1. Color mixing Suppose a system of colorants (lights, inks,…). Given two colors with spectra c 1 ( ) and c 2 ( ). This may be reflectance spectra, transmittance spectra, emission spectra,…Let d be a mix of c 1 and c 2. The system is additive if d( ) = c 1 ( ) + c 2 ( ) no matter what c 1 and c 2 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 (  is k(c( )) for each  the system is scalable

3. Scalability Consider a color production system and a colors c 1,c 2 with c 2 =kc 1. Let m i =max(c i ( )) and d i =(1/m i )c i. Highschool algebra shows that the system is scalable if and only if d 1 ( )=d 2 ( ) for all, no matter what c 1 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. kL 0 L0L0 k*kL 0 knL0knL0 n ddd Scalability in subtractive systems 0<=k<=1

7. L0L0 knL0knL0 n L(nd) = k n L 0 n integer; L(bd) = k b L 0 b arbitrary L(b) = k b L 0 when d = 1; L(b)/L 0 = k b Scalability in subtractive systems T  = t b where T is total transmittance at wavelength, t transmittance of unit thickness and b is thickness 0<=k<=1

8. Linearity in subtractive systems Absorbance A = -log(T ) defn = -log(t b ) = -blog(t ) = -ba  where a =absorbance 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 T  and S  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 [X mix Y mix Z mix ] = [X 1 Y 1 Z 1 ] + [X 2 Y 2 Z 2 ] c [X Y Z] = [cX cY cZ] This is true because –r( ) g( ) b( ) 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 R 3

10. Monitor (non)Linearity L 1 (A,B,C) L 2 (A,B,C) L 3 (A,B,C) f 2 (L 1, L 2, L 3 ) ABCABC Linear stage Non-linear stage f 1 (L 1, L 2, L 3 ) f 3 (L 1, L 2, L 3 )

11. Monitor (non)Linearity In = [A,B,C] --> L = [L 1, L 2, L 3 ]--> Out = [O 1 O 2 O 3 ] = [f 1 (L 1, L 2, L 3 ) f 2 (L 1, L 2, L 3 ) f 3 (L 1, L 2, L 3 )] Interesting monitor cases to consider: –In = [dr dg db] where d r, d g, d b are integers 0…255 or numbers 0…1 describing the programming API for red, green, blue channels –Out = [X Y Z] tristimulus coords or monitor intensities in each channel –Typically: f i depends only on L i f i are all the same f i (u) = u  for some  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  =2.4-2.7) Somewhat related: Most LCD displays are built with analog instead of digital inputs, in order to function as SVGA monitors. This is changing.

13. Monitor (non)Linearity drdgdbdrdgdb RGBRGB a 0 0 0 a 0 0 0 a = + b 0 0 0 b 0 0 0 b where a=1.02/255, b= -.02 (CRT Colorimetry example of Berns, p. 168-169) Non-linearity is f(u)=u ,  = 2.7, same for all output channels. Linearity is diagonal: