Abstract

Multiple degree of freedom object recognition concerns objects with no stable rest position with all scale, rotation, and aspect distortions possible. We assume that the objects are in a fairly benign background, so that feature extractors are usable. In-plane distortion invariance is provided by use of a polar–log coordinate transform feature space, and out-of-plane distortion invariance is provided by linear discriminant function design. Relational graph decision nets are considered for multiple degree of freedom pattern recognition. The design of Fisher linear discriminant functions and synthetic discriminant functions for use at the nodes of binary and multidecision nets is discussed. Case studies are detailed for two-class and multiclass problems. Simulation results demonstrate the robustness of the processors to quantization of the filter coefficients and to noise.

© 1988 Optical Society of America

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References

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  1. D. Casasent, A. J. Lee, “A Feature Space Rule-Based Optical Relational Graph Processor,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 234 (1986); D. P. Casasent, A. J. Lee, “Optical Relational-Graph Rule-Based Processor for Structural-Attribute Knowledge Bases,” Appl. Opt. 25, 3065 (1986).
    [Crossref] [PubMed]
  2. R. A. Fisher, “The Use of Multiple Measurements in Taxonomic Problems,” Ann. Eugenics 7, 179 (1936).
    [Crossref]
  3. D. Casasent, “Unified Synthetic Discriminant Function Computational Formulation,” Appl. Opt. 23, 1620 (1984).
    [Crossref] [PubMed]
  4. J. R. Leger, S. H. Lee, “Image Classification by an Optical Implementation of the Fukunaga-Koontz Transform,” J. Opt. Soc. Am. 72, 556 (1982).
    [Crossref]
  5. Z-H. Gu, J. R. Leger, S. H. Lee, “Optical Implementation of the Least-Squares Linear Mapping Technique for Image Classification,” J. Opt. Soc. Am. 72, 787 (1982).
    [Crossref]
  6. Z-H. Gu, S. H. Lee, “Optical Implementation of the Hotelling Trace Criterion for Image Classification,” Opt. Eng. 23, 727 (1984).
    [Crossref]
  7. D. Casasent, H. Okuyama, “A High Dimensionality Pattern Recognition Feature Space,” Proc. Soc. Photo-Opt. Instrum. Eng. 579, 245 (1985).
  8. A. Mahalanobis, D. Casasent, “Large Class Iconic Pattern Recognition: an OCR Case Study,” Proc. Soc. Photo-Opt. Instrum. Eng. 726, 2 (1986).

1986 (2)

D. Casasent, A. J. Lee, “A Feature Space Rule-Based Optical Relational Graph Processor,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 234 (1986); D. P. Casasent, A. J. Lee, “Optical Relational-Graph Rule-Based Processor for Structural-Attribute Knowledge Bases,” Appl. Opt. 25, 3065 (1986).
[Crossref] [PubMed]

A. Mahalanobis, D. Casasent, “Large Class Iconic Pattern Recognition: an OCR Case Study,” Proc. Soc. Photo-Opt. Instrum. Eng. 726, 2 (1986).

1985 (1)

D. Casasent, H. Okuyama, “A High Dimensionality Pattern Recognition Feature Space,” Proc. Soc. Photo-Opt. Instrum. Eng. 579, 245 (1985).

1984 (2)

Z-H. Gu, S. H. Lee, “Optical Implementation of the Hotelling Trace Criterion for Image Classification,” Opt. Eng. 23, 727 (1984).
[Crossref]

D. Casasent, “Unified Synthetic Discriminant Function Computational Formulation,” Appl. Opt. 23, 1620 (1984).
[Crossref] [PubMed]

1982 (2)

1936 (1)

R. A. Fisher, “The Use of Multiple Measurements in Taxonomic Problems,” Ann. Eugenics 7, 179 (1936).
[Crossref]

Casasent, D.

D. Casasent, A. J. Lee, “A Feature Space Rule-Based Optical Relational Graph Processor,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 234 (1986); D. P. Casasent, A. J. Lee, “Optical Relational-Graph Rule-Based Processor for Structural-Attribute Knowledge Bases,” Appl. Opt. 25, 3065 (1986).
[Crossref] [PubMed]

A. Mahalanobis, D. Casasent, “Large Class Iconic Pattern Recognition: an OCR Case Study,” Proc. Soc. Photo-Opt. Instrum. Eng. 726, 2 (1986).

D. Casasent, H. Okuyama, “A High Dimensionality Pattern Recognition Feature Space,” Proc. Soc. Photo-Opt. Instrum. Eng. 579, 245 (1985).

D. Casasent, “Unified Synthetic Discriminant Function Computational Formulation,” Appl. Opt. 23, 1620 (1984).
[Crossref] [PubMed]

Fisher, R. A.

R. A. Fisher, “The Use of Multiple Measurements in Taxonomic Problems,” Ann. Eugenics 7, 179 (1936).
[Crossref]

Gu, Z-H.

Z-H. Gu, S. H. Lee, “Optical Implementation of the Hotelling Trace Criterion for Image Classification,” Opt. Eng. 23, 727 (1984).
[Crossref]

Z-H. Gu, J. R. Leger, S. H. Lee, “Optical Implementation of the Least-Squares Linear Mapping Technique for Image Classification,” J. Opt. Soc. Am. 72, 787 (1982).
[Crossref]

Lee, A. J.

D. Casasent, A. J. Lee, “A Feature Space Rule-Based Optical Relational Graph Processor,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 234 (1986); D. P. Casasent, A. J. Lee, “Optical Relational-Graph Rule-Based Processor for Structural-Attribute Knowledge Bases,” Appl. Opt. 25, 3065 (1986).
[Crossref] [PubMed]

Lee, S. H.

Leger, J. R.

Mahalanobis, A.

A. Mahalanobis, D. Casasent, “Large Class Iconic Pattern Recognition: an OCR Case Study,” Proc. Soc. Photo-Opt. Instrum. Eng. 726, 2 (1986).

Okuyama, H.

D. Casasent, H. Okuyama, “A High Dimensionality Pattern Recognition Feature Space,” Proc. Soc. Photo-Opt. Instrum. Eng. 579, 245 (1985).

Ann. Eugenics (1)

R. A. Fisher, “The Use of Multiple Measurements in Taxonomic Problems,” Ann. Eugenics 7, 179 (1936).
[Crossref]

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

Opt. Eng. (1)

Z-H. Gu, S. H. Lee, “Optical Implementation of the Hotelling Trace Criterion for Image Classification,” Opt. Eng. 23, 727 (1984).
[Crossref]

Proc. Soc. Photo-Opt. Instrum. Eng. (3)

D. Casasent, H. Okuyama, “A High Dimensionality Pattern Recognition Feature Space,” Proc. Soc. Photo-Opt. Instrum. Eng. 579, 245 (1985).

A. Mahalanobis, D. Casasent, “Large Class Iconic Pattern Recognition: an OCR Case Study,” Proc. Soc. Photo-Opt. Instrum. Eng. 726, 2 (1986).

D. Casasent, A. J. Lee, “A Feature Space Rule-Based Optical Relational Graph Processor,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 234 (1986); D. P. Casasent, A. J. Lee, “Optical Relational-Graph Rule-Based Processor for Structural-Attribute Knowledge Bases,” Appl. Opt. 25, 3065 (1986).
[Crossref] [PubMed]

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Figures (3)

Fig. 1
Fig. 1

Optical relational graph processor.

Fig. 2
Fig. 2

Binary relational graph decision net.

Fig. 3
Fig. 3

Multidecision relational graph decision net.

Tables (2)

Tables Icon

Table I Results of Fisher LDF Binary Decision Net Tests (Number on Left is Number Correct Out of Number on Right)

Tables Icon

Table II Results of SDF Multidecision Net Tests (Number on Left is Number Correct Out of Number on Right)

Equations (11)

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

m x = 1 N x i = 1 N x x i , m y = 1 N y j = 1 N y y j ,
S x = x = 1 N x ( x i m x ) ( x i m x ) T , S y = j = 1 N y ( y j m y ) ( y j m y ) T .
w F = k ( S x + S y ) 1 ( m x m y ) ,
i = 1 N x ( x i m x ) ( x i m x ) T w F + j = 1 N y ( y j m y ) ( y j m y ) T w F .
i = 1 N x ( x i m x ) ( w F T x i w F T m x ) + j = 1 N y ( y j m y ) ( w F T y j w F T m y ) = k ( m x m y ) .
w S = i = 1 N x a i x i + j = 1 N y b j y j ,
w S T x i = α , i = 1 , 2 , , N x , w S T y j = β , j = 1 , 2 , , N y ,
w F T m x = 1 N x i = 1 N x w F T x i = α , w F T m y = 1 N y j = 1 N y w F T y j = β .
i = 1 N x ( x i m x ) ( α α ) + j = 1 N y ( y j m y ) ( β β ) = k ( m x m y )
k ( m x m y ) = 0 .
S B w i = λ i S W w i ,

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