Color: Cohen’s intriguing F matrix again

Minor edits, 4 Jan 2010.

Let me run thru the derivation of Figure 20 again, still with the CIE 1964 tables, but at 20 nm intervals. Then I show two alternative calculations. And one of them will show us why Cohen switched the sign on f2.

There is interesting linear algebra and vector algebra in this post, even if you’re not interested in where these problems came from.

Review:

Color: Cohen Figure 20, “an intriguing F matrix”

Typographic edits, 4 Jan 2010.

I struggled to make sense of pages 94-101 of Cohen’s “Visual Color and Color Mixture”.

I understand them now. Mostly.

The challenge was to derive a drawing of three basis vectors (Figure 20, p. 95), shown here :

He calls these three vectors both “an intriguing F matrix” and “This canonical orthonormal configuration….” But as far as I can tell, he never said where he got this particular one. He made it sound important, so I had to derive it. Figuring it out, as usual, was very educational.

What follows is a detective story.

Transition matrix: to be or not to be

Cohen (“Visual Color & Color Mixture”, see the bibliography) did something very interesting. In fact, he did something useful which I had never seen before.

Although this post uses some matrices which we saw in the color posts, I think this can stand on its own: you need not have read the color posts. But if you are specifically interested in color, or in Cohen’s work, this post is very relevant.

He was trying to describe how to find a transition matrix between two given data matrices. This will come in handy — very handy! — whenever people give the alleged result of an unspecified linear transformation of a data matrix.

Color: re-doing Cohen’s example

Cohen’s example again

Let me now show you how I would do Cohen’s example. (His computations, pretty much, were the previous post.) I cannot over-emphasize that he deserves a lot of credit for getting the mathematics right, even if he didn’t name it correctly or do it beautifully.

I start with the A matrix:

Cohen: “Visual Color and Color Mixture”

Introduction

Oct 10 edit: the third heading has been changed to “Computing E from A”

I want to work an example from Cohen, “Visual Color and Color Mixture” (see the bibliography, and this “books added” post). I am not, however, going to do it exactly the way he did. Nevertheless, I will show you everything he calculated.

Because I want to get everything of his into one post, I will break this into small sections. I expect that my next post will show you what I would have done instead.

All of his matrices can be found on p. 70 of his book.

What we have here is an extremely small example to illustrate “color matching functions” applied to light spectra, resulting in three real numbers which we call R,G,B or X,Y,Z. This is a prelude to using real color matching functions on real spectra. I will refer to the 3 numbers I get during the course of this example as “R,G,B”, in quotes because this is a toy example, and because down the road I’ll be computing “XYZ tristimulus values” in preference to RGB.

The A and E matrices: computing A from E