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

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Fourier Series and Fourier Transform

I want to show you something that I think is really neat. (Whether you think it is neat is your decision, not mine. I don’t like it when people tell me what I will think about something they’re about to show me.)

I do not know why this calculation is not readily available. All I know is that I had to search high and low to find an example with sufficient detail that I could duplicate it. In fact, it is difficult to find a text which actually states the result at all, never mind one which works it out for a particular case.

  • I intend to take a function which is zero outside the interval [-1/2, 1/2]…
  • I will compute several terms in the associated Fourier series…
  • I will compute its Fourier transform…
  • and I will hit you between the eyes with the relationship between them.

I won’t even keep it a secret:

  • the coefficients in the Fourier series are samples of the Fourier transform.
  • alternatively, the Fourier transform of the Fourier series is discrete samples of the original continuous Fourier transform.

I think that’s marvelous. But even knowing that it’s true, I don’t see it stated very often.

There are several things I will not do in this post.
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Linear Algebra: the four fundamental subspaces

Edit 18 Sep 2009. Purely cosmetic. I have highlighted in blue the result describing the spans of the parts of the u and v matrices. I have found that I still need to refer to it rather than reconstruct it when I want it. In other words, it’s what I want to pick out quickly from this post.

Given a linear operator A, we are told that there are four fundamental subspaces associated with it. They are

  • the nullspace of A
  • the column space of A
  • the nullspace of A^T\ (i.e. of the transpose of A)
  • the column space of A^T\ .

Okay, they are easy enough to list.

Next, we need to realize that the column space of A is the range of A, and the column space of A^T\ is the range of A^T\ . That’s what we really care about — the ranges.

But why is A^T\ involved? In particular, why do we care about the nullspace and the range of A^T\ ?

Let me show you what they “really” are. (That is, this is how I understand them.)
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