Abstract

Various linear combinations of simple matched spatial filters have been proposed in the literature to improve the discrimination in multiclass pattern recognition. It has been shown that all such approaches based on deterministic constraints can be modeled as similar matrix/vector problems, the only differences arising in the individual constraint vectors. Since the design of any of these linear combination filters (LCF) can be posed as a common matrix/vector problem, efficient iterative methods can be used to determine the LCFs. The application of one such method called the modified hyperplane (MHP) method for determining the LCF is described and its convergence behavior is numerically investigated for a set of seven patterns. It is shown that the MHP method yields correct LCFs (with rms error <0.1%) in less than ten iterations.

© 1983 Optical Society of America

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References

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1982

D. Casasent, B. V. K. Vijaya Kumar, V. Sharma, “Synthetic Discriminant Functions for 3 Dimensional Object Recognition,” Proc. Soc. Photo-Opt. Instrum. Eng. 360 (1982).

B. V. K. Vijaya Kumar, IEEE Trans. Antennas Propag. AP-30, 512 (1982).
[CrossRef]

1980

1979

1978

D. C. Youla, IEEE Trans. Circuits Syst. CS-25, 694 (1978).
[CrossRef]

1969

1964

A. VanderLugt, IEEE Trans. Inf. Theory IF-10, 139 (1964).

Braunecker, B.

Casasent, D.

D. Casasent, B. V. K. Vijaya Kumar, V. Sharma, “Synthetic Discriminant Functions for 3 Dimensional Object Recognition,” Proc. Soc. Photo-Opt. Instrum. Eng. 360 (1982).

C. F. Hester, D. Casasent, Appl. Opt. 19, 1758 (1980).
[CrossRef] [PubMed]

Caulfield, H. J.

Hageman, L. A.

L. A. Hageman, D. M. Young, Applied Iterative Methods (Academic, New York, 1981).

Haimes, R.

Hauck, R.

Hester, C. F.

Lohmann, A. W.

Maloney, W. T.

Mullick, S. K.

Ramakrishnam, R. S.

Rathore, R. K. S.

Sharma, V.

D. Casasent, B. V. K. Vijaya Kumar, V. Sharma, “Synthetic Discriminant Functions for 3 Dimensional Object Recognition,” Proc. Soc. Photo-Opt. Instrum. Eng. 360 (1982).

Subramanian, R.

VanderLugt, A.

A. VanderLugt, IEEE Trans. Inf. Theory IF-10, 139 (1964).

Vijaya Kumar, B. V. K.

D. Casasent, B. V. K. Vijaya Kumar, V. Sharma, “Synthetic Discriminant Functions for 3 Dimensional Object Recognition,” Proc. Soc. Photo-Opt. Instrum. Eng. 360 (1982).

B. V. K. Vijaya Kumar, IEEE Trans. Antennas Propag. AP-30, 512 (1982).
[CrossRef]

Youla, D. C.

D. C. Youla, IEEE Trans. Circuits Syst. CS-25, 694 (1978).
[CrossRef]

Young, D. M.

L. A. Hageman, D. M. Young, Applied Iterative Methods (Academic, New York, 1981).

Appl. Opt.

IEEE Trans. Antennas Propag.

B. V. K. Vijaya Kumar, IEEE Trans. Antennas Propag. AP-30, 512 (1982).
[CrossRef]

IEEE Trans. Circuits Syst.

D. C. Youla, IEEE Trans. Circuits Syst. CS-25, 694 (1978).
[CrossRef]

IEEE Trans. Inf. Theory

A. VanderLugt, IEEE Trans. Inf. Theory IF-10, 139 (1964).

Proc. Soc. Photo-Opt. Instrum. Eng.

D. Casasent, B. V. K. Vijaya Kumar, V. Sharma, “Synthetic Discriminant Functions for 3 Dimensional Object Recognition,” Proc. Soc. Photo-Opt. Instrum. Eng. 360 (1982).

Other

L. A. Hageman, D. M. Young, Applied Iterative Methods (Academic, New York, 1981).

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Tables (2)

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Table I The rms Errors Vs Iteration Number

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Table II The rms Errors Vs Iteration Number

Equations (14)

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Y = a 1 X 1 + a 2 X 2 + + a N X N ,
Y i X j = δ i j ,
Y X i = 1             for i = 1 , 2 , , N .
Y 1 X 1 = 0 ; Y 2 X 1 = 1 ; Y 1 X 2 = 1 ; Y 2 X 2 = 0 ; Y 1 X 3 = 1 ; Y 2 X 3 = 1.
Ra i = u n i .
S i T a = u ( i ) ,
S 1 # = S 1 ,
S n # = S n - { ( S n S T n - 1 # ) / ( S n - 1 # S T n - 1 # ) } S n - 1 # ,             n = 2 , 3 , , N .
u # ( 1 ) = u ( 1 ) ,
u # ( n ) = u ( n ) - { ( S n S T n - 1 # ) / ( S n - 1 # S T n - 1 # ) } u # ( n - 1 ) ,             n = 2 , 3 , , N .
R # a = u # ,
a n = a n - 1 - { [ a n - 1 S T n # - u # ( n ) ] / ( S n # S T n # ) } S n # ,             n = 1 , 2 , , N .
a 0 = [ 1 1 1 1 1 1 1 ] T .
err = a n - a * ,

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