Abstract

We describe an efficient algorithm based on the Hilbert transform for reconstructing cross-sectional or three-dimensional images from the input images acquired by an interference microscope. First the design of this filter is presented, and cross-sectional images of an integrated circuit constructed with this algorithm are demonstrated. It is shown that this Hilbert transform algorithm can be easily implemented with a low-cost frame grabber so that the computation time required for image reconstruction is drastically reduced.

© 1992 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. S. C. Chim, P. A. Beck, G. S. Kino, “A novel thin film interferometer,” Rev. Sci. Instrum. 61, 980–983 (1990).
    [CrossRef]
  2. S. S. C. Chim, G. S. Kino, “Correlation microscope,” Opt. Lett. 15, 579–581 (1990).
    [CrossRef] [PubMed]
  3. G. S. Kino, S. S. C. Chim, “The Mirau correlation microscope,” Appl. Opt. 29, 3775–3783 (1990).
    [CrossRef] [PubMed]
  4. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986).
  5. Image Processing Handbook, (Data Translation, Inc., Marlboro, Mass., 19902).
  6. L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

1990 (3)

Beck, P. A.

S. S. C. Chim, P. A. Beck, G. S. Kino, “A novel thin film interferometer,” Rev. Sci. Instrum. 61, 980–983 (1990).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986).

Chim, S. S. C.

Gold, B.

L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Kino, G. S.

Rabiner, L. R.

L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Appl. Opt. (1)

Opt. Lett. (1)

Rev. Sci. Instrum. (1)

S. S. C. Chim, P. A. Beck, G. S. Kino, “A novel thin film interferometer,” Rev. Sci. Instrum. 61, 980–983 (1990).
[CrossRef]

Other (3)

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986).

Image Processing Handbook, (Data Translation, Inc., Marlboro, Mass., 19902).

L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Schematic of the Mirau correlation microscope.

Fig. 2
Fig. 2

Comparison between the frequency responses—(top) magnitude and (bottom) phase—of an ideal Hilbert filter (solid line) and those of finite length (solid and dashed curves).

Fig. 3
Fig. 3

Comparison between (a) the simulated and (b) the experimental amplitude responses of the unprocessed signal and those obtained by the Hilbert and Fourier transform algorithms.

Fig. 4a
Fig. 4a

Intensity images of a multilayered (complementary metal-oxide semiconductor) integrated circuit at different foci [left, (a) and (b)]. The corresponding phase images of the same circuit [right, (a) and (b)].

Equations (12)

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

I x y ( n Δ z ) = A x y 2 + B 2 + 2 A x y B γ x y ( n Δ z ) .
γ x y ( n Δ z ) ~ g x y ( n Δ z ) cos [ Φ x y ( n Δ z ) ] .
V x y ( n Δ z ) = 2 A x y B g x y ( n Δ z ) exp [ j Φ x y ( n Δ z ) ] .
V x y ( n ) = i x y ( n ) + j i x y * ( n ) ,
i x y ( n ) = I x y ( n ) - ( A x y 2 + B 2 ) = 2 A x y B g x y ( n ) cos [ Φ x y ( n ) ] ,
i x y * ( n ) = 2 A x y B g x y ( n ) sin [ Φ x y ( n ) ] ,
H [ exp ( j w ) ] = { - j 0 ω < π j π ω < 2 π .
h ( n ) = { 2 / π n n odd 0 otherwise .
h norm ( n ) = { 1 n if n M and n is odd 0 otherwise
i x y * ( n ) = 2 π m = - M m i x y ( m - n ) h norm ( m ) .
i x y * ( n ) = 2 π m = 1 m odd M i x y ( m - n ) - i x y ( m + n ) m .
V x y ( n ) = { [ i x y ( n ) ] 2 + [ i x y * ( n ) ] 2 } 1 / 2 , arg [ V x y ( n ) ] = tan - 1 [ i x y * ( n ) i x y ( n ) ] .

Metrics