Abstract

An equation to describe the granularity of an electrophotographic imaging system has been developed to include the density variance attributed to the reflectance of paper and toner surfaces. Granularity measurements from two conventional electrophotographic copiers were used to validate the relationship. The results show good agreement with the overall shape of the granularity-versus-density curve; however, the values are not consistent with estimated toner particle size. Clustering of the toner particles is suggested as the cause of the high granularity values.

© 1986 Optical Society of America

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References

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  1. E. W. H. Selwyn, “A theory of graininess,” Photogr. J. 75, 571–580 (1935).
  2. J. C. Dainty, R. Shaw, Image Science (Academic, New York, 1974), p. 222.
  3. H. Siedentopf, “Graininess, density variations, and enlargement capability of photographic negatives,”Z. Phys. 38, 454–459 (1937).
  4. R. N. Goren, J. F. Szczepanik, “Image noise of magnetic brush xerographic development,” Photogr. Sci. Eng. 22, 235–239 (1978).
  5. F. Bestenreiner, J. Freund, D. Giglberger, U. Greis, J. Helmberger, A. Rheude, K. Stadler, J. Van Engeland, W. Simm, “Studies on special problems of color electrophotography,” in SPSE Second International Conference of Electrophotography (Society of Photographic Scientists and Engineers, Spring-field, Va., 1974), pp. 9–28.
  6. F. Bestenreiner, “Pictorial electrophotography,” Photogr. Sci. Eng. 23, 92–99 (1979).
  7. R. Shaw, “Image noise evaluation,” Proc. Soc. Inf. Displ. 21, 293–304 (1980).
  8. P. E. Castro, W. C. Lu, “Reflection of light from toned paper,” Photogr. Sci. Eng. 22, 154–159 (1978).
  9. E. A. Trabka, W. H. Lawton, “Color granularity: a layered model,”J. Photogr. Sci. 22, 131–136 (1974).
  10. K. Tanaka, S. Uchida, “Extended random-dot model,”J. Opt. Soc. Am. 73, 1312–1319 (1983).
    [CrossRef]
  11. M. Maltz, “Light scattering in xerographic images,”J. Appl. Photogr. Eng. 9, 83–89 (1983).
  12. R. Shaw, “The statistical analysis of detector limitations,” in Image Science Mathematics (Western, Hollywood, Calif., 1977), pp. 1–9.

1983 (2)

M. Maltz, “Light scattering in xerographic images,”J. Appl. Photogr. Eng. 9, 83–89 (1983).

K. Tanaka, S. Uchida, “Extended random-dot model,”J. Opt. Soc. Am. 73, 1312–1319 (1983).
[CrossRef]

1980 (1)

R. Shaw, “Image noise evaluation,” Proc. Soc. Inf. Displ. 21, 293–304 (1980).

1979 (1)

F. Bestenreiner, “Pictorial electrophotography,” Photogr. Sci. Eng. 23, 92–99 (1979).

1978 (2)

R. N. Goren, J. F. Szczepanik, “Image noise of magnetic brush xerographic development,” Photogr. Sci. Eng. 22, 235–239 (1978).

P. E. Castro, W. C. Lu, “Reflection of light from toned paper,” Photogr. Sci. Eng. 22, 154–159 (1978).

1974 (1)

E. A. Trabka, W. H. Lawton, “Color granularity: a layered model,”J. Photogr. Sci. 22, 131–136 (1974).

1937 (1)

H. Siedentopf, “Graininess, density variations, and enlargement capability of photographic negatives,”Z. Phys. 38, 454–459 (1937).

1935 (1)

E. W. H. Selwyn, “A theory of graininess,” Photogr. J. 75, 571–580 (1935).

Bestenreiner, F.

F. Bestenreiner, “Pictorial electrophotography,” Photogr. Sci. Eng. 23, 92–99 (1979).

F. Bestenreiner, J. Freund, D. Giglberger, U. Greis, J. Helmberger, A. Rheude, K. Stadler, J. Van Engeland, W. Simm, “Studies on special problems of color electrophotography,” in SPSE Second International Conference of Electrophotography (Society of Photographic Scientists and Engineers, Spring-field, Va., 1974), pp. 9–28.

Castro, P. E.

P. E. Castro, W. C. Lu, “Reflection of light from toned paper,” Photogr. Sci. Eng. 22, 154–159 (1978).

Dainty, J. C.

J. C. Dainty, R. Shaw, Image Science (Academic, New York, 1974), p. 222.

Freund, J.

F. Bestenreiner, J. Freund, D. Giglberger, U. Greis, J. Helmberger, A. Rheude, K. Stadler, J. Van Engeland, W. Simm, “Studies on special problems of color electrophotography,” in SPSE Second International Conference of Electrophotography (Society of Photographic Scientists and Engineers, Spring-field, Va., 1974), pp. 9–28.

Giglberger, D.

F. Bestenreiner, J. Freund, D. Giglberger, U. Greis, J. Helmberger, A. Rheude, K. Stadler, J. Van Engeland, W. Simm, “Studies on special problems of color electrophotography,” in SPSE Second International Conference of Electrophotography (Society of Photographic Scientists and Engineers, Spring-field, Va., 1974), pp. 9–28.

Goren, R. N.

R. N. Goren, J. F. Szczepanik, “Image noise of magnetic brush xerographic development,” Photogr. Sci. Eng. 22, 235–239 (1978).

Greis, U.

F. Bestenreiner, J. Freund, D. Giglberger, U. Greis, J. Helmberger, A. Rheude, K. Stadler, J. Van Engeland, W. Simm, “Studies on special problems of color electrophotography,” in SPSE Second International Conference of Electrophotography (Society of Photographic Scientists and Engineers, Spring-field, Va., 1974), pp. 9–28.

Helmberger, J.

F. Bestenreiner, J. Freund, D. Giglberger, U. Greis, J. Helmberger, A. Rheude, K. Stadler, J. Van Engeland, W. Simm, “Studies on special problems of color electrophotography,” in SPSE Second International Conference of Electrophotography (Society of Photographic Scientists and Engineers, Spring-field, Va., 1974), pp. 9–28.

Lawton, W. H.

E. A. Trabka, W. H. Lawton, “Color granularity: a layered model,”J. Photogr. Sci. 22, 131–136 (1974).

Lu, W. C.

P. E. Castro, W. C. Lu, “Reflection of light from toned paper,” Photogr. Sci. Eng. 22, 154–159 (1978).

Maltz, M.

M. Maltz, “Light scattering in xerographic images,”J. Appl. Photogr. Eng. 9, 83–89 (1983).

Rheude, A.

F. Bestenreiner, J. Freund, D. Giglberger, U. Greis, J. Helmberger, A. Rheude, K. Stadler, J. Van Engeland, W. Simm, “Studies on special problems of color electrophotography,” in SPSE Second International Conference of Electrophotography (Society of Photographic Scientists and Engineers, Spring-field, Va., 1974), pp. 9–28.

Selwyn, E. W. H.

E. W. H. Selwyn, “A theory of graininess,” Photogr. J. 75, 571–580 (1935).

Shaw, R.

R. Shaw, “Image noise evaluation,” Proc. Soc. Inf. Displ. 21, 293–304 (1980).

J. C. Dainty, R. Shaw, Image Science (Academic, New York, 1974), p. 222.

R. Shaw, “The statistical analysis of detector limitations,” in Image Science Mathematics (Western, Hollywood, Calif., 1977), pp. 1–9.

Siedentopf, H.

H. Siedentopf, “Graininess, density variations, and enlargement capability of photographic negatives,”Z. Phys. 38, 454–459 (1937).

Simm, W.

F. Bestenreiner, J. Freund, D. Giglberger, U. Greis, J. Helmberger, A. Rheude, K. Stadler, J. Van Engeland, W. Simm, “Studies on special problems of color electrophotography,” in SPSE Second International Conference of Electrophotography (Society of Photographic Scientists and Engineers, Spring-field, Va., 1974), pp. 9–28.

Stadler, K.

F. Bestenreiner, J. Freund, D. Giglberger, U. Greis, J. Helmberger, A. Rheude, K. Stadler, J. Van Engeland, W. Simm, “Studies on special problems of color electrophotography,” in SPSE Second International Conference of Electrophotography (Society of Photographic Scientists and Engineers, Spring-field, Va., 1974), pp. 9–28.

Szczepanik, J. F.

R. N. Goren, J. F. Szczepanik, “Image noise of magnetic brush xerographic development,” Photogr. Sci. Eng. 22, 235–239 (1978).

Tanaka, K.

Trabka, E. A.

E. A. Trabka, W. H. Lawton, “Color granularity: a layered model,”J. Photogr. Sci. 22, 131–136 (1974).

Uchida, S.

Van Engeland, J.

F. Bestenreiner, J. Freund, D. Giglberger, U. Greis, J. Helmberger, A. Rheude, K. Stadler, J. Van Engeland, W. Simm, “Studies on special problems of color electrophotography,” in SPSE Second International Conference of Electrophotography (Society of Photographic Scientists and Engineers, Spring-field, Va., 1974), pp. 9–28.

J. Appl. Photogr. Eng. (1)

M. Maltz, “Light scattering in xerographic images,”J. Appl. Photogr. Eng. 9, 83–89 (1983).

J. Opt. Soc. Am. (1)

J. Photogr. Sci. (1)

E. A. Trabka, W. H. Lawton, “Color granularity: a layered model,”J. Photogr. Sci. 22, 131–136 (1974).

Photogr. J. (1)

E. W. H. Selwyn, “A theory of graininess,” Photogr. J. 75, 571–580 (1935).

Photogr. Sci. Eng. (3)

R. N. Goren, J. F. Szczepanik, “Image noise of magnetic brush xerographic development,” Photogr. Sci. Eng. 22, 235–239 (1978).

F. Bestenreiner, “Pictorial electrophotography,” Photogr. Sci. Eng. 23, 92–99 (1979).

P. E. Castro, W. C. Lu, “Reflection of light from toned paper,” Photogr. Sci. Eng. 22, 154–159 (1978).

Proc. Soc. Inf. Displ. (1)

R. Shaw, “Image noise evaluation,” Proc. Soc. Inf. Displ. 21, 293–304 (1980).

Z. Phys. (1)

H. Siedentopf, “Graininess, density variations, and enlargement capability of photographic negatives,”Z. Phys. 38, 454–459 (1937).

Other (3)

F. Bestenreiner, J. Freund, D. Giglberger, U. Greis, J. Helmberger, A. Rheude, K. Stadler, J. Van Engeland, W. Simm, “Studies on special problems of color electrophotography,” in SPSE Second International Conference of Electrophotography (Society of Photographic Scientists and Engineers, Spring-field, Va., 1974), pp. 9–28.

J. C. Dainty, R. Shaw, Image Science (Academic, New York, 1974), p. 222.

R. Shaw, “The statistical analysis of detector limitations,” in Image Science Mathematics (Western, Hollywood, Calif., 1977), pp. 1–9.

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Figures (6)

Fig. 1
Fig. 1

Normalized G-versus-D curve for Gparticles with various values of toner maximum density, according to Eq. (9).

Fig. 2
Fig. 2

Weighting factor for Gtoner as a function of image density and various values of toner maximum density. Paper reflectance, 1.0. These factors are multiplied by Gtoner to realize the toner reflectance GD curve.

Fig. 3
Fig. 3

Weighting factor versus density for Gpaper and various values of maximum density. Paper reflectance, 1.0. Multiplication of these factors by the Gpaper will yield the paper reflectance GD relationship.

Fig. 4
Fig. 4

GD model predictions and measured data for the Xerox 1075 copier. Solid line is the curve predicted by the model when the effective area is determined from the data.

Fig. 5
Fig. 5

GD model predictions and measured data for the Xerox 8200 copier. Solid line is the model prediction when the effective area is determined from the data.

Fig. 6
Fig. 6

Normalized G-versus-D curve for the Trabka–Lawton layered granularity model, parametric in imaging-element reflection density.

Equations (18)

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G = K a ¯ D [ 1 + ( σ 2 a / a ¯ 2 ) ] ,
G total = G particles + G toner + G paper ,
R ¯ = R p exp ( - n ¯ a ¯ / A ) + R t [ 1 - exp ( - n ¯ a ¯ / A ) ] ,
σ 2 R = ( R / a ) 2 a = a ¯ σ 2 a + ( R / n ) 2 n = n ¯ σ 2 n + ( R / R p ) 2 R p = R ¯ p σ 2 R p + ( R / R t ) 2 R t = R ¯ t σ 2 R t ,
σ 2 D = K 2 σ 2 R / R ¯ 2 ,
G particles = K 2 ( 1 - R t / R ¯ ) 2 ln [ ( R p - R t ) / ( R ¯ - R t ) ] × a ¯ ( 1 + σ 2 a / a ¯ 2 ) .
G particles = K a ¯ D ¯ ( 1 + σ 2 a / a ¯ 2 ) ,
G toner = ( A K 2 σ 2 R t / R ¯ t 2 ) [ 1 - ( R ¯ / R ¯ p ) ] 2 / [ 1 - ( R ¯ t / R ¯ p ) ] 2 ,
G paper = ( A K 2 σ 2 R p / R ¯ p 2 ) [ 1 - ( R ¯ / R ¯ t ) ] 2 / [ 1 - ( R ¯ p / R ¯ t ) ] 2 ,
G total = K 2 ( 1 - R t / R ¯ ) 2 ln [ ( R p - R t ) / ( R ¯ - R t ) ] a ¯ ( 1 + σ 2 a / a ¯ 2 ) + ( A K 2 σ 2 R t / R ¯ t 2 ) [ 1 - ( R ¯ / R ¯ p ) ] 2 / [ 1 - ( R ¯ t / R ¯ p ) ] 2 + ( A K 2 σ 2 R p / R ¯ p 2 ) [ 1 - ( R ¯ / R ¯ t ) ] 2 / [ 1 - ( R ¯ p / R ¯ t ) ] 2 .
G min = A ( σ 2 electronics + σ 2 quantizer ) ,
a * = ( G measured - G toner - G paper ) / { K 2 ( 1 - R t / R ¯ ) 2 × ln [ ( R p - R t ) / ( R ¯ - R t ) ] }
G = K a ¯ D m ¯ [ 1 + 1 / m ¯ + ( 1 / m ¯ ) ( σ 2 a / a ¯ 2 ) ] .
T = t + ( 1 - t ) exp ( - n ¯ a / m A ) ,
σ 2 D = K 2 6 m ( a / A ) [ 1 - ( t / T ) ] 2 Q ( z ) ,
G = K 2 6 m a [ 1 - ( t / T ) ] 2 Q ( z ) .
G = K 2 6 m a [ 1 - ( t / T ) ] 2 ( z / 3 ! + z 2 / 4 ! + z 3 / 5 ! + ) .
G = K 2 m a [ 1 - ( t / T ) ] 2 ln [ ( 1 - t ) / ( T - t ) ] { 1 + 1 / 4 × ln [ ( 1 - t ) / ( T - t ) ] + 1 / 20 ln 2 [ ( 1 - t ) / ( T - t ) ] + } .

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