

Fig. 4
Fig. 4

Sample realization of unconstrained reconstruction in the presence of 5% measurement noise with q = 0.01. Compare with Fig. 2.

Fig. 5
Fig. 5

Sample realization of unconstrained reconstruction in the presence of 5% measurement noise with q = 0.05. Compare with Fig. 3.

Equations (31)

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

Iv=ab tv-vovdv,
tv=sin vv2.
Iv=n=1N Hntv-vnovn,
Tmnon=Im,
Im=Ivm,on=ovn,Tmn=Hntvm-vn.
To=I,
on=cn,
TO=,
TmnTmn,  nn,
On=on,  nn,
mIm-n=1L Tmncn.
=Φ+Φ= minimum,
ΦTO-.
T=UΣV+
Σ=D0,
D=σ1σ200σN-L.
σ1σ2σ30.
O=T,
TVΣ-1U+,
Σ-1=D-1000,
D-1= σ1-1σ2-100σN-L-1.
O=l=1N-Lvlσl-1ul+,
O=l=1N-Lvlfσl-1ul+,
fσlσlσl2+q,  0<q  1.
O=lvlfσl-1ul+c+lvlfσl-1ul+d,
=c+d,
Iv=Icv1+δuv,
fuv=12,  |uv|1=0,  |uv|>1.
TO=,
=Φ+WΦ= minimum,
Φ=TO-,

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