Abstract

Various analytical or diagnostic techniques (e.g., evaluation of 2-D electrophoretograms, discrimination, and counting of objects) deal with pictures containing many similar patterns or objects and a few which are dissimilar. In this paper an efficient method is described which analyzes such pictures. It proceeds in three stages: removal of film background, separation into regions enclosing a single object or pattern, and classification of these regions into groups containing objects that are similar. The classification is based on moment analysis of the separated regions. The limitations due to noise (film granularity and fluctuations in object shape) on the selection of moment orders are discussed. Using selected integral functions (moments) and a corresponding choice of variables for the representation of 2-D space, adapted to the particular characteristics of the patterns to be classified, leads to a fast method for discriminating patterns that are difficult to distinguish by visual inspection because of their low texture content. The intended application of the method is automated data extraction from 2-D electrophoretograms. The performance of the procedure is shown in two examples.

© 1984 Optical Society of America

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References

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  1. J. Bossinger et al., “Quantitative Analysis of 2-D Electrophoretograms,” J. Biol. Chem. 254, 7986 (1979).
    [PubMed]
  2. H. Kronberg et al., “Photometric Evaluation of Slab Gels,” in Electrophoresis ’79 (de Gruyter, Berlin, 1980), p. 27.
    [CrossRef]
  3. J. Taylor et al., “Estimation of 2-D Electrophoretic Spot Intensities and Positions by Modeling,” in Electrophoresis ’79 (de Gruyter, Berlin, 1980), p. 329.
  4. J. Garrels, “2-D Gel Electrophoresis and Computer Analysis of Proteins Synthesized by Clonal Cell Lines,” J. Biol. Chem. 254, 7961 (1979).
    [PubMed]
  5. S. R. Sternberg, “Biomedical Image Processing,” Proc. IEEE 71, 22 (Jan.1983).
  6. M. Capel et al., “Quantitative, Comparative Analysis of Complex 2-D Electropherograms,” Anal. Biochem. 97, 210 (1979).
    [CrossRef] [PubMed]
  7. M. K. Hu, “Visual Pattern Recognition by Moment Invariants,” IRE Trans. Inf. Theory IT-8, 179 (1962).
  8. D. Casasent, D. Psaltis, “Optical Pattern Recognition using Normalized Invariant Moments,” Proc. Soc. Photo-Opt. Instrum. Eng. 201, 107 (1979).
  9. M. R. Teague, “Image Analysis via the General Theory of Moments,” J. Opt. Soc. Am. 70, 920 (1980).
    [CrossRef]
  10. F. L. Alt, “Digital Pattern Recognition by Moments,” JACM 9, 2, 240 (1962).
    [CrossRef]
  11. F. W. Smith, M. H. Wright, “Automatic Ship Photo Interpretation by the Method of Moments,” IEEE Trans. Comput. C-20, 1089 (1971).
    [CrossRef]
  12. D. Casasent, J. Pauly, D. R. Fetterly, “Infrared Ship Classification Using a New Moment Pattern Recognition Concept,” Proc. Soc. Photo-Opt. Instrum. Eng. 302, 126 (1981).
  13. K. Nottbohm, Diplomarbeit, Universitaet Goettingen (1982).
  14. F. G. Tricomi, Vorlesungen ueber Orthogonalreihen (Springer, New York, 1955), p. 60ff.
  15. D. A. Agard, “Quantitative Analysis of Electrophoretograms: A Mathematical Approach to Super-Resolution,” Anal. Biochem. 111, 257 (1981).
    [CrossRef] [PubMed]

1983 (1)

S. R. Sternberg, “Biomedical Image Processing,” Proc. IEEE 71, 22 (Jan.1983).

1981 (2)

D. Casasent, J. Pauly, D. R. Fetterly, “Infrared Ship Classification Using a New Moment Pattern Recognition Concept,” Proc. Soc. Photo-Opt. Instrum. Eng. 302, 126 (1981).

D. A. Agard, “Quantitative Analysis of Electrophoretograms: A Mathematical Approach to Super-Resolution,” Anal. Biochem. 111, 257 (1981).
[CrossRef] [PubMed]

1980 (1)

1979 (4)

D. Casasent, D. Psaltis, “Optical Pattern Recognition using Normalized Invariant Moments,” Proc. Soc. Photo-Opt. Instrum. Eng. 201, 107 (1979).

M. Capel et al., “Quantitative, Comparative Analysis of Complex 2-D Electropherograms,” Anal. Biochem. 97, 210 (1979).
[CrossRef] [PubMed]

J. Bossinger et al., “Quantitative Analysis of 2-D Electrophoretograms,” J. Biol. Chem. 254, 7986 (1979).
[PubMed]

J. Garrels, “2-D Gel Electrophoresis and Computer Analysis of Proteins Synthesized by Clonal Cell Lines,” J. Biol. Chem. 254, 7961 (1979).
[PubMed]

1971 (1)

F. W. Smith, M. H. Wright, “Automatic Ship Photo Interpretation by the Method of Moments,” IEEE Trans. Comput. C-20, 1089 (1971).
[CrossRef]

1962 (2)

M. K. Hu, “Visual Pattern Recognition by Moment Invariants,” IRE Trans. Inf. Theory IT-8, 179 (1962).

F. L. Alt, “Digital Pattern Recognition by Moments,” JACM 9, 2, 240 (1962).
[CrossRef]

Agard, D. A.

D. A. Agard, “Quantitative Analysis of Electrophoretograms: A Mathematical Approach to Super-Resolution,” Anal. Biochem. 111, 257 (1981).
[CrossRef] [PubMed]

Alt, F. L.

F. L. Alt, “Digital Pattern Recognition by Moments,” JACM 9, 2, 240 (1962).
[CrossRef]

Bossinger, J.

J. Bossinger et al., “Quantitative Analysis of 2-D Electrophoretograms,” J. Biol. Chem. 254, 7986 (1979).
[PubMed]

Capel, M.

M. Capel et al., “Quantitative, Comparative Analysis of Complex 2-D Electropherograms,” Anal. Biochem. 97, 210 (1979).
[CrossRef] [PubMed]

Casasent, D.

D. Casasent, J. Pauly, D. R. Fetterly, “Infrared Ship Classification Using a New Moment Pattern Recognition Concept,” Proc. Soc. Photo-Opt. Instrum. Eng. 302, 126 (1981).

D. Casasent, D. Psaltis, “Optical Pattern Recognition using Normalized Invariant Moments,” Proc. Soc. Photo-Opt. Instrum. Eng. 201, 107 (1979).

Fetterly, D. R.

D. Casasent, J. Pauly, D. R. Fetterly, “Infrared Ship Classification Using a New Moment Pattern Recognition Concept,” Proc. Soc. Photo-Opt. Instrum. Eng. 302, 126 (1981).

Garrels, J.

J. Garrels, “2-D Gel Electrophoresis and Computer Analysis of Proteins Synthesized by Clonal Cell Lines,” J. Biol. Chem. 254, 7961 (1979).
[PubMed]

Hu, M. K.

M. K. Hu, “Visual Pattern Recognition by Moment Invariants,” IRE Trans. Inf. Theory IT-8, 179 (1962).

Kronberg, H.

H. Kronberg et al., “Photometric Evaluation of Slab Gels,” in Electrophoresis ’79 (de Gruyter, Berlin, 1980), p. 27.
[CrossRef]

Nottbohm, K.

K. Nottbohm, Diplomarbeit, Universitaet Goettingen (1982).

Pauly, J.

D. Casasent, J. Pauly, D. R. Fetterly, “Infrared Ship Classification Using a New Moment Pattern Recognition Concept,” Proc. Soc. Photo-Opt. Instrum. Eng. 302, 126 (1981).

Psaltis, D.

D. Casasent, D. Psaltis, “Optical Pattern Recognition using Normalized Invariant Moments,” Proc. Soc. Photo-Opt. Instrum. Eng. 201, 107 (1979).

Smith, F. W.

F. W. Smith, M. H. Wright, “Automatic Ship Photo Interpretation by the Method of Moments,” IEEE Trans. Comput. C-20, 1089 (1971).
[CrossRef]

Sternberg, S. R.

S. R. Sternberg, “Biomedical Image Processing,” Proc. IEEE 71, 22 (Jan.1983).

Taylor, J.

J. Taylor et al., “Estimation of 2-D Electrophoretic Spot Intensities and Positions by Modeling,” in Electrophoresis ’79 (de Gruyter, Berlin, 1980), p. 329.

Teague, M. R.

Tricomi, F. G.

F. G. Tricomi, Vorlesungen ueber Orthogonalreihen (Springer, New York, 1955), p. 60ff.

Wright, M. H.

F. W. Smith, M. H. Wright, “Automatic Ship Photo Interpretation by the Method of Moments,” IEEE Trans. Comput. C-20, 1089 (1971).
[CrossRef]

Anal. Biochem. (2)

M. Capel et al., “Quantitative, Comparative Analysis of Complex 2-D Electropherograms,” Anal. Biochem. 97, 210 (1979).
[CrossRef] [PubMed]

D. A. Agard, “Quantitative Analysis of Electrophoretograms: A Mathematical Approach to Super-Resolution,” Anal. Biochem. 111, 257 (1981).
[CrossRef] [PubMed]

IEEE Trans. Comput. (1)

F. W. Smith, M. H. Wright, “Automatic Ship Photo Interpretation by the Method of Moments,” IEEE Trans. Comput. C-20, 1089 (1971).
[CrossRef]

IRE Trans. Inf. Theory (1)

M. K. Hu, “Visual Pattern Recognition by Moment Invariants,” IRE Trans. Inf. Theory IT-8, 179 (1962).

J. Biol. Chem. (2)

J. Bossinger et al., “Quantitative Analysis of 2-D Electrophoretograms,” J. Biol. Chem. 254, 7986 (1979).
[PubMed]

J. Garrels, “2-D Gel Electrophoresis and Computer Analysis of Proteins Synthesized by Clonal Cell Lines,” J. Biol. Chem. 254, 7961 (1979).
[PubMed]

J. Opt. Soc. Am. (1)

JACM (1)

F. L. Alt, “Digital Pattern Recognition by Moments,” JACM 9, 2, 240 (1962).
[CrossRef]

Proc. IEEE (1)

S. R. Sternberg, “Biomedical Image Processing,” Proc. IEEE 71, 22 (Jan.1983).

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

D. Casasent, J. Pauly, D. R. Fetterly, “Infrared Ship Classification Using a New Moment Pattern Recognition Concept,” Proc. Soc. Photo-Opt. Instrum. Eng. 302, 126 (1981).

D. Casasent, D. Psaltis, “Optical Pattern Recognition using Normalized Invariant Moments,” Proc. Soc. Photo-Opt. Instrum. Eng. 201, 107 (1979).

Other (4)

K. Nottbohm, Diplomarbeit, Universitaet Goettingen (1982).

F. G. Tricomi, Vorlesungen ueber Orthogonalreihen (Springer, New York, 1955), p. 60ff.

H. Kronberg et al., “Photometric Evaluation of Slab Gels,” in Electrophoresis ’79 (de Gruyter, Berlin, 1980), p. 27.
[CrossRef]

J. Taylor et al., “Estimation of 2-D Electrophoretic Spot Intensities and Positions by Modeling,” in Electrophoresis ’79 (de Gruyter, Berlin, 1980), p. 329.

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Figures (11)

Fig. 1
Fig. 1

Electrophoretic picture (RNA analysis, radioactively marked nucleotide diphosphate), autoradiograph.

Fig. 2
Fig. 2

Example 1, electrophoretic picture (RNA analysis, radioactively marked nucleotide diphosphate), original data.

Fig. 3
Fig. 3

Example 1 after background removal.

Fig. 4
Fig. 4

Example 1 after clustering.

Fig. 5
Fig. 5

Moments n30,n40,n50,n60,n03,n04,n05,n06 of the clusters of example 1. Vector numbers refer to the numbers of Fig. 4.

Fig. 6
Fig. 6

Distance matrix and row sums of example 1.

Fig. 7
Fig. 7

Example 2, simulated picture, consisting of six Gaussian functions with added noise and background of 25.

Fig. 8
Fig. 8

Example 2 after background removal.

Fig. 9
Fig. 9

Example 2 after clustering.

Fig. 10
Fig. 10

Moments n30,n40,n50,n60,n03,n04,n05,n06 of example 2; numbers refer to Fig. 9.

Fig. 11
Fig. 11

Distance matrix and row sums of example 2.

Equations (19)

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m i j = 1 / 8 k , 1 = - 1 1 f ( i + k , j + 1 ) - f ( i , j ) ,
a , b , c , x 0 , y 0 IR , a , b , c ± 0 exist ,
f ( x , y ) = a · g [ b ( x - x 0 ) , c ( y - y 0 ) ] .
m ^ p q = - x p y q f ( x , y ) d x d y             p , q = 0 , 1 , 2 , .
μ p q = 1 / m ^ 00 - ( x - m 10 ) p ( y - m 01 ) q f ( x , y ) d x d y ,
n p q = μ p q / ( μ 20 p / 2 μ 02 q / 2 )
f ^ ( x , y ) = f ( x , y ) + e ( x , y )             ( x , y ) G IR 2
B e ( x , y ) d x d y < ɛ for every region B G .
a b g ( x ) f ( x ) d x = g ( a ) a ξ f ( x ) d x + g ( b ) ξ b f ( x ) d x .
y 0 y 1 x 0 x 1 g 1 ( x ) g 2 ( y ) f ( x , y ) d x d y = g 1 ( x 0 ) g 2 ( y 0 ) × y 0 ξ 2 x 0 ξ 1 f ( x , y ) d x d y + g 1 ( x 1 ) g 2 ( y 1 ) ξ 2 y 1 ξ 1 x 1 f ( x , y ) d x d y .
R 1 = [ x 0 , 0 ] × [ y 0 , 0 ] , R 2 = [ x 0 , 0 ] × [ 0 , y 1 ] , R 3 = [ 0 , x 1 ] × [ y 0 , 0 ] , R 4 = [ 0 , x 1 ] × [ 0 , y 1 ] .
m ^ p q - m p q = | 1 m 00 i = 1 4 R i x p y q e ( x , y ) d x d y | .
g 1 ( x ) : = x p , g 2 ( y ) : = y q , f ( x , y ) : = e ( x , y ) ,
m ^ p q - m p q = | 1 m 00 y 0 y 1 x 0 x 1 x p y q e ( x , y ) d x d y | = | 1 m 00 i = 1 4 R i x p y q e ( x , y ) d x d y | 1 m 00 y 0 q x 0 p + y 0 q x 1 p + y 1 q x 0 p + y 1 q x 1 p · ɛ 1 m 00 L x p L y q ɛ ,             where L x = x 1 - x 0 , L y = y 1 - y 0 .
f ( x , y ) = f 1 ( x ) · f 2 ( y ) .
E i = ( n 30 , n 40 , n 50 , n 60 , n 03 , n 04 , n 05 , n 06 ) i IR 8 ,
Δ i : = j = 1 N d i j ,
f ( x ) = P · exp [ - ( x - m 10 ) 2 / σ 1 2 ] · exp [ - ( y - m 01 ) 2 / σ 2 2 ] .
p ( n = 0 ) = 8 19 , p ( n = 1 ) = 8 19 , p ( n = 2 ) = 2 19 , p ( n = 3 ) = 1 19 .

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