D.2 Transparent Hardcopy Devices
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## D.2 Transparent Hardcopy Devices

### D.2.1 Measuring the System Characteristic Curve

A transparent hardcopy device is exemplified by a laser printer (including processor) that prints (exposes and processes) one or more images on a sheet of transparent film (typically a 14" x 17" film). This film is eventually placed over a high Luminance light-box in a darkened room for viewing.

The Characteristic Curve for such a transparent hardcopy device is obtained by printing a test image consisting of a pattern of n bars, each bar having a specific numeric value (DDL). The optical density of each printed bar is then measured, using a transmission densitometer, for each of the printed bars.

To accurately define a printer's Characteristic Curve, it is desirable that n be as large as possible (to capture as many points as possible on the Characteristic Curve). However, the limitations on absolute quantitative repeatability imposed by the printer, processor, or media technologies may dictate that a much smaller value of n be used (to prevent a conformance metric that is sensitive to differences from becoming unstable and meaningless, as the density differences between adjacent bars become "in the noise" as the number of bars becomes large).

One example of a test image is a pattern of 32 approximately equal-height bars, spanning the usable printable region of the film, having 32 approximately equi-spaced DDLs as follows:

Figure D.2-1. Layout of a Test Pattern for Transparent Hardcopy Media

To define a test pattern with n DDLs for a printer with an N-bit input, the DDL of step # i can be set to

Equation D.2-1.

DDLi = (2N-1)i/(n-1)

rounded to the nearest integer.

The tabulated values of DDLi and the corresponding measured optical densities ODi constitute a Characteristic Curve of the printer.

### D.2.2 Application of the Grayscale Standard Display Function

The films that are produced by transparent hardcopy printers are often brought to a variety of locations, where they may be viewed on different light-boxes and under a variety of viewing conditions. Accordingly, the approach of PS3.14 is to define, for hardcopy transparent printers, what densities (rather than Luminances) should be produced, and to provide here a method of applying the Grayscale Standard Display Function to the transparent hardcopy case, based on parameters that are typical of the expected range of light-box Luminances and other viewing parameters.

The specific parameters that are used in the following example are as follows:

- L0 (Luminance of light-box with no film present): 2000 cd/m2

- La (ambient room light reflected by film): 10 cd/m2

- Dmin (minimum optical density obtainable on film): 0.20

- Dmax (maximum optical density desirable on film): 3.00.

The process of constructing a table of desired OD values from the Grayscale Standard Display Function begins with defining the Luminance Range and the corresponding range of the Just-Noticeable Difference Index, j. The minimum and maximum Luminance values are given respectively by

Equation D.2-2.

Lmin= La + L010-Dmax = 12.0 cd/m2

Equation D.2-3.

Lmax= La + L010-Dmin = 1271.9 cd/m2

Next, calculate the corresponding Just-Noticeable Difference Index values, jmin and jmax. For the current example, we obtain

Equation D.2-4.

jmin = 233.32

Equation D.2-5.

jmax = 848.75

This gives us the range of j-values that the printer should cover. The printer should map its minimum input (P-Value = 0) to jmin and the corresponding Lmin. It should map its maximum input (P-Value = 2N-1 where N is the number of input bits) to jmax and the corresponding Lmax. At any intermediate input it should map its input proportionately:

Equation D.2-6.

j(PV) = jmin + (jmax-jmin)

and target values for the Luminance given by the Standard's formula: L(j(P-Value)). This "targeting" consists of producing an optical density OD for this P-Value that will give the desired Luminance L(j(P-Value)) under the conditions of L0 and La previously defined. The required density can thus be calculated as follows:

Equation D.2-7.

### D.2.3 Implementation of the Grayscale Standard Display Function

Carrying this example into the even more specific case of a printer with an 8-bit input leads to the following table, which defines the OD's to be generated for each of the 256 possible P-Values.

Table D.2-1. Optical Densities for Each P-Value for an 8-Bit Printer

P-Value | Optical Density (OD) |  | P-Value | Optical Density (OD) |  | P-Value | Optical Density (OD) |  | P-Value | Optical Density (OD)
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
0 | 3.000 |  | 1 | 2.936 |  | 2 | 2.880 |  | 3 | 2.828
4 | 2.782 |  | 5 | 2.739 |  | 6 | 2.700 |  | 7 | 2.662
8 | 2.628 |  | 9 | 2.595 |  | 10 | 2.564 |  | 11 | 2.534
12 | 2.506 |  | 13 | 2.479 |  | 14 | 2.454 |  | 15 | 2.429
16 | 2.405 |  | 17 | 2.382 |  | 18 | 2.360 |  | 19 | 2.338
20 | 2.317 |  | 21 | 2.297 |  | 22 | 2.277 |  | 23 | 2.258
24 | 2.239 |  | 25 | 2.221 |  | 26 | 2.203 |  | 27 | 2.185
28 | 2.168 |  | 29 | 2.152 |  | 30 | 2.135 |  | 31 | 2.119
32 | 2.103 |  | 33 | 2.088 |  | 34 | 2.073 |  | 35 | 2.058
36 | 2.043 |  | 37 | 2.028 |  | 38 | 2.014 |  | 39 | 2.000
40 | 1.986 |  | 41 | 1.973 |  | 42 | 1.959 |  | 43 | 1.946
44 | 1.933 |  | 45 | 1.920 |  | 46 | 1.907 |  | 47 | 1.894
48 | 1.882 |  | 49 | 1.870 |  | 50 | 1.857 |  | 51 | 1.845
52 | 1.833 |  | 53 | 1.821 |  | 54 | 1.810 |  | 55 | 1.798
56 | 1.787 |  | 57 | 1.775 |  | 58 | 1.764 |  | 59 | 1.753
60 | 1.742 |  | 61 | 1.731 |  | 62 | 1.720 |  | 63 | 1.709
64 | 1.698 |  | 65 | 1.688 |  | 66 | 1.677 |  | 67 | 1.667
68 | 1.656 |  | 69 | 1.646 |  | 70 | 1.636 |  | 71 | 1.626
72 | 1.616 |  | 73 | 1.605 |  | 74 | 1.595 |  | 75 | 1.586
76 | 1.576 |  | 77 | 1.566 |  | 78 | 1.556 |  | 79 | 1.547
80 | 1.537 |  | 81 | 1.527 |  | 82 | 1.518 |  | 83 | 1.508
84 | 1.499 |  | 85 | 1.490 |  | 86 | 1.480 |  | 87 | 1.471
88 | 1.462 |  | 89 | 1.453 |  | 90 | 1.444 |  | 91 | 1.434
92 | 1.425 |  | 93 | 1.416 |  | 94 | 1.407 |  | 95 | 1.398
96 | 1.390 |  | 97 | 1.381 |  | 98 | 1.372 |  | 99 | 1.363
100 | 1.354 |  | 101 | 1.346 |  | 102 | 1.337 |  | 103 | 1.328
104 | 1.320 |  | 105 | 1.311 |  | 106 | 1.303 |  | 107 | 1.294
108 | 1.286 |  | 109 | 1.277 |  | 110 | 1.269 |  | 111 | 1.260
112 | 1.252 |  | 113 | 1.244 |  | 114 | 1.235 |  | 115 | 1.227
116 | 1.219 |  | 117 | 1.211 |  | 118 | 1.202 |  | 119 | 1.194
120 | 1.186 |  | 121 | 1.178 |  | 122 | 1.170 |  | 123 | 1.162
124 | 1.154 |  | 125 | 1.146 |  | 126 | 1.138 |  | 127 | 1.130
128 | 1.122 |  | 129 | 1.114 |  | 130 | 1.106 |  | 131 | 1.098
132 | 1.090 |  | 133 | 1.082 |  | 134 | 1.074 |  | 135 | 1.066
136 | 1.058 |  | 137 | 1.051 |  | 138 | 1.043 |  | 139 | 1.035
140 | 1.027 |  | 141 | 1.020 |  | 142 | 1.012 |  | 143 | 1.004
144 | 0.996 |  | 145 | 0.989 |  | 146 | 0.981 |  | 147 | 0.973
148 | 0.966 |  | 149 | 0.958 |  | 150 | 0.951 |  | 151 | 0.943
152 | 0.935 |  | 153 | 0.928 |  | 154 | 0.920 |  | 155 | 0.913
156 | 0.905 |  | 157 | 0.898 |  | 158 | 0.890 |  | 159 | 0.883
160 | 0.875 |  | 161 | 0.868 |  | 162 | 0.860 |  | 163 | 0.853
164 | 0.845 |  | 165 | 0.838 |  | 166 | 0.831 |  | 167 | 0.823
168 | 0.816 |  | 169 | 0.808 |  | 170 | 0.801 |  | 171 | 0.794
172 | 0.786 |  | 173 | 0.779 |  | 174 | 0.772 |  | 175 | 0.764
176 | 0.757 |  | 177 | 0.750 |  | 178 | 0.742 |  | 179 | 0.735
180 | 0.728 |  | 181 | 0.721 |  | 182 | 0.713 |  | 183 | 0.706
184 | 0.699 |  | 185 | 0.692 |  | 186 | 0.684 |  | 187 | 0.677
188 | 0.670 |  | 189 | 0.663 |  | 190 | 0.656 |  | 191 | 0.648
192 | 0.641 |  | 193 | 0.634 |  | 194 | 0.627 |  | 195 | 0.620
196 | 0.613 |  | 197 | 0.606 |  | 198 | 0.598 |  | 199 | 0.591
200 | 0.584 |  | 201 | 0.577 |  | 202 | 0.570 |  | 203 | 0.563
204 | 0.556 |  | 205 | 0.549 |  | 206 | 0.542 |  | 207 | 0.534
208 | 0.527 |  | 209 | 0.520 |  | 210 | 0.513 |  | 211 | 0.506
212 | 0.499 |  | 213 | 0.492 |  | 214 | 0.485 |  | 215 | 0.478
216 | 0.471 |  | 217 | 0.464 |  | 218 | 0.457 |  | 219 | 0.450
220 | 0.443 |  | 221 | 0.436 |  | 222 | 0.429 |  | 223 | 0.422
224 | 0.415 |  | 225 | 0.408 |  | 226 | 0.401 |  | 227 | 0.394
228 | 0.387 |  | 229 | 0.380 |  | 230 | 0.373 |  | 231 | 0.366
232 | 0.359 |  | 233 | 0.352 |  | 234 | 0.345 |  | 235 | 0.338
236 | 0.331 |  | 237 | 0.324 |  | 238 | 0.317 |  | 239 | 0.311
240 | 0.304 |  | 241 | 0.297 |  | 242 | 0.290 |  | 243 | 0.283
244 | 0.276 |  | 245 | 0.269 |  | 246 | 0.262 |  | 247 | 0.255
248 | 0.248 |  | 249 | 0.241 |  | 250 | 0.234 |  | 251 | 0.228
252 | 0.221 |  | 253 | 0.214 |  | 254 | 0.207 |  | 255 | 0.200

Plotting these values gives the curve of Figure D.2-3.

Figure D.2-3. Plot of OD vs P-Value for an 8-Bit Printer

### D.2.4 Measures of Conformance

As an example, a bar pattern with 32 optical densities was printed on transmissive media (film). Beforehand, the printer had been set up to print over a density range from 0.2 (Dmin) to 3.0 (Dmax) and had been pre-configured by the manufacturer to use the Grayscale Standard Display Function, converted by the manufacturer into the table of target density values vs. P-Values described earlier.

The test pattern that was used for this was an 8-bit image consisting essentially of 32 horizontal bars. The 32 P-Values used for the bars were as follows: 0, 8, 16, 25, 33, 41, 49, 58, 66, 74, 82, 90,99, 107, 115, 123, 132, 140, 148, 156, 165, 173, 181, 189, 197, 206, 214,222, 230, 239, 247, 255.

For a given film, the 32 bars' optical densities were measured (near the middle of the film), converted to Luminances (using the standard parameters of light-box Luminance and reflected ambient light described earlier),and converted to Just-Noticeable Difference Indices by mathematically computing j(L) from L(j), where L(j) is the Grayscale Standard Display Function of Luminance L as a function of the Just-Noticeable Difference Index j. For each of the 31 intervals between consecutive measured values, a calculated value of "JNDs per increment in P-Values" was obtained by dividing the difference in Just-Noticeable Difference Index by the difference in P-Values for that interval. (In these calculations, density, L, and j are all floating-point variables. No rounding to integer values is done, so no truncation error is introduced.)

In this example, the film's data could be reasonably well fit by a horizontal straight line. That is, the calculated "JNDs per increment in P-Values was essentially constant at 2.4. A mathematical fit yielded a slight non-zero slope (specifically, dropping from 2.5 to 2.3 as the P-Value went from 0 to 255), but the 0.2 total difference was considerably smaller than the noise that was present in the 31 individual values of "JNDs per increment in P-Value" so is of doubtful significance. (The "noise" referred to here consists of the random, non-repeatable variations that are seen if a new set of measured data (e.g., from a second print of the same test pattern) is compared with a previous set of measurements.)

No visual tests were done to see if a slope that small could be detected by a human observer in side-by-side film comparisons.

Incidentally, if one considers just the 32 original absolute measured densities (rather than differential values based on small differences), one finds, in this case, quite reasonable agreement between the target and measured optical densities (within the manufacturer's norms for density accuracy, at a given density). But if one uses any metric that is based on differential information over small intervals, the results must be considered more cautiously, since they can be strongly affected by (and may be dominated by) various imperfections that are independent of a device's "true" (or averaged over many cases) characteristic behavior.

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