Abstract

General expressions are derived for the degradation in the signal-to-noise ratio (SNR) as a function of rotation and scale distortions for modified matched spatial filters. These are numerically evaluated for image classes with Gaussian- and exponential-shaped autocorrelation functions to demonstrate the effects of training set size, input noise level, and image space–bandwidth product (SBWP) on the resulting SNR. The SNR for distorted input images is shown to improve, whereas the SNR for undistorted inputs degrades, as the number of training set images is increased. If the number of training set images is increased beyond a certain point, the SNR becomes constant for any input distortion. This SNR and the number of training set images required to attain it increase with the SBWP. The optimum training set image SBWP for a fixed training set size is also determined.

© 1986 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. B. Vander Lugt, “Signal detection by complex matched spatial filtering,”IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
    [Crossref]
  2. H. L. Van Trees, Detection, Estimation and Modulation Theory: Part I (Wiley, New York, 1968).
  3. B. V. K. Vijaya Kumar, C. Carroll, “Loss of optimality in cross correlators,” J. Opt. Soc. Am. A 1, 392–397 (1984).
    [Crossref]
  4. D. Casasent, A. Furman, “Sources of correlation degradation,” Appl. Opt. 16, 1652–1661 (1977).
    [Crossref] [PubMed]
  5. H. Mostafavi, F. W. Smith, “Image correlation with geometric distortion—Part I: acquisition performance,”IEEE Trans. Aerosp. Electron. Syst. AES-14, 487–493 (1978).
    [Crossref]
  6. D. Casasent, D. Psaltis, “New optical transforms for pattern recognition,” Proc. IEEE 65, 77 (1977).
    [Crossref]
  7. H. J. Caulfield, M. H. Weinberg, “Computer recognition of 2-D patterns using generalized matched filters,” Appl. Opt. 21, 1699–1704 (1982).
    [Crossref] [PubMed]
  8. B. Braunecker, R. Hauck, A. W. Lohmann, “Optical character recognition based on nonredundant correlation measurements,” Appl. Opt. 18, 2746–2753 (1979).
    [Crossref] [PubMed]
  9. C. F. Hester, D. Casasent, “Multivariant technique for multi-class pattern recognition,” Appl. Opt. 19, 1758–1761 (1980).
    [Crossref] [PubMed]
  10. Y. Yang, Y. N. Hsu, H. S. Arsenault, “Optimum circular symmetrical filters and their uses in pattern recognition,” Opt. Acta 29, 627–644 (1982).
    [Crossref]
  11. Y. N. Hsu, H. S. Arsenault, “Optical character recognition using circular harmonic expansion,” Appl. Opt. 21, 4016–4019 (1982).
    [Crossref] [PubMed]
  12. B. V. K. Vijaya Kumar, “Efficient approach to designing linear combination filters,” Appl. Opt. 22, 1445–1448 (1983).
    [Crossref]
  13. D. Casasent, “Unified synthetic discriminant function computational formulation,” Appl. Opt. 23, 1620–1627(1984).
    [Crossref] [PubMed]
  14. H. L. Larson, B. O. Shubert, Random Noise, Signals, and Dynamic Systems, Vol. II of Probabilistic Models in Engineering Sciences (Wiley, New York, 1979).
  15. R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).
  16. M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (U.S. Government Printing Office, Washington, D.C., 1964).

1984 (2)

1983 (1)

1982 (3)

1980 (1)

1979 (1)

1978 (1)

H. Mostafavi, F. W. Smith, “Image correlation with geometric distortion—Part I: acquisition performance,”IEEE Trans. Aerosp. Electron. Syst. AES-14, 487–493 (1978).
[Crossref]

1977 (2)

D. Casasent, D. Psaltis, “New optical transforms for pattern recognition,” Proc. IEEE 65, 77 (1977).
[Crossref]

D. Casasent, A. Furman, “Sources of correlation degradation,” Appl. Opt. 16, 1652–1661 (1977).
[Crossref] [PubMed]

1964 (1)

A. B. Vander Lugt, “Signal detection by complex matched spatial filtering,”IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[Crossref]

Arsenault, H. S.

Y. Yang, Y. N. Hsu, H. S. Arsenault, “Optimum circular symmetrical filters and their uses in pattern recognition,” Opt. Acta 29, 627–644 (1982).
[Crossref]

Y. N. Hsu, H. S. Arsenault, “Optical character recognition using circular harmonic expansion,” Appl. Opt. 21, 4016–4019 (1982).
[Crossref] [PubMed]

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).

Braunecker, B.

Carroll, C.

Casasent, D.

Caulfield, H. J.

Furman, A.

Hauck, R.

Hester, C. F.

Hsu, Y. N.

Y. Yang, Y. N. Hsu, H. S. Arsenault, “Optimum circular symmetrical filters and their uses in pattern recognition,” Opt. Acta 29, 627–644 (1982).
[Crossref]

Y. N. Hsu, H. S. Arsenault, “Optical character recognition using circular harmonic expansion,” Appl. Opt. 21, 4016–4019 (1982).
[Crossref] [PubMed]

Larson, H. L.

H. L. Larson, B. O. Shubert, Random Noise, Signals, and Dynamic Systems, Vol. II of Probabilistic Models in Engineering Sciences (Wiley, New York, 1979).

Lohmann, A. W.

Mostafavi, H.

H. Mostafavi, F. W. Smith, “Image correlation with geometric distortion—Part I: acquisition performance,”IEEE Trans. Aerosp. Electron. Syst. AES-14, 487–493 (1978).
[Crossref]

Psaltis, D.

D. Casasent, D. Psaltis, “New optical transforms for pattern recognition,” Proc. IEEE 65, 77 (1977).
[Crossref]

Shubert, B. O.

H. L. Larson, B. O. Shubert, Random Noise, Signals, and Dynamic Systems, Vol. II of Probabilistic Models in Engineering Sciences (Wiley, New York, 1979).

Smith, F. W.

H. Mostafavi, F. W. Smith, “Image correlation with geometric distortion—Part I: acquisition performance,”IEEE Trans. Aerosp. Electron. Syst. AES-14, 487–493 (1978).
[Crossref]

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation and Modulation Theory: Part I (Wiley, New York, 1968).

Vander Lugt, A. B.

A. B. Vander Lugt, “Signal detection by complex matched spatial filtering,”IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[Crossref]

Vijaya Kumar, B. V. K.

Weinberg, M. H.

Yang, Y.

Y. Yang, Y. N. Hsu, H. S. Arsenault, “Optimum circular symmetrical filters and their uses in pattern recognition,” Opt. Acta 29, 627–644 (1982).
[Crossref]

Appl. Opt. (7)

IEEE Trans. Aerosp. Electron. Syst. (1)

H. Mostafavi, F. W. Smith, “Image correlation with geometric distortion—Part I: acquisition performance,”IEEE Trans. Aerosp. Electron. Syst. AES-14, 487–493 (1978).
[Crossref]

IEEE Trans. Inf. Theory (1)

A. B. Vander Lugt, “Signal detection by complex matched spatial filtering,”IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[Crossref]

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

Opt. Acta (1)

Y. Yang, Y. N. Hsu, H. S. Arsenault, “Optimum circular symmetrical filters and their uses in pattern recognition,” Opt. Acta 29, 627–644 (1982).
[Crossref]

Proc. IEEE (1)

D. Casasent, D. Psaltis, “New optical transforms for pattern recognition,” Proc. IEEE 65, 77 (1977).
[Crossref]

Other (4)

H. L. Van Trees, Detection, Estimation and Modulation Theory: Part I (Wiley, New York, 1968).

H. L. Larson, B. O. Shubert, Random Noise, Signals, and Dynamic Systems, Vol. II of Probabilistic Models in Engineering Sciences (Wiley, New York, 1979).

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).

M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (U.S. Government Printing Office, Washington, D.C., 1964).

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 (8)

Fig. 1
Fig. 1

SNR versus in-plane rotation angle for N = 1, 12, and 72 images for (a) Gaussian and (b) exponential image models.

Fig. 2
Fig. 2

SNR versus rotation distortion for SBWP = 1000 and 10,000, for the case of 12 training set images.

Fig. 3
Fig. 3

Worst-case SNR versus the number of training set images using (a) Gaussian and (b) exponential image models.

Fig. 4
Fig. 4

Limiting worst-case SNR versus SBWP for Gaussian and exponential image models.

Fig. 5
Fig. 5

Optimum SBWP versus the number of training set images.

Fig. 6
Fig. 6

Optimum worst-case SNR versus the number of training set images.

Fig. 7
Fig. 7

SNR versus scale-distortion factor using 1, 5 and 25 training set images.

Fig. 8
Fig. 8

Worst-case SNR versus the number of training set images, for scale-distorted input images.

Equations (47)

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

s ( x ) = r ( x ) + n ( x ) .
SNR OPT = E [ C ( 0 , 0 ) ] 2 Var [ C ( 0 , 0 ) ] = S r 2 ( x ) d x σ 2 ,
( h , r i ) = s h ( x ) r i ( x ) d x = 1 ,             i = 1 , 2 , , N ,
h ( x ) = a j r j ( x ) ,
j = 1 N a j ( r i , r j ) = 1 ,             i = 1 , 2 , , N .
Ra = u ,
R i j = ( r i , r j ) .
R = S r ( x ) r T ( x ) d x ,
a = R - 1 u ,
h ( x ) = a T r ( x ) = r T ( x ) a .
A = α [ cos θ sin θ - sin θ cos θ ] .
E [ C ( 0 , 0 ) ] = E [ ( h , s ) ] = E [ ( h , n ) ] + E [ ( h , r ) ] = E [ S n ( x ) r T ( x ) a d x ] + E [ S r ( Ax ) r T ( x ) a d x ] = S E [ n ( x ) ] E [ r T ( x ) a ] d x + S E [ r ( Ax ) r T ( x ) a ] d x = S E [ r ( Ax ) r T ( x ) a ] d x = S E [ r ( Ax ) r T ( x ) R - 1 ] u d x ,
Var [ C ( 0 , 0 ) ] = Var [ ( h , n ) ] + Var [ ( h , r ) ] ,
Var [ ( h , n ) ] = E [ S n ( x ) r T ( x ) a d x S n ( y ) r T ( y ) a d y ] = S S E [ n ( x ) n ( y ) ] E [ r T ( x ) a · r T ( y ) a ] d x d y = σ 2 S S δ ( x - y ) E [ r T ( x ) a · r T ( y ) a ] d x d y = σ 2 E [ a T { S r ( x ) r T ( x ) d x } a ] = σ 2 E [ a T Ra ] = σ 2 E [ u T R - 1 u ]
Var [ ( h , r ) ] = E [ S r ( Ax ) r T ( x ) a d x S r ( Ay ) r T ( y ) a d y ] - E [ ( h , r ) ] 2 = S S E [ r ( Ax ) r T ( x ) R - 1 × u · r ( Ay ) r T ( y ) R - 1 u ] d x d y - E [ ( h , r ) ] 2 .
SNR { S E [ r ( Ax ) r T ( x ) a ] d x } 2 σ 2 E [ a T Ra ] .
A i = α [ cos θ i sin θ i - sin θ i cos θ i ] ,             θ i = 2 π i / N rad .
R i j = ( r i , r j ) = C r ( A i x ) r ( A j x ) d x = 1 α 2 C r ( A i A j - 1 y ) r ( y ) d y ,
SNR = { C E [ r ( Ax ) r T ( x ) ] u d x } 2 σ 2 u T E [ R ] u .
R g ( τ ) = E [ r ( x ) r ( x + τ ) ] = E 0 exp ( - τ x 2 + τ y 2 2 Δ 2 ) .
R e ( τ ) = E 0 exp ( - τ x - τ y Δ ) ,
BW = - - S ( f ) d f S ( 0 , 0 ) = R ( 0 , 0 ) S ( 0 , 0 ) ,
SBWP g = T 2 2 π Δ 2 ,
SBWP e = T 2 4 Δ 2 .
E [ ( r i , r j ) ] E [ ( r i , r i ) ] = 4 2 π SBWP g d 2 [ 0 ( 2 π SBWP g ) 1 / 2 ( d / 2 ) exp ( - y 2 / 2 ) d y ] 2 ,
d = ( A i - A j ) 1 / 2 .
d = ( I - A ) 1 / 2 = ( 1 + α 2 - 2 α cos θ ) 1 / 2 .
E [ ( r i , r j ) ] E [ ( r i , r i ) ] = 1 SBWP e d 2 T Δ exp ( - x 1 - x 2 ) d x ,
F { exp [ - π ( τ x 2 + τ y 2 ) ] } = exp [ - π ( f x 2 + f y 2 ) ] ,
F [ R g ( τ ) ] = F { E 0 exp [ - τ x 2 + τ y 2 ) / 2 Δ ] } = 2 π Δ 2 E 0 exp [ - 2 Δ 2 π 2 ( f x 2 + f y 2 ) ] = S g ( f x , f y ) .
F [ E 0 exp ( - τ x / Δ - τ y / Δ ) ] = E 0 4 Δ 2 [ 1 + ( 2 π Δ f x ) 2 ] [ 1 + ( 2 π Δ f y ) 2 ] = S e ( f x , f y ) .
E [ ( r i , r j ) ] = E [ S r ( A i x ) r ( A j x ) d x ]
E [ ( r i , r j ) ] = S E [ r ( A i x ) r ( A j x ) ] d x = S R [ ( A i - A j ) x ] d x ,
z = ( A i - A j ) x
E [ ( r i , r j ) ] = 1 A i - A j T R ( z ) d z ,
( τ x , τ y ) = 1 Δ z
E [ ( r i , r j ) ] = Δ 2 A i - A j T Δ R ( Δ τ x , Δ τ y ) d τ x d τ y
E [ ( r i , r j ) ] = Δ 2 d 2 T Δ R g ( Δ τ x , Δ τ y ) d τ x d τ y = E 0 Δ 2 d 2 ( - T d ) / 2 Δ T d / 2 Δ ( - T d ) / 2 Δ T d / 2 Δ × exp [ - ( τ x 2 + τ y ) 2 / 2 ] d τ x d τ y = 4 E 0 Δ 2 d 2 0 T d / 2 Δ exp ( - τ x 2 2 ) d τ x × 0 T d / 2 Δ exp ( - τ y 2 / 2 ) d τ y = 2 E 0 π Δ 2 d 2 erf 2 ( T d / 8 Δ ) ,
E [ ( r i , r j ) ] E [ ( r i , r j ) ] = 2 π Δ 2 T 2 d 2 erf 2 ( T d / 8 Δ ) = 1 SBWP g d 2 erf 2 [ ( π SBWP g ) 1 / 2 ( d / 2 ) ] ,
E [ ( r i , r j ) ] = Δ 2 d 2 T Δ exp ( - τ x - τ y ) d τ x d τ y .
( X , Y ) T = ( A i - A j ) ( T / 2 Δ , T / 2 Δ ) T = ( A i - A j ) [ ( SBWP e ) 1 / 2 , ( SBWP e ) 1 / 2 ] T .
τ y = m 1 τ x + b 1
τ y = m 2 τ x + b 2 ,
m 1 = Y - X X + Y , b 1 = X 2 + Y 2 X + Y , m 2 = X + Y X - Y ,
b 2 = X 2 + Y 2 Y - X .
E [ ( r i , r j ) ] = 4 Δ 2 d 2 0 X 0 m 1 τ x + b 1 exp ( - τ y ) exp ( - τ x ) d τ y d τ x + 4 Δ 2 d 2 X b 1 0 m 2 τ x + b 2 exp ( - τ y ) exp ( - τ x ) d τ y d τ x = 4 Δ 2 d 2 ( 1 - exp ( - b 1 ) + exp ( - b 1 ) m 1 + 1 × { exp [ - ( m 1 + 1 ) X ] - 1 } + exp [ - b 2 - ( m 2 + 1 ) ] m 2 + 1 × [ exp ( - b 1 - exp ( - X ) - 1 ] ) ,
E [ ( r i , r j ) ] = 4 Δ 2 d 2 [ 1 - exp ( - X ) - X exp ( - X ) ] .

Metrics