Abstract

A procedure is presented for generating a small bank of optical correlation filters that can recognize a large number of perspective views of an object. The method applies to general kinds of image distortions in addition to those generated by different perspective views. The holographic filters are also invariant to image intensity and position (translation invariance). The method of design is to decompose the entire set of object variations into a set of eigenimages. These eigenimages contain complete information about the target set. An iterative procedure combines the eigenimages with different relative phases, so that complete target information can be extracted in an optical implementation. An example illustrates that a set of only 20 holographic filters recognizes a three-dimensional target over a continuous range of viewing angles.

© 1988 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Casasent, “Unified synthetic discriminant function computational formulation,” Appl. Opt. 23, 1620–1627 (1984).
    [CrossRef] [PubMed]
  2. B. Braunecker, R. Hauck, A. W. Lohmann, “Optical character recognition based on nonredundant correlation measurements,” Appl. Opt. 18, 2746–2753 (1979).
    [CrossRef] [PubMed]
  3. H. J. Caulfield, “Linear combinations of filters for character recognition: a unified treatment,” Appl. Opt. 19, 3877–3878 (1980).
    [CrossRef] [PubMed]
  4. B. V. K. Vijaya Kumar, E. Pochapsky, “Signal-to-noise ratio considerations in modified matched spatial filters,” J. Opt. Soc. Am. A 3, 777–786 (1986).
    [CrossRef]
  5. A. Vander Lugt, “Signal detection by complex spatial filtering,”IRE Trans. Inform. Theory IT-10, 139–145 (1964).
    [CrossRef]
  6. A. W. Naylor, G. R. Sell, Linear Operator Theory in Engineering and Science (Springer-Verlag, New York, 1982).
    [CrossRef]
  7. F. R. Gantmacher, The Theory of Matrices (Chelsea, New York, 1959), Vol. I.
  8. See for example, D. Casasent, W.-T. Chang, “Correlation synthetic discriminant functions,” Appl. Opt. 25, 2343–2350 (1986), and the many references contained therein.
    [CrossRef] [PubMed]
  9. H. L. Van Trees, Detection, Estimation, and Linear Modulation Theory, Part I (Wiley, New York, 1968).
  10. C. R. Rao, Linear Statistical Inference and Its Applications, 2nd ed. (Wiley, New York, 1973).
    [CrossRef]
  11. R. W. Gerchberg, W. O. Saxton, “Phase determination from image and diffraction plane pictures in the electron microscope,” Optik 34, 275–283 (1971).
  12. P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of making an object dependent diffuser,” U.S. Patent3,619,022 (November9, 1971).
  13. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  14. N. C. Gallagher, B. Liu, “Method for computing kinoforms that reduces image reconstruction error,” Appl. Opt. 12, 2328–2335 (1973).
    [CrossRef] [PubMed]
  15. R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
    [CrossRef]
  16. A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,”IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
    [CrossRef]
  17. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  18. J. R. Fienup, “Reconstruction and synthesis applications of an iterative algorithm,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, eds., Proc. Soc. Photo-Opt. Instrum. Eng.373, 147–160 (1981).
    [CrossRef]
  19. D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1. Theory,”IEEE Trans. Med. Imag. MI-1, 81–94 (1982).
    [CrossRef]
  20. D. W. Sweeney, E. Ochoa, G. F. Schils, “Experimental use of iteratively designed rotation invariant correlation filters,” Appl. Opt. 26, 3458–3465 (1987).
    [CrossRef] [PubMed]
  21. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).
  22. Y. N. Hsu, H. H. Arsenault, “Optical pattern recognition using circular harmonic expansion,” Appl. Opt. 21, 4016–4019 (1982).
    [CrossRef] [PubMed]
  23. G. F. Schils, D. W. Sweeney, “Iterative technique for the synthesis of optical-correlation filters,”J. Opt. Soc. Am. 3, 1433–1442 (1986).
    [CrossRef]
  24. Y. Sheng, H. H. Arsenault, “Experiments on pattern recognition using invariant Fourier–Mellin descriptors,” J. Opt. Soc. Am. A 3, 771–776 (1986).
    [CrossRef] [PubMed]
  25. D. Casasent, D. Psaltis, “Deformation invariant, space-variant optical pattern recognition,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. 16, pp. 289–356.
    [CrossRef]
  26. D. Casasent, S. A. Liebowitz, A. Mahalanobis, “Parameter selection for iconic and symbolic pattern recognition filters,” in Optical and Digital Pattern Recognition, P. S. Schenker, H. Liu, eds., Proc. Soc. Photo-Opt. Instrum. Eng.754, 284–303 (1987).
    [CrossRef]
  27. IMSL Library, fortran Subroutines for Mathematics and Statistics (IMSL, Houston, Tex., 1984).
  28. G. F. Schils, D. W. Sweeney, “Iterative technique for the synthesis of distortion-invariant optical correlation filters,” Opt. Lett. 12, 307–309 (1987).
    [CrossRef] [PubMed]
  29. G. F. Schils, D. W. Sweeney, “Rotationally invariant correlation filtering,” J. Opt. Soc. Am. A 2, 1411–1418 (1985).
    [CrossRef]
  30. Y. Yang, Y.-N. Hsu, H. H. Arsenault, “Optimum circular symmetrical filters and their uses in optical pattern recognition,” Opt. Acta 29, 627–644 (1982).
    [CrossRef]

1987

1986

1985

1984

1982

Y. N. Hsu, H. H. Arsenault, “Optical pattern recognition using circular harmonic expansion,” Appl. Opt. 21, 4016–4019 (1982).
[CrossRef] [PubMed]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1. Theory,”IEEE Trans. Med. Imag. MI-1, 81–94 (1982).
[CrossRef]

Y. Yang, Y.-N. Hsu, H. H. Arsenault, “Optimum circular symmetrical filters and their uses in optical pattern recognition,” Opt. Acta 29, 627–644 (1982).
[CrossRef]

1980

1979

1975

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,”IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

1974

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

1973

1972

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1971

R. W. Gerchberg, W. O. Saxton, “Phase determination from image and diffraction plane pictures in the electron microscope,” Optik 34, 275–283 (1971).

1964

A. Vander Lugt, “Signal detection by complex spatial filtering,”IRE Trans. Inform. Theory IT-10, 139–145 (1964).
[CrossRef]

Arsenault, H. H.

Braunecker, B.

Casasent, D.

See for example, D. Casasent, W.-T. Chang, “Correlation synthetic discriminant functions,” Appl. Opt. 25, 2343–2350 (1986), and the many references contained therein.
[CrossRef] [PubMed]

D. Casasent, “Unified synthetic discriminant function computational formulation,” Appl. Opt. 23, 1620–1627 (1984).
[CrossRef] [PubMed]

D. Casasent, D. Psaltis, “Deformation invariant, space-variant optical pattern recognition,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. 16, pp. 289–356.
[CrossRef]

D. Casasent, S. A. Liebowitz, A. Mahalanobis, “Parameter selection for iconic and symbolic pattern recognition filters,” in Optical and Digital Pattern Recognition, P. S. Schenker, H. Liu, eds., Proc. Soc. Photo-Opt. Instrum. Eng.754, 284–303 (1987).
[CrossRef]

Caulfield, H. J.

Chang, W.-T.

Fienup, J. R.

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

J. R. Fienup, “Reconstruction and synthesis applications of an iterative algorithm,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, eds., Proc. Soc. Photo-Opt. Instrum. Eng.373, 147–160 (1981).
[CrossRef]

Gallagher, N. C.

Gantmacher, F. R.

F. R. Gantmacher, The Theory of Matrices (Chelsea, New York, 1959), Vol. I.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

Gerchberg, R. W.

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

R. W. Gerchberg, W. O. Saxton, “Phase determination from image and diffraction plane pictures in the electron microscope,” Optik 34, 275–283 (1971).

Hauck, R.

Hirsch, P. M.

P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of making an object dependent diffuser,” U.S. Patent3,619,022 (November9, 1971).

Hsu, Y. N.

Hsu, Y.-N.

Y. Yang, Y.-N. Hsu, H. H. Arsenault, “Optimum circular symmetrical filters and their uses in optical pattern recognition,” Opt. Acta 29, 627–644 (1982).
[CrossRef]

Jordan, J. A.

P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of making an object dependent diffuser,” U.S. Patent3,619,022 (November9, 1971).

Lesem, L. B.

P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of making an object dependent diffuser,” U.S. Patent3,619,022 (November9, 1971).

Liebowitz, S. A.

D. Casasent, S. A. Liebowitz, A. Mahalanobis, “Parameter selection for iconic and symbolic pattern recognition filters,” in Optical and Digital Pattern Recognition, P. S. Schenker, H. Liu, eds., Proc. Soc. Photo-Opt. Instrum. Eng.754, 284–303 (1987).
[CrossRef]

Liu, B.

Lohmann, A. W.

Mahalanobis, A.

D. Casasent, S. A. Liebowitz, A. Mahalanobis, “Parameter selection for iconic and symbolic pattern recognition filters,” in Optical and Digital Pattern Recognition, P. S. Schenker, H. Liu, eds., Proc. Soc. Photo-Opt. Instrum. Eng.754, 284–303 (1987).
[CrossRef]

Naylor, A. W.

A. W. Naylor, G. R. Sell, Linear Operator Theory in Engineering and Science (Springer-Verlag, New York, 1982).
[CrossRef]

Ochoa, E.

Papoulis, A.

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,”IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

Pochapsky, E.

Psaltis, D.

D. Casasent, D. Psaltis, “Deformation invariant, space-variant optical pattern recognition,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. 16, pp. 289–356.
[CrossRef]

Rao, C. R.

C. R. Rao, Linear Statistical Inference and Its Applications, 2nd ed. (Wiley, New York, 1973).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

R. W. Gerchberg, W. O. Saxton, “Phase determination from image and diffraction plane pictures in the electron microscope,” Optik 34, 275–283 (1971).

Schils, G. F.

Sell, G. R.

A. W. Naylor, G. R. Sell, Linear Operator Theory in Engineering and Science (Springer-Verlag, New York, 1982).
[CrossRef]

Sheng, Y.

Sweeney, D. W.

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation, and Linear Modulation Theory, Part I (Wiley, New York, 1968).

Vander Lugt, A.

A. Vander Lugt, “Signal detection by complex spatial filtering,”IRE Trans. Inform. Theory IT-10, 139–145 (1964).
[CrossRef]

Vijaya Kumar, B. V. K.

Webb, H.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1. Theory,”IEEE Trans. Med. Imag. MI-1, 81–94 (1982).
[CrossRef]

Yang, Y.

Y. Yang, Y.-N. Hsu, H. H. Arsenault, “Optimum circular symmetrical filters and their uses in optical pattern recognition,” Opt. Acta 29, 627–644 (1982).
[CrossRef]

Youla, D. C.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1. Theory,”IEEE Trans. Med. Imag. MI-1, 81–94 (1982).
[CrossRef]

Appl. Opt.

IEEE Trans. Circuits Syst.

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,”IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

IEEE Trans. Med. Imag.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1. Theory,”IEEE Trans. Med. Imag. MI-1, 81–94 (1982).
[CrossRef]

IRE Trans. Inform. Theory

A. Vander Lugt, “Signal detection by complex spatial filtering,”IRE Trans. Inform. Theory IT-10, 139–145 (1964).
[CrossRef]

J. Opt. Soc. Am.

G. F. Schils, D. W. Sweeney, “Iterative technique for the synthesis of optical-correlation filters,”J. Opt. Soc. Am. 3, 1433–1442 (1986).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Acta

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Y. Yang, Y.-N. Hsu, H. H. Arsenault, “Optimum circular symmetrical filters and their uses in optical pattern recognition,” Opt. Acta 29, 627–644 (1982).
[CrossRef]

Opt. Lett.

Optik

R. W. Gerchberg, W. O. Saxton, “Phase determination from image and diffraction plane pictures in the electron microscope,” Optik 34, 275–283 (1971).

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Other

J. R. Fienup, “Reconstruction and synthesis applications of an iterative algorithm,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, eds., Proc. Soc. Photo-Opt. Instrum. Eng.373, 147–160 (1981).
[CrossRef]

P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of making an object dependent diffuser,” U.S. Patent3,619,022 (November9, 1971).

A. W. Naylor, G. R. Sell, Linear Operator Theory in Engineering and Science (Springer-Verlag, New York, 1982).
[CrossRef]

F. R. Gantmacher, The Theory of Matrices (Chelsea, New York, 1959), Vol. I.

H. L. Van Trees, Detection, Estimation, and Linear Modulation Theory, Part I (Wiley, New York, 1968).

C. R. Rao, Linear Statistical Inference and Its Applications, 2nd ed. (Wiley, New York, 1973).
[CrossRef]

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

D. Casasent, D. Psaltis, “Deformation invariant, space-variant optical pattern recognition,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. 16, pp. 289–356.
[CrossRef]

D. Casasent, S. A. Liebowitz, A. Mahalanobis, “Parameter selection for iconic and symbolic pattern recognition filters,” in Optical and Digital Pattern Recognition, P. S. Schenker, H. Liu, eds., Proc. Soc. Photo-Opt. Instrum. Eng.754, 284–303 (1987).
[CrossRef]

IMSL Library, fortran Subroutines for Mathematics and Statistics (IMSL, Houston, Tex., 1984).

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 (13)

Fig. 1
Fig. 1

Illustration of the constraints on the parameter-response function C f g ( k ) ( p ). The amplitude | C f g k ( p ) | is permitted to vary with the distortion parameter p but is required to be constant as the filter number k varies.

Fig. 2
Fig. 2

Illustration of the constraints on the eigenimage weighting coefficients amk. The amplitude |amk| is required to be constant with respect to the harmonic number m but is permitted to vary over the different filters k. The constraints on | C f g k ( p ) | of Fig. 1 and on |amk| of this figure are seen to be crossed; that is, the directions of constancy are orthogonal.

Fig. 3
Fig. 3

Iterative method for producing a bank of holographic filters that are invariant to a particular distortion p.

Fig. 4
Fig. 4

Method of processing the output intensity data. As different correlation filters gk(x) are used in the optical correlator shown, the output correlation intensities Ik and intensities squared Ik2 are summed into two frame buffers as shown. Typically only about 20 correlation filters are necessary. The two output frame buffers are used to determine at which pixels the intensity is constant. This determination is made by computing the mean intensity 〈I 〉and the intensity standard deviation σI from the data in the two frame buffers; then the ratio r = 〈I〉/σI is formed. Targets are located at those points where the ratio r is large, that is, where the intensity is nearly constant as the filters are changed.

Fig. 5
Fig. 5

Sixteen views of a model tank object taken from different perspective points of view. These views are gathered over a 180-deg azimuthal range and over a 60-deg range of depression angles. A bank of 20 holographic filters was designed to recognize the tank over this continuous cone of views.

Fig. 6
Fig. 6

Plot of the eigenvalues λm for the problem of recognition of the different perspective views of the tank object.

Fig. 7
Fig. 7

Several of the eigenimages for the tank-recognition example. From left to right and then down, these are the eigenimages fm(x) for m = 2, 3, …, 17. The eigenimages fm(x) are an orthogonal set of fundamental image components that contain all of the information about the images to be recognized; that is, any of the object views can be constructed from these fundamental components by performing the proper linear combination [see Eq. (10)]. The filters that we synthesized are composed of these eigenimages.

Fig. 8
Fig. 8

The dominant eigenimage f1(x). This eigenimage represents the information that is common to all elements of the target set. This spatial mode is approximately an average over the target set. Because this mode correlates so strongly with almost all other images, this mode was excluded from the filters gk(x).

Fig. 9
Fig. 9

Solutions for the eigenimage weighting coefficients amk produced by the spectral-iteration algorithm. A set of solutions amk for k = 1, 2, …, 20 was produced. The amplitudes of the first four solutions |amk| for k = 1, 2, 3, 4 are shown in (a), (b), (c), and (d), respectively. It is seen that the design constraint that |amk| be constant over m is satisfied well, except for the large values of m, which correspond to weak modes (small values of λm).

Fig. 10
Fig. 10

Amplitude of 16 of the 20 LAT filters for the example of perspective-invariant recognition of the tank-type object. These filters can be seen to be different, yet each embodies a complete set of target information. The LAT filters gk(x) are produced by using the am coefficients found from the spectral-iteration algorithm. The filters gk(x) are determined first by inverse spectral transforming the am coefficients into the weighting function a(p) by using Eq. (18). This weighting function a(p) is then used to form the weighted sum of the target images according to Eq. (17).

Fig. 11
Fig. 11

Phase of 16 of the 20 LAT filters whose amplitudes are shown in Fig. 10. Even though each LAT consists of all the eigenimages fm(x), these eigenimages are incorporated into each LAT filter with a different phase. This phase is determined so that the output parameter-response function exhibits a constant amplitude response as the different filters gk(x) are applied. Because this information is laid down with different phases in the different filters, the phases of the filters themselves are different.

Fig. 12
Fig. 12

Input scene containing both target and nontarget images. The objective is to recognize the tank image over a large solid cone of continuous views. The leftmost tank image is a member of the training set of 380 images used to build the filter. The other two tank images are not members of the training set and are views taken in between those of the training set. The object at the upper right is the image of a bulldozer and is a false target that should not be recognized.

Fig. 13
Fig. 13

Plot of the ratio r = 〈I〉/σI for the input scene of Fig. 11 as the 20 holographic filters are applied. The peaks identify the locations of the target images. One of the target images is a member of the training set used to build the filter, and the other two target images are intermediate views that are not in the training set. It is seen that all three targets are located, even though they are close to one another. The bulldozer image at the upper right is a nontarget image; it is not recognized by the filter bank of 20 filters.

Equations (37)

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

f ( x ; p ) , x R 2 , p P .
A = [ a b c d ] .
f ( x ; p ) = f ( A x ) = f [ x ; ( a , b , c , d ) ] .
f ( x ; p ) = f [ x ; ( θ , ϕ ) ] .
C f f ( p , q ) = R 2 f ( x ; p ) f * ( x ; q ) d x , p P , q P .
P C f f ( p , q ) ϕ m ( q ) d q = λ m ϕ m ( p ) .
C f f ( p , q ) = m z m ( q ) ϕ m ( p ) ,
z m ( q ) = P C f f ( p , q ) ϕ m * ( p ) d p = [ P C f f ( q , p ) ϕ m ( p ) d p ] * = λ m ϕ m * ( q ) .
C f f ( p , q ) = m λ m ϕ m ( p ) ϕ m * ( q ) .
f ( x ; p ) = m f m ( x ) ϕ m ( p ) , p P .
f m ( x ) = P f ( x ; p ) ϕ m * ( p ) d p .
R 2 f m ( x ) f n * ( x ) d x = R 2 P f ( x ; p ) ϕ m * ( p ) d p × P f * ( x ; q ) ϕ n ( q ) d q d x = P P ϕ n ( q ) ϕ m * ( p ) × [ R 2 f ( x ; p ) f * ( x ; q ) d x ] d p d q = P P ϕ n ( q ) ϕ m * ( p ) C f f ( p , q ) d p d q = P ϕ m * ( p ) [ P C f f ( p , q ) ϕ n ( q ) d q ] d p = P λ n ϕ n ( p ) ϕ m * ( p ) d p = λ m δ m n .
R 2 | f m ( x ) | 2 d x = λ m .
C ˆ f f ( x , y ) = P f ( x ; p ) f * ( y ; p ) d p , x , y R 2 .
R 2 C ˆ f f ( x , y ) f m ( y ) d y = λ m f m ( x ) .
C f g ( p ) = R 2 f ( x ; p ) g * ( x ) d x .
g ( x ) = P f ( x ; p ) a * ( p ) d p ,
a ( p ) = m a m ϕ m ( p ) , p P .
g ( x ) = P f ( x ; p ) m a m * ϕ m * ( p ) d p = m a m * P f ( x ; p ) ϕ m * ( p ) d p = m a m * f m ( x ) .
C f g ( p ) = R 2 f ( x ; p ) P f * ( x ; q ) a ( q ) d p d x = P [ R 2 f ( x ; p ) f * ( x ; q ) d x ] a ( q ) d q = P C f f ( p , q ) a ( q ) d q .
C f g ( p ) = P C f f ( p , q ) m a m ϕ m ( q ) d q = m a m P C f f ( p , q ) ϕ m ( q ) d q = m λ m a m ϕ m ( p ) = m c m ϕ m ( p ) ,
c m = λ m a m .
c m = P C f g ( p ) ϕ m * ( p ) d p .
| C f g k ( p ) | = b ( p ) for k = 1 , , M , p P .
| a m | ~ 1
| a m k | ~ d k , k = 1 , , M .
( P | ϕ m ( p ) | 2 d p ) 1 / 2 = 1 for all m ,
{ a m k } constraints , { C f g k ( p ) } constraints , a m k C f g k ( p ) .
φ k ( p ) = phase [ C f g k ( p ) ] , b ( p ) = [ 1 M k = 1 M | C f g k ( p ) | 2 ] 1 / 2 , C f g k ( p ) b ( p ) exp [ i φ k ( p ) ]
g k ( n + 1 ) ( x ) g k ( n ) ( x ) 2 = m [ a m k ( n + 1 ) a m k ( n ) ] f m ( x ) 2 = m [ a m k ( n + 1 ) a m k ( n ) ] 2 λ m ,
r = r / σ I
C f f ( p , q ) = C f f ( α 1 , α 2 ) = 0 0 2 π f ( r , θ + α 1 ) f * ( r , θ + α 2 ) d θ r d r = 0 0 2 π f ( r , θ + α 1 α 2 ) f * ( r , θ ) d θ r d r = C f f ( α 1 α 2 ) .
0 2 π C f f ( α 1 α 2 ) ϕ m ( α 2 ) d α 2 = λ m ϕ m ( α 1 ) .
ϕ m ( α ) = 1 2 π exp ( i m α ) ,
C f f ( s 1 , s 2 ) = R 2 s 1 f ( s 1 x ) s 2 f * ( s 2 x ) d x = R 2 f ( s 1 s 2 y ) f * ( y ) s 1 s 2 d y .
C f f ( u 1 , u 2 ) = C f f ( u 1 u 2 ) = R 2 f [ exp ( u 1 u 2 ) y ] f * ( y ) exp ( u 1 u 2 ) d y .
ϕ μ ( u ) = exp ( i μ u ) , ϕ μ ( s ) = exp [ i μ ln ( s ) ] = s i μ ,

Metrics