Printing in Color: Optimizing the Print Sequence for Expanded Gamut

Results & Discussion

Figure 3: Gamut volumes for the print sequences
Figure 3: Gamut volumes for the print sequences

Figure 3 shows the resulting volumes expressed as a percentage as compared to the smallest gamut. The cubic L*a*b* units are indicated on the y-axis.

When initiating this project, our first assumption was the optimal sequence would be determined by the opacity of the primary colors. The assumption was that more opaque inks should be printed first, with higher transparency inks overprinting them. The opacity of the primaries, from high to low, are as follows:

  • V: 15.71
  • O: 10.48
  • M: 6.34
  • Y: 4.83
  • C: 4.06
  • G: 3.76

The closest sequence printed in the study to this scenario was the KOVGCMY, which has violet and orange printing before their process color pairs. Green was the least opaque ink, but the difference in opacity between it and its analogous process colors—cyan and yellow—is far less than that of violet and orange, which are two to three times as opaque as the other inks. Opacity was determined from the printed samples by measuring XYZ tristimulus values with Illuminate C and a 2 degree observer function, and using the CIE opacity formula:

Opacity = 100(Y0/Y∞)

Where: Y0 = Black backing and Y∞ = White backing

However, as Figure 3 illustrates, placing orange and violet first yielded one of the smaller gamuts, which suggests sequencing based on the opacity of the primaries does not yield the optimal gamut. This led us to look for other means to model and predict an optimized EG print sequence.

Table 2: L*C*h˚ of the EG overprints
Table 2: L*C*h˚ of the EG overprints

We next examined the changes in L*C*h˚ of the various 2-color overprints of the CMYOGV color pairs depending on sequence. Table 2 shows the overprint pairs and the resulting Delta L*, Delta C* and Delta h˚ results, as well as the overall Delta Eab of the print sequences. In addition, the average difference for the various color pairs is shown on the right; it can be seen hue angle is the color attribute most affected by the print sequence.

The change in chroma due to sequence is more pronounced for some pairs than others. A sequence based on C*, in which the sequenced pair yielding the highest chroma is selected, would yield a sequence of KYGVOCM. Of the sequences actually performed on press, this is closest to the KYOMGVC sequence, which yielded the highest gamut.

Figure 4: Hypothetical triangle area of primaries and overprints
Figure 4: Hypothetical triangle area of primaries and overprints

However, chroma doesn’t tell the whole story, as the hue angle also plays a critical role. If the chroma is the same between two overprints, the overprint that is more centered on hue angle between the primaries will create the greatest area. This is illustrated in Figure 4, which depicts a hypothetical set of data in which the chroma is equal but the hue angle changes. Two overprint sequences are depicted, one of which is closer to its primary than the other. When the primaries and the overprints are connected, it is clear the area of the triangle of the centrally located overprint (OM) is greater than the one that is closer to one of its primaries (MO).

Again, Figure 4 depicts hypothetical data for illustrative purposes. The actual triangle areas can be calculated by determining the length of the sides with a simple Delta Eab equation, and then once the lengths of each side are known, the area of the triangle can be calculated using Heron’s formula:clemson-ohara-congdon-gasque-equation-1

Where p = half the perimeter of the triangle, and the perimeter of the triangle is the sum of sides a, b and c, in which:

  • a = Delta Eab of Primary 1 to Overprint (i.e. M to MO)
  • b = Delta Eab of Primary 2 to Overprint (i.e. O to MO)
  • c = Delta Eab of Primary 1 to Primary 2 (i.e. M to O)

Table 3 shows the resulting areas of triangles formed between the primaries and their overprints, and the differences (denoted as Delta Area) between the print sequences. The largest area for each sequence is highlighted in gray. Some pairs yield much larger changes in area than others; these are highlighted in yellow.

Table 3: Delta Area for overprint print sequence
Table 3: Delta Area for overprint print sequence

The differences in sequence for the others were marginal. This suggests the sequences for orange in relation to magenta, and green in relation to yellow, are critical, but the sequence for violet in relation to cyan and magenta is not. The resulting print sequence based on triangle area is KYOMVCG; this is close to the KYOMGVC that had yielded our highest printed gamut volume.

The triangle area method is instructive, but the real issue in gamut optimization is volume. In order to predict volume based on 2-color overprints, the three points of the triangles are combined with an L*min and L*max to create a three dimensional (3-D) shape. L*max is defined by the white point of the paper, but L*min can be a moving point due to the fact that it’s commonly a build of black and another color (a rich black rather than 100 percent black). This could result in the black point of the various color pairs overlapping or leaving “gaps” in the overall gamut.

So for this model, a common, neutral black point was selected: L* = 9.5, with a* and b* being zero. We recognize the true black point may not be captured in this fashion, but the goal of EG is to expand chroma, not the black point, so we assume a common black point will serve the purpose of optimizing for gamut expansion.

To calculate the volume of each gamut sector, the sector can be divided into two tetrahedrons:

  • L*max, L*min, Primary 1, Overprint
  • L*max, L*min, Primary 2, Overprint
Figure 5: Tetrahedrons that form an overprint sector volume
Figure 5: Tetrahedrons that form an overprint sector volume

Figure 5 shows the individual tetrahedrons and how they combine to create the volume of an overprint sector, in this case MO. Each of the tetrahedrons shares a common face, L*max, L*min and the overprint, MO.

To calculate the volume of each tetrahedron within the overprint sector, one can use the four L*a*b* points of the tetrahedron to extrapolate a parallelepiped, as illustrated in Figure 6.

Figure 6: Extrapolating a parallelepiped from the four points of a tetrahedron
Figure 6: Extrapolating a parallelepiped from
the four points of a tetrahedron

The volume of a parallelepiped (Vp) can be calculated from the four L*a*b* points:

Vp = (L4-L1)[(a2-a1)(b3-b1)-(b2-b1)(a3-a1)] +(a4-a1)[(b2-b1)(L3-L1)-(L2-L1)(b3-b1)] +(b4-b1)[(L2-L1)(a3-a1)-(a2-a1)(L3-L1)]

Once the volume of the extrapolated parallelepiped has been determined, the volume of the tetrahedron can be determined:

Vt = Vp/6

To determine the volume of the overprint sector, simply add the two tetrahedrons’ volumes together. Once the volume of each sector dependent on the print sequence of the overprint has been determined, the larger volume is the optimal sequence for that particular color pair. Of course, this is a simplified model for the gamut sector, as it reduces the shape to straight lines and flat planes, whereas the actual shape of a gamut sector is a curved geometry due to factors such as TVI, trapping and hue error, but the goal of the model is to be able to optimize a sequence based on the primaries and overprints. To calculate the curved geometry would require far more data, which could be prohibitively complex.

Working from the printed overprints from the various experimental sequences, the overprint sector volumes for each color pair and their differences (denoted as Delta V) were determined and shown in Table 4. The largest volume for each sequence is highlighted in gray. Some pairs yield much larger changes in area than others; these are highlighted in yellow. The differences in sequence for the others were marginal. As was true of the triangle area method, this suggests the sequence for orange in relation to magenta and green in relation to yellow has a relatively high impact, but the sequence for violet in relation to cyan and magenta does not. The resulting print sequence based on overprint sector volume is KOYGCMV.

Table 4: Delta V for overprint sector volumes
Table 4: Delta V for overprint sector volumes

Thus, of the four print sequences performed on press, none of them actually match the result of the predictive model. However, the predictive model does accurately rank the volume of the four printed sequences in the same fashion, and a comparison of the predicted volume and the printed volume shows they are relatively similar. Again, the predictive model is a simplified geometry of flat planes, as opposed to the curved contours of the actual gamut. Figure 7 shows the calculated volumes and actual printed volumes for comparison.

Figure 7: Graphs comparing actual volume vs. model volume
Figure 7: Graphs comparing actual volume vs. model volume

In looking at the results of the four printed sequences, it is interesting to isolate the gamut expansion of the KCMY sequence as well. There was a 106.2 percent increase in gamut volume between running KCMY and KYMC.

Figure 8 shows the gamut expansion between the best of the printed sequences (depicted in green) and the worst printed sequence (depicted in red) in terms of gamut area of a*b* plots at various L* levels. The brown in Figure 8 is where they overlap. It can be seen that the greatest expansion is in the green and orange sectors, with little difference in the violet sector.

Figure 8: a*b* gamut views at various L* levels
Figure 8: a*b* gamut views at various L* levels

This is due primarily to the sequence of MO and YG (which are shown to have the greatest impact on volume in Table 4), and reinforces the observation that violet’s sequence with cyan and magenta has minimal impact on the gamut volume. The gamut expansions are most prevalent at L* values less than 50.