Abstract

Utilizing the phase-coded optical processor, the least-squares linear mapping technique (LSLMT) has been optically implemented to classify large-dimensional images. The LSLMT is useful for performing a transform from large-dimensional observation or feature space to small-dimensional decision space for separating multiple image classes by maximizing the interclass differences while minimizing the intraclass variations. As an example, the classifier designed for handwritten letters was studied. The performance of the LSLMT was compared also with those of a matched filter and an average filter.

© 1982 Optical Society of America

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References

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  1. A. Vander Lugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964); A. Vander Lugt, F. B. Rotz, and A. Klooster, “Character reading by optical spatial filtering,” in Optical and Electrooptical Information Processing, J. T. Tippet and et al., eds. (MIT Press, Cambridge, Mass., 1965), p. 125.
    [Crossref]
  2. A. D. Gara, “Real-time tracking of moving objects by optical correlation,” Appl. Opt. 18, 172–174 (1979).
    [Crossref] [PubMed]
  3. S. P. Almeida and J. K. T. Eu, “Water pollution monitoring using matched spatial filters,” Appl. Opt. 15, 510–515 (1976).
    [Crossref] [PubMed]
  4. D. Casasent and D. Psaltis, “Position, rotation, and scale invariant optical correlation,” Appl. Opt. 15, 1795–1799 (1976).
    [Crossref] [PubMed]
  5. C. F. Hester and D. Casasent, “Optical pattern recognition using average filters to produce discriminant hypersurfaces,” Proc. Soc. Photo-Opt. Instrumen. Eng. 201, 77–82 (1979).
  6. H. J. Caufield, R. Haimes, and D. Casasent, “Beyond matched filtering,” Opt. Eng. 19, 152–156 (1980).
  7. J. R. Leger and S. H. Lee, “A hybrid optical processor for pattern recognition and classification using a generalized set of pattern functions,” Appl. Opt. 21, 274–287 (1982).
    [Crossref] [PubMed]
  8. J. Duvernoy, “Optical pattern recognition and clustering: Karhunen–Loéve analysis,” Appl. Opt. 15, 1584–1590 (1976).
    [Crossref] [PubMed]
  9. J. R. Leger and S. H. Lee, “Image classification by an optical implementation of Fukunaga–Koontz transform,” J. Opt. Soc. Am. 72, 556–564 (1982).
    [Crossref]
  10. N. Ahmed and K. R. Rao, Orthogonal Transforms for Digital Signal Processing (Springer-Verlag, New York, 1975), pp. 225–253.
    [Crossref]
  11. K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972), pp. 40–42.
  12. J. R. Leger and S. H. Lee, “Coherent optical implementation of generalized two-dimensional transforms,” Opt. Eng. 18, 518–523 (1979).
    [Crossref]
  13. C. P. Leibel, Change Your Handwriting, Change Your Life (Stein and Day, New York, 1972), pp. 87–101.
  14. J. Cederquist and S. H. Lee, “Coherent optical feedback for the analog solution of partial differential equations,” J. Opt. Soc. Am. 70, 944–953 (1980).
    [Crossref]
  15. J. R. Leger, J. Cederquist, and S. H. Lee, “Micro-computer based hybrid processor at UCSD,” Opt. Eng. (to be published, May/June1982).
    [Crossref]
  16. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 179–181.

1982 (2)

1980 (2)

1979 (3)

J. R. Leger and S. H. Lee, “Coherent optical implementation of generalized two-dimensional transforms,” Opt. Eng. 18, 518–523 (1979).
[Crossref]

A. D. Gara, “Real-time tracking of moving objects by optical correlation,” Appl. Opt. 18, 172–174 (1979).
[Crossref] [PubMed]

C. F. Hester and D. Casasent, “Optical pattern recognition using average filters to produce discriminant hypersurfaces,” Proc. Soc. Photo-Opt. Instrumen. Eng. 201, 77–82 (1979).

1976 (3)

1964 (1)

A. Vander Lugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964); A. Vander Lugt, F. B. Rotz, and A. Klooster, “Character reading by optical spatial filtering,” in Optical and Electrooptical Information Processing, J. T. Tippet and et al., eds. (MIT Press, Cambridge, Mass., 1965), p. 125.
[Crossref]

Ahmed, N.

N. Ahmed and K. R. Rao, Orthogonal Transforms for Digital Signal Processing (Springer-Verlag, New York, 1975), pp. 225–253.
[Crossref]

Almeida, S. P.

Casasent, D.

H. J. Caufield, R. Haimes, and D. Casasent, “Beyond matched filtering,” Opt. Eng. 19, 152–156 (1980).

C. F. Hester and D. Casasent, “Optical pattern recognition using average filters to produce discriminant hypersurfaces,” Proc. Soc. Photo-Opt. Instrumen. Eng. 201, 77–82 (1979).

D. Casasent and D. Psaltis, “Position, rotation, and scale invariant optical correlation,” Appl. Opt. 15, 1795–1799 (1976).
[Crossref] [PubMed]

Caufield, H. J.

H. J. Caufield, R. Haimes, and D. Casasent, “Beyond matched filtering,” Opt. Eng. 19, 152–156 (1980).

Cederquist, J.

J. Cederquist and S. H. Lee, “Coherent optical feedback for the analog solution of partial differential equations,” J. Opt. Soc. Am. 70, 944–953 (1980).
[Crossref]

J. R. Leger, J. Cederquist, and S. H. Lee, “Micro-computer based hybrid processor at UCSD,” Opt. Eng. (to be published, May/June1982).
[Crossref]

Duvernoy, J.

Eu, J. K. T.

Fukunaga, K.

K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972), pp. 40–42.

Gara, A. D.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 179–181.

Haimes, R.

H. J. Caufield, R. Haimes, and D. Casasent, “Beyond matched filtering,” Opt. Eng. 19, 152–156 (1980).

Hester, C. F.

C. F. Hester and D. Casasent, “Optical pattern recognition using average filters to produce discriminant hypersurfaces,” Proc. Soc. Photo-Opt. Instrumen. Eng. 201, 77–82 (1979).

Lee, S. H.

Leger, J. R.

J. R. Leger and S. H. Lee, “Image classification by an optical implementation of Fukunaga–Koontz transform,” J. Opt. Soc. Am. 72, 556–564 (1982).
[Crossref]

J. R. Leger and S. H. Lee, “A hybrid optical processor for pattern recognition and classification using a generalized set of pattern functions,” Appl. Opt. 21, 274–287 (1982).
[Crossref] [PubMed]

J. R. Leger and S. H. Lee, “Coherent optical implementation of generalized two-dimensional transforms,” Opt. Eng. 18, 518–523 (1979).
[Crossref]

J. R. Leger, J. Cederquist, and S. H. Lee, “Micro-computer based hybrid processor at UCSD,” Opt. Eng. (to be published, May/June1982).
[Crossref]

Leibel, C. P.

C. P. Leibel, Change Your Handwriting, Change Your Life (Stein and Day, New York, 1972), pp. 87–101.

Psaltis, D.

Rao, K. R.

N. Ahmed and K. R. Rao, Orthogonal Transforms for Digital Signal Processing (Springer-Verlag, New York, 1975), pp. 225–253.
[Crossref]

Vander Lugt, A.

A. Vander Lugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964); A. Vander Lugt, F. B. Rotz, and A. Klooster, “Character reading by optical spatial filtering,” in Optical and Electrooptical Information Processing, J. T. Tippet and et al., eds. (MIT Press, Cambridge, Mass., 1965), p. 125.
[Crossref]

Appl. Opt. (5)

IEEE Trans. Inf. Theory (1)

A. Vander Lugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964); A. Vander Lugt, F. B. Rotz, and A. Klooster, “Character reading by optical spatial filtering,” in Optical and Electrooptical Information Processing, J. T. Tippet and et al., eds. (MIT Press, Cambridge, Mass., 1965), p. 125.
[Crossref]

J. Opt. Soc. Am. (2)

Opt. Eng. (2)

H. J. Caufield, R. Haimes, and D. Casasent, “Beyond matched filtering,” Opt. Eng. 19, 152–156 (1980).

J. R. Leger and S. H. Lee, “Coherent optical implementation of generalized two-dimensional transforms,” Opt. Eng. 18, 518–523 (1979).
[Crossref]

Proc. Soc. Photo-Opt. Instrumen. Eng. (1)

C. F. Hester and D. Casasent, “Optical pattern recognition using average filters to produce discriminant hypersurfaces,” Proc. Soc. Photo-Opt. Instrumen. Eng. 201, 77–82 (1979).

Other (5)

N. Ahmed and K. R. Rao, Orthogonal Transforms for Digital Signal Processing (Springer-Verlag, New York, 1975), pp. 225–253.
[Crossref]

K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972), pp. 40–42.

C. P. Leibel, Change Your Handwriting, Change Your Life (Stein and Day, New York, 1972), pp. 87–101.

J. R. Leger, J. Cederquist, and S. H. Lee, “Micro-computer based hybrid processor at UCSD,” Opt. Eng. (to be published, May/June1982).
[Crossref]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 179–181.

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Figures (12)

Fig. 1
Fig. 1

Optical implementation of LSLMT.

Fig. 2
Fig. 2

Schematic diagrams of LSLMT for three classes: (a) feature space, (b) decision space.

Fig. 3
Fig. 3

(a) Class 1 training set consisting of 10 characters of m, (b) Class 2 training set consisting of 10 characters of t, (c) Class 3 training set consisting of 10 characters of a.

Fig. 4
Fig. 4

Three pattern functions fpq for LSLMT. Since the pattern functions fpq contain both positive and negative values in general, bias levels (indicated by the small gray squares below fpq) are added to fpq to display them.

Fig. 5
Fig. 5

Computer-generated spatial filter, made with the laser scanning system: (a) 4 × 4 filter array, (b) one filter, (c) the central part of one filter.

Fig. 6
Fig. 6

Test set consisting of two new m characters, three new a characters, and three new t characters.

Fig. 7
Fig. 7

The result of optical implementation of the LSLMT for three classes. Three of the eight test images used are shown on the left-hand side. The outputs of the optical processor are displayed on the right-hand side. The brightest point shows to which class the input image belongs.

Fig. 8
Fig. 8

Classification result with LSLMT for three classes (m, t, a). The filled symbols of triangle, circle, and square are for the training sets. The open symbols of triangle, circle, and square are for the test images.

Fig. 9
Fig. 9

(a) Test class of m and t characters, (b) classification using the matched filter for two classes (m, t). The solid line represents a restrictive linear classification boundary. The error rate is 9 of 30, or 30%.

Fig. 10
Fig. 10

Classification result with LSLMT of two classes (m and t characters).

Fig. 11
Fig. 11

Three average filters of the training sets (m, t, a).

Fig. 12
Fig. 12

Classification using average filters for two classes (m, t). The error rate is 6 of 30, or 20%.

Tables (1)

Tables Icon

Table 1 Measured Values of V1, V2, and V3 for the Eight Test Patterns shown in Fig. 6

Equations (41)

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i j = A X j ( i ) - V i = L i j - V i .
i = 1 M j = 1 M i j 2 .
i = 1 M j = 1 m [ X j ( i ) + A + A X j ( i ) - 2 X j ( 1 ) + A + V i + V i 2 ] .
A i = 0 ,
A { 2 M j = 1 M [ X j ( i ) X j ( i ) + ] } = 2 M j = 1 M V i X j ( i ) + .
A 2 M { i = 1 K j = 1 M [ X j ( i ) X j ( i ) + ] } = 2 M i = 1 K j = 1 M V i X j ( i ) +
A = [ i = 1 K j = 1 M V i X j ( i ) + ] [ i = 1 K j = 1 M X j ( i ) X j ( i ) + ] - 1 .
V 1 = [ 1 0 0 ] ,             V 2 = [ 0 1 0 ] ,             V 3 = [ 0 0 1 ] ,
[ i = 1 K j = 1 M X j ( i ) X j ( i ) + ] - 1
Φ ^ - 1 [ i = 1 K j = 1 M X j ( i ) X j ( i ) + ] - 1 = W Ψ Λ - 2 Ψ + W + ,
W = ( X 1 ( 1 ) , X 2 ( 1 ) , , X M ( 1 ) , X 1 ( 2 ) , X 2 ( 2 ) , , X M ( 2 ) , X 1 ( K ) , X 2 ( K ) , , X M ( K ) ) ,
Φ ^ p W + W = Ψ Λ Ψ + .
A X j ( i ) = V i + i j
[ a 11 a 12 a 1 N a 21 a 22 a 2 N a K 1 a K 2 a K N ] [ X j 1 ( i ) X j 2 ( i ) X j l ( i ) X j N ( i ) ] = [ V 1 V 2 V K ] + i j ,
h * ( - x , - y ) = p , q K f * p q ( x + p Δ , y + q Δ ) × exp [ - j ϕ r ( x + p Δ , y + q Δ ) ] .
C ( ζ , η ) = g ( x , y ) × exp [ j ϕ r ( x , y ) ] × p , q f * p , q ( x - ζ + p Δ , y - η + q Δ ) × exp [ - j ϕ r ( x - ζ + p Δ , y - η + q Δ ) ] d x d y = p , q K [ g ( x , y ) f * p q ( x , y ) d x d y ] × δ ( ζ - p Δ , η - q Δ ) ,
V 1 = g m 1 * / m 1 m 1 * , V 2 = g t 1 * / t 1 t 1 * ,
V 1 = g j m j * / | j m j | 2 , V 2 = g j t j * / | j t j | 2 .
X j ( i ) + = ( x 1 , x 2 , , x N )
A = [ a 11 a 12 a 1 N a 21 a 22 a 2 N a K 1 a K 2 a K N ] .
1 M j = 1 M { A [ X j ( i ) + A + A X j ( i ) ] - 2 A [ X j ( i ) + A + V i ] + A V i 2 } = 0.
A [ X j ( i ) + A + A X j ( i ) ] = 2 A ( X j ( i ) X j ( i ) + ) .
P = X j ( i ) + A + A X j ( i ) = [ x 1 x 2 ] [ a 11 a 21 a 12 a 22 ] [ a 11 a 12 a 21 a 22 ] [ x 1 x 2 ] = ( a 11 x 1 + a 12 x 2 ) 2 + ( a 21 x 1 + a 22 x 2 ) 2 ,
A P = [ P a 11 P a 12 P a 21 P a 22 ] .
P a 11 = 2 ( a 11 x 1 + a 12 x 2 ) x 1 , P a 12 = 2 ( a 11 x 1 + a 12 x 2 ) x 2 , P a 21 = 2 ( a 21 x 1 + a 22 x 2 ) x 1 , P a 22 = 2 ( a 21 x 1 + a 22 x 2 ) x 2 .
A P = 2 [ a 11 a 12 a 21 a 22 ] [ x 1 2 x 1 x 2 x 1 x 2 x 2 2 ] = 2 [ a 11 a 12 a 21 a 22 ] [ x 1 x 2 ] [ x 1 x 2 ]
= 2 A [ X j ( i ) X j ( i ) + ] .
A [ X j ( i ) + A + V i ] = V i X j ( i ) +
A { V i + V i } = 0.
A { 2 M j = 1 M [ X j ( i ) X j ( i ) + ] } - 2 M j = 1 M V i X j ( i ) + = 0 ,
Φ ^ = i = 1 K j = 1 M X j ( i ) X j ( i ) + = ( W W + ) ,
W = [ X 1 ( 1 ) , X 2 ( 1 ) X M ( 1 ) , X 1 ( 2 ) , X M ( 2 ) , X 1 ( K ) , X M ( K ) ] .
Φ ^ p = W + W ,
Φ ^ p = Ψ Λ Ψ + .
Φ ^ p Ψ = Ψ Λ ,
( W + W ) Ψ = Ψ Λ .
( W W + ) ( W Ψ ) = ( W Ψ ) Λ .
U = ( W Ψ Λ - 1 / 2 ) .
U + U = Λ - 1 / 2 Ψ + W + W Ψ Λ - 1 / 2 = Λ - 1 / 2 Ψ + Ψ Λ Λ - 1 / 2 = I .
( W W + ) ( W Ψ Λ - 1 / 2 ) = ( W Ψ Λ - 1 / 2 ) Λ , ( W W + ) = ( W Ψ Λ - 1 / 2 ) Λ ( W Ψ Λ - 1 / 2 ) + , Φ ^ = ( W W + ) = ( W Ψ Λ - 1 / 2 ) Λ ( Λ - 1 / 2 Ψ + W + ) ,
Φ ^ - 1 = ( W Ψ Λ - 1 / 2 ) Λ - 1 ( Λ - 1 / 2 Ψ + W + ) = W Ψ Λ - 2 Ψ + W + .