Abstract

We investigate the problem of detecting and localizing a known signal in a photon-limited image, where Poisson noise is the dominant source of image degradation. For this purpose we developed and evaluated three new algorithms. The first two are based on the impulse restoration (IR) principle and the third is based on the generalized likelihood ratio test (GLRT). In the IR approach, the problem is formulated as one of restoring a delta function at the location of the desired object. In the GLRT approach, which is a well-known variation on the optimal likelihood ratio test, the problem is formulated as a hypothesis testing problem, in which the unknown background intensity of the image and the intensity scale of the object are obtained by maximum-likelihood estimation. We used Monte Carlo simulations and localization receiver operating characteristic (LROC) curves to evaluate the proposed algorithms quantitatively. LROC curves demonstrate the ability of an algorithm to detect and locate objects in a scene correctly. Our simulations demonstrate that the GLRT approach is superior to all other tested algorithms.

© 2006 Optical Society of America

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  1. S. M. Kay, Fundamentals of Statistical Signal Processing: Detection Theory (Prentice Hall, 1998).
  2. A. K. Jain, Fundamentals of Digital Image Processing (Prentice Hall, 1989).
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    [CrossRef]
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    [CrossRef]
  5. R. E. Sequira, J. Gubner, and B. E. A. Saleh, "Image detection under low-level illumination," IEEE Trans. Image Process. 2, 18-26 (1993).
    [CrossRef]
  6. M.N.Wernick and J.N.Aarsvold, eds., Emission Tomography: The Fundamentals of PET and SPECT (Academic, 2004).
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    [CrossRef]
  8. J. Ben-Arie and K. Rao, "Optimal template matching by non-orthogonal image expansion using restoration," Mach. Vision Appl. 7, 69-81 (1994).
    [CrossRef]
  9. C. Chatwin, R. Wang, and R. Young, "Assessment of a Wiener filter synthetic discriminant function for optical correlation," Opt. Lasers Eng. 22, 33-51 (1995).
    [CrossRef]
  10. E. Marom and H. Inbar, "New interpretations of Wiener filters for image recognition," J. Opt. Soc. Am. A 13, 1325-1330 (1996).
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    [CrossRef]
  13. R. Dufour, E. Miller, and N. P. Galatsanos, "Template matching based object recognition with unknown geometry," IEEE Trans. Image Process. 11, 1385-1396 (2002).
    [CrossRef]
  14. A. Abu-Naser, "Object recognition based on impulse restoration for images in Gaussian noise," M.S. thesis (Illinois Institute of Technology, 1996).
  15. A. Abu-Naser, N. P. Galatsanos, M. N. Wernick, and D. Schonfeld, "Object recognition based on impulse restoration with use of the expectation maximization algorithm," J. Opt. Soc. Am. A 15, 2327-2339 (1998).
    [CrossRef]
  16. A. Abu-Naser, N. P. Galatsanos, and M. N. Wernick, "ML and Bayesian impulse restoration based object recognition in photon limited noise," in Proceedings of the 2002 IEEE International Conference on Image Processing (IEEE Press, 2002), Vol. 2, pp. 837-840.
  17. A. Dempster, N. Laird, and D. Rubin, "Maximum-likelihood from incomplete data," J. R. Stat. Soc. Ser. B. Methodol. 39, 1-38 (1977).
  18. L. Shepp and Y. Vardi, "Maximum likelihood reconstruction for emission tomography," IEEE Trans. Med. Imaging 1, 113-122 (1982).
    [CrossRef] [PubMed]
  19. K. Lange and R. Carson, "EM reconstruction algorithms for emission and transmission tomography," J. Comput. Assist. Tomogr. 8, 306-316 (1984).
    [PubMed]
  20. D. S. Lalush and M. N. Wernick, "Iterative image reconstruction," in Ref. , pp. 443-472.
  21. S. Geman and D. Geman, "Stochastic distributions, and the Bayesian restoration of images," IEEE Trans. Pattern Anal. Mach. Intell. 6, 721-741 (1984).
    [CrossRef]
  22. P. J. Green, "Bayesian reconstructions from emission tomography data using a modified EM algorithm," IEEE Trans. Med. Imaging 9, 84-93 (1990).
    [CrossRef] [PubMed]
  23. R. Swensson, "Unified measurement of observer performance and localizing target objects on images," Med. Phys. 23, 1709-1725 (1996).
    [CrossRef] [PubMed]
  24. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
    [CrossRef]

2002 (1)

R. Dufour, E. Miller, and N. P. Galatsanos, "Template matching based object recognition with unknown geometry," IEEE Trans. Image Process. 11, 1385-1396 (2002).
[CrossRef]

2000 (1)

R. Dufour, E. Miller, and N. P. Galatsanos, "Geometric parameter estimation with a multiscale template library," in Automatic Target Recognition X, F. A. Sadjadi, ed., Proc. SPIE 4050, 397-407 (2000).
[CrossRef]

1998 (1)

1996 (3)

1995 (1)

C. Chatwin, R. Wang, and R. Young, "Assessment of a Wiener filter synthetic discriminant function for optical correlation," Opt. Lasers Eng. 22, 33-51 (1995).
[CrossRef]

1994 (1)

J. Ben-Arie and K. Rao, "Optimal template matching by non-orthogonal image expansion using restoration," Mach. Vision Appl. 7, 69-81 (1994).
[CrossRef]

1993 (2)

R. E. Sequira, J. Gubner, and B. E. A. Saleh, "Image detection under low-level illumination," IEEE Trans. Image Process. 2, 18-26 (1993).
[CrossRef]

J. Ben-Arie and K. Rao, "A novel approach to template matching by non-orthogonal image expansion," IEEE Trans. Circuits Syst. Video Technol. 3, 71-84 (1993).
[CrossRef]

1990 (1)

P. J. Green, "Bayesian reconstructions from emission tomography data using a modified EM algorithm," IEEE Trans. Med. Imaging 9, 84-93 (1990).
[CrossRef] [PubMed]

1986 (1)

1984 (3)

G. M. Morris, "Scene matching using photon-limited images," J. Opt. Soc. Am. A 1, 482-488 (1984).
[CrossRef]

K. Lange and R. Carson, "EM reconstruction algorithms for emission and transmission tomography," J. Comput. Assist. Tomogr. 8, 306-316 (1984).
[PubMed]

S. Geman and D. Geman, "Stochastic distributions, and the Bayesian restoration of images," IEEE Trans. Pattern Anal. Mach. Intell. 6, 721-741 (1984).
[CrossRef]

1982 (1)

L. Shepp and Y. Vardi, "Maximum likelihood reconstruction for emission tomography," IEEE Trans. Med. Imaging 1, 113-122 (1982).
[CrossRef] [PubMed]

1977 (1)

A. Dempster, N. Laird, and D. Rubin, "Maximum-likelihood from incomplete data," J. R. Stat. Soc. Ser. B. Methodol. 39, 1-38 (1977).

Abu-Naser, A.

A. Abu-Naser, N. P. Galatsanos, M. N. Wernick, and D. Schonfeld, "Object recognition based on impulse restoration with use of the expectation maximization algorithm," J. Opt. Soc. Am. A 15, 2327-2339 (1998).
[CrossRef]

A. Abu-Naser, "Object recognition based on impulse restoration for images in Gaussian noise," M.S. thesis (Illinois Institute of Technology, 1996).

A. Abu-Naser, N. P. Galatsanos, and M. N. Wernick, "ML and Bayesian impulse restoration based object recognition in photon limited noise," in Proceedings of the 2002 IEEE International Conference on Image Processing (IEEE Press, 2002), Vol. 2, pp. 837-840.

Ben-Arie, J.

J. Ben-Arie and K. Rao, "Optimal template matching by non-orthogonal image expansion using restoration," Mach. Vision Appl. 7, 69-81 (1994).
[CrossRef]

J. Ben-Arie and K. Rao, "A novel approach to template matching by non-orthogonal image expansion," IEEE Trans. Circuits Syst. Video Technol. 3, 71-84 (1993).
[CrossRef]

Carson, R.

K. Lange and R. Carson, "EM reconstruction algorithms for emission and transmission tomography," J. Comput. Assist. Tomogr. 8, 306-316 (1984).
[PubMed]

Chatwin, C.

C. Chatwin, R. Wang, and R. Young, "Assessment of a Wiener filter synthetic discriminant function for optical correlation," Opt. Lasers Eng. 22, 33-51 (1995).
[CrossRef]

Dempster, A.

A. Dempster, N. Laird, and D. Rubin, "Maximum-likelihood from incomplete data," J. R. Stat. Soc. Ser. B. Methodol. 39, 1-38 (1977).

Dufour, R.

R. Dufour, E. Miller, and N. P. Galatsanos, "Template matching based object recognition with unknown geometry," IEEE Trans. Image Process. 11, 1385-1396 (2002).
[CrossRef]

R. Dufour, E. Miller, and N. P. Galatsanos, "Geometric parameter estimation with a multiscale template library," in Automatic Target Recognition X, F. A. Sadjadi, ed., Proc. SPIE 4050, 397-407 (2000).
[CrossRef]

Galatsanos, N. P.

R. Dufour, E. Miller, and N. P. Galatsanos, "Template matching based object recognition with unknown geometry," IEEE Trans. Image Process. 11, 1385-1396 (2002).
[CrossRef]

R. Dufour, E. Miller, and N. P. Galatsanos, "Geometric parameter estimation with a multiscale template library," in Automatic Target Recognition X, F. A. Sadjadi, ed., Proc. SPIE 4050, 397-407 (2000).
[CrossRef]

A. Abu-Naser, N. P. Galatsanos, M. N. Wernick, and D. Schonfeld, "Object recognition based on impulse restoration with use of the expectation maximization algorithm," J. Opt. Soc. Am. A 15, 2327-2339 (1998).
[CrossRef]

A. Abu-Naser, N. P. Galatsanos, and M. N. Wernick, "ML and Bayesian impulse restoration based object recognition in photon limited noise," in Proceedings of the 2002 IEEE International Conference on Image Processing (IEEE Press, 2002), Vol. 2, pp. 837-840.

Geman, D.

S. Geman and D. Geman, "Stochastic distributions, and the Bayesian restoration of images," IEEE Trans. Pattern Anal. Mach. Intell. 6, 721-741 (1984).
[CrossRef]

Geman, S.

S. Geman and D. Geman, "Stochastic distributions, and the Bayesian restoration of images," IEEE Trans. Pattern Anal. Mach. Intell. 6, 721-741 (1984).
[CrossRef]

Green, P. J.

P. J. Green, "Bayesian reconstructions from emission tomography data using a modified EM algorithm," IEEE Trans. Med. Imaging 9, 84-93 (1990).
[CrossRef] [PubMed]

Gubner, J.

R. E. Sequira, J. Gubner, and B. E. A. Saleh, "Image detection under low-level illumination," IEEE Trans. Image Process. 2, 18-26 (1993).
[CrossRef]

Inbar, H.

Jain, A. K.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice Hall, 1989).

Javidi, B.

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing: Detection Theory (Prentice Hall, 1998).

Laird, N.

A. Dempster, N. Laird, and D. Rubin, "Maximum-likelihood from incomplete data," J. R. Stat. Soc. Ser. B. Methodol. 39, 1-38 (1977).

Lalush, D. S.

D. S. Lalush and M. N. Wernick, "Iterative image reconstruction," in Ref. , pp. 443-472.

Lange, K.

K. Lange and R. Carson, "EM reconstruction algorithms for emission and transmission tomography," J. Comput. Assist. Tomogr. 8, 306-316 (1984).
[PubMed]

Marom, E.

Miller, E.

R. Dufour, E. Miller, and N. P. Galatsanos, "Template matching based object recognition with unknown geometry," IEEE Trans. Image Process. 11, 1385-1396 (2002).
[CrossRef]

R. Dufour, E. Miller, and N. P. Galatsanos, "Geometric parameter estimation with a multiscale template library," in Automatic Target Recognition X, F. A. Sadjadi, ed., Proc. SPIE 4050, 397-407 (2000).
[CrossRef]

Morris, G. M.

Parchekani, F.

Rao, K.

J. Ben-Arie and K. Rao, "Optimal template matching by non-orthogonal image expansion using restoration," Mach. Vision Appl. 7, 69-81 (1994).
[CrossRef]

J. Ben-Arie and K. Rao, "A novel approach to template matching by non-orthogonal image expansion," IEEE Trans. Circuits Syst. Video Technol. 3, 71-84 (1993).
[CrossRef]

Rubin, D.

A. Dempster, N. Laird, and D. Rubin, "Maximum-likelihood from incomplete data," J. R. Stat. Soc. Ser. B. Methodol. 39, 1-38 (1977).

Saleh, B. E.

R. E. Sequira, J. Gubner, and B. E. A. Saleh, "Image detection under low-level illumination," IEEE Trans. Image Process. 2, 18-26 (1993).
[CrossRef]

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

Schonfeld, D.

Sequira, R. E.

R. E. Sequira, J. Gubner, and B. E. A. Saleh, "Image detection under low-level illumination," IEEE Trans. Image Process. 2, 18-26 (1993).
[CrossRef]

Shepp, L.

L. Shepp and Y. Vardi, "Maximum likelihood reconstruction for emission tomography," IEEE Trans. Med. Imaging 1, 113-122 (1982).
[CrossRef] [PubMed]

Swensson, R.

R. Swensson, "Unified measurement of observer performance and localizing target objects on images," Med. Phys. 23, 1709-1725 (1996).
[CrossRef] [PubMed]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

Vardi, Y.

L. Shepp and Y. Vardi, "Maximum likelihood reconstruction for emission tomography," IEEE Trans. Med. Imaging 1, 113-122 (1982).
[CrossRef] [PubMed]

Wang, R.

C. Chatwin, R. Wang, and R. Young, "Assessment of a Wiener filter synthetic discriminant function for optical correlation," Opt. Lasers Eng. 22, 33-51 (1995).
[CrossRef]

Wernick, M. N.

A. Abu-Naser, N. P. Galatsanos, M. N. Wernick, and D. Schonfeld, "Object recognition based on impulse restoration with use of the expectation maximization algorithm," J. Opt. Soc. Am. A 15, 2327-2339 (1998).
[CrossRef]

M. N. Wernick and G. M. Morris, "Image classification at low light levels," J. Opt. Soc. Am. A 3, 2179-2187 (1986).
[CrossRef]

A. Abu-Naser, N. P. Galatsanos, and M. N. Wernick, "ML and Bayesian impulse restoration based object recognition in photon limited noise," in Proceedings of the 2002 IEEE International Conference on Image Processing (IEEE Press, 2002), Vol. 2, pp. 837-840.

D. S. Lalush and M. N. Wernick, "Iterative image reconstruction," in Ref. , pp. 443-472.

Young, R.

C. Chatwin, R. Wang, and R. Young, "Assessment of a Wiener filter synthetic discriminant function for optical correlation," Opt. Lasers Eng. 22, 33-51 (1995).
[CrossRef]

Zhang, G.

Appl. Opt. (1)

IEEE Trans. Circuits Syst. Video Technol. (1)

J. Ben-Arie and K. Rao, "A novel approach to template matching by non-orthogonal image expansion," IEEE Trans. Circuits Syst. Video Technol. 3, 71-84 (1993).
[CrossRef]

IEEE Trans. Image Process. (2)

R. E. Sequira, J. Gubner, and B. E. A. Saleh, "Image detection under low-level illumination," IEEE Trans. Image Process. 2, 18-26 (1993).
[CrossRef]

R. Dufour, E. Miller, and N. P. Galatsanos, "Template matching based object recognition with unknown geometry," IEEE Trans. Image Process. 11, 1385-1396 (2002).
[CrossRef]

IEEE Trans. Med. Imaging (2)

L. Shepp and Y. Vardi, "Maximum likelihood reconstruction for emission tomography," IEEE Trans. Med. Imaging 1, 113-122 (1982).
[CrossRef] [PubMed]

P. J. Green, "Bayesian reconstructions from emission tomography data using a modified EM algorithm," IEEE Trans. Med. Imaging 9, 84-93 (1990).
[CrossRef] [PubMed]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

S. Geman and D. Geman, "Stochastic distributions, and the Bayesian restoration of images," IEEE Trans. Pattern Anal. Mach. Intell. 6, 721-741 (1984).
[CrossRef]

J. Comput. Assist. Tomogr. (1)

K. Lange and R. Carson, "EM reconstruction algorithms for emission and transmission tomography," J. Comput. Assist. Tomogr. 8, 306-316 (1984).
[PubMed]

J. Opt. Soc. Am. A (4)

J. R. Stat. Soc. Ser. B. Methodol. (1)

A. Dempster, N. Laird, and D. Rubin, "Maximum-likelihood from incomplete data," J. R. Stat. Soc. Ser. B. Methodol. 39, 1-38 (1977).

Mach. Vision Appl. (1)

J. Ben-Arie and K. Rao, "Optimal template matching by non-orthogonal image expansion using restoration," Mach. Vision Appl. 7, 69-81 (1994).
[CrossRef]

Med. Phys. (1)

R. Swensson, "Unified measurement of observer performance and localizing target objects on images," Med. Phys. 23, 1709-1725 (1996).
[CrossRef] [PubMed]

Opt. Lasers Eng. (1)

C. Chatwin, R. Wang, and R. Young, "Assessment of a Wiener filter synthetic discriminant function for optical correlation," Opt. Lasers Eng. 22, 33-51 (1995).
[CrossRef]

Proc. SPIE (1)

R. Dufour, E. Miller, and N. P. Galatsanos, "Geometric parameter estimation with a multiscale template library," in Automatic Target Recognition X, F. A. Sadjadi, ed., Proc. SPIE 4050, 397-407 (2000).
[CrossRef]

Other (7)

A. Abu-Naser, "Object recognition based on impulse restoration for images in Gaussian noise," M.S. thesis (Illinois Institute of Technology, 1996).

A. Abu-Naser, N. P. Galatsanos, and M. N. Wernick, "ML and Bayesian impulse restoration based object recognition in photon limited noise," in Proceedings of the 2002 IEEE International Conference on Image Processing (IEEE Press, 2002), Vol. 2, pp. 837-840.

M.N.Wernick and J.N.Aarsvold, eds., Emission Tomography: The Fundamentals of PET and SPECT (Academic, 2004).

S. M. Kay, Fundamentals of Statistical Signal Processing: Detection Theory (Prentice Hall, 1998).

A. K. Jain, Fundamentals of Digital Image Processing (Prentice Hall, 1989).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

D. S. Lalush and M. N. Wernick, "Iterative image reconstruction," in Ref. , pp. 443-472.

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Figures (7)

Fig. 1
Fig. 1

Plot of the derivative of the clique energy V as a function of r for several values of λ.

Fig. 2
Fig. 2

Target objects that were embedded within the simulated test scenes.

Fig. 3
Fig. 3

Example noise-free scene from which noisy test images were generated.

Fig. 4
Fig. 4

Additional background images used in the simulations. Test scenes were generated in equal numbers using these four backgrounds and the one shown in Fig. 3.

Fig. 5
Fig. 5

Examples of simulated photon-limited test scenes for mean total photon counts of 25,000 (left), 50,000 (center), and 100,000 (right). The object is virtually impossible to detect visually; however, the GLRT finds it easily.

Fig. 6
Fig. 6

Examples of the output of tested algorithms for the 50,000 photon-count case: EM (upper left), MAP (upper right), exact LRT (lower left), GLRT (lower right). Arrows point to the target location.

Fig. 7
Fig. 7

LROC curves for mean total photon counts of 25,000 (left), 50,000 (center) and 100,000 (right). The proposed GLRT performed best among the methods tested, and its performance improves consistently with an increasing number of photons.

Equations (33)

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

E [ g ( m ) ] = p = 1 P f ( m m p ) ,
E [ g ( m ) ] = f ( m ) δ ̃ ( m ) ,
δ ̃ ( m ) = p = 1 P δ ( m m p ) .
E [ g ] = F δ ,
p ( g δ ) = i = 0 N 1 [ F δ ] i g i e ( [ F δ ] i ) g i ! ,
δ ̂ = arg max δ { ln p ( g δ ) } .
ln [ p ( g δ ) ] = i = 0 N 1 [ F δ ] i + g i ln ( [ F δ ] i ) ln ( g i ! ) .
g i = j = 0 N 1 z i j , i = 0 , 1 , , N 1 ,
z i j = F i j δ j .
δ ̂ j ( l + 1 ) = δ ̂ j ( l ) i = 0 N 1 F i j i = 0 N 1 { F i j g i k = 0 N 1 F i k δ ̂ k ( l ) } , j = 0 , , N 1 .
δ ̂ = arg max δ { ln p ( δ g ) } = arg max δ { ln p ( g δ ) + ln p ( δ ) } .
p ( δ ) = 1 Z exp [ β U ( δ ) ] ,
U ( δ ) = j , k C j V ( δ j , δ k ) .
δ ̂ j ( l + 1 ) = δ ̂ j ( l ) { i = 0 N 1 F i j } + β U ( δ ) δ j ( l ) i = 0 N 1 { F i j g i k = 0 N 1 F i k δ k ( l ) } , j = 0 , , N 1 .
V ( r ) = λ ln [ cosh ( r λ ) ] ,
V ( r ) r = exp ( r λ ) exp ( r λ ) exp ( r λ ) + exp ( r λ ) ,
r = δ ̂ j δ ̂ i ,
Λ ( g ) = p ( g H 1 ; θ ̂ 1 ) p ( g H 0 ; θ ̂ 0 ) ,
ln Λ ( g ) d 0 d 1 T ,
H 0 : g i Poisson ( b i 1 ) ,
H 1 : g i Poisson ( a i f i ) ,
p ( g i H 0 ; b i ) = j W i b i g i , j exp ( b i ) g i , j ! ,
p ( g i H 1 ; a i ) = j W i ( a i f i , j ) g i , j exp ( a i f i , j ) g i , j ! ,
Λ ( g ) = j W i ( a ̂ i f i , j b ̂ i ) g i , j exp ( a ̂ i f i , j + b ̂ i ) ,
ln Λ ( g ) = j W i [ g i , j ln ( a ̂ i f i , j b ̂ i ) a ̂ i f i , j + b ̂ i ] .
a i p ( g i H 1 ; a i ) = 0 ,
b i p ( g i H 0 ; b i ) = 0 .
a ̂ i = j W i g i , j j W i f i , j ,
b ̂ i = 1 N W j W i g i , j .
j W i g i , j ln ( N W f ̃ i , j ) d 0 d 1 T ,
j W g i , j ln ( f i , j ) d 0 d 1 T .
P DL = T p ( t H 1 ) d t ,
P FA = T p ( t H 0 ) d t ,

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