Abstract

Characteristic vector analysis of multivariate response data has been successfully applied to a variety of optical and photographic response functions. The method provides a simple characterization of complex response functions. The philosophy of the method is discussed.

© 1963 Optical Society of America

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References

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  1. H. Hotelling, J. Educational Psychol. 24, 417–441, 498–520 (1933).
    [Crossref]
  2. W. C. Krumbein and T. W. Tukey, J. Sedimentary Petrol. 26, 322–337 (1956).
  3. R. Harper, Appl. Statistics 5, 32–48 (1956).
    [Crossref]
  4. H. Gulliksen, Am. Scientist 47, 178–201 (1959).
  5. R. H. Morris and J. H. Morrissey, J. Opt. Soc. Am. 44, 530–534 (1954).
    [Crossref]
  6. J. L. Simonds, Phot. Sci. Eng. 2, 205–209 (1958).
  7. J. L. Simonds, Phot. Sci. Eng. 5, 270–277 (1961).
  8. T. W. Anderson, Introduction to Multivariate Analysis (John Wiley & Sons, Inc., New York, 1958).
  9. H. Hotelling, Psychometrik 1, 27–35 (1936).
    [Crossref]
  10. H. Hotelling, “Multivariate Quality Control,” Technique of Statistical Analysis, edited by Eisenhart, Hastay, and Wallis (McGraw-Hill Book Company, Inc., New York, 1947).
  11. J. E. Jackson and R. H. Morris, J. Am. Statistical Assoc. 52, 186 (1957).
    [Crossref]
  12. F. Grum and T. Wightman, Tappi 43, 400 (1960).

1961 (1)

J. L. Simonds, Phot. Sci. Eng. 5, 270–277 (1961).

1960 (1)

F. Grum and T. Wightman, Tappi 43, 400 (1960).

1959 (1)

H. Gulliksen, Am. Scientist 47, 178–201 (1959).

1958 (1)

J. L. Simonds, Phot. Sci. Eng. 2, 205–209 (1958).

1957 (1)

J. E. Jackson and R. H. Morris, J. Am. Statistical Assoc. 52, 186 (1957).
[Crossref]

1956 (2)

W. C. Krumbein and T. W. Tukey, J. Sedimentary Petrol. 26, 322–337 (1956).

R. Harper, Appl. Statistics 5, 32–48 (1956).
[Crossref]

1954 (1)

1936 (1)

H. Hotelling, Psychometrik 1, 27–35 (1936).
[Crossref]

1933 (1)

H. Hotelling, J. Educational Psychol. 24, 417–441, 498–520 (1933).
[Crossref]

Anderson, T. W.

T. W. Anderson, Introduction to Multivariate Analysis (John Wiley & Sons, Inc., New York, 1958).

Grum, F.

F. Grum and T. Wightman, Tappi 43, 400 (1960).

Gulliksen, H.

H. Gulliksen, Am. Scientist 47, 178–201 (1959).

Harper, R.

R. Harper, Appl. Statistics 5, 32–48 (1956).
[Crossref]

Hotelling, H.

H. Hotelling, Psychometrik 1, 27–35 (1936).
[Crossref]

H. Hotelling, J. Educational Psychol. 24, 417–441, 498–520 (1933).
[Crossref]

H. Hotelling, “Multivariate Quality Control,” Technique of Statistical Analysis, edited by Eisenhart, Hastay, and Wallis (McGraw-Hill Book Company, Inc., New York, 1947).

Jackson, J. E.

J. E. Jackson and R. H. Morris, J. Am. Statistical Assoc. 52, 186 (1957).
[Crossref]

Krumbein, W. C.

W. C. Krumbein and T. W. Tukey, J. Sedimentary Petrol. 26, 322–337 (1956).

Morris, R. H.

J. E. Jackson and R. H. Morris, J. Am. Statistical Assoc. 52, 186 (1957).
[Crossref]

R. H. Morris and J. H. Morrissey, J. Opt. Soc. Am. 44, 530–534 (1954).
[Crossref]

Morrissey, J. H.

Simonds, J. L.

J. L. Simonds, Phot. Sci. Eng. 5, 270–277 (1961).

J. L. Simonds, Phot. Sci. Eng. 2, 205–209 (1958).

Tukey, T. W.

W. C. Krumbein and T. W. Tukey, J. Sedimentary Petrol. 26, 322–337 (1956).

Wightman, T.

F. Grum and T. Wightman, Tappi 43, 400 (1960).

Am. Scientist (1)

H. Gulliksen, Am. Scientist 47, 178–201 (1959).

Appl. Statistics (1)

R. Harper, Appl. Statistics 5, 32–48 (1956).
[Crossref]

J. Am. Statistical Assoc. (1)

J. E. Jackson and R. H. Morris, J. Am. Statistical Assoc. 52, 186 (1957).
[Crossref]

J. Educational Psychol. (1)

H. Hotelling, J. Educational Psychol. 24, 417–441, 498–520 (1933).
[Crossref]

J. Opt. Soc. Am. (1)

J. Sedimentary Petrol. (1)

W. C. Krumbein and T. W. Tukey, J. Sedimentary Petrol. 26, 322–337 (1956).

Phot. Sci. Eng. (2)

J. L. Simonds, Phot. Sci. Eng. 2, 205–209 (1958).

J. L. Simonds, Phot. Sci. Eng. 5, 270–277 (1961).

Psychometrik (1)

H. Hotelling, Psychometrik 1, 27–35 (1936).
[Crossref]

Tappi (1)

F. Grum and T. Wightman, Tappi 43, 400 (1960).

Other (2)

H. Hotelling, “Multivariate Quality Control,” Technique of Statistical Analysis, edited by Eisenhart, Hastay, and Wallis (McGraw-Hill Book Company, Inc., New York, 1947).

T. W. Anderson, Introduction to Multivariate Analysis (John Wiley & Sons, Inc., New York, 1958).

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Figures (5)

Fig. 1
Fig. 1

A hypothetical response curve.

Fig. 2
Fig. 2

Family of hypothetical response curves generated as specified in Table I.

Fig. 3
Fig. 3

Set of basis response vectors which, in linear combination with the mean response vector, account for the curve-shape differences among the family of curves in Fig. 2.

Fig. 4
Fig. 4

Plot of the mean D-log E curve and of the four characteristic vectors of density variability.

Fig. 5
Fig. 5

Plot of the mean D-log E curve and of combinations of the mean curve with each of the four characteristic vectors.

Tables (1)

Tables Icon

Table I Amounts of basis curves of Fig. 3 to be added to the response curve of Fig. 1 to generate the response curves of Fig. 2.

Equations (20)

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| S L I | = 0 ,
z 1 = z ¯ 1 + Y 1 V 1 , 1 + Y 2 V 2 , 1 + Y 3 V 3 , 1 + Y p V p , 1 , z 2 = z ¯ 2 + Y 1 V 1 , 2 + Y 2 V 2 , 2 + Y 3 V 3 , 2 + Y p V p , 2 , p r z r = z ¯ r + Y 1 V 1 , r + Y 2 V 2 , r + Y 3 V 3 , r + Y p V p , r .
i = 1 r V a , x i V b , x i = 0 , a b .
[ 0.134 0.166 0.384 0.883 1.446 ]
( 0.014 0.006 0.024 0.203 0.386 0.024 0.046 0.094 0.143 0.206 0.006 0.014 0.016 0.013 0.046 0.014 0.026 0.054 0.063 0.066 0.026 0.034 0.076 0.117 0.174 0.014 0.026 0.044 0.017 0.074 0.036 0.054 0.126 0.287 0.454 ) .
( 0.003172 0.004828 0.010572 0.020197 0.030828 0.004828 0.007772 0.016628 0.028286 0.040912 0.010572 0.016628 0.036172 0.065814 0.098628 0.020197 0.028286 0.065814 0.162143 0.264486 0.030828 0.040912 0.098628 0.264486 0.439772 ) .
u 0 = [ 1 1 1 1 1 ] .
[ 0.069597 0.098426 0.227814 0.540926 0.874626 ] .
u 1 = [ 0.079573 0.112534 0.260470 0.618465 1.000000 ] .
u 1 P P = [ 0.046868 0.063996 0.151466 0.386699 0.636094 ] , u 2 = [ 0.073680 0.100607 0.238118 0.607927 1.000000 ] , u 2 P P = [ 0.046343 0.063205 0.149703 0.383062 0.630433 ] , u 3 = [ 0.073509 0.100256 0.237460 0.607617 1.000000 ] , u 3 P P = [ 0.046328 0.063182 0.149651 0.382955 0.630266 ] , u 4 = [ 0.073505 0.100246 0.237441 0.607608 1.000000 ] , u 4 P P = [ 0.046327 0.063181 0.149650 0.382952 0.630261 ] .
L 1 = 0.630261.
V 1 = [ 0.048612 0.066297 0.157030 0.401836 0.661341 ] .
0.630261 / 0.649031 = 0.9711.
V 1 V 1 = ( 0.002363 0.003223 0.007634 0.019534 0.032149 0.003223 0.004395 0.010411 0.026640 0.043845 0.007634 0.010411 0.024658 0.063100 0.103850 0.019534 0.026640 0.063100 0.161472 0.265751 0.032149 0.043845 0.103850 0.265751 0.437372 ) .
P P V 1 V 1 = ( 0.000809 0.001605 0.002938 0.000663 0.001321 0.001605 0.003377 0.006217 0.001646 0.002933 0.002938 0.006217 0.011514 0.002714 0.005222 0.000663 0.001646 0.002714 0.000671 0.001265 0.001321 0.002933 0.005222 0.001265 0.002400 ) .
[ 0.004694 0.009912 0.018161 0.004429 0. 008341 ] , u 1 = [ 0.258465 0.545784 1.000000 0.243874 0.459280 ] , u 1 ( P P V 1 V 1 ) = [ 0.004791 0.010223 0.018727 0.004528 0.008575 ] , u 2 = [ 0.255833 0.545896 1.000000 0.241789 0.457895 ] , u 2 ( P P V 1 V 1 ) = [ 0.004786 0.010212 0.018707 0.004524 0.008566 ] , u 3 = [ 0.255840 0.545891 1.000000 0.241834 0.457903 ] , u 3 ( P P V 1 V 1 ) = [ 0.004786 0.010212 0.018707 0.004524 0.008566 ] .
L 2 = 0.018707.
V 2 = [ 0.027394 0.058452 0.107076 0.025895 0.049031 ] .
Y 1 = i = 1 r W 1 , i ( D i D ¯ i ) , Y 2 = i = 1 r W 2 , i ( D i D ¯ i ) .
W 1 = [ 0.077130 0.105190 0.249151 0.637571 1.049313 ] , W 2 = [ 1.464372 3.124606 5.723847 1.384241 2.620998 ] .