Abstract

The Fukunaga–Koontz (F–K) transform is a linear transformation that performs image-feature extraction for a two-class image classification problem. It has the property that the most important basis functions for representing one class of image data (in a least-squares sense) are also the least important for representing a second image class. We present a new method of calculating the F–K basis functions for large dimensional imagery by using a small digital computer, when the intraclass variation can be approximated by correlation matrices of low rank. Having calculated the F–K basis functions, we use a coherent optical processor to obtain the coefficients of the F–K transform in parallel. Finally, these coefficients are detected electronically, and a classification is performed by the small digital computer.

© 1982 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. R. Leger and S. H. Lee, “Signal processing using hybrid systems,” in Applications of the Optical Fourier Transform, H. Stark, ed. (Academic, New York, 1982).
    [Crossref]
  2. J. R. Leger and S. H. Lee, “Coherent optical implementation of generalized two-dimensional transforms,” Opt. Eng. 18, 518–523 (1979).
    [Crossref]
  3. K. Fukunaga and W. L. G. Koontz, “Application of the Karhunen–Loève expansion to feature selection and ordering,” IEEE Trans. Comput. C-19, 311–318 (1970).
    [Crossref]
  4. K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972).
  5. P. Sanyal and D. H. Foley, “Feature selection by a modified Fukunaga–Koontz transform and its graphical interpretation,” presented at the Milwaukee Symposium on Automatic Computation and Control, April24, 1976.

1979 (1)

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

1970 (1)

K. Fukunaga and W. L. G. Koontz, “Application of the Karhunen–Loève expansion to feature selection and ordering,” IEEE Trans. Comput. C-19, 311–318 (1970).
[Crossref]

Foley, D. H.

P. Sanyal and D. H. Foley, “Feature selection by a modified Fukunaga–Koontz transform and its graphical interpretation,” presented at the Milwaukee Symposium on Automatic Computation and Control, April24, 1976.

Fukunaga, K.

K. Fukunaga and W. L. G. Koontz, “Application of the Karhunen–Loève expansion to feature selection and ordering,” IEEE Trans. Comput. C-19, 311–318 (1970).
[Crossref]

K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972).

Koontz, W. L. G.

K. Fukunaga and W. L. G. Koontz, “Application of the Karhunen–Loève expansion to feature selection and ordering,” IEEE Trans. Comput. C-19, 311–318 (1970).
[Crossref]

Lee, S. H.

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 and S. H. Lee, “Signal processing using hybrid systems,” in Applications of the Optical Fourier Transform, H. Stark, ed. (Academic, New York, 1982).
[Crossref]

Leger, J. R.

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 and S. H. Lee, “Signal processing using hybrid systems,” in Applications of the Optical Fourier Transform, H. Stark, ed. (Academic, New York, 1982).
[Crossref]

Sanyal, P.

P. Sanyal and D. H. Foley, “Feature selection by a modified Fukunaga–Koontz transform and its graphical interpretation,” presented at the Milwaukee Symposium on Automatic Computation and Control, April24, 1976.

IEEE Trans. Comput. (1)

K. Fukunaga and W. L. G. Koontz, “Application of the Karhunen–Loève expansion to feature selection and ordering,” IEEE Trans. Comput. C-19, 311–318 (1970).
[Crossref]

Opt. Eng. (1)

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

Other (3)

J. R. Leger and S. H. Lee, “Signal processing using hybrid systems,” in Applications of the Optical Fourier Transform, H. Stark, ed. (Academic, New York, 1982).
[Crossref]

K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972).

P. Sanyal and D. H. Foley, “Feature selection by a modified Fukunaga–Koontz transform and its graphical interpretation,” presented at the Milwaukee Symposium on Automatic Computation and Control, April24, 1976.

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 (11)

Fig. 1
Fig. 1

(a) Class 1 training set consisting of 10 song birds; (b) class 2 training set consisting of 10 fish.

Fig. 2
Fig. 2

(a) F–K basis functions. Basis function 3 is best for class 2 (fish), and basis function 8 is best for class 1 (birds). Small square to the left of image number indicates gray level corresponding to zero. (b) Test images consisting of 5 new birds and 5 new fish.

Fig. 3
Fig. 3

Comparisons of basis functions 3 (fish) and 8 (birds) with filters and formed by the arithmetic average of the training sets.

Fig. 4
Fig. 4

Hybrid implementation of the coded-phase optical processor. A computer hologram of a coded-phase array is shown as CGH # 1. A second computer hologram containing the F–K basis functions in coded-phase form is shown as CGH #2. LCLV is a liquid crystal light valve for converting an incoherent (test) image into a coherent image. The resultant F–K coefficients are detected by the vidicon and analyzed by a digital computer. With the same CGH #2 but new test images, new F–K coefficients are obtained and new classifications are achieved in real time.

Fig. 5
Fig. 5

Optical implementation of the F–K transform by using the coded-phase optical processor. The six basis functions with greatest separation power were used. Basis functions 3, 4, and 1 are best for fish, and 8, 9, and 10 are best for birds. Five of the ten images are used as inputs (a)–(e), in which the input is reproduced in the upper-left-hand corner of the television screen. The output of the optical processor is detected by a television camera, displayed in the lower-right-hand corner of the screen, measured by the video digitizer, and graphed in the lower left. A linear classifier using basis functions 3 and 8 is shown in the upper-right-hand corner of the screen, with the dotted line separating birds (below and to the right of the line) from fish (above and to the left of the line).

Fig. 6
Fig. 6

Classification of birds and fish by using coefficients of basis functions 3 and 8. Ten members of each training set, as well as ten members of the test set, were classified.

Fig. 7
Fig. 7

(a) Class 1 training set consisting of random numbers run through a low-pass filter. (b) Class 2 training set, in which a high-pass filter was used. (c) F–K basis functions. Basis function 1 is best for class 2 images, and basis function 9 is best for class 1 images.

Fig. 8
Fig. 8

Classification using coefficients of basis functions 1 and 9. Member of the training sets, as well as new class members, were transformed and plotted. The dotted line represents a classification boundary based on the ratio of the two coefficients. The solid line represents a general linear classification boundary.

Fig. 9
Fig. 9

Failure of classifier resulting from excessive intraclass variation. Members of the training set are linearly separable, but test members are not.

Fig. 10
Fig. 10

(a) Class 1 training set. Four bandpass filters were applied to different corners of a two-dimensional array of random numbers. (b) Class 2 training set. The same bandpass filters were used but were applied to different corners of the random array. (c) F–K basis functions. Basis function 2 is best for class 1, and basis function 8 is best for class 2.

Fig. 11
Fig. 11

Classification using coefficients of basis functions 2 and 8.

Tables (2)

Tables Icon

Table 1 Eigenvalues Corresponding to Ten Basis Functions

Tables Icon

Table 2 Frequency Cutoffs of the Four Filters

Equations (21)

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

W 1 = [ ( P 1 M 1 ) 1 / 2 X 1 ( 1 ) ( P 1 M 1 ) 1 / 2 X 2 ( 1 ) ( P 1 M 1 ) 1 / 2 X 3 ( 1 ) ( P 1 M 1 ) 1 / 2 X M 1 ( 1 ) ] , W 2 = [ ( P 2 M 2 ) 1 / 2 X 1 ( 2 ) ( P 2 M 2 ) X 2 ( 2 ) ( P 2 M 2 ) 1 / 2 X 3 ( 2 ) ( P 2 M 2 ) 1 / 2 X M 2 ( 2 ) ] ,
W t = [ ( P 1 M 1 ) 1 / 2 X 1 ( 1 ) ( P 1 M 1 ) 1 / 2 X 2 ( 1 ) ( P 1 M 1 ) 1 / 2 X M 1 ( 1 ) ( P 2 M 2 ) 1 / 2 X 1 ( 2 ) ( P 2 M 2 ) 1 / 2 X 2 ( 2 ) ( P 2 M 2 ) 1 / 2 X M 2 ( 2 ) ] ,
W t W t + = W 1 W 1 + + W 2 W 2 + ,
W i W i + = p i M i j = 1 M i X j ( i ) X j ( i ) + ,             i = 1 , 2
( W t + W t ) E = E Λ
( W t W t + ) ( W t E ) = ( W t E ) Λ .
( W t W t + ) ( W t E Λ - 1 / 2 ) = ( W t E Λ - 1 / 2 ) Λ .
( W t E Λ - 1 / 2 ) + ( W t E Λ - 1 / 2 ) = Λ - 1 / 2 E + ( W t + W t E ) Λ - 1 / 2 = Λ - 1 / 2 E + E Λ Λ - 1 / 2 = Λ - 1 / 2 Λ Λ - 1 / 2 = I .
( Λ - 1 / 2 E + W t + ) ( W t W t + ) ( W t E Λ - 1 / 2 ) = Λ .
Λ - 1 E + W t + ( W t W t + ) W t E Λ - 1 = I .
[ Λ - 1 E + W t + ( W 1 W 1 + ) W t E Λ - 1 ] Ψ = Ψ Γ ,
[ Λ - 1 E + W t + ( W 2 W 2 + ) W t E Λ - 1 ] Θ = Θ M .
Ψ + Λ - 1 E + W t + ( W 1 W 1 + ) W t E Λ - 1 Ψ = Γ .
( ψ i + Λ - 1 E + W t + ) ,
Λ - 1 E + W t + ( W t W t + - W 2 W 2 + ) W t E Λ - 1 Ψ = Ψ Γ
Λ - 1 E + W t + ( W 2 W 2 + ) W t E Λ - 1 Ψ = Ψ ( I - Γ ) .
Θ = Ψ
M = I - Γ .
μ i = 1 - γ i ,
h * ( - x , - y ) = p q b f p q * ( x + p Δ , y + q Δ ) × exp [ - i ϕ r ( x + p Δ , y + q Δ ) ] ,
C ( x , y ) = g ( x , y ) exp [ i ϕ r ( x , y ) ] × p , q M f p q * ( x - x + p Δ , y - y + q Δ ) × exp [ - i ϕ r ( x - x + p Δ , y - y + q Δ ) ] d x d y = p , q M [ g ( x , y ) f p q * ( x , y ) d x d y ] × δ ( x - p Δ , y - q Δ ) .