Abstract

We address the design of optimal correlation filters for pattern detection and recognition in the presence of signal-dependent image noise sources. The particular examples considered are film-grain noise and speckle. Two basic approaches are investigated: (1) deriving the optimal matched filters for the signal-dependent noise models and comparing their performances with those derived for traditional signal-independent noise models and (2) first nonlinearly transforming the signal-dependent noise to signal-independent noise followed by the use of a classical filter matched to the transformed signal. We present both theoretical and computer simulation results that demonstrate the generally superior performance of the second approach in terms of the correlation peak signal-to-noise ratio.

© 1994 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. F. Walkup, R. C. Choens, “Image processing in signal-dependent noise,” Opt. Eng. 13, 258–266 (1974).
    [CrossRef]
  2. 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).
    [CrossRef]
  3. D. R. Cunningham, R. D. Laramore, E. Barrett, “Detection in image dependent noise,” IEEE Trans. Inf. Theory IT-22, 603–609 (1976).
    [CrossRef]
  4. G. Froehlich, J. F. Walkup, R. B. Asher, “Optimal estimation in signal-dependent noise,” J. Opt. Soc. Am. 68, 1665–1672 (1978).
    [CrossRef]
  5. F. Naderi, A. A. Sawchuk, “Detection of low-contrast images in film-grain noise,” Appl. Opt. 17, 2883–2891 (1978).
    [CrossRef] [PubMed]
  6. 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).
    [CrossRef] [PubMed]
  7. 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).
    [CrossRef]
  8. G. M. Morris, “Pattern recognition using photon-limited images,” in Optical Processing and Computing, H. H. Arsenault, T. Szoplik, B. Macukow, eds. (Academic, Boston, Mass., 1989).
    [CrossRef]
  9. G. M. Morris, “Scene matching using photon-limited images,” J. Opt. Soc. Am. A 1, 482–488 (1984).
    [CrossRef]
  10. B. Javidi, P. Refregier, P. Willett, “Optimum receiver design for pattern recognition with nonoverlapping target and scene noise,” Opt. Lett. 18, 1660–1662 (1993).
    [CrossRef] [PubMed]
  11. J. B. Thomas, An Introduction to Statistical Communication Theory (Wiley, New York, 1969).
  12. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).
  13. J. C. Dainty, ed., Laser Speckle and Related Phenomena (Springer-Verlag, New York, 1975).
  14. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  15. H. Kato, J. W. Goodman, “Nonlinear filtering in coherent optical systems through halftone screen processes,” Appl. Opt. 14, 1813–1824 (1975).
    [CrossRef] [PubMed]
  16. H. H. Arsenault, G. April, “Properties of speckle integrated with a finite aperture and logarithmically transformed,” J. Opt. Soc. Am. 66, 1160–1163 (1976).
    [CrossRef]
  17. J. S. Lim, H. Nawab, “Techniques for speckle noise removal,” Opt. Eng. 20, 472–480 (1981).
    [CrossRef]
  18. M. Tur, K. C. Chin, J. W. Goodman, “When is speckle noise multiplicative?” Appl. Opt. 21, 1157–1159 (1982).
    [CrossRef] [PubMed]
  19. B. V. K. Vijaya Kumar, R. D. Juday, D. W. Carlson, “Bias in correlation peak location,” in Optical Pattern Recognition III, D. P. Casasent, T.-H. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1701, 149–158 (1992).
    [CrossRef]
  20. H. H. Arsenault, M. Denis, “Integral expression for transforming signal-dependent noise into signal-independent noise,” Opt. Lett. 6, 210–212 (1981).
    [CrossRef] [PubMed]
  21. P. R. Prucnal, B. E. A. Saleh, “Transformation of image-signal-dependent noise into image-signal-independent noise,” Opt. Lett. 6, 316–318 (1981).
    [CrossRef] [PubMed]
  22. 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).
    [CrossRef] [PubMed]
  23. H. H. Arsenault, G. April, “Properties of speckle integrated with a finite aperture and logarithmically transformed,” J. Opt. Soc. Am. 66, 1160–1163 (1976).
    [CrossRef]

1993 (1)

1985 (1)

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).
[CrossRef]

1984 (2)

1983 (1)

1982 (1)

1981 (4)

1978 (2)

1976 (3)

1975 (1)

1974 (1)

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

April, G.

Arsenault, H. H.

Asher, R. B.

Barrett, E.

D. R. Cunningham, R. D. Laramore, E. Barrett, “Detection in image dependent noise,” IEEE Trans. Inf. Theory IT-22, 603–609 (1976).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).

Carlson, D. W.

B. V. K. Vijaya Kumar, R. D. Juday, D. W. Carlson, “Bias in correlation peak location,” in Optical Pattern Recognition III, D. P. Casasent, T.-H. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1701, 149–158 (1992).
[CrossRef]

Chavel, P.

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).
[CrossRef]

Chin, K. C.

Choens, R. C.

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

Cunningham, D. R.

D. R. Cunningham, R. D. Laramore, E. Barrett, “Detection in image dependent noise,” IEEE Trans. Inf. Theory IT-22, 603–609 (1976).
[CrossRef]

Denis, M.

Froehlich, G.

Gendron, C.

Goodman, J. W.

Javidi, B.

Juday, R. D.

B. V. K. Vijaya Kumar, R. D. Juday, D. W. Carlson, “Bias in correlation peak location,” in Optical Pattern Recognition III, D. P. Casasent, T.-H. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1701, 149–158 (1992).
[CrossRef]

Kasturi, R.

Kato, H.

Krile, T. F.

Kuan, D. T.

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).
[CrossRef]

Laramore, R. D.

D. R. Cunningham, R. D. Laramore, E. Barrett, “Detection in image dependent noise,” IEEE Trans. Inf. Theory IT-22, 603–609 (1976).
[CrossRef]

Levesque, M.

Lim, J. S.

J. S. Lim, H. Nawab, “Techniques for speckle noise removal,” Opt. Eng. 20, 472–480 (1981).
[CrossRef]

Morris, G. M.

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

G. M. Morris, “Pattern recognition using photon-limited images,” in Optical Processing and Computing, H. H. Arsenault, T. Szoplik, B. Macukow, eds. (Academic, Boston, Mass., 1989).
[CrossRef]

Naderi, F.

Nawab, H.

J. S. Lim, H. Nawab, “Techniques for speckle noise removal,” Opt. Eng. 20, 472–480 (1981).
[CrossRef]

Prucnal, P. R.

Refregier, P.

Saleh, B. E. A.

Sawchuk, A. A.

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).
[CrossRef]

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

Strand, T. C.

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).
[CrossRef]

Thomas, J. B.

J. B. Thomas, An Introduction to Statistical Communication Theory (Wiley, New York, 1969).

Tur, M.

Vijaya Kumar, B. V. K.

B. V. K. Vijaya Kumar, R. D. Juday, D. W. Carlson, “Bias in correlation peak location,” in Optical Pattern Recognition III, D. P. Casasent, T.-H. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1701, 149–158 (1992).
[CrossRef]

Walkup, J. F.

Willett, P.

Appl. Opt. (5)

IEEE Trans. Inf. Theory (1)

D. R. Cunningham, R. D. Laramore, E. Barrett, “Detection in image dependent noise,” IEEE Trans. Inf. Theory IT-22, 603–609 (1976).
[CrossRef]

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

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).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (1)

Opt. Eng. (2)

J. S. Lim, H. Nawab, “Techniques for speckle noise removal,” Opt. Eng. 20, 472–480 (1981).
[CrossRef]

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

Opt. Lett. (3)

Other (6)

B. V. K. Vijaya Kumar, R. D. Juday, D. W. Carlson, “Bias in correlation peak location,” in Optical Pattern Recognition III, D. P. Casasent, T.-H. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1701, 149–158 (1992).
[CrossRef]

G. M. Morris, “Pattern recognition using photon-limited images,” in Optical Processing and Computing, H. H. Arsenault, T. Szoplik, B. Macukow, eds. (Academic, Boston, Mass., 1989).
[CrossRef]

J. B. Thomas, An Introduction to Statistical Communication Theory (Wiley, New York, 1969).

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).

J. C. Dainty, ed., Laser Speckle and Related Phenomena (Springer-Verlag, New York, 1975).

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

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 (9)

Fig. 1
Fig. 1

64 × 64 pixel image of an eye-bolt object with 256 gray levels used for simulations.

Fig. 2
Fig. 2

SNR of correlations for images with both film-grain SDN and SIN, for both the optimal filter and the classical matched filter for SIN.

Fig. 3
Fig. 3

SNR of correlations for images with both speckle SDN and SIN, for both the optimal filter and the classical matched filter for SIN.

Fig. 4
Fig. 4

SNR versus the standard deviation of SIN for two filters designed to guarantee a central correlation peak for the uncorrupted signal s(x). The images are assumed to have speckle noise and SIN. h1(x): silhouette filter b(x); H2(x): filter hϕ(x) with Fourier transform Hϕ(u) designed to have the same phase as that of the target image while optimizing the absolute value to minimize the distance measure to the optimal filter.

Fig. 5
Fig. 5

Variance-stabilizing transformation applied to images with film-grain SDN. The standard deviation of n1(x) = 1.0. The statistics were generated from 104 samples for each mean image value.

Fig. 6
Fig. 6

Variance-stabilizing transformation applied to images with speckle SDN. The statistics were generated from 104 samples for each mean image value.

Fig. 7
Fig. 7

SNR versus film-grain noise strength for two approaches to correlation of images with film-grain SDN. The solid curves are theoretical results, and the individual data points are results from computer simulations of 3000 noisy image realizations and their correlation with the filters.

Fig. 8
Fig. 8

SNR versus relative image intensity for two approaches to correlation of images with speckle SIN. An intensity magnitude of 1.0 corresponds to the image shown in Fig. 1; all the lower values correspond to dimmer versions of the image. The solid line and curve are theoretical results, and the individual data points are results from computer simulations of 3000 noisy image realizations and their correlation with the filters.

Fig. 9
Fig. 9

Theoretical results for SNR versus σ n 2 2 / σ n 1 2 for two approaches to correlation of images with film-grain SDN + SIN.

Tables (1)

Tables Icon

Table 1 SNR Results for Optimal Filters Designed for SDN (SNRmax) and Matched Filter Designed for SIN (SNRmf)

Equations (77)

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

r ( x ) = s ( x ) + κ s p ( x ) n 1 ( x ) ,
c 0 = r ( x ) h ( x ) d x .
SNR = | E ( c 0 ) | 2 var ( c 0 ) .
E ( c 0 ) = s ( x ) h ( x ) d x
var ( c 0 ) = R n 1 ( ξ ) s p ( x ) h ( x ) s p ( x + ξ ) h ( x + ξ ) d x d ξ ,
R n 1 ( ξ ) = E [ n 1 ( x ) n 1 ( x + ξ ) ] .
var ( c 0 ) = P n 1 ( u ) | G ( u ) H ( u ) | 2 d u ,
SNR f g = | S ( u ) H * ( u ) d u | 2 P n 1 ( u ) | G ( u ) H ( u ) | 2 d u .
SNR f g = | s ( x ) h ( x ) d x | 2 σ n 1 2 s 2 p ( x ) h 2 ( x ) d x ,
r sp ( x ) = κ s ( x ) n sp ( x ) ,
p n sp [ n sp ( x ) ] = exp [ n sp ( x ) ] ,
E ( c 0 ) = E [ r sp ( x ) h ( x ) x ] = s ( x ) h ( x ) d x ,
E ( c 0 ) 2 = R n sp ( ξ ) s ( x ) h ( x ) s ( x + ξ ) h ( x + ξ ) d x d ξ ,
E [ n sp ( x ) n sp ( x + ξ ) ] = E [ n sp ( x ) ] E [ n sp ( x + ξ ) ] = ( 1 ) ( 1 ) = 1 for all ξ 0 .
E [ n sp 2 ( x ) ] = var [ n sp ( x ) ] + { E [ n sp ( x ) ] } 2 = 1 2 + 1 2 = 2 .
R n sp ( ξ ) = 1 + δ ( ξ ) ,
E [ c 0 2 ] = [ s ( x ) h ( x ) d x ] 2 + s 2 ( x ) h 2 ( x ) d x
var ( c 0 ) = s 2 ( x ) h 2 ( x ) d x .
SNR sp = | s ( x ) h ( x ) d x | 2 s 2 ( x ) h 2 ( x ) d x .
SNR sp = | S ( u ) H * ( u ) d u | 2 | S ( u ) H ( u ) d u | 2 d u .
SNR SIN = | S ( u ) H ( u ) d u | 2 P n ( u ) | H ( u ) | 2 d u ,
H mf ( u ) = S * ( u ) P n ( u ) .
h fg ( x ) = s ( x ) s 2 p ( x ) = s 1 2 p ( x ) .
h fg ( x ) = s 0 ( x ) = b ( x ) ,
b ( x ) = { 1 s ( x ) > 0 0 otherwise .
h sp ( x ) = s ( x ) s 2 ( x ) = 1 s ( x ) .
h sp ( x ) = b ( x ) s ( x ) { 1 / s ( x ) s ( x ) > 0 0 otherwise .
SNR fg , max = | s ( x ) b ( x ) d x | 2 σ n 1 2 s 2 p ( x ) b 2 ( x ) d x ,
SNR fg , max = s ( x ) d x σ n 1 2 .
SNR fg , mf = | s 2 ( x ) d x | 2 σ n 1 2 s 3 ( x ) d x .
SNR fg , max = | s ( x ) b ( x ) s ( x ) d x | 2 s 2 ( x ) b 2 ( x ) s 2 ( x ) d x = b ( x ) d x .
SNR sp , mf = | s 2 ( x ) d x | 2 s 4 ( x ) d x .
r ( x ) = s ( x ) + s p ( x ) n 1 ( x ) + n 2 ( x ) ,
r ( x ) = s ( x ) n sp ( x ) + n 2 ( x ) .
SNR fg + SIN = | s ( x ) h ( x ) d x | 2 [ σ n 1 2 s 2 p ( x ) + σ n 2 2 ] h 2 ( x ) d x ,
SNR sp + SIN = | s ( x ) h ( x ) d x | 2 [ s 2 ( x ) + σ n 2 2 ] h 2 ( x ) d x ,
h fg + SIN ( x ) = s ( x ) σ n 1 2 s ( x ) + σ n 2 2
h sp + SIN ( x ) = s ( x ) s 2 ( x ) + σ n 2 2 .
S ( u ) = A ( u ) exp [ i ϕ ( u ) ] .
H ( u ) = W ( u ) exp [ - i φ ( u ) ]
H sp + SIN ( u ) = M ( u ) exp [ i α ( u ) ] .
Q = u | H ϕ ( u ) H sp + SIN ( u ) | 2 = u | W ( u ) exp [ i ϕ ( u ) ] M ( u ) exp [ i α ( u ) | 2 .
W ( k ) = M ( k ) cos [ ϕ ( k ) α ( k ) ] .
σ r = D ( μ r ) ,
y = g ( r ) = K d r D ( r )
r ( x ) = s ( x ) + s p ( x ) n 1 ( x ) ,
μ r = E [ r ( x ) ] = s ( x ) ,
σ r = σ n 1 s p ( x ) = σ n 1 μ r p .
D ( r ) = σ n 1 r p ,
y ( x ) = K σ n 1 ( 1 p ) r 1 p ( x ) .
y ( x ) = 2 K σ n 1 r 1 / 2 ( x ) .
r ( x ) = s ( x ) n sp ( x ) ,
μ r = E [ r ( x ) ] = s ( x ) ,
σ r = s ( x ) = μ r .
y ( x ) = K ln [ r ( x ) ] .
h fg ( x ) = s 1 / 2 ( x ) ,
c 0 = y ( x ) h fg ( x ) d x .
c 0 = [ s ( x ) + s 1 / 2 ( x ) n 1 ( x ) ] 1 / 2 s 1 / 2 ( x ) d x ,
c 0 = R s ( x ) [ 1 + n 1 ( x ) s 1 / 2 ( x ) ] 1 / 2 d x ,
n 1 ( x ) s 1 / 2 ( x ) 1 for all x .
c 0 R s ( x ) [ 1 + 1 2 n 1 ( x ) s 1 / 2 ( x ) ] d x = R s ( x ) d x + 1 2 R s 1 / 2 ( x ) n 1 ( x ) d x .
E ( c 0 ) = R s ( x ) d x ,
var ( c 0 ) = 1 4 σ n 1 2 R s ( x ) d x .
SNR fg = 4 σ n 1 2 R s ( x ) d x ,
h sp ( x ) = { ln [ s ( x ) ] s ( x ) > 0 0 otherwise .
c 0 = R y ( x ) h sp ( x ) d x = R ln [ r ( x ) ] ln [ s ( x ) ] d x = R ln [ s ( x ) n sp ( x ) ] ln [ s ( x ) ] d x .
c 0 = R { ln [ s ( x ) ] } 2 d x + R ln [ s ( x ) ] ln [ n sp ( x ) ] d x ,
E { ln [ n sp ( x ) ] } = 0 ln ( n sp ) exp ( n sp ) d n sp ,
E ( c 0 ) = R { ln [ s ( x ) ] 2 d x γ R ln [ s ( x ) ] d x .
var ( c 0 ) = π 2 6 R { ln [ s ( x ) ] } 2 d x ,
SNR sp = [ R { ln [ s ( x ) ] } 2 d x γ R ln [ s ( x ) ] d x ] 2 π 2 6 R { ln [ s ( x ) ] } 2 d x .
μ r = s ( x ) ,
σ r 2 = σ n 1 2 s ( x ) + σ n 2 2 ,
σ r = ( σ n 1 2 μ r + σ n 2 2 ) 1 / 2 .
y ( x ) = K [ r ( x ) + σ n 2 2 σ n 1 2 ] 1 / 2
h fg + SIN ( x ) = [ s ( x ) + σ n 2 2 σ n 1 2 ] 1 / 2
SNR fg + SIN = 2 σ n 1 2 [ R s ( x ) d x + σ n 2 2 σ n 1 2 b ( x ) d x + σ n 2 4 2 σ n 1 4 R 1 s ( x ) d x ] 2 1 2 R s ( x ) d x + 3 σ n 2 2 2 σ n 1 2 R b ( x ) d x + σ n 2 4 2 σ n 1 4 R 1 s ( x ) d x + 3 σ n 2 6 2 σ n 1 6 R 1 s 2 ( x ) d x ,

Metrics