A real example (part #1)

We have two printing devices.  Let's call them "A" and "B".

In the "industrial world" A and B have a defined relationship.  A is coming off lease soon and B is taking over for A.  Since A has been in place for several years millions of impressions a month pass through it.  The owner's of A have many customers who supply this work and they rely on A to produce the output they know and love and, most importantly, they are heavily invested in the process with proofing, approvals, etc.

So after B is set up we run some tests and we discover some interesting differences between A and B (remember, B is years new than A in terms of technology).  The gradient shades are scans of the color output for a set of quantized color values.  The vertical bars are indicators that tell us where each quantized value changes.  Both machines are properly calibrated and use identical profiles, paper and input.

Here Source #1 is printer A and Source #2 is printer B (though it doesn't really matter which is which as we will see in a minute).  This image represents a span of some color C from a value of 100% to 0% (and again, what C is is not important right now).

So what does this tell us?  In general we see that printer A takes about 20 steps to go from say 50% color to 0% color and printer B takes only about 10 steps (marked by #5).  We generally see that B's color output relative to the steps is "compressed" toward the left.

So if we imagine that this represents "gray" output (say the shaded gray's around boxes on statements and so forth) then B's shaded boxes, all things considered, will be lighter than A's.  We can also imagine that this represents a colorant (say yellow) then we would expect weakness in yellow in lighter shades of color using yellow.  Finally, this could be any specific color - say a red used in a sign for a cola company.  In any case the output of B is not going to match A.

So we have identified a problem with B's output.  Note that B is not broken and is in fact working as advertised.  We are simply examining the usability of B's output relative to A.

The next step is to quantify what we see here visually in a mathematical sense.  For this purpose we collect this data via a scanner.

Now we have a characterization of the differences (A on the left, B on the right).  We illustrate some of the differences (note these scales are not normalized) between the two:  #3 indicates a bump in A's output which B does not have, #4 indicates a sharper "knee" in B's output which A does not have.

What's important to note here is that all things being equal for acquiring this information the relative differences between A and B we show here are, for the most part, completely independent of paper, lighting, etc.

Next we normalize the scaling and remove the comb-like spikes introduced by the "tick marks".  Once we do that we can display things more clearly.

Here on the left top we see A's output and on the bottom left B's.  If we superimpose A and B we get the differences between the two devices for color C (the area marked by the bars and #6).

Now remember, the amount of color C is 100% at the left and 0% at the right.

The next step is to characterize the difference between A and  B with a mathematical function.  Before we do that I am going to try and motivate for you what we want to accomplish so hopefully the math, even if its not clear, will not get in the way of understanding.

So let's imagine the large gray area under #6 (the plot of B's output) to be made of rubber so that it can be stretched (or shrunk).  So, naively, if we want to make B's output look more like A's we would want to stretch B's curve to the right and make B's curve taller on the left.   The reverse, making A like B, also works by shrinking - in this case shrinking A's output on the right and left.  So if we could somehow link the stretching to somehow altering A or B we would be able to eliminate the difference between the machines.

In a more formal mathematical sense two things are important here.  First, that the stretching and shrinking between the curves is continuous.  That means we don't use a knife or scissors to get the job done - we only stretch or shrink these areas.  Second that we can "reverse" the stretching and shrinking operations we do for the transition from A to B to create a B to A transition.

We will call these stretching and shrinking operations transformations.