Abstract

Binary images appear in various pattern recognition applications. It is thus important to design pattern recognition algorithms that are optimal for such images. We address the problem of target location in binary images perturbed with nonhomogeneous background noise. The proposed algorithm optimizes the likelihood ratio between the hypotheses that the target is present within a small subwindow of the image and that it is not present in this subwindow. The algorithm is shown to consist of two correlation operations and a few pointwise nonlinear transformations. With numerical simulations, we illustrate the efficiency of this technique, especially in the presence of strongly nonhomogeneous background noise.

© 1998 Optical Society of America

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    [CrossRef]
  8. H. Sjoberg, B. Noharet, L. Wosinski, R. Hey, “A compact optical correlator: preprocessing and filter encoding strategies applied to images with varying illumination,” Opt. Eng. (Bellingham) 37, 1316–1324 (1998).
    [CrossRef]
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  19. R. N. McDonough, A. D. Whalen, Detection of Signals in Noise (Academic, San Diego, Calif., 1995), pp. 151–196.
  20. L. L. Scharf, Statistical Signal Processing (Addison-Wesley, Reading, Mass., 1991), pp. 179–205.
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    [CrossRef]
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1998 (2)

H. Sjoberg, B. Noharet, L. Wosinski, R. Hey, “A compact optical correlator: preprocessing and filter encoding strategies applied to images with varying illumination,” Opt. Eng. (Bellingham) 37, 1316–1324 (1998).
[CrossRef]

F. Guérault, P. Réfrégier, “Unified statistically independent region processor for deterministic and fluctuating targets in nonoverlapping background,” Opt. Lett. 23, 412–414 (1998).
[CrossRef]

1997 (2)

1996 (2)

F. Goudail, P. Réfrégier, “Optimal and suboptimal detection of a target with random gray levels imbedded in non-overlapping noise,” Opt. Commun. 125, 211–216 (1996).
[CrossRef]

F. Goudail, P. Réfrégier, “Optimal detection of a target with random gray levels on a spatially disjoint noise,” Opt. Lett. 21, 495–497 (1996).
[CrossRef] [PubMed]

1994 (1)

1993 (1)

1990 (1)

1988 (1)

P. K. Sahoo, S. Soltani, A. K. C. Wong, “A survey of thresholding techniques,” Comput. Vision Graph. Image Process. 41, 233–260 (1988).
[CrossRef]

1984 (2)

1981 (1)

1979 (1)

Y. Nakagawa, A. Rosenfeld, “Some experiments on variable thresholding,” Pattern Recog. 11, 191–204 (1979).
[CrossRef]

1960 (1)

G. L. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theory 6, 311–329 (1960).
[CrossRef]

Azzalini, A.

A. Azzalini, Statistical Inference—Based on the Likelihood (Chapman & Hall, New York, 1996), pp. 116–122.

Bock, B.

S. Lindell, B. Bock, W. Hahn, D. Homan, G. Shapiro, “Optical processing at Lockheed Martin,” in Optical Pattern Recognition VIII, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE3073, 138–155 (1997).
[CrossRef]

Casasent, D.

Dydyk, B.

J. Karins, S. Mills, J. Ryan, B. Dydyk, J. Lucas, “Second generation miniature ruggedized optical correlator (mroc) module,” in Optical Pattern Recognition VIII, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE3073, 128–137 (1997).
[CrossRef]

Evans, M.

M. Evans, N. Hastings, B. Peacock, Statistical Distributions (Wiley, New York, 1993), p. 28.

Fazlollahi, A. H.

Garthwaite, P.

P. Garthwaite, I. Jolliffe, B. Jones, Statistical Inference (Prentice-HallEurope, London, 1995).

Gianino, P. D.

Goudail, F.

F. Goudail, P. Réfrégier, “Optimal detection of a target with random gray levels on a spatially disjoint noise,” Opt. Lett. 21, 495–497 (1996).
[CrossRef] [PubMed]

F. Goudail, P. Réfrégier, “Optimal and suboptimal detection of a target with random gray levels imbedded in non-overlapping noise,” Opt. Commun. 125, 211–216 (1996).
[CrossRef]

Guérault, F.

Hahn, W.

S. Lindell, B. Bock, W. Hahn, D. Homan, G. Shapiro, “Optical processing at Lockheed Martin,” in Optical Pattern Recognition VIII, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE3073, 138–155 (1997).
[CrossRef]

Hastings, N.

M. Evans, N. Hastings, B. Peacock, Statistical Distributions (Wiley, New York, 1993), p. 28.

Hey, R.

H. Sjoberg, B. Noharet, L. Wosinski, R. Hey, “A compact optical correlator: preprocessing and filter encoding strategies applied to images with varying illumination,” Opt. Eng. (Bellingham) 37, 1316–1324 (1998).
[CrossRef]

Homan, D.

S. Lindell, B. Bock, W. Hahn, D. Homan, G. Shapiro, “Optical processing at Lockheed Martin,” in Optical Pattern Recognition VIII, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE3073, 138–155 (1997).
[CrossRef]

Horner, J. L.

Jain, A. K.

A. K. Jain, Fundamentals of Digital Image Processing, Prentice-Hall Information and System Science Series (Prentice-Hall, N.J., 1989).

Javidi, B.

Jolliffe, I.

P. Garthwaite, I. Jolliffe, B. Jones, Statistical Inference (Prentice-HallEurope, London, 1995).

Jones, B.

P. Garthwaite, I. Jolliffe, B. Jones, Statistical Inference (Prentice-HallEurope, London, 1995).

Karins, J.

J. Karins, S. Mills, J. Ryan, B. Dydyk, J. Lucas, “Second generation miniature ruggedized optical correlator (mroc) module,” in Optical Pattern Recognition VIII, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE3073, 128–137 (1997).
[CrossRef]

Kumar, V.

Lindell, S.

S. Lindell, B. Bock, W. Hahn, D. Homan, G. Shapiro, “Optical processing at Lockheed Martin,” in Optical Pattern Recognition VIII, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE3073, 138–155 (1997).
[CrossRef]

Lucas, J.

J. Karins, S. Mills, J. Ryan, B. Dydyk, J. Lucas, “Second generation miniature ruggedized optical correlator (mroc) module,” in Optical Pattern Recognition VIII, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE3073, 128–137 (1997).
[CrossRef]

McDonough, R. N.

R. N. McDonough, A. D. Whalen, Detection of Signals in Noise (Academic, San Diego, Calif., 1995), pp. 151–196.

Mills, S.

J. Karins, S. Mills, J. Ryan, B. Dydyk, J. Lucas, “Second generation miniature ruggedized optical correlator (mroc) module,” in Optical Pattern Recognition VIII, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE3073, 128–137 (1997).
[CrossRef]

Nakagawa, Y.

Y. Nakagawa, A. Rosenfeld, “Some experiments on variable thresholding,” Pattern Recog. 11, 191–204 (1979).
[CrossRef]

Noharet, B.

H. Sjoberg, B. Noharet, L. Wosinski, R. Hey, “A compact optical correlator: preprocessing and filter encoding strategies applied to images with varying illumination,” Opt. Eng. (Bellingham) 37, 1316–1324 (1998).
[CrossRef]

Peacock, B.

M. Evans, N. Hastings, B. Peacock, Statistical Distributions (Wiley, New York, 1993), p. 28.

Réfrégier, P.

Rosenfeld, A.

Y. Nakagawa, A. Rosenfeld, “Some experiments on variable thresholding,” Pattern Recog. 11, 191–204 (1979).
[CrossRef]

Ryan, J.

J. Karins, S. Mills, J. Ryan, B. Dydyk, J. Lucas, “Second generation miniature ruggedized optical correlator (mroc) module,” in Optical Pattern Recognition VIII, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE3073, 128–137 (1997).
[CrossRef]

Sahoo, P. K.

P. K. Sahoo, S. Soltani, A. K. C. Wong, “A survey of thresholding techniques,” Comput. Vision Graph. Image Process. 41, 233–260 (1988).
[CrossRef]

Scharf, L. L.

L. L. Scharf, Statistical Signal Processing (Addison-Wesley, Reading, Mass., 1991), pp. 179–205.

Shapiro, G.

S. Lindell, B. Bock, W. Hahn, D. Homan, G. Shapiro, “Optical processing at Lockheed Martin,” in Optical Pattern Recognition VIII, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE3073, 138–155 (1997).
[CrossRef]

Sjoberg, H.

H. Sjoberg, B. Noharet, L. Wosinski, R. Hey, “A compact optical correlator: preprocessing and filter encoding strategies applied to images with varying illumination,” Opt. Eng. (Bellingham) 37, 1316–1324 (1998).
[CrossRef]

Soltani, S.

P. K. Sahoo, S. Soltani, A. K. C. Wong, “A survey of thresholding techniques,” Comput. Vision Graph. Image Process. 41, 233–260 (1988).
[CrossRef]

Turin, G. L.

G. L. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theory 6, 311–329 (1960).
[CrossRef]

Wang, J.

Whalen, A. D.

R. N. McDonough, A. D. Whalen, Detection of Signals in Noise (Academic, San Diego, Calif., 1995), pp. 151–196.

Willet, P.

Wong, A. K. C.

P. K. Sahoo, S. Soltani, A. K. C. Wong, “A survey of thresholding techniques,” Comput. Vision Graph. Image Process. 41, 233–260 (1988).
[CrossRef]

Wosinski, L.

H. Sjoberg, B. Noharet, L. Wosinski, R. Hey, “A compact optical correlator: preprocessing and filter encoding strategies applied to images with varying illumination,” Opt. Eng. (Bellingham) 37, 1316–1324 (1998).
[CrossRef]

Appl. Opt. (3)

Comput. Vision Graph. Image Process. (1)

P. K. Sahoo, S. Soltani, A. K. C. Wong, “A survey of thresholding techniques,” Comput. Vision Graph. Image Process. 41, 233–260 (1988).
[CrossRef]

IRE Trans. Inf. Theory (1)

G. L. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theory 6, 311–329 (1960).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

F. Goudail, P. Réfrégier, “Optimal and suboptimal detection of a target with random gray levels imbedded in non-overlapping noise,” Opt. Commun. 125, 211–216 (1996).
[CrossRef]

Opt. Eng. (Bellingham) (1)

H. Sjoberg, B. Noharet, L. Wosinski, R. Hey, “A compact optical correlator: preprocessing and filter encoding strategies applied to images with varying illumination,” Opt. Eng. (Bellingham) 37, 1316–1324 (1998).
[CrossRef]

Opt. Lett. (4)

Pattern Recog. (1)

Y. Nakagawa, A. Rosenfeld, “Some experiments on variable thresholding,” Pattern Recog. 11, 191–204 (1979).
[CrossRef]

Other (8)

J. Karins, S. Mills, J. Ryan, B. Dydyk, J. Lucas, “Second generation miniature ruggedized optical correlator (mroc) module,” in Optical Pattern Recognition VIII, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE3073, 128–137 (1997).
[CrossRef]

S. Lindell, B. Bock, W. Hahn, D. Homan, G. Shapiro, “Optical processing at Lockheed Martin,” in Optical Pattern Recognition VIII, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE3073, 138–155 (1997).
[CrossRef]

M. Evans, N. Hastings, B. Peacock, Statistical Distributions (Wiley, New York, 1993), p. 28.

A. Azzalini, Statistical Inference—Based on the Likelihood (Chapman & Hall, New York, 1996), pp. 116–122.

P. Garthwaite, I. Jolliffe, B. Jones, Statistical Inference (Prentice-HallEurope, London, 1995).

R. N. McDonough, A. D. Whalen, Detection of Signals in Noise (Academic, San Diego, Calif., 1995), pp. 151–196.

L. L. Scharf, Statistical Signal Processing (Addison-Wesley, Reading, Mass., 1991), pp. 179–205.

A. K. Jain, Fundamentals of Digital Image Processing, Prentice-Hall Information and System Science Series (Prentice-Hall, N.J., 1989).

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Figures (8)

Fig. 1
Fig. 1

(a) Example of a gray-scale image containing an object of interest. (b) Binarized version of (a). (c) Reference object w to be searched for in (b). (d) Edge-enhanced version of (a). (e) Binarized version of (d). (f) Reference object w to be searched for in (e).

Fig. 2
Fig. 2

(a) Scene without noise. (b) Noisy version of (a) with μ=0.3 and η=0.2. (c) Noisy version of (a) with μ=0.5 and η=0.2.

Fig. 3
Fig. 3

Diagram of the two hypotheses considered in the image model.

Fig. 4
Fig. 4

Probability of correct location for the ML processor with known noise parameters rMLk(j) [see Eq. (22)], the ML processor with estimated parameters rML(j) [see Eq. (20)], and the MLRT processor r(j) [see Eq. (17)] as a function of the target noise parameter μ. The criterion for good location was a 1×1 pixel window. Probability of correct location was estimated on 100 realizations of the noise for each value of μ. The value of η was set to 0.2.

Fig. 5
Fig. 5

(a) Binary reference target w used in the simulation. (b) Window F used in the simulation. (c) Example of an image with nonhomogeneous noise for which the ML processor rML(j) fails. (d) Output plane of the ML processor rML(j) applied to image (c). (d) Output plane of the MLRT processor r(j) applied to image (c).

Fig. 6
Fig. 6

Probability of correct location for the ML processor rML(j) [see Eq. (20)] and for the MLRT processor r(j) [see Eq. (17)] for increasing values f of the nonhomogeneous background noise. f is the parameter of the power law on which the background noise is based [see Eq. (24)]. The value of μ is set to 0.1.

Fig. 7
Fig. 7

(a) Example of a realistic scene. (b) Image (a) edge enhanced with a Sobel operator and binarized with a threshold of 0.005 (after the edge-enhanced image has been rescaled so that its levels go from 0 to 1). (c) Image (a) edge enhanced with a Sobel operator and binarized with a threshold of 0.001. (d) Reference object w used in the simulation. (e) Result of processing image (c) with the ML processor rML(j) [see Eq. (20)]: The object is not correctly located. (f) Result of processing image (c) with the MLRT processor r(j) [see Eq. (17)]. The position of the maximum of this plane corresponds to the true position of the object in scene (c): The object is correctly located.

Fig. 8
Fig. 8

Probability of correct location for the MLRT processor compared with that for the ML processor when only hypothesis H1 is used (named “ML with F” in the figure).

Tables (1)

Tables Icon

Table 1 Rate of Correct Location for the ML Processor with Estimated Parameters rML(j) [See Eq. (20)] and the MLRT Processor r(j) [See Eq. (17)] for Two Different Binarization Thresholdsa

Equations (39)

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

si=nitwi-j+nib(1-wi-j),
nit=1withprobability1-μ0withprobabilityμ,
nib=1withprobabilityη0withprobability1-η.
iFj,si=nitwi-j+nib(1-wi-j).
iFj,si=nib.
L1(μ, η, j)=iwj[si(1-μ)+(1-si)μ]×iFj-wj[(siη+(1-si)(1-η)],
L2(ρ, j)=iFj[(siρ+(1-si)(1-ρ)],
l˜1(μ, η, j)=i=0N-1[si ln(1-μ)+(1-si)ln μ]wi-j+i=0N-1[si ln η+(1-si)ln(1-η)]×(Fi-j-wi-j),
l˜2(ρ, j)=i=0N-1[si ln ρ+(1-si)ln(1-ρ)]Fi-j.
μˆ(j)=1-I(j)withI(j)=isiwi-jNI,
ηˆ(j)=O(j)withO(j)=isiFi-j-isiwi-jNO,
ρˆ(j)=A(j)withA(j)=isiFi-jNA,
NI isthenumberofpixelsof w,
NO isthenumberofpixelsof F-w,
NA isthenumberofpixelsof F,
NAA(j)=NII(j)+NOO(j).
l1(j)=l˜1[μˆ(j),ηˆ(j), j]=NIf[I(j)]+NOf[O(j)]
l2(j)=l˜2[ρˆ(j), j]=NAf[A(j)],
f(x)=x ln x+(1-x)ln(1-x).
r(j)=NIf[I(j)]+NOf[O(j)]-NAf[A(j)].
(s * w)j=i=0N-1siwi-j,
(s * F)j=i=0N-1siFi-j.
NIf(m1)+NOf(m2)-NAfm1+m22NIf(m1)+NOf(m2)-NAfm1+m22.
rML(j)=NIf[I(j)]+NOf[O(j)],
I(j)=1NI(s * w)j,
O(j)=1NO i=0N-1si-(s * w)j.
rMLk(j)=[ln(1-μ)-ln μ-ln η+ln(1-η)]iwi-jsi,
CMF(j)=C * (s * w)j,
S(kx, ky)=S0 if(kx, ky)=(0, 0)S0(kx2+ky2)f/2 otherwise,
l2(j)=l˜2[ρˆ(j), j]=ln[ρˆ(j)]i=0N-1siFi-j+ln[1-ρˆ(j)]NA-i=0N-1siFi-j,
NA=i=0N-1Fi-j.
l2(j)=ln[A(j)]NAA(j)+ln[1-A(j)]NA[1-A(j)].
f(x)=x ln x+(1-x)ln(1-x).
l2(j)=NAf[A(j)],
l1(j)=l˜2[μˆ(j),ηˆ(j), j]=ln[1-μˆ(j)]i=0N-1siwi-j+ln[μˆ(j)]NI-i=0N-1siwi-j+ln[ηˆ(j)]i=0N-1siFi-j-i=0N-1siwi-j+ln[1-ηˆ(j)]NA-NI-i=0N-1siFi-j+i=0N-1siwi-j,
NI=i=0N-1wi-j.
l1(j)=ln[I(j)]NII(j)+ln[1-I(j)]NI[1-I(j)]+ln[O(j)]NOO(j)+ln[1-O(j)]×[NA-NI-NOO(j)].
l1(j)=NI ln[I(j)]I(j)+NI ln[1-I(j)][1-I(j)]+NO ln[O(j)]O(j)+NO ln[1-O(j)][1-O(j)].
l1(j)=NIf[I(j)]+NOf[O(j)],

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