© James H Nobbs 



The way in which a coloured printed image is created by the fourcolour process is described in an earlier document.
The colour seen by the eye in each part of the image arises from the blending of the light reflected by a mosaic of dots, as shown in Figure 1.
Each type of dot will contribute a proportion of light in the red, the green and the blue bands of the spectrum and the net effect of the local mosaic of dots is determined by the sum of these contributions. 
The steps in the conversion of the intended colour of a printed area (XYZ) to the relative areas of the substrate covered by dots of cyan, yellow and magenta ink (CYM) is not straightforward.
It is simpler to consider the reverse transformation, taking a set of known values for CYM and determining the colour of the print (XYZ). The steps involved are summarised by Figure 2.
The relationships used to model each of the steps are given in the following section. 
Factors involved
The fourcolour printing process creates tonal scales by printing patterns of coloured dots. There is set of dots for each of the process inks, cyan (C), yellow (Y), magenta (M) and black (K).
Each position on the print can have a dot created by a single layer of ink or two layers of ink. The need to print three layers of ink is replaced by substituting a dot from the black ink. Each dot position can be one of the eight colours shown in Table 1.
Different fractional areas of the each type of coloured dot produce a range of colours within the print; the variation in dot area and dot colour within each small region on the print creates the impression of a coloured image with a wide tonal range. 

Dot Colour 
Ink Layers 
Dot Type 

White 
None 
wht 

Cyan 
Cyan 
cyn 

Yellow 
Yellow 
ylw 

Magenta 
Magenta 
mgt 

Red 
Yellow + Magenta 
red 

Green 
Yellow + Cyan 
grn 

Blue 
Cyan + Magenta 
blu 

Black 
Black 
blk 
The amounts of light reflected by an area of the print can be predicted from the fractional areas of the each type of dot by the Neugebauer equations using the principles of additive colour mixing.
Intended print area (S) and effective area (A)
The instructions sent to the inkjet printer or to the printing plate production process are the fraction of the local geometric area to be printed with dots of process ink. In this analysis the intended printed fractional areas are represented by SC, SY, SM, (and SK). The term S is used since this area was once known as “Screen Area”.
The dot area can absorb light as if it has a greater area than the intended printed area; this arises from the boundary/edge region of the dots having slightly different light absorbing properties to those of the dot centre. The change is called the dotgain. The magnitude of the dot gain depends on the value of SC, SY or SM. There is zero dot gain when the fractional area is zero and when it is 1.0 (full area coverage, no edges).
The effective dot areas AC, AY, and AM are given by:


Equation 2 


Where G is the gain factor that has values typically in the range 10% to 25%.
Dot likelihood, e.g. P(cyn)
The probability of finding a point on the print having just a cyan ink layer is the fraction area AC, however the same point may also be have a yellow ink layer or a magenta ink layer or both. The probability of a cyan ink only layer is given by:
Equation 3 
(probability of cyan ) × (probability of not yellow ) × (probability of not magenta) 
Equation 4 
P(cyn) = AC(1 – AY)(1 – AM) 
Where the probability
of 
not finding a magenta layer is (1 – AM) 
and of 
not finding a yellow layer is (1 – AY). 
The probabilities, or fractional areas, of each of the eight types of coloured dot are given in Table 2. 

Dot Colour 
Ink layers 
Fractional area of mosaic 

White 
None 
P(wht) = (1 – AC)(1 – AY)(1 – AM) 

Cyan 
Cyan 
P(cyn) = AC (1 – AY)(1 – AM) 

Yellow 
Yellow 
P(ylw) = AY (1 – AC)(1 – AM) 

Magenta 
Magenta 
P(mgt) = AM (1 – AC)(1 – AY) 

Red 
Yellow + Magenta 
P(red) = AY AM (1 – AC) 

Green 
Cyan + Yellow 
P(grn) = AC AY (1 – AM) 

Blue 
Cyan + Magenta 
P(blu) = AC AM (1 – AY) 

Black 
Cyan + Yellow + Magenta 
P(blk) = AC AY AM 
Dot R, G, B
Print RGB, colour of the mosaic
A summation is made of the red contributions from each of the eight types of dot in the reproduction mosaic to give an estimate of R, the amount of light in the red wavelength band that is reflected by that area of the print.
Similar equations estimate the amount of green (G) and blue (B) light reflected by the area of print.
Although there are eight terms on the right hand side of each equation they are derived from only three quantities, the fractional dot areas of the print of C, Y and M.
Print XYZ, colour of the mosaic
The values of R G and B can be reexpressed in terms of X Y and Z by a set of three linear equations.





Inverting the process, X Y Z to dot areas C Y M K
Equation 5 shows that the amount of red light from an area on the printed image is linked to the fractional dot areas of all four of the inks. In addition there are nonlinear effects such as dotgain, As a result, a number of corrections and adjustments of the dot areas must be made in order to obtain an acceptable quality of colour reproduction.


© James H Nobbs 