Abstract

The Hough transform (HT) is an efficient shape detector that maps straight lines into a two-parameter feature space. Recently it has been pointed out that the forward Radon transform (FRT), well known from the theory of computed tomography, and the HT are equivalent for binary images. In this paper, analog coherent optical implementation of the FRT is discussed. The FRT will not only be of use in implementing the HT shape descriptors but also act as a coherent optical preprocessor for the implementation of multidimensional convolution, correlation, and spectral analysis using 1-D acoustooptical signal processing devices. Several different coherent optical FRT architectures are presented. Experimental results using conventional coherent Fourier transform configuration are given. The relationship between the coherent optical implementation of the FRT and the inverse Radon transform, an important tool in computed tomography, is also detailed.

© 1983 Optical Society of America

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References

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  1. P. V. C. Hough, “Methods and Means for Recognizing Complex Patterns,” U.S. Patent3,069,654 (1962).
  2. W. K. Pratt, Digital Image Processing (Wiley-Interscience, New York, 1978).
  3. S. D. Shapiro, A. Iannino, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-1, 310 (1979).
    [CrossRef]
  4. R. O. Duda, P. E. Hart, Commun. ACM 15, 11 (1972).
    [CrossRef]
  5. S. R. Deans, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-2, 185(1981).
    [CrossRef]
  6. J. Radon, Ber. Saechs. Akad. Wiss. Leipzig 69, 262 (1917).
  7. K. R. Sloan, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-4, 87(1982).
    [CrossRef]
  8. R. Gordon, R. Bender, G. T. Herman, J. Theor. Biol. 29, 471 (1970).
    [CrossRef] [PubMed]
  9. W. M. Boerner, C. M. Ho, B. Y. Foo, IEEE Trans. Antennas Propag. AP-29, 336 (1981).
    [CrossRef]
  10. R. C. Chase, F. H. Seguin, M. Gerassimenko, R. Petrasso, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 265 (1980).
  11. R. M. Mersereau, A. V. Oppenheim, Proc. IEEE 62, 1319 (1974).
    [CrossRef]
  12. Technical Digest, Topical Meeting on Imaging Processing for 2-D and 3-D Reconstructions from Projections (Optical Society of America, Washington, D.C., 1975).
  13. Z. H. Cho, Ed., Special issue on Physical and Computational Aspects of 3-D Image Reconstruction, IEEE Trans. Nucl. SciNS-21 (June1971).
  14. B. K. Gilbert, A. Chu, D. E. Atkins, E. E. Swartzlander, E. L. Ritman, Comput. Biomed. Res. 12, 17 (1979).
    [CrossRef] [PubMed]
  15. H. H. Barrett, W. Swindell, Proc. IEEE 65, 89 (1977).
    [CrossRef]
  16. E. W. Hansen, J. W. Goodman, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 222 (1980).
  17. J. Hofer, Opt. Commun. 29, 22 (1979).
    [CrossRef]
  18. P. Edholm, L. G. Hellstrom, B. J. Jacobson, Phys. Med Biol 23, 90(1978).
    [CrossRef] [PubMed]
  19. H. Platzer, H. Glunder, “Tomogram—Reconstruction by Holographic Methods,” in Holography in Medicine and Biology, G. V. Bally, Ed. (Springer, New York, 1979), pp. 117–123.
  20. A. F. Gmitro, J. E. Greivenkamp, W. Swindell, H. H. Barrett, M. Y. Chiu, S. K. Gordon, Opt. Eng. 19, 260 (1980).
    [CrossRef]
  21. H. H. Barrett, Opt. Lett. 7, 248 (1982).
    [CrossRef] [PubMed]
  22. I. M. Gelfand, M. I. Graev, N. Ya. Vilkin, Generalized Functions (McGraw-Hill, New York, 1966), Vol. 5.
  23. S. D. Shapiro, Comput. Graph. Image Process. 8, 219 (1978).
    [CrossRef]
  24. J. Sklansky, IEEE Trans. Comput. COM-27, 923 (1978).
    [CrossRef]
  25. J. W. Goodman, “Linear Space-Variant Optical Processing,” in Optical Data Processing, Fundamentals, S. Lee, Ed. (Springer, New York, 1980).
  26. R. J. Marks, J. F. Walkup, M. O. Hagler, T. F. Krile, Appl Opt. 16, 739(1977).
    [CrossRef]
  27. D. Casasent, D. Psaltis, Appl. Opt. 15, 1795 (1976).
    [CrossRef] [PubMed]
  28. D. Casasent, D. Psaltis, “Deformation Invariant, Space-Variant Optica) Pattern Recognition,” in Progress in Optics, Vol. 18, E. Wolf, Ed., (North-Holland, Amsterdam, 1978).
    [CrossRef]
  29. G. Tyras, Radiation and Propagation of Electromagnetic Waves (Academic, New York, 1969), Chap. 8.
  30. G. Eichmann, R. Stirbl, R. Mammone, Proc. Soc. Photo-Opt. Instrum. Eng. 209, 24 (1979).

1982 (2)

K. R. Sloan, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-4, 87(1982).
[CrossRef]

H. H. Barrett, Opt. Lett. 7, 248 (1982).
[CrossRef] [PubMed]

1981 (2)

W. M. Boerner, C. M. Ho, B. Y. Foo, IEEE Trans. Antennas Propag. AP-29, 336 (1981).
[CrossRef]

S. R. Deans, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-2, 185(1981).
[CrossRef]

1980 (3)

R. C. Chase, F. H. Seguin, M. Gerassimenko, R. Petrasso, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 265 (1980).

E. W. Hansen, J. W. Goodman, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 222 (1980).

A. F. Gmitro, J. E. Greivenkamp, W. Swindell, H. H. Barrett, M. Y. Chiu, S. K. Gordon, Opt. Eng. 19, 260 (1980).
[CrossRef]

1979 (4)

G. Eichmann, R. Stirbl, R. Mammone, Proc. Soc. Photo-Opt. Instrum. Eng. 209, 24 (1979).

J. Hofer, Opt. Commun. 29, 22 (1979).
[CrossRef]

B. K. Gilbert, A. Chu, D. E. Atkins, E. E. Swartzlander, E. L. Ritman, Comput. Biomed. Res. 12, 17 (1979).
[CrossRef] [PubMed]

S. D. Shapiro, A. Iannino, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-1, 310 (1979).
[CrossRef]

1978 (3)

P. Edholm, L. G. Hellstrom, B. J. Jacobson, Phys. Med Biol 23, 90(1978).
[CrossRef] [PubMed]

S. D. Shapiro, Comput. Graph. Image Process. 8, 219 (1978).
[CrossRef]

J. Sklansky, IEEE Trans. Comput. COM-27, 923 (1978).
[CrossRef]

1977 (2)

R. J. Marks, J. F. Walkup, M. O. Hagler, T. F. Krile, Appl Opt. 16, 739(1977).
[CrossRef]

H. H. Barrett, W. Swindell, Proc. IEEE 65, 89 (1977).
[CrossRef]

1976 (1)

1974 (1)

R. M. Mersereau, A. V. Oppenheim, Proc. IEEE 62, 1319 (1974).
[CrossRef]

1972 (1)

R. O. Duda, P. E. Hart, Commun. ACM 15, 11 (1972).
[CrossRef]

1970 (1)

R. Gordon, R. Bender, G. T. Herman, J. Theor. Biol. 29, 471 (1970).
[CrossRef] [PubMed]

1917 (1)

J. Radon, Ber. Saechs. Akad. Wiss. Leipzig 69, 262 (1917).

Atkins, D. E.

B. K. Gilbert, A. Chu, D. E. Atkins, E. E. Swartzlander, E. L. Ritman, Comput. Biomed. Res. 12, 17 (1979).
[CrossRef] [PubMed]

Barrett, H. H.

H. H. Barrett, Opt. Lett. 7, 248 (1982).
[CrossRef] [PubMed]

A. F. Gmitro, J. E. Greivenkamp, W. Swindell, H. H. Barrett, M. Y. Chiu, S. K. Gordon, Opt. Eng. 19, 260 (1980).
[CrossRef]

H. H. Barrett, W. Swindell, Proc. IEEE 65, 89 (1977).
[CrossRef]

Bender, R.

R. Gordon, R. Bender, G. T. Herman, J. Theor. Biol. 29, 471 (1970).
[CrossRef] [PubMed]

Boerner, W. M.

W. M. Boerner, C. M. Ho, B. Y. Foo, IEEE Trans. Antennas Propag. AP-29, 336 (1981).
[CrossRef]

Casasent, D.

D. Casasent, D. Psaltis, Appl. Opt. 15, 1795 (1976).
[CrossRef] [PubMed]

D. Casasent, D. Psaltis, “Deformation Invariant, Space-Variant Optica) Pattern Recognition,” in Progress in Optics, Vol. 18, E. Wolf, Ed., (North-Holland, Amsterdam, 1978).
[CrossRef]

Chase, R. C.

R. C. Chase, F. H. Seguin, M. Gerassimenko, R. Petrasso, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 265 (1980).

Chiu, M. Y.

A. F. Gmitro, J. E. Greivenkamp, W. Swindell, H. H. Barrett, M. Y. Chiu, S. K. Gordon, Opt. Eng. 19, 260 (1980).
[CrossRef]

Chu, A.

B. K. Gilbert, A. Chu, D. E. Atkins, E. E. Swartzlander, E. L. Ritman, Comput. Biomed. Res. 12, 17 (1979).
[CrossRef] [PubMed]

Deans, S. R.

S. R. Deans, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-2, 185(1981).
[CrossRef]

Duda, R. O.

R. O. Duda, P. E. Hart, Commun. ACM 15, 11 (1972).
[CrossRef]

Edholm, P.

P. Edholm, L. G. Hellstrom, B. J. Jacobson, Phys. Med Biol 23, 90(1978).
[CrossRef] [PubMed]

Eichmann, G.

G. Eichmann, R. Stirbl, R. Mammone, Proc. Soc. Photo-Opt. Instrum. Eng. 209, 24 (1979).

Foo, B. Y.

W. M. Boerner, C. M. Ho, B. Y. Foo, IEEE Trans. Antennas Propag. AP-29, 336 (1981).
[CrossRef]

Gelfand, I. M.

I. M. Gelfand, M. I. Graev, N. Ya. Vilkin, Generalized Functions (McGraw-Hill, New York, 1966), Vol. 5.

Gerassimenko, M.

R. C. Chase, F. H. Seguin, M. Gerassimenko, R. Petrasso, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 265 (1980).

Gilbert, B. K.

B. K. Gilbert, A. Chu, D. E. Atkins, E. E. Swartzlander, E. L. Ritman, Comput. Biomed. Res. 12, 17 (1979).
[CrossRef] [PubMed]

Glunder, H.

H. Platzer, H. Glunder, “Tomogram—Reconstruction by Holographic Methods,” in Holography in Medicine and Biology, G. V. Bally, Ed. (Springer, New York, 1979), pp. 117–123.

Gmitro, A. F.

A. F. Gmitro, J. E. Greivenkamp, W. Swindell, H. H. Barrett, M. Y. Chiu, S. K. Gordon, Opt. Eng. 19, 260 (1980).
[CrossRef]

Goodman, J. W.

E. W. Hansen, J. W. Goodman, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 222 (1980).

J. W. Goodman, “Linear Space-Variant Optical Processing,” in Optical Data Processing, Fundamentals, S. Lee, Ed. (Springer, New York, 1980).

Gordon, R.

R. Gordon, R. Bender, G. T. Herman, J. Theor. Biol. 29, 471 (1970).
[CrossRef] [PubMed]

Gordon, S. K.

A. F. Gmitro, J. E. Greivenkamp, W. Swindell, H. H. Barrett, M. Y. Chiu, S. K. Gordon, Opt. Eng. 19, 260 (1980).
[CrossRef]

Graev, M. I.

I. M. Gelfand, M. I. Graev, N. Ya. Vilkin, Generalized Functions (McGraw-Hill, New York, 1966), Vol. 5.

Greivenkamp, J. E.

A. F. Gmitro, J. E. Greivenkamp, W. Swindell, H. H. Barrett, M. Y. Chiu, S. K. Gordon, Opt. Eng. 19, 260 (1980).
[CrossRef]

Hagler, M. O.

R. J. Marks, J. F. Walkup, M. O. Hagler, T. F. Krile, Appl Opt. 16, 739(1977).
[CrossRef]

Hansen, E. W.

E. W. Hansen, J. W. Goodman, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 222 (1980).

Hart, P. E.

R. O. Duda, P. E. Hart, Commun. ACM 15, 11 (1972).
[CrossRef]

Hellstrom, L. G.

P. Edholm, L. G. Hellstrom, B. J. Jacobson, Phys. Med Biol 23, 90(1978).
[CrossRef] [PubMed]

Herman, G. T.

R. Gordon, R. Bender, G. T. Herman, J. Theor. Biol. 29, 471 (1970).
[CrossRef] [PubMed]

Ho, C. M.

W. M. Boerner, C. M. Ho, B. Y. Foo, IEEE Trans. Antennas Propag. AP-29, 336 (1981).
[CrossRef]

Hofer, J.

J. Hofer, Opt. Commun. 29, 22 (1979).
[CrossRef]

Hough, P. V. C.

P. V. C. Hough, “Methods and Means for Recognizing Complex Patterns,” U.S. Patent3,069,654 (1962).

Iannino, A.

S. D. Shapiro, A. Iannino, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-1, 310 (1979).
[CrossRef]

Jacobson, B. J.

P. Edholm, L. G. Hellstrom, B. J. Jacobson, Phys. Med Biol 23, 90(1978).
[CrossRef] [PubMed]

Krile, T. F.

R. J. Marks, J. F. Walkup, M. O. Hagler, T. F. Krile, Appl Opt. 16, 739(1977).
[CrossRef]

Mammone, R.

G. Eichmann, R. Stirbl, R. Mammone, Proc. Soc. Photo-Opt. Instrum. Eng. 209, 24 (1979).

Marks, R. J.

R. J. Marks, J. F. Walkup, M. O. Hagler, T. F. Krile, Appl Opt. 16, 739(1977).
[CrossRef]

Mersereau, R. M.

R. M. Mersereau, A. V. Oppenheim, Proc. IEEE 62, 1319 (1974).
[CrossRef]

Oppenheim, A. V.

R. M. Mersereau, A. V. Oppenheim, Proc. IEEE 62, 1319 (1974).
[CrossRef]

Petrasso, R.

R. C. Chase, F. H. Seguin, M. Gerassimenko, R. Petrasso, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 265 (1980).

Platzer, H.

H. Platzer, H. Glunder, “Tomogram—Reconstruction by Holographic Methods,” in Holography in Medicine and Biology, G. V. Bally, Ed. (Springer, New York, 1979), pp. 117–123.

Pratt, W. K.

W. K. Pratt, Digital Image Processing (Wiley-Interscience, New York, 1978).

Psaltis, D.

D. Casasent, D. Psaltis, Appl. Opt. 15, 1795 (1976).
[CrossRef] [PubMed]

D. Casasent, D. Psaltis, “Deformation Invariant, Space-Variant Optica) Pattern Recognition,” in Progress in Optics, Vol. 18, E. Wolf, Ed., (North-Holland, Amsterdam, 1978).
[CrossRef]

Radon, J.

J. Radon, Ber. Saechs. Akad. Wiss. Leipzig 69, 262 (1917).

Ritman, E. L.

B. K. Gilbert, A. Chu, D. E. Atkins, E. E. Swartzlander, E. L. Ritman, Comput. Biomed. Res. 12, 17 (1979).
[CrossRef] [PubMed]

Seguin, F. H.

R. C. Chase, F. H. Seguin, M. Gerassimenko, R. Petrasso, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 265 (1980).

Shapiro, S. D.

S. D. Shapiro, A. Iannino, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-1, 310 (1979).
[CrossRef]

S. D. Shapiro, Comput. Graph. Image Process. 8, 219 (1978).
[CrossRef]

Sklansky, J.

J. Sklansky, IEEE Trans. Comput. COM-27, 923 (1978).
[CrossRef]

Sloan, K. R.

K. R. Sloan, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-4, 87(1982).
[CrossRef]

Stirbl, R.

G. Eichmann, R. Stirbl, R. Mammone, Proc. Soc. Photo-Opt. Instrum. Eng. 209, 24 (1979).

Swartzlander, E. E.

B. K. Gilbert, A. Chu, D. E. Atkins, E. E. Swartzlander, E. L. Ritman, Comput. Biomed. Res. 12, 17 (1979).
[CrossRef] [PubMed]

Swindell, W.

A. F. Gmitro, J. E. Greivenkamp, W. Swindell, H. H. Barrett, M. Y. Chiu, S. K. Gordon, Opt. Eng. 19, 260 (1980).
[CrossRef]

H. H. Barrett, W. Swindell, Proc. IEEE 65, 89 (1977).
[CrossRef]

Tyras, G.

G. Tyras, Radiation and Propagation of Electromagnetic Waves (Academic, New York, 1969), Chap. 8.

Vilkin, N. Ya.

I. M. Gelfand, M. I. Graev, N. Ya. Vilkin, Generalized Functions (McGraw-Hill, New York, 1966), Vol. 5.

Walkup, J. F.

R. J. Marks, J. F. Walkup, M. O. Hagler, T. F. Krile, Appl Opt. 16, 739(1977).
[CrossRef]

Appl Opt. (1)

R. J. Marks, J. F. Walkup, M. O. Hagler, T. F. Krile, Appl Opt. 16, 739(1977).
[CrossRef]

Appl. Opt. (1)

Ber. Saechs. Akad. Wiss. Leipzig (1)

J. Radon, Ber. Saechs. Akad. Wiss. Leipzig 69, 262 (1917).

Commun. ACM (1)

R. O. Duda, P. E. Hart, Commun. ACM 15, 11 (1972).
[CrossRef]

Comput. Biomed. Res. (1)

B. K. Gilbert, A. Chu, D. E. Atkins, E. E. Swartzlander, E. L. Ritman, Comput. Biomed. Res. 12, 17 (1979).
[CrossRef] [PubMed]

Comput. Graph. Image Process. (1)

S. D. Shapiro, Comput. Graph. Image Process. 8, 219 (1978).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

W. M. Boerner, C. M. Ho, B. Y. Foo, IEEE Trans. Antennas Propag. AP-29, 336 (1981).
[CrossRef]

IEEE Trans. Comput. (1)

J. Sklansky, IEEE Trans. Comput. COM-27, 923 (1978).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (3)

S. D. Shapiro, A. Iannino, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-1, 310 (1979).
[CrossRef]

K. R. Sloan, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-4, 87(1982).
[CrossRef]

S. R. Deans, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-2, 185(1981).
[CrossRef]

J. Theor. Biol. (1)

R. Gordon, R. Bender, G. T. Herman, J. Theor. Biol. 29, 471 (1970).
[CrossRef] [PubMed]

Opt. Commun. (1)

J. Hofer, Opt. Commun. 29, 22 (1979).
[CrossRef]

Opt. Eng. (1)

A. F. Gmitro, J. E. Greivenkamp, W. Swindell, H. H. Barrett, M. Y. Chiu, S. K. Gordon, Opt. Eng. 19, 260 (1980).
[CrossRef]

Opt. Lett. (1)

Phys. Med Biol (1)

P. Edholm, L. G. Hellstrom, B. J. Jacobson, Phys. Med Biol 23, 90(1978).
[CrossRef] [PubMed]

Proc. IEEE (2)

H. H. Barrett, W. Swindell, Proc. IEEE 65, 89 (1977).
[CrossRef]

R. M. Mersereau, A. V. Oppenheim, Proc. IEEE 62, 1319 (1974).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (3)

E. W. Hansen, J. W. Goodman, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 222 (1980).

R. C. Chase, F. H. Seguin, M. Gerassimenko, R. Petrasso, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 265 (1980).

G. Eichmann, R. Stirbl, R. Mammone, Proc. Soc. Photo-Opt. Instrum. Eng. 209, 24 (1979).

Other (9)

J. W. Goodman, “Linear Space-Variant Optical Processing,” in Optical Data Processing, Fundamentals, S. Lee, Ed. (Springer, New York, 1980).

D. Casasent, D. Psaltis, “Deformation Invariant, Space-Variant Optica) Pattern Recognition,” in Progress in Optics, Vol. 18, E. Wolf, Ed., (North-Holland, Amsterdam, 1978).
[CrossRef]

G. Tyras, Radiation and Propagation of Electromagnetic Waves (Academic, New York, 1969), Chap. 8.

P. V. C. Hough, “Methods and Means for Recognizing Complex Patterns,” U.S. Patent3,069,654 (1962).

W. K. Pratt, Digital Image Processing (Wiley-Interscience, New York, 1978).

Technical Digest, Topical Meeting on Imaging Processing for 2-D and 3-D Reconstructions from Projections (Optical Society of America, Washington, D.C., 1975).

Z. H. Cho, Ed., Special issue on Physical and Computational Aspects of 3-D Image Reconstruction, IEEE Trans. Nucl. SciNS-21 (June1971).

H. Platzer, H. Glunder, “Tomogram—Reconstruction by Holographic Methods,” in Holography in Medicine and Biology, G. V. Bally, Ed. (Springer, New York, 1979), pp. 117–123.

I. M. Gelfand, M. I. Graev, N. Ya. Vilkin, Generalized Functions (McGraw-Hill, New York, 1966), Vol. 5.

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Figures (6)

Fig. 1
Fig. 1

Cartesian input and radon transform plane coordinate geometries. The shortest distance to the straight line is p, while θ is the angle formed by the shortest distance cord and the x axis.

Fig. 2
Fig. 2

Coherent Fourier optical lens configuration used to record the FRT of a 2-D input. The input is located in plane P1. A vertical slit is placed in the Fourier transform plane P2, while the output FRT is recorded in the inverse Fourier transform plane P3. For a fixed angle of the input in the plane P1, a Cartesian coordinate slice, with FRT coordinates as a Cartesian grid, is generated at the output plane P3. Rotating the input in plane P1 by an incremental angle while simultaneously linearly translating the output storage material in the output plane P3, a new slice of the FRT picture is generated.

Fig. 3
Fig. 3

Experimental results generated using the optical setup described in Fig. 2. In (a) the input while in (b) the FRT slices of (a) are shown. The vertical axis in (b) is p, while the horizontal is the θ axis. Here the angle is measured from the Cartesian y axis.

Fig. 4
Fig. 4

Computer generated FRT slices for the input of Fig. 3(a).

Fig. 5
Fig. 5

Experimental results generated using the optical arrangement shown in Fig. 2. In (a) the straight lines input, while in (b) the FRT output slices of the input of (a) are shown.

Fig. 6
Fig. 6

Computer generated FRT slices for the input of Fig. 5(a).

Equations (36)

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

f ˆ ( p , θ ) = f ( x ) δ ( p x · ξ ) d x ,
x = x a x + y a y ,
ξ = cos θ a x + sin θ a y = ξ 1 a x + ξ 2 a y .
p = x 0 cos θ + y 0 sin θ
f x 0 ( x ) = f ( x + x 0 , y + y 0 ) ,
f ˆ x 0 ( p , θ ) = f ˆ ( p + ξ · x 0 , θ ) .
f 2 ( x ) = a · f ( x ) ,
f ˆ 2 ( p , θ ) = ( a · ξ ) f ˆ / p ,
f 3 ( x ) = ( a · x ) f ( x )
f ˆ 3 / p = ( a · ξ ) f ˆ .
f ( x ) = f 1 ( y ) f 2 ( x y ) d y
f ˆ ( p , θ ) = f ˆ 1 ( θ , t ) f ˆ 2 ( θ , p t ) d t ,
f ˆ ( p , θ ) = f ( x , y ) δ ( p x cos θ y sin θ ) d x d y ,
u = x cos θ + y sin θ υ = x sin θ + y cos θ .
f ˆ ( p , θ ) = f ( u , υ ) δ ( p u ) d u d υ .
f ˆ ( p , θ ) = F ( z , w ) H ( z , w ; p , θ ) d z d w ,
H ( z , w ; p , θ ) = ( 1 / 2 π ) δ ( w ) exp ( j z p ) .
x = r cos ϕ y = r sin ϕ .
h ( r , ϕ ; p , θ ) = δ [ p r cos ( θ ϕ ) ] .
h ( r , ϕ ; p , θ ) = δ [ r p / cos ( θ ϕ ) ] / | cos ( θ ϕ ) |
h ( r , ϕ ; p , θ ) = δ [ ϕ θ + cos 1 ( p / r ) ] / r sqr [ 1 ( p / r ) 2 ] .
f ( r , ϕ ) = n = f n ( r ) exp ( j n ϕ ) ,
f n ( r ) = 1 2 π 0 2 π f ( r , ϕ ) exp ( j n ϕ ) d ϕ .
f ˆ ( p , θ ) = n = exp ( j n θ ) f ˆ n ( p ) ,
f ˆ n = 0 f n ( r ) [ 1 ( p / r ) 2 ] 1 / 2 exp [ j n cos 1 ( p / r ) ] d r .
f ( r , ϕ ) = 1 2 π j Br G ( w , ϕ ) exp ( w r ) d w ,
G ( w , ϕ ) = 0 f ( r , ϕ ) exp ( w r ) d r
f ˆ ( p , θ ) = 1 2 π j Br G ( w ; p , θ ) d w ,
G ( w ; p , θ ) = p ξ 1 1 ξ ¯ 1 1 ( 1 u 2 ) 1 / 2 / u G [ w , cos 1 ( 1 / u ) + θ ] × exp ( p u w ) d u .
δ ( p x cos θ y sin θ ) = 1 2 π exp [ j w ( p x cos θ y × sin θ ) ] d w .
f ˆ ( p , θ ) = 1 2 π F ( w cos θ , w sin θ ) exp ( j w p ) d w ,
F ( w cos θ , w sin θ ) = f ( x , y ) exp [ j w ( x cos θ + y sin θ ) ] d x d y .
δ ( x x ) = n = 0 A n cos n π x / L cos n π x / L ,
A n = 1 / L n = 0 and 2 / L n 0.
f ˆ ( p , θ ) = n = 0 A n cos n π p / L g n ( θ ) ,
g n ( θ ) = f ( x , y ) cos n π / L [ x cos θ + y sin θ ] d x d y

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