Abstract

I propose a new method that ensures efficient rotation-invariant pattern recognition in the presence of signal-dependent noise by combining the application of rotation-invariant correlation filters with preprocessing of the noisy input images. The preprocessing uses local suboptimal estimators derived from estimation theory and implies an a priori knowledge of a model describing the noise source. The image noise sources considered are speckle and film-grain noise. Four different metrics are used to analyze the correlation performance of the circular-harmonic filter, the phase-only circular-harmonic filter, and the binary phase-only circular-harmonic filter, with and without a preprocessing. Computer simulations show that signal-dependent noise can seriously degrade the performance of the phase-only circular-harmonic filter and the binary phase-only circular-harmonic filter. The most severe indication of correlation-performance degradation is the occurrence of false alarms in 15% to 20% of noise realizations of the correlation. Preprocessing increases the correlation-peak signal-to-noise ratio significantly and reduces the false-alarm probability by one to two orders of magnitude.

© 1996 Optical Society of America

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  1. J. F. Walkup, R. C. Choens, “Image processing in signal-dependent noise,” Opt. Eng. 13, 258–266 (1974).
  2. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  3. H. Siedentopf, “Concerning granularity, density fluctuations and the enlargement of photographic negatives,” Phys. Z. 38, 454 (1937).
  4. C. M. Lo, “Estimation of image signals with Poisson noise,” Rep. 890 (Image Processing Institute, University of California, Los Angeles, Calif., 1979).
  5. J. Marron, G. M. Morris, “Image recognition in the presence of laser speckle,” J. Opt. Soc. Am. A 3, 964–971 (1986).
  6. E. I. Shubnikov, “Effect of additive and multiplicative noise in the correlation comparison of images,” Opt. Spectrosc. (USSR) 62, 389–392 (1987).
  7. G. M. Morris, “Pattern recognition using photon-limited images,” in Optical processing and Computing, H. H. Arse-nault, T. Szoplik, B. Macukow, eds. (Academic, Boston, Mass., 1989).
  8. T. A. Isberg, G. M. Morris, “Rotation-invariant image recognition at low light levels,” J. Opt. Soc. Am. A 3, 954–963 (1986).
  9. G. K. Froehlich, J. F. Walkup, R. B. Asher, “Optimal estimation in signal-dependent noise,” J. Opt. Soc. Am. 68, 1665–1672 (1978).
  10. F. Naderi, A. A. Sawchuk, “Detection of low-contrast images in film-grain noise,” Appl. Opt. 17, 2883–2891 (1978).
  11. D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive noise smoothing filter for images with signal-dependent noise,” IEEE Trans. Pattern Anal. Mach. Intell.PAMI-7, 165–177 (1985).
  12. D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive restoration of images with speckle,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 373–382 (1987).
  13. H. H. Arsenault, G. April, “Properties of speckle integrated with a finite aperture and logarithmically transformed,” J. Opt. Soc. Am. 66, 1160–1163 (1976).
  14. H. H. Arsenault, C. Gendron, M. Denis, “Transformation of film-grain noise into signal-independent additive Gaussian noise,” J. Opt. Soc. Am. 71, 91–94 (1981).
  15. H. H. Arsenault, M. Denis, “Integral expression for transforming signal-dependent noise into signal-independent noise,” Opt. Lett. 6, 210–212 (1981).
  16. P. R. Prucnal, B. E. A. Saleh, “Transformation of image-sinal-dependent noise into image-signal-independent noise,” Opt. Lett. 6, 316–318 (1981).
  17. P. R. Prucnal, E. L. Goldstein, “Exact variance stabilizing transformations for image signal-dependent Rayleigh and other Weibull noise sources,” Appl. Opt. 26, 1038–1041 (1987).
  18. A. K. Jain, C. R. Christensen, “Digital processing of images in speckle noise,” in Applications of Speckle Phenomena, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 243, 46–50 (1980).
  19. H. H. Arsenault, M. Denis, “Image processing in signal-dependent noise,” Can. J. Phys. 61, 309–317 (1983).
  20. R. Kasturi, J. F. Walkup, T. F. Krile, “Image restoration by transformation of signal-dependent noise to signal-independent noise,” Appl. Opt. 22, 3537–3542 (1983).
  21. H. H. Arsenault, M. Levesque, “Combined homomorphic and local-statistics processing for restoration of images degraded by signal-dependent noise,” Appl. Opt. 23, 845–850 (1984).
  22. J. S. Lee, “Digital image enhancement and noise filtering by use of local statistics,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-2, 165–168 (1980).
  23. G. K. Froehlich, J. F. Walkup, T. F. Krile, “Estimation in signal-dependent film-grain noise,” Appl. Opt. 20, 3619–3626 (1981).
  24. J. D. Downie, J. F. Walkup, “Optimal correlation filters for images with signal-dependent noise,” J. Opt. Soc. Am. A 11, 1599–1609 (1994).
  25. M. Tur, C. Chin, J. W. Goodman, “When is speckle noise multiplicative?,” Appl. Opt. 21, 1157–1159 (1982).
  26. Research and Education Association, Handbook of Mathematical, Scientific, and Engineering Formulas, Tables, Functions, Graphs, Transforms, (Research and Education Association, N.J., 1994), Chap. 16, p. 595.
  27. M. Denis, H. H. Arsenault, “On the accuracy of a method to make film-grain noise independent of the signal,” Opt. Commun. 38, 166–169 (1981).
  28. R. Wallis, “An approach to the space variant restoration and enhancement of images,” in Proceedings of the Symposium on Current Mathematical Problems in Image Science (Naval Postgraduate School, Monterey, Calif., 1976).
  29. Y. N. Hsu, H. H. Arsenault, G. April, “Rotation-invariant digital pattern recognition using circular harmonic expansion,” Appl. Opt. 21, 4012–4015 (1982).
  30. Y. N. Hsu, H. H. Arsenault, “Optical pattern recognition using circular harmonic expansion,” Appl. Opt. 21, 4016–4019 (1982).
  31. L. Leclerc, Y. Sheng, H. H. Arsenault, “Rotation-invariant phase-only and binary phase-only correlation,” Appl. Opt. 28, 1251–1256 (1989).
  32. Y. Sheng, H. H. Arsenault, “Method for determining expansion centers and predicting sidelobe levels for circular harmonic filters,” J. Opt. Soc. Am. A 1, 1793–1797 (1987).
  33. G. Premont, Y. Sheng, “Fast design of circular-harmonic filters using simulated annealing,” Appl. Opt. 32, 3116–3121 (1993).
  34. Y. N. Hsu, H. H. Arsenault, “Pattern discrimination by multiple circular harmonic components,” Appl. Opt. 23, 841–844 (1984).
  35. L. Leclerc, Y. Sheng, H. H. Arsenault, “Optical binary phase-only filters for circular harmonic correlations,” Appl. Opt. 30, 4643–4649 (1991).
  36. B. V. K. Vijaya Kumar, L. Hassebrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 (1990).
  37. J. L. Horner, J. R. Leger, “Pattern recognition with binary phase-only filters,” Appl. Opt. 24, 609–611 (1985).
  38. J. L. Horner, H. O. Bartelt, “Two-bit correlation,” Appl. Opt. 24, 2889–2893 (1985).
  39. M. A. Flavin, J. L. Horner, “Amplitude encoded phase-only filters,” Appl. Opt. 28, 1692–1696 (1989).
  40. B. Javidi, J. Wang, Q. Tang, “Multiple-object binary joint transform correlation using multiple-level threshold cross-ing,” Appl. Opt. 30, 4234–4244 (1991).
  41. J. L. Horner, “Metrics for assessing pattern-recognition perfor-mance,” Appl. Opt. 31, 165–166 (1992).

1994

1993

1992

1991

1990

1989

1987

P. R. Prucnal, E. L. Goldstein, “Exact variance stabilizing transformations for image signal-dependent Rayleigh and other Weibull noise sources,” Appl. Opt. 26, 1038–1041 (1987).

E. I. Shubnikov, “Effect of additive and multiplicative noise in the correlation comparison of images,” Opt. Spectrosc. (USSR) 62, 389–392 (1987).

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive restoration of images with speckle,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 373–382 (1987).

Y. Sheng, H. H. Arsenault, “Method for determining expansion centers and predicting sidelobe levels for circular harmonic filters,” J. Opt. Soc. Am. A 1, 1793–1797 (1987).

1986

1985

1984

1983

R. Kasturi, J. F. Walkup, T. F. Krile, “Image restoration by transformation of signal-dependent noise to signal-independent noise,” Appl. Opt. 22, 3537–3542 (1983).

H. H. Arsenault, M. Denis, “Image processing in signal-dependent noise,” Can. J. Phys. 61, 309–317 (1983).

1982

1981

1980

J. S. Lee, “Digital image enhancement and noise filtering by use of local statistics,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-2, 165–168 (1980).

1978

1976

1974

J. F. Walkup, R. C. Choens, “Image processing in signal-dependent noise,” Opt. Eng. 13, 258–266 (1974).

1937

H. Siedentopf, “Concerning granularity, density fluctuations and the enlargement of photographic negatives,” Phys. Z. 38, 454 (1937).

April, G.

Arsenault, H. H.

L. Leclerc, Y. Sheng, H. H. Arsenault, “Optical binary phase-only filters for circular harmonic correlations,” Appl. Opt. 30, 4643–4649 (1991).

L. Leclerc, Y. Sheng, H. H. Arsenault, “Rotation-invariant phase-only and binary phase-only correlation,” Appl. Opt. 28, 1251–1256 (1989).

Y. Sheng, H. H. Arsenault, “Method for determining expansion centers and predicting sidelobe levels for circular harmonic filters,” J. Opt. Soc. Am. A 1, 1793–1797 (1987).

Y. N. Hsu, H. H. Arsenault, “Pattern discrimination by multiple circular harmonic components,” Appl. Opt. 23, 841–844 (1984).

H. H. Arsenault, M. Levesque, “Combined homomorphic and local-statistics processing for restoration of images degraded by signal-dependent noise,” Appl. Opt. 23, 845–850 (1984).

H. H. Arsenault, M. Denis, “Image processing in signal-dependent noise,” Can. J. Phys. 61, 309–317 (1983).

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

Y. N. Hsu, H. H. Arsenault, G. April, “Rotation-invariant digital pattern recognition using circular harmonic expansion,” Appl. Opt. 21, 4012–4015 (1982).

H. H. Arsenault, M. Denis, “Integral expression for transforming signal-dependent noise into signal-independent noise,” Opt. Lett. 6, 210–212 (1981).

H. H. Arsenault, C. Gendron, M. Denis, “Transformation of film-grain noise into signal-independent additive Gaussian noise,” J. Opt. Soc. Am. 71, 91–94 (1981).

M. Denis, H. H. Arsenault, “On the accuracy of a method to make film-grain noise independent of the signal,” Opt. Commun. 38, 166–169 (1981).

H. H. Arsenault, G. April, “Properties of speckle integrated with a finite aperture and logarithmically transformed,” J. Opt. Soc. Am. 66, 1160–1163 (1976).

Asher, R. B.

Bartelt, H. O.

Chavel, P.

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive restoration of images with speckle,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 373–382 (1987).

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive noise smoothing filter for images with signal-dependent noise,” IEEE Trans. Pattern Anal. Mach. Intell.PAMI-7, 165–177 (1985).

Chin, C.

Choens, R. C.

J. F. Walkup, R. C. Choens, “Image processing in signal-dependent noise,” Opt. Eng. 13, 258–266 (1974).

Christensen, C. R.

A. K. Jain, C. R. Christensen, “Digital processing of images in speckle noise,” in Applications of Speckle Phenomena, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 243, 46–50 (1980).

Denis, M.

H. H. Arsenault, M. Denis, “Image processing in signal-dependent noise,” Can. J. Phys. 61, 309–317 (1983).

M. Denis, H. H. Arsenault, “On the accuracy of a method to make film-grain noise independent of the signal,” Opt. Commun. 38, 166–169 (1981).

H. H. Arsenault, C. Gendron, M. Denis, “Transformation of film-grain noise into signal-independent additive Gaussian noise,” J. Opt. Soc. Am. 71, 91–94 (1981).

H. H. Arsenault, M. Denis, “Integral expression for transforming signal-dependent noise into signal-independent noise,” Opt. Lett. 6, 210–212 (1981).

Downie, J. D.

Flavin, M. A.

Froehlich, G. K.

Gendron, C.

Goldstein, E. L.

Goodman, J. W.

Hassebrook, L.

Horner, J. L.

Hsu, Y. N.

Isberg, T. A.

Jain, A. K.

A. K. Jain, C. R. Christensen, “Digital processing of images in speckle noise,” in Applications of Speckle Phenomena, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 243, 46–50 (1980).

Javidi, B.

Kasturi, R.

Krile, T. F.

Kuan, D. T.

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive restoration of images with speckle,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 373–382 (1987).

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive noise smoothing filter for images with signal-dependent noise,” IEEE Trans. Pattern Anal. Mach. Intell.PAMI-7, 165–177 (1985).

Leclerc, L.

Lee, J. S.

J. S. Lee, “Digital image enhancement and noise filtering by use of local statistics,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-2, 165–168 (1980).

Leger, J. R.

Levesque, M.

Lo, C. M.

C. M. Lo, “Estimation of image signals with Poisson noise,” Rep. 890 (Image Processing Institute, University of California, Los Angeles, Calif., 1979).

Marron, J.

Morris, G. M.

J. Marron, G. M. Morris, “Image recognition in the presence of laser speckle,” J. Opt. Soc. Am. A 3, 964–971 (1986).

T. A. Isberg, G. M. Morris, “Rotation-invariant image recognition at low light levels,” J. Opt. Soc. Am. A 3, 954–963 (1986).

G. M. Morris, “Pattern recognition using photon-limited images,” in Optical processing and Computing, H. H. Arse-nault, T. Szoplik, B. Macukow, eds. (Academic, Boston, Mass., 1989).

Naderi, F.

Premont, G.

Prucnal, P. R.

Saleh, B. E. A.

Sawchuk, A. A.

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive restoration of images with speckle,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 373–382 (1987).

F. Naderi, A. A. Sawchuk, “Detection of low-contrast images in film-grain noise,” Appl. Opt. 17, 2883–2891 (1978).

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive noise smoothing filter for images with signal-dependent noise,” IEEE Trans. Pattern Anal. Mach. Intell.PAMI-7, 165–177 (1985).

Sheng, Y.

Shubnikov, E. I.

E. I. Shubnikov, “Effect of additive and multiplicative noise in the correlation comparison of images,” Opt. Spectrosc. (USSR) 62, 389–392 (1987).

Siedentopf, H.

H. Siedentopf, “Concerning granularity, density fluctuations and the enlargement of photographic negatives,” Phys. Z. 38, 454 (1937).

Strand, T. C.

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive restoration of images with speckle,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 373–382 (1987).

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive noise smoothing filter for images with signal-dependent noise,” IEEE Trans. Pattern Anal. Mach. Intell.PAMI-7, 165–177 (1985).

Tang, Q.

Tur, M.

Vijaya Kumar, B. V. K.

Walkup, J. F.

Wallis, R.

R. Wallis, “An approach to the space variant restoration and enhancement of images,” in Proceedings of the Symposium on Current Mathematical Problems in Image Science (Naval Postgraduate School, Monterey, Calif., 1976).

Wang, J.

Appl. Opt.

F. Naderi, A. A. Sawchuk, “Detection of low-contrast images in film-grain noise,” Appl. Opt. 17, 2883–2891 (1978).

G. K. Froehlich, J. F. Walkup, T. F. Krile, “Estimation in signal-dependent film-grain noise,” Appl. Opt. 20, 3619–3626 (1981).

M. Tur, C. Chin, J. W. Goodman, “When is speckle noise multiplicative?,” Appl. Opt. 21, 1157–1159 (1982).

Y. N. Hsu, H. H. Arsenault, G. April, “Rotation-invariant digital pattern recognition using circular harmonic expansion,” Appl. Opt. 21, 4012–4015 (1982).

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

R. Kasturi, J. F. Walkup, T. F. Krile, “Image restoration by transformation of signal-dependent noise to signal-independent noise,” Appl. Opt. 22, 3537–3542 (1983).

Y. N. Hsu, H. H. Arsenault, “Pattern discrimination by multiple circular harmonic components,” Appl. Opt. 23, 841–844 (1984).

H. H. Arsenault, M. Levesque, “Combined homomorphic and local-statistics processing for restoration of images degraded by signal-dependent noise,” Appl. Opt. 23, 845–850 (1984).

P. R. Prucnal, E. L. Goldstein, “Exact variance stabilizing transformations for image signal-dependent Rayleigh and other Weibull noise sources,” Appl. Opt. 26, 1038–1041 (1987).

L. Leclerc, Y. Sheng, H. H. Arsenault, “Rotation-invariant phase-only and binary phase-only correlation,” Appl. Opt. 28, 1251–1256 (1989).

M. A. Flavin, J. L. Horner, “Amplitude encoded phase-only filters,” Appl. Opt. 28, 1692–1696 (1989).

B. V. K. Vijaya Kumar, L. Hassebrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 (1990).

B. Javidi, J. Wang, Q. Tang, “Multiple-object binary joint transform correlation using multiple-level threshold cross-ing,” Appl. Opt. 30, 4234–4244 (1991).

L. Leclerc, Y. Sheng, H. H. Arsenault, “Optical binary phase-only filters for circular harmonic correlations,” Appl. Opt. 30, 4643–4649 (1991).

G. Premont, Y. Sheng, “Fast design of circular-harmonic filters using simulated annealing,” Appl. Opt. 32, 3116–3121 (1993).

J. L. Horner, H. O. Bartelt, “Two-bit correlation,” Appl. Opt. 24, 2889–2893 (1985).

J. L. Horner, J. R. Leger, “Pattern recognition with binary phase-only filters,” Appl. Opt. 24, 609–611 (1985).

J. L. Horner, “Metrics for assessing pattern-recognition perfor-mance,” Appl. Opt. 31, 165–166 (1992).

Can. J. Phys.

H. H. Arsenault, M. Denis, “Image processing in signal-dependent noise,” Can. J. Phys. 61, 309–317 (1983).

IEEE Trans. Acoust. Speech Signal Process.

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive restoration of images with speckle,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 373–382 (1987).

IEEE Trans. Pattern Anal. Mach. Intell.

J. S. Lee, “Digital image enhancement and noise filtering by use of local statistics,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-2, 165–168 (1980).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

M. Denis, H. H. Arsenault, “On the accuracy of a method to make film-grain noise independent of the signal,” Opt. Commun. 38, 166–169 (1981).

Opt. Eng.

J. F. Walkup, R. C. Choens, “Image processing in signal-dependent noise,” Opt. Eng. 13, 258–266 (1974).

Opt. Lett.

Opt. Spectrosc. (USSR)

E. I. Shubnikov, “Effect of additive and multiplicative noise in the correlation comparison of images,” Opt. Spectrosc. (USSR) 62, 389–392 (1987).

Phys. Z.

H. Siedentopf, “Concerning granularity, density fluctuations and the enlargement of photographic negatives,” Phys. Z. 38, 454 (1937).

Other

C. M. Lo, “Estimation of image signals with Poisson noise,” Rep. 890 (Image Processing Institute, University of California, Los Angeles, Calif., 1979).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

G. M. Morris, “Pattern recognition using photon-limited images,” in Optical processing and Computing, H. H. Arse-nault, T. Szoplik, B. Macukow, eds. (Academic, Boston, Mass., 1989).

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive noise smoothing filter for images with signal-dependent noise,” IEEE Trans. Pattern Anal. Mach. Intell.PAMI-7, 165–177 (1985).

A. K. Jain, C. R. Christensen, “Digital processing of images in speckle noise,” in Applications of Speckle Phenomena, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 243, 46–50 (1980).

R. Wallis, “An approach to the space variant restoration and enhancement of images,” in Proceedings of the Symposium on Current Mathematical Problems in Image Science (Naval Postgraduate School, Monterey, Calif., 1976).

Research and Education Association, Handbook of Mathematical, Scientific, and Engineering Formulas, Tables, Functions, Graphs, Transforms, (Research and Education Association, N.J., 1994), Chap. 16, p. 595.

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Figures (5)

Fig. 1
Fig. 1

Input images of the letter E of dimensions 64 × 64 pixels: LE000 (upper left) and LE045 (upper right). Input images of an X-29 aircraft of dimensions 128 × 128 pixels: X29-000 (lower left) and X29-090 (lower right).

Fig. 2
Fig. 2

Detection of LE045 with the POCHF (generated from LE000, m = 2): (a) the original input image and the corresponding correlation (three-dimensional view and top view); (b) particular noise realization of the input image degraded with speckle SDN (M = 1) and the corresponding noisy correlation showing one false alarm; (c) the input image restored by the homomorphic processing with a window of dimensions 5 × 5 pixels and the resulting correlation. The correlation-peak intensity is normalized in each case.

Fig. 3
Fig. 3

Detection of the aircraft X29-000 with the POCHF (m = 2): (a) original input image and corresponding correlation (three-dimensional view and top view); (b) particular noise realization of the input image degraded with speckle SDN (M = 1) and the corresponding noisy correlation showing two false alarms; (c) the input image restored with the LLMMSE estimator designed for multiplicative noise with a processing window of dimensions 5 × 5 pixels and the resulting correlation. The correlation-peak intensity is normalized in each case.

Fig. 4
Fig. 4

Correlation results as a function of the parameter M of the POCHF (m = 2) of the aircraft X29-000 degraded with speckle SDN and preprocessed by the LLMMSE estimator with a window of dimensions 3 × 3 pixels and the homomorphic processing with a window of dimensions 5 × 5 pixels. Each data point is generated from 104 noise realizations or preprocessed realizations of the correlation. (a) Correlation-peak SNR and (b) false-alarm probability P FA. SNRin is the SNR measured in the input image.

Fig. 5
Fig. 5

Correlation results for the object LE000 degraded with film-grain SDN (p = 0.5) with the CHF, the POCHF, and the BPOCHF (m = 2). Each data point is generated from 104 noise realizations of the correlation. (a) Correlation-peak SNR versus the square of the SNR in the input image and (b) false-alarm probability P FA versus the SNR in the input image. Here, SNR in = S 0 / k [please see Eq. (6)#x0005D; with S 0 = 255.

Tables (6)

Tables Icon

Table 1 Correlation Results for Three of the Objects in Fig. 1 without Noise and with Speckle SDN (M = 1) for the CHF, the POCHF, and the BPOCHF (m = 2)a

Tables Icon

Table 2 Correlation Results of the POCHF (m = 2) of the Letter E Degraded with Speckle SDN (N = 1), and Preprocessed with the LLMMSE Estimator Designed for Multiplicative Noise or the Homomorphic Processing (H.T. + J.–S.) with a Window of Dimensions 3 × 3 or 5 × 5 Pixelsa

Tables Icon

Table 3 Correlation Results of the POCHF (m = 2) of the Object LE045 Degraded with Speckle SDN (M = 1) for Different Values of the Radial Cutoff Frequency ρ0 of the Filter, and Preprocessed (without a Reduction of ρ0) by the LLMMSE Estimator Designed for Multiplicative Noise or the Homomorphic Processing (H.T. + J.-S.) with a Window of Dimensions 3 × 3 or 5 × 5 Pixelsa

Tables Icon

Table 4 Correlation Results of the BPOCHF (m = 2) of the Object LE000 Degraded with Speckle SDN (M = 1), and Preprocessed by the LLMMSE Estimator Designed for Multiplicative Noise or the Homomorphic Processing (H.T. + J.-S.) with a Window of Dimensions 3 × 3 Pixels a

Tables Icon

Table 5 Correlation Results of the POCHF (m = 2) of the X-29 Aircraft Degraded with Speckle SDN (M = 1), and Preprocessed by the LLMMSE Estimator Designed for Multiplicative Noise or the Homomorphic Processing (H.T. + J.-S.) with a Window of Dimensions 3 × 3 or 5 × 5 Pixelsa

Tables Icon

Table 6 Correlation Results of the POCHF (m = 2) of the Object LE000 Degraded with Film-Grain SDN p = 0.5, SNR in = S 0 / k = 1.0 for Different Values of the Radial Cutoff Frequency ρ0 of the Filter, and Preprocessed (without a Reduction of ρ0) by the Homomorphic Processing (H.T. + J.-S.) with a Window of Dimensions 3 × 3 Pixelsa

Equations (39)

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

R _ sp ( x ) = S ( x ) N _ sp ( x ) ,
P R _ s p ( R _ s p ) = { M M Γ ( M ) S ( R sp S ) M - 1 exp ( - M R sp S ) R s p > 0 0 otherwise ,
SNR in = S ( x ) σ R ,
( SNR in ) sp = M .
R _ ( x ) = S ( x ) + k S P ( x ) N _ ( x ) ,
( SNR in ) fg = S 1 - p ( x ) k ,
S ^ ( i , j ) = R ¯ i j + Q i j + [ R _ ( i , j ) - R ¯ i j ] ,
Q i j + = Max [ ( 1 - R ¯ i j 2 σ N sp 2 V i j 1 + σ N sp 2 ) , 0 ] ,
R ¯ i j = 1 ( 2 M + 1 ) ( 2 N + 1 ) k = i - M i + M l = j - n j + N R _ ( k , l ) ,
V i j = 1 [ ( 2 M + 1 ) ( 2 N + 1 ) - 1 ] × k = i - M i + M l = j - N j + N [ R _ ( k , l ) - R ¯ k l ] 2 .
R _ = g ( R _ ) = C d R σ R ( R ) ,
R _ sp ( x ) = C M ln [ R _ sp ( x ) ] = C M { ln [ S ( x ) ] + ln [ N _ sp ( x ) ] } .
R _ sp ( x ) = C M { ln [ S ( x ) ] + Γ ( M ) Γ ( M ) - ln M } ,
σ R sp 2 = C 2 M d d x [ Γ ( x ) Γ ( x ) ] | x = M = C 2 M n = 1 1 ( n + M - 1 ) 2 ,
N _ sp ( x ) = C M [ Γ ( M ) Γ ( M ) - ln M ] ,
R _ sp ( x ) M = 1 = C { ln [ S ( x ) ] - γ } ,
σ R sp 2 M = 1 = C 2 π 2 6 ,
R _ sp ( i , j ) = 255 ln ( 256 ) ln [ R _ sp ( i , j ) + 1 ] ,
R _ sp ( i , j ) = exp [ R _ sp ( i , j ) ln ( 256 ) 255 ] - 1 ,
σ R sp M = 1 = σ N sp M = 1 = π 6 255 ln ( 256 ) 59.0 ,
R _ ( x ) = C k ( 1 - p ) R _ 1 - p ( x ) ,
R _ ( i , j ) = 255 R _ ( i , j ) ,
R _ ( i , j ) = R _ 2 ( i , j ) 255 .
R _ = 255 S ,
σ R = σ N = C = k 255 2 ,
Q i j + = Max [ ( 1 - σ N 2 V i j ) , 0 ] .
S ^ ( i , j ) = ( R ¯ i j + γ C ) + Q i j + [ R _ ( i , j ) - R ¯ i j ] .
MSD _ = 1 M N i = 1 M j = 1 N [ R _ ( i , j ) - S ( i , j ) ] 2 ,
f ( r , θ ) = m = - f m ( r ) exp ( j m θ ) ,
f m ( r ) = 1 2 π 0 2 π f ( r , θ ) exp ( - j m θ ) d θ .
F m ( ρ ) F m ( ρ ) exp [ j α m ( ρ ) ] = 2 π ( - j ) m 0 f m ( r ) J m ( 2 π r ρ ) r d r .
H φ , m ( ρ , ϕ ) = exp { j [ α m ( ρ ) + m ϕ ] } .
H φ BC , m ( ρ , ϕ ) = { 1 if cos [ α m ( ρ ) + m ϕ ] > 0 - 1 otherwise ,
H φ BS , m ( ρ , ϕ ) = { 1 if sin [ α m ( ρ ) + m ϕ ] > 0 - 1 otherwise .
SNR = C _ ( 0 ) 2 var [ C _ ( 0 ) ] ,
SNR sp = M | - - F ( u ) H * ( u ) d 2 u | 2 - - | F ( u ) H ( u ) | 2 d 2 u ,
SNR fg = | - - f ( x ) h ( x ) d 2 x | 2 k 2 - - f p ( x ) 2 h ( x ) 2 d 2 x ,
SNRI = I p 1 N Ω i Ω [ I ( ξ i , η i ) - I ¯ ] 2 ,
I ¯ = 1 N Ω i Ω I ( ξ i , η i ) .

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