Abstract

We consider the optimal likelihood algorithm for the estimation of a target location when the images are corrupted by substitutive noise. We show the relationship between the optimal algorithm and the sliced orthogonal nonlinear generalized (SONG) correlation. The SONG correlation is based on the application of a linear correlation to corresponding binary slices of both the input scene and the reference object with appropriate weight factors. For a particular case, we show that the optimal strategy is a function of only the number of pixels for which the gray values in the noisy image match the ones of the reference image when the substitutive noise is uniformly distributed. This is exactly what a particular definition of the SONG correlation does.

© 2001 Optical Society of America

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References

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  1. I. Pitas, A. N. Venetsanopoulos, Nonlinear Digital Filters: Principles and Applications (Kluwer Academic, Boston, 1990).
  2. J. P. Fitch, E. J. Coyle, N. C. Gallagher, “Median filtering by threshold decomposition,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 1183–1188 (1984).
    [CrossRef]
  3. P. Garcia-Martinez, H. H. Arsenault, “A correlation matrix representation using sliced orthogonal nonlinear generalized decomposition,” Opt. Commun. 174, 503–515 (2000).
    [CrossRef]
  4. P. Maragos, “Morphological approaches to image matching and object detection,” in ICASSP-89: 1989 International Conference on Acoustic, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1989), Vol 3, pp. 1568–1571.
  5. P. Garcia-Martinez, D. Mas, J. Garcia, C. Ferreira, “Nonlinear morphological correlation: optoelectronic implementation,” Appl. Opt. 37, 2112–2118 (1998).
    [CrossRef]
  6. P. Garcia-Martinez, H. H. Arsenault, S. Roy, “Optical implementation of the sliced orthogonal nonlinear generalized correlation for images degrades by nonoverlapping background noise,” Opt. Commun. 173, 185–193 (2000).
    [CrossRef]
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    [CrossRef] [PubMed]
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  9. G. L. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theor. IT-6, 311–319 (1960).
    [CrossRef]
  10. A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
  11. Ph. Réfrégier, V. Laude, B. Javidi, “Basic properties of nonlinear global filtering techniques and optimal discriminant solutions,” Appl. Opt. 34, 3915–3923 (1995).
    [CrossRef] [PubMed]
  12. Ph. Réfrégier, F. Goudail, “Decision theory approach to nonlinear joint-tranform correlation,” J. Opt. Soc. Am. A 15, 61–67 (1998).
    [CrossRef]
  13. P. Garcia-Martinez, H. H. Arsenault, C. Ferreira, “Improved rotation invariant pattern recognition using circular harmonics of binary gray-level slices,” Opt. Commun. 185, 41–48 (2000).
    [CrossRef]

2000

P. Garcia-Martinez, H. H. Arsenault, “A correlation matrix representation using sliced orthogonal nonlinear generalized decomposition,” Opt. Commun. 174, 503–515 (2000).
[CrossRef]

P. Garcia-Martinez, H. H. Arsenault, S. Roy, “Optical implementation of the sliced orthogonal nonlinear generalized correlation for images degrades by nonoverlapping background noise,” Opt. Commun. 173, 185–193 (2000).
[CrossRef]

P. Garcia-Martinez, H. H. Arsenault, C. Ferreira, “Improved rotation invariant pattern recognition using circular harmonics of binary gray-level slices,” Opt. Commun. 185, 41–48 (2000).
[CrossRef]

1998

1995

1994

1989

1984

J. P. Fitch, E. J. Coyle, N. C. Gallagher, “Median filtering by threshold decomposition,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 1183–1188 (1984).
[CrossRef]

1964

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

1960

G. L. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theor. IT-6, 311–319 (1960).
[CrossRef]

Arsenault, H. H.

P. Garcia-Martinez, H. H. Arsenault, “A correlation matrix representation using sliced orthogonal nonlinear generalized decomposition,” Opt. Commun. 174, 503–515 (2000).
[CrossRef]

P. Garcia-Martinez, H. H. Arsenault, S. Roy, “Optical implementation of the sliced orthogonal nonlinear generalized correlation for images degrades by nonoverlapping background noise,” Opt. Commun. 173, 185–193 (2000).
[CrossRef]

P. Garcia-Martinez, H. H. Arsenault, C. Ferreira, “Improved rotation invariant pattern recognition using circular harmonics of binary gray-level slices,” Opt. Commun. 185, 41–48 (2000).
[CrossRef]

Coyle, E. J.

J. P. Fitch, E. J. Coyle, N. C. Gallagher, “Median filtering by threshold decomposition,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 1183–1188 (1984).
[CrossRef]

Ferreira, C.

P. Garcia-Martinez, H. H. Arsenault, C. Ferreira, “Improved rotation invariant pattern recognition using circular harmonics of binary gray-level slices,” Opt. Commun. 185, 41–48 (2000).
[CrossRef]

P. Garcia-Martinez, D. Mas, J. Garcia, C. Ferreira, “Nonlinear morphological correlation: optoelectronic implementation,” Appl. Opt. 37, 2112–2118 (1998).
[CrossRef]

Fitch, J. P.

J. P. Fitch, E. J. Coyle, N. C. Gallagher, “Median filtering by threshold decomposition,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 1183–1188 (1984).
[CrossRef]

Gallagher, N. C.

J. P. Fitch, E. J. Coyle, N. C. Gallagher, “Median filtering by threshold decomposition,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 1183–1188 (1984).
[CrossRef]

Garcia, J.

Garcia-Martinez, P.

P. Garcia-Martinez, H. H. Arsenault, S. Roy, “Optical implementation of the sliced orthogonal nonlinear generalized correlation for images degrades by nonoverlapping background noise,” Opt. Commun. 173, 185–193 (2000).
[CrossRef]

P. Garcia-Martinez, H. H. Arsenault, C. Ferreira, “Improved rotation invariant pattern recognition using circular harmonics of binary gray-level slices,” Opt. Commun. 185, 41–48 (2000).
[CrossRef]

P. Garcia-Martinez, H. H. Arsenault, “A correlation matrix representation using sliced orthogonal nonlinear generalized decomposition,” Opt. Commun. 174, 503–515 (2000).
[CrossRef]

P. Garcia-Martinez, D. Mas, J. Garcia, C. Ferreira, “Nonlinear morphological correlation: optoelectronic implementation,” Appl. Opt. 37, 2112–2118 (1998).
[CrossRef]

Goudail, F.

Javidi, B.

Laude, V.

Maragos, P.

P. Maragos, “Morphological approaches to image matching and object detection,” in ICASSP-89: 1989 International Conference on Acoustic, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1989), Vol 3, pp. 1568–1571.

Mas, D.

Pitas, I.

I. Pitas, A. N. Venetsanopoulos, Nonlinear Digital Filters: Principles and Applications (Kluwer Academic, Boston, 1990).

Réfrégier, Ph.

Roy, S.

P. Garcia-Martinez, H. H. Arsenault, S. Roy, “Optical implementation of the sliced orthogonal nonlinear generalized correlation for images degrades by nonoverlapping background noise,” Opt. Commun. 173, 185–193 (2000).
[CrossRef]

Turin, G. L.

G. L. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theor. IT-6, 311–319 (1960).
[CrossRef]

VanderLugt, A.

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

Venetsanopoulos, A. N.

I. Pitas, A. N. Venetsanopoulos, Nonlinear Digital Filters: Principles and Applications (Kluwer Academic, Boston, 1990).

Appl. Opt.

IEEE Trans. Acoust. Speech Signal Process.

J. P. Fitch, E. J. Coyle, N. C. Gallagher, “Median filtering by threshold decomposition,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 1183–1188 (1984).
[CrossRef]

IEEE Trans. Inf. Theory

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

IRE Trans. Inf. Theor.

G. L. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theor. IT-6, 311–319 (1960).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

P. Garcia-Martinez, H. H. Arsenault, C. Ferreira, “Improved rotation invariant pattern recognition using circular harmonics of binary gray-level slices,” Opt. Commun. 185, 41–48 (2000).
[CrossRef]

P. Garcia-Martinez, H. H. Arsenault, “A correlation matrix representation using sliced orthogonal nonlinear generalized decomposition,” Opt. Commun. 174, 503–515 (2000).
[CrossRef]

P. Garcia-Martinez, H. H. Arsenault, S. Roy, “Optical implementation of the sliced orthogonal nonlinear generalized correlation for images degrades by nonoverlapping background noise,” Opt. Commun. 173, 185–193 (2000).
[CrossRef]

Opt. Lett.

Other

I. Pitas, A. N. Venetsanopoulos, Nonlinear Digital Filters: Principles and Applications (Kluwer Academic, Boston, 1990).

P. Maragos, “Morphological approaches to image matching and object detection,” in ICASSP-89: 1989 International Conference on Acoustic, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1989), Vol 3, pp. 1568–1571.

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Figures (3)

Fig. 1
Fig. 1

Input scene with the reference object, located in the lower part of the figure, and another object to be rejected.

Fig. 2
Fig. 2

Input scene highly degraded with uniform distribution random noise.

Fig. 3
Fig. 3

(a) Three-dimensional plot of the SONG (Ω P ) correlation. (b) Three-dimensional plot of the common phase-only filter. This method is not able to detect the reference object.

Tables (1)

Tables Icon

Table 1 Discrimination Capability (DC) of Several Pattern-Recognition Operations and the New SONG Correlation When Different Noise Degrees Are Considered

Equations (32)

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

emfx, yenfx, y=0  if mn,emfx, yenfx, y=1  if m=n.
fx, y=i=0Q-1 Fieifx, y,
eifx, y=1fx, y=i0otherwise.
gx, yfx, y=i=0Q-1j=0Q-1 GiFjeigx, yejfx, y,
Ωgfx, y=i=0Q-1j=0Q-1 Wijeigx, yejfx, y.
Wij=0  for ij,Wij=1  for i=j.
ΩgfPx, y=i=0Q-1 eigx, yeifx, y.
gi=1-bifi+bixi.
L=Pgi|fi.
l=i=1NlnPgi|fi,
l=i=1N δgi-fiGfi+Bg,
gi=1-bifi-k+bixi.
lk=i=1NlnPgi|fi-k.
lk=i=1N δgi-fi-kGfi-k+Bg.
lk=i=1N δgi-fi-kGfi-k.
Gfi=lnq+pa-lnpa,
l˜k=i=1N δgi-fi-k,
SNR=Ni/Nc-1,
DC=1-CrossCorrAutoCorr.
Pgi|fi=Pgi, bi=0|fi+Pgi, bi=1|fi.
Pgi, bi|fi=Pgi|fi, biPBbi,
Pgi|fi=qPgi|fi, bi=0+pPgi|fi, bi=1.
Pgi|fi, bi=1=PXgi.
Pgi|fi, bi=0=δgi-fi,
δy=1if y=00otherwise.
Pgi|fi=qδgi-fi+pPXgi
l=i=1Nlnqδgi-fi+pPXgi.
lnqδgi-fi+pPXgi=δgi-filnq+pPXgi+1-δgi-filnpPXgi,
lnqδgi-fi+pPXgi=δgi-fi[lnq+pPXfi-lnpPXfi]+lnpPXgi.
Gfi=lnq+pPXfi-lnpPXfi
Bg=i=1NlnpPXgi,
l=i=1N δgi-fiGfi+Bg.

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