Presenting a possible colour image rendering method applicable for the pixels of the above multi-primary pixel architectures (Figure II/7 and Figure II/8) with 6 or 7 primaries the following constraints are made. The method is described in terms of device independent CIE XYZ tristimulus values. No real binary input data or signal will be dealt with only the need for exhibiting a desired colour and intensity output at a desired location of the screen.
Despite the continuous mathematical representation of an object to be rendered, extra information to perform sub-pixel rendering is given by the following: The input for a given area of the sub-pixel architecture is the desired luminance at the level of sub-pixels and the desired colour information at the level of whole (logical or physical) pixels. This means that it is the luminance information that is given at higher resolution corresponding to the high luminance resolution of the HVS. The method can also be adapted to any multi-primary system featuring n primaries (n>3).85,86 The method is formulated for colour image rendering with and without sub-pixel rendering.
First the notations are described. Ci stands for the relative driving value or weight of sub-pixel Pi in a pixel. Ci is in the interval of [0, 1] and i is the index of the primaries, i=1…n. The problem of multi-primary colour image rendering is that the decomposition of a three dimensional input vector (X,Y,Z)T is not direct since the (3×n) matrix containing the tristimulus values of the colour primaries is not square (n>3) hence it is not directly
where M is a 3×n matrix, actually the ‘phosphor matrix’. For instance for n=6, to render the colour of any arbitrary (X,Y,Z)Torig desired tri-stimulus value, the following algorithm
can be applied. Let us divide the 6 colour primaries of the pixel into two groups of the same size: e.g. {P1, P2, P3} and {P4, P5, P6}. Change the driving values of the first group (Ci, i=1, 2, 3) and calculate the corresponding (X,Y,Z)T output in every optimization step by summing up the output of the first group. A so-called remainder (X,Y,Z)Trem is also calculated as the difference vector between the desired colour (X,Y,Z)Torig and the current value of (X,Y,Z)T.
This remainder can be rendered unambiguously in the 3-dimensional space of relative driving values of the second group of primaries {P4, P5, P6} by using a linear matrix transform. The solution is valid only if Ci is in the interval of [0, 1] for i=4, 5, and 6.
Therefore, the so-called out-of-gamut error term Ecol is also calculated:
col [0, 1] to minimize Ecol until Ecol becomes less than a colour tolerance value or equals zero.
Any general n dimensional optimization technique can be used.
An example for permuting the weights of the first group in the optimization is a so-called cube-scanning method which chooses 7 equidistant discrete values in the interval [0, 1] including the boundaries for Ci (i=1, 2, 3) and repeats the scanning in a tighter
There may be identical colour primaries among the values in the first group but the second set of primaries { P4, P5, P6} has to span a real 3 dimensional space in the CIE XYZ space of tri-stimulus values. The method introduced above85,86 yields the weight factors of each colour primary to represent a desired colour output within a single pixel without any sub-pixel level rendering.
To include sub-pixel rendering, luminance information considerations are also required: luminance information is represented at higher resolution than colour information, namely, at the sub-pixel level. The method shown for this case represents fine intensity resolution at the cost of some colour error (the principle of sub-pixel rendering).
Decomposition of (X,Y,Z)Torig is mathematically not unique for 6 primary colours. There are an infinite number of linear combinations of the weights of the six primary colours (if they span a real 6 dimensional space). Based on linear algebraic considerations it is possible to choose one single combination of weights with such a luminance distribution of the sub-pixels that yields the best approximation of the luminance distribution of the original image.
This multi-primary sub-pixel rendering algorithm can be chosen similar to the first algorithm described above (Eqs. (II/3)-(II/6)) but this one incorporates sub-pixel rendering by adding a second error function term, the so-called sub-pixel luminance error term (Elum), to the out-of-gamut error term Ecol. The sub-pixel luminance error term Elum can be defined for instance as the difference of the intensity ratios of the sub-pixels (Yi) within a logical pixel (Yi/Y) and the same luminance ratios in the original image (denoted by Yi, orig/Yorig):
,orig lum
1 orig
n i i i
Y Y
E = Y Y
=
∑
− , (II/7)The total error function that should be minimized is:
tot col lum
E =αE +βE , (II/8)
where α and β are suitable weight parameters. If α=0 then no colour information is taken into account (only luminance), if β=0 then the case without sub-pixel rendering is returned to (Eq. (II/3)).
During the optimization procedure, each weight of the set {P1, P2, P3} is changed in the interval [0, 1] to minimize Etot until it becomes less than a suitable tolerance value. Any general three-dimensional optimization technique can be used again.
For the case of the 7-primary architecture, a similar method can be used. The only modification is that we have to select four primaries to optimize in the first group. The second group must contain three elements again which should be linearly independent in a three dimensional space.