Color mixing

1
Color mixing

• Suppose a system of colorants (lights, inks,).
  Given two colors with spectra c1(l) and c2(l).
  This may be reflectance spectra, transmittance
  spectra, emission spectra,Let d be a mix of
  c1and c2. The system is additive if d(l)
  c1(l) c2(l)no matter what c1 and c2 are.

2
Scalability

• Suppose the system has some way of scaling the
  intensity of the color by a scalar k.
• Examples
• CRT increase intensity by k.
• halftone printing make dots k times bigger
• colored translucent materials make k times as
  thick
• If c is a color, denote the scaled color as d. If
  the spectrum d (l) is k(c(l)) for each l, the
  system is scalable

3
Scalability

• Consider a color production system and a colors
  c1,c2 with c2kc1. Let mimax(ci(l))and
  di(1/mi)ci. Highschool algebra shows that the
  system is scalable if and only if d1(l )d2 (l)
  for all l, no matter what c1 and k.

4
Control in color mixing systems

• Normally we control some variable to control
  intensity
• CRT
• voltage on electron gun
• integer 0...255
• Translucent materials (liquids, plastics...)
• thickness
• Halftone printing
• dot size

5
Linearity

• A color production system is linear if it is
  additive and scalable.
• Linearity is good it means that model
  computations involving only linear algebra make
  good predictions.
• Interesting systems are typically additive over
  some range, but rarely scalable.
• A simple compensation can restore often restore
  linearity by considering a related mixing system.

6
Scalability in subtractive systems
n
0ltklt1
kL0
L0
kkL0
knL0
d
d
d
7
Scalability in subtractive systems
n
0ltklt1
L0
knL0
Tl tlb where Tl is total transmittance at
wavelength l, tl transmittance of unit thickness
and b is thickness
L(nd) knL0 n integer L(bd) kbL0 b
arbitrary L(b) kbL0 when d 1 L(b)/L0
kb
8
Linearity in subtractive systems

• Absorbance
• Al -log(Tl) defn
• -log(tlb)
• -blog(tl)
• -bal where alabsorbance of unit
  thickness
• so absorbance is scalable when thickness b is the
  control variable
• By same argument as for scalability, the
  transmittance of the "sum" of colors Tl and Sl
  will be their product and so the absorbance of
  the sum will be the sum of the absorbances.
• Thus absorbance as a function of thickness is a
  linear mixture system

9
Tristimulus Linearity

• Xmix Ymix Zmix X1 Y1 Z1 X2 Y2 Z2
• c X Y Z cX cY cZ
• This is true because
• r(l) g(l) b(l) are the basis of a 3-d linear
  space (of functions on wavelength) describing
  lights
• Grassman's laws are precisely the linearity of
  light when described in that space.
• X Y Z is a linear transformation from this
  space to R3

10
Monitor (non)Linearity
L1(A,B,C)
L2(A,B,C)
f2(L1, L2, L3)
L3(A,B,C)
11
Monitor (non)Linearity

• In A,B,C --gt L L1, L2, L3 --gt Out
  O1 O2 O3 f1(L1, L2, L3) f2(L1, L2, L3)
  f3(L1, L2, L3)
• Interesting monitor cases to consider
• In dr dg db where dr, dg, db are integers
  0255 or numbers 01 describing the programming
  API for red, green, blue channels
• Out X Y Z tristimulus coords or monitor
  intensities in each channel
• Typically
• fi depends only on Li
• fi are all the same
• fi(u) ug for some g characteristic of the
  monitor

12
Monitor (non)Linearity

• Warning
• LCD non-linearity is logistic, not exponential
  but flat panel displays are usually built to
  mimic CRT because much software is
  gamma-corrected (with typical g2.4-2.7)
• Somewhat related Most LCDdisplays are built
  with analoginstead of digital inputs, in
  orderto function as SVGA monitors.This is
  changing.

13
Monitor (non)Linearity

• (CRT Colorimetry example of Berns, p. 168-169)
• Non-linearity is f(u)ug , g 2.7, same for all
  output channels.
• Linearity is diagonal

a 0 00 a 00 0 a
b 0 00 b 00 0 b


where a1.02/255, b -.02