**1A. (5 points) Describe the physics (in terms of changes to various
spectra) when two colored light sources are combined. What's the
difference between combining light sources and combining two colored filters?**

A colored light source can be represented as a spectral energy distribution,
a graph of how much energy the light source puts out at each wavelength.
When we combine two light sources, we are combining their energies.
To find the spectral energy distribution of the result, we sum the energies
of the two sources at each wavelength. Since we are adding light
energies, we call this **additive** color mixing.

A colored filter can also be thought of in terms of a spectral energy
distribution, but this time in terms of how much energy at each wavelength
the filter blocks. When we combine two filters, we combine their
blocking powers multiplicatively: if filter A passes 40% of the energy
at a given wavelength and filter B passes 50%, the combination of filters
A and B will pass 20% of the energy at that wavelength. Since each
filter subtracts energies from our original light source, we call this
**subtractive** color mixing.

*Scoring:*

**1B. (5 points) Most printing systems use cyan, magenta and yellow
as the primaries. Why do printers use CMY and computers (displaying
images on monitors) use RGB?**

Monitors work by starting with blackness and adding colored light in proportion to how many electrons are fired at the phosphors in the screen. Since a monitor adds light energies to blackness, it is natural to use an additive color system like RGB.

Printers work by depositing a blob of ink on paper (usually white). The blob of ink acts as a filter, allowing only certain wavelengths to be reflected from the surface of the paper. Since a printer subtracts light energies from white, it is natural to use a subtractive color system like CMY.

*Scoring:*

**1C. (10 points) Just like black and white printers, color printers
use halftone patterns to place colored dots on the paper. Instead
of a single spot of black ink, three spots: one cyan, one magenta and one
yellow spot are put in each cell. Suppose one dot of each color is
placed inside each grid cell on the page; suppose also that the size, and
hence coverage, of each colored dot relative to the unit cell is a _{c}
(for alpha of cyan), a_{m} and a_{y}, and that these dots
overlap randomly as follows:**

**(7 points) Assume you measure the color of a single ink spot and
the 4 other combinations (cyan on magenta, etc.) of spots. Write
an expression for the final color of the unit cell in terms of primary
ink colors, the colors of the various combinations, and the coverage of
each primary ink spot.**

The final color of the cell will be the sum of all the different colors
in the cell times their respective areas. There are a total of **eight**
colors in the cell: the primaries CMY, the three pair-wise combinations,
the region where all three colors overlap, and the white of the empty part
of the cell (most people forgot the white).

Let's examine how to calculate the coverage of one of these regions,
where cyan and magenta overlap. Where these two colors overlap, there
will be a new color C_{b} (the b is for blue). What is the
area of this region? Since we treat the cell as having an area of
1, we can view the quantity a_{c} as the percentage of the cell
that contains cyan ink. Therefore, the area of the blue region will
be the percentage of the cell that has both cyan and magenta ink, but no
yellow. Since we assume that the overlap of our three dots is random,
we can calculate this area but multiplying the three factors together.
So, a_{b} = a_{c}a_{m}(1-a_{y}).

The math is similar for each of the eight regions, which yields the expression:

a_{c}a_{m}a_{y}C_{k} + (1-a_{c})a_{m}a_{y}C_{r}
+ a_{c}(1-a_{m})a_{y}C_{g} + a_{c}a_{m}(1-a_{y})C_{b}
+ a_{c}(1-a_{m})(1-a_{y})C_{c} + (1-a_{c})a_{m}(1-a_{y})C_{m}
+ (1-a_{c})(1-a_{m})a_{y}C_{y} + (1-a_{c})(1-a_{m})(1-a_{y})C_{w}

where C_{k} is the color where all three dots overlap, C_{r}
is the color where the magenta and yellow dots overlap, C_{c} is
the color of the cyan dot, C_{w} is the color of the page, etc.

*Scoring:*

**(3 points) Assume the printer allows you to control the coverage
values. Is a coordinate system based on coverage a linear color coordinate
system?**

A color coordinate in the coverage-based system would be a triple (a_{c
}a_{m }a_{y}). Since there are multiplicative
terms of the coverage values in the above formula, this system would not
be linear.

*Scoring:*