Hi Ben,
at that point
______________
x + 2y = 5
2x + 3x = 8
From here, you can get x = 5 - 2y, work the rest of the way through, etc., to
eventually find that x = 1 and y = 2.
_____________
I couldn't follow any more with the math I learned… ;-)))
Best regards
Andreas
Von: argyllcms-bounce@xxxxxxxxxxxxx <argyllcms-bounce@xxxxxxxxxxxxx> Im Auftrag
von Ben Goren
Gesendet: Montag, 14. Juni 2021 22:10
An: argyllcms@xxxxxxxxxxxxx
Betreff: [argyllcms] Re: Spectral data for building Monitor profiles
On Jun 14, 2021, at 11:12 AM, Ben Goren <ben@xxxxxxxxxxxxxxxx
<mailto:ben@xxxxxxxxxxxxxxxx> > wrote:
If you don’t have a colorimeter…I don’t think the spectral data is going to do
much to improve things.
Just to elaborate on that…I “just happen” to be 2/3 of the way through a Linear
Algebra summer class, with a big test tomorrow on linear dependence, basis,
etc. And that “just happens” to get right to the heart of why more spectral
bands isn’t going to make an improvement.
So…as part of my preparation for tomorrow’s test…let me expound upon that.
First, the physics.
A colorimeter is conceptually equivalent to a low spectral resolution
spectrometer. The colorimeter has three wide-band spectral filters, roughly
matching up with the biological analogues in the human eye. There’s a red
filter over one sensor, a green filter over another, and a blue filter over the
last. There’s overlap; something cyan-colored is going to register roughly
equally on the green and blue sensors, and just barely (if at all) on the red
filter.
Instead of colored filters, a spectrometer uses a diffraction grating (it could
use a prism, but those are bigger, heavier, and more expensive) to split the
light into a rainbow, and that gets projected onto unfiltered sensors. I think
the i1Pro has 21 sensors. How the spectrometer records that same cyan light
depends on the spectral composition of the light. If it’s a monochromatic
source, such as from a, say, 490 nm laser, then just that one single sensor
that spans that part of the spectrum will get triggered; all others will record
negligible noise. But, if it’s from a display, it’ll record basically a
double-humped camel in profile, with one hump aligning with the peak of the
display’s blue phosphors and the other the display’s green phosphors.
Imagine recording a sequence of red-only, blue-only, and green-only patches of
varying intensities, exactly like what you see Argyll do when calibrating the
display. On the spectrometer, you’ll see single humps get bigger or smaller,
but those humps won’t move around or change shape.
In an ideal world, the filters on your colorimeter will be exact matches for
the spectra of the display. But that’s not only impractical to manufacture in
the first place, it would mean the colorimeter could only be used with that one
display.
So, here’s where the math comes in.
We normally think of RGB colors as being made of three numbers. To make it
simple, think of each being a simple zero to nine. Now, instead of, say, R=2,
G=1, B=7, we could write that color as 217. Pretty clearly, if we do that
consistently, we get the same thousand colors whether you write them separated
by commas or as a single three-digit number.
Right off the bat, I need to note that, where this is heading, it’s not going
to form what’s called a “subspace” in Linear Algebra. If you add together 777
and 777, you get a number bigger than 999, the largest number in our system.
That means it’s not closed for addition. (By similar reasoning, it’s also not
closed for scalar multiplication, the other big test for subspaces.) In the
graphic arts world, that means that clipping is a concern. How to deal with
that is beyond the scope of this email. But we can still do lots of very useful
math regardless.
Anyway, we can put that together to write some systems of equations. For the
colorimeter, we might have three variables: r, g, and b. For a low-resolution
spectrometer, we might have v, b, c, g, y, o, and r for the famous spectral
colors. For the i1Pro, you might have x_1, x_2, x_3, …, x_21.
I’m now going to step away from examples that obviously map to the instruments
we’re actually working with. Hopefully, I’ve convinced you that it’s logically
reasonable to represent things this way, with a bit of hand-waving that the
actual math can quickly turn into mind-numbing walls of numbers.
So, let’s wave our hands and make life about as simple as we can. Our
colorimeter is actually just two channels, x and y; and our spectrometer is
three channels, x, y, and z. Our display only has two colors.
We go through all sorts of algebraic adventures to boil everything for the
colorimeter down to this system of equations:
x + 2y = 5
2x + 3x = 8
From here, you can get x = 5 - 2y, work the rest of the way through, etc., to
eventually find that x = 1 and y = 2.
For the spectrometer, we get:
16x + 2y + 3z = 13
5x + 11y + 10z = 8
11x - 9y -7z = 5
Looks like you need all three of them, right?
Wrong. Subtract the second equation from the first, and you get the third. As
it turns out, you can use any value you want for z and you’ll get the exact
same answer. You’ve got three equations and three variables, sure…but you can
get the exact same answers with two variables and two equations. (In this case,
the simplification involves some nasty fractions; I’ll spare you the details.)
In essence, the three-equation system is only two dimensional, with the third
dimension having gotten flattened all the way down to a plane. There’s nothing
you can do to the third variable that will move you perpendicular to the plane.
The same thing (in principle) happens with the spectrometer. Yes, you’ll
capture all that extra data…but the math works out to being just the same as if
you only use the colorimeter. The 21 dimensions of the i1Pro get flattened to
the 3-dimensional space of the display. And the CCMX skews / stretches /
whatever the 3-dimensional space of the colorimeter so that it matches that of
the display.
There’s still a possibility worthy of a footnote. There’s noise in any
real-world system. If the CCMX isn’t spot-on, the errors will get “baked into”
all the measurements. With the spectrometer, you’re re-sampling the display’s
spectral response with every measurement, so any one glitch has much less
impact. But, in the real world…I very much doubt that sort of error will be
meaningfully measurable — and the gain in low-light performance from the
colorimeter can potentially be quite noticeable. (A visual analogue…imagine not
a zero-height two-dimensional plane, but a piece of paper with measurable but
probably meaningless thickness.)
A last footnote. Mathematicians will note that I used systems of equations, as
opposed to vectors, to demonstrate the notion of linear dependence. This is
especially noteworthy since the systems above have singular solutions.
But…novices probably know enough about systems of equations to be able to
recognize how the “spectrometer” example reduces, whereas it took me a month of
summer classes to get to the point where I understand enough about vectors to
be able to describe linear dependence.
Cheers,
b&
