## (abuse of) terminology

Sometimes I get tired of writing “xbar, ybar, zbar tables” — and I just write “xyz bar tables” or even “XYZ tables”. Similarly for rbar, gbar, bbar tables — rgb bar. I’m not talking about anything new, just abbreviating the names.

## Introduction

This is the third post about the example on p. 160 of W&S. Once again, I am going to decompose the reflected spectrum into its fundamental and its residual.

This time, however, I’m going to use the rbar, gbar, bbar tables (RGB) instead of the xbar, ybar, zbar (XYZ) tables. They did not do this.

I’ll tell you now there is one little twist in these calculations. We will need the ybar table, because we still need to use it to scale our results.

In addition to showing, as I did previously, the dual basis spectra and the orthonormal basis spectra for the non-nullspace, I will display the orthonormal basis for the nullspace.

As usual, I am using the CIE 1931 tables, and I am working at 20 nm intervals.

Our ingredients were: a reflectance spectrum, an illiminant spectrum, and some choice of color matching functions. For this post, I am choosing a different set of color matching functions.

Our results in the previous post were a few sets of 3 numbers. The final result was called xyz — meaning the triple (x, y, z) — for plotting a point on the CIE chromaticity chart. We will not do this again.

That result came from a triple denoted XYZ, which are values in XYZ color-space. That space has the advantage of being device-independent. We could — and did — convert those values to an RGB color space, specifically the one defined by the rgb bar (i.e. rbar, gbar, bbar) color-matching tables.

This post will compute the RGB values directly, and we will find that we get almost exactly the same RGB values. (We really ought to.)

The XYZ result in turn came from a triple for which I have no convenient notation, but I keep saying without proof, and confirming by computation, that that triple is the components of the fundamental of the reflected spectrum wrt the basis dual to the chosen color-matching functions.

Let us go see that, this time for the rbar, gbar, bbar tables instead of xbar, ybar, zbar.

## The same old calculations, but with a twist

But first, let me review the calculations. Feel free to skip to the next section.

Our first computation is to get the reflected spectrum, which is the pointwise product of the illuminant spectrum and the reflectance spectrum. For this problem, the reflected spectrum is the perceived spectrum, the input to the eye of “the standard observer”. The data itself is in the fourth case of the first post about this example.

Here is what the three of those spectra looked like:

As before, I’ve decided to avoid red, green, and blue — except where they are appropriate. For this post, they will be appropriate in some cases. The illuminant is shown in white, the reflectance spectrum is shown in purple, and the reflected spectrum is shown in yellow.

Having said that, I am going to use red, green, blue colors for the three columns of our A matrix, because these entires are rbar, gbar, bbar values. These are the 1931 color matching functions (i.e. a subset, at 20 nm intervals)

BTW, we can tell that we are using RGB (i.e. rbar, gbar, bbar tables) because one of these functions is clearly negative over part of its range. In fact all of them are negative somewhere, but it’s only obvious for the red one.

Since I’ve just introduced new data, let me display it:

$A = \left(\begin{array}{ccc} 0.00003 & -0.00001 & 0.00117 \\ 0.0003 & -0.00014 & 0.01214 \\ 0.00211 & -0.0011 & 0.11541 \\ -0.00261 & 0.00149 & 0.31228 \\ -0.02608 & 0.01485 & 0.29821 \\ -0.04939 & 0.03914 & 0.14494 \\ -0.07173 & 0.08536 & 0.04776 \\ -0.09264 & 0.17468 & 0.01221 \\ -0.03152 & 0.21466 & 0.00146 \\ 0.0906 & 0.19702 & -0.0013 \\ 0.24526 & 0.1361 & -0.00108 \\ 0.34429 & 0.06246 & -0.00049 \\ 0.29708 & 0.01828 & -0.00015 \\ 0.15968 & 0.00334 & -0.00003 \\ 0.05932 & 0.00037 & 0 \\ 0.01687 & 0.00003 & 0 \\ 0.0041 & 0 & 0 \\ 0.00105 & 0 & 0 \\ 0.00025 & 0 & 0 \\ 0.00006 & 0 & 0 \\ 0 & 0 & 0\end{array}\right)$

Our second computation (or third, this order doesn’t matter) should be to apply the matrix A^T — the (transpose of) the color matching functions A — to the reflected spectrum S:

$A^T\ S = \{70.5889,\ 61.249,\ 47.8412\}\$.

As before these numbers are components of (the fundamental of) the reflected spectrum wrt the basis dual to $A^T\$. In Cohen’s notation, these 3 numbers are the components wrt the columns of the E matrix.

The other computation, whether we do it second or third, is to find the constant for normalizing things, using the illuminant spectrum; that is, to set the white point (1,1,1) in our color space.

Here’s the twist. We still have to use the second column of the XYZ (xyz bar) tables; not one of the columns of our new A is the total cone response. (Nor is the sum of them, and I’ll confess that I don’t know why. My WAG (my guess) is that the three color matching functions each have to be weighted differently in order to sum to the total cone response.)

The number we got by applying the second column of the XYZ tables to the illuminant was 529.764 .

Now we move on to the fourth computation: we divide our 3 values $A^T\ S$ by that to get RGB.

RGB = {0.133246, 0.115616, 0.0903067}.

It is encouraging to see that these RGB values are in [0, 1]. These are tristimulus values in an RGB color space, namely the one I keep showing on the chromaticity chart.

Let me be clear. The red, green, and blue dots on this diagram are the projections of (1,0,0), (0,1,0) and (0,0,1) onto xy-space. The values I just got are in this RGB space.

Oh, in the previous post we transformed the computed XYZ values into RGB; what did we get?

{0.133246, 0.115616, 0.0903103}

So. Almost exactly the same answers: only the B value differs. That’s good. I said I would confirm this equivalence.

The blue value is off a tiny bit. (The simple fact is that the two sets of tables — xyz bar and rgb bar — and the transition matrix between them are not consistent. In fact, it appears that the columns of the two tables do not quite span the same 3D space. If I’m right, I could fix this by using one table and the transition matrix to construct a very slightly different version of the second table. But bear in mind that I am printing far more digits than W&S showed.)

## the dual basis

Now, let’s get the dual basis E. We recall that instead of

$E^T = A^{-1}\$,

we use the pseudo inverse and write

$E^T = (A'A)^{-1}\ A'\$

It is easier to work with $E^T$ but easier to display E:

$E = \left(\begin{array}{ccc} 0.000561675 & -0.000849316 & 0.00535311 \\ 0.00583075 & -0.00906171 & 0.0555629 \\ 0.0527362 & -0.0836627 & 0.52778 \\ 0.110419 & -0.184984 & 1.42203 \\ 0.0130902 & -0.0537785 & 1.34031 \\ -0.149518 & 0.244672 & 0.616947 \\ -0.311401 & 0.664664 & 0.135783 \\ -0.492205 & 1.35497 & -0.0912093 \\ -0.339342 & 1.5831 & -0.142044 \\ 0.0806418 & 1.32314 & -0.0962321 \\ 0.656427 & 0.713092 & 0.00326589 \\ 1.0634 & 0.072223 & 0.0910496 \\ 0.956881 & -0.194913 & 0.10246 \\ 0.521479 & -0.151607 & 0.0594277 \\ 0.194688 & -0.0626132 & 0.0226816 \\ 0.0554502 & -0.0183495 & 0.00649827 \\ 0.0134844 & -0.00451219 & 0.00158394 \\ 0.00345331 & -0.00115556 & 0.000405644 \\ 0.000822217 & -0.000275134 & 0.0000965819 \\ 0.000197332 & -0.0000660321 & 0.0000231797 \\ 0. & 0. & 0.\end{array}\right)$

Check that we do have $E^T\ A = I\$. I didn’t say it last time, so let me emphasize: I didn’t prove that the pseudo-inverse would give me the dual basis. This easy “check” is the real confirmation that my recipe actually works. This check is crucial, not merely convenient.

Let me call the basis vectors E1, E2, E3, and let’s look at them. In fact, let me juxtapose the dual basis (right) with the color matching functions (left).

If we multiply each of those dual basis vectors (E1, E2, E3) by our original three numbers…

{70.5889, 61.249, 47.8412}

and add the results.. we get numbers…

{0.243727,2.51476,23.848,64.4959,61.7521,33.9471,25.2246,
43.8826,66.214,82.1295,90.1689,83.8434,60.5088,30.3679,
10.993,3.10116,0.751257,0.192395,0.0458083,0.010994,0.}

and a picture…

where I have shown the sum in white.

Let’s look quickly at the reflected spectrum (in yellow) and that sum (still in white):

Yes, we’ve seen that before. And as before, that white curve is the fundamental. It differs from the yellow curve by a metameric black, a spectrum in the null space of $A^T\$.

As before, we could stop here. We have gotten RGB values from the reflected spectrum… we have found the dual basis… we have found the fundamental of the reflected spectrum… we know that the first three numbers we computed ($A^T\ S\$) are the components of the fundamental wrt the dual basis.

## The SVD, orthonormal basis, and projection operator

Now let’s do the SVD (Singular Value Decomposition), because we can get a nice orthonormal basis for the entire domain of $A^T\$ including its null space, i.e. for spectra including the metameric blacks.

As before, we will get an orthonormal basis v1 for the pre-image of $A^T\$, then a projection operator R onto that pre-image, then recompute the fundamental by applying the projection operator to the reflected spectrum.

First, let

$A^T = u\ w\ v^T\$.

(The first time I did this, I let $B = A^T\$. Now we’re more experienced, but if you want to use B instead of $A^T\$, feel free.)

As always, our vectors have length 21 because we used 20 nm intervals from 380 to 780 nm. Then, the leftmost 3 columns of v are an orthonormal basis for the pre-image of $A^T\$, and the rightmost 18 (i.e. 21-3) are an orthonormal basis for the nullspace of $A^T\$.

Here are the leftmost 3 columns of v, denoted v1 as is my habit:

$v1 = \left(\begin{array}{ccc} 0.000242117 & 0.00237304 & 0.000773655 \\ 0.00254521 & 0.0246016 & 0.00811386 \\ 0.0253371 & 0.23387 & 0.0760077 \\ 0.0806111 & 0.633731 & 0.188135 \\ 0.110853 & 0.608184 & 0.12797 \\ 0.102268 & 0.306133 & -0.04696 \\ 0.0972532 & 0.129667 & -0.243487 \\ 0.0880827 & 0.101349 & -0.512971 \\ -0.0317461 & 0.111471 & -0.580417 \\ -0.228205 & 0.117862 & -0.449608 \\ -0.460923 & 0.112998 & -0.181177 \\ -0.596203 & 0.0923988 & 0.0811638 \\ -0.500375 & 0.0613641 & 0.162946 \\ -0.266387 & 0.0296408 & 0.104503 \\ -0.0986158 & 0.0105714 & 0.0410965 \\ -0.0280157 & 0.00296645 & 0.0118833 \\ -0.00680593 & 0.000717077 & 0.00290704 \\ -0.00174298 & 0.000183642 & 0.000744487 \\ -0.000414996 & 0.0000437242 & 0.000177259 \\ -0.000099599 & 0.0000104938 & 0.0000425421 \\ 0. & 0. & 0.\end{array}\right)$

OK, I have to know: did v1 change? Yes. I’ll show you, soon.

Let me juxtapose the orthonormal basis v1 (right) with the dual basis E vectors. Both bases span the same space, the pre-image of $A^T\$, but they are different.

And in this case, I can’t see that any vectors in one basis look similar to vectors in the other basis.

Let me recall an image from the previous post: here are the dual basis E and the o/n basis v1 from the previous post. This is the old counterpart to the preceding drawing.

And the right hand pieces of those two drawings are the new v1 and the old v1. We can see that they are quite different.

Having an orthonormal basis for the preimage of $A^T\$, we know that we can construct a projection operator R from the domain of $A^T\$ onto the preimage; we just take the product of v1 and $v1^T\$ in the other order:

$R = v1\ v1^T\$.

It’s 21×21, as before, so I’m not going to show it to you. But, for checking, I’ll show you the first column:

And I’ll tell you that the last column, as before, is identically 0.

## the fundamental and the residual

So. Apply the projection operator R to the reflected spectrum S to get the fundamental f…

f = {0.243727,2.51476,23.848,64.4959,61.7521,33.9471,25.2246,
43.8826,66.214,82.1295,90.1689,83.8434,60.5088,30.3679,
10.993,3.10116,0.751257,0.192395,0.0458083,0.010994,0.}

and subtract to get the residual n = S – f…

n = {4.85627,26.2996,19.116,-14.6684,-7.32845,18.6715,27.0686,
15.2246,3.00319,-13.0295,-23.7795,-12.7434,13.5977,41.0282,
59.1236,63.6887,60.2519,52.106,59.2832,33.0258,39.625}

Did we get the same fundamental? They are very close. Again, I think the real problem is that the two tables are not exactly consistent. Here’s

old f minus new f =

{-0.00362232,0.000221807,0.00281357,-0.00101889,-0.00138667,
-0.00346164,-0.00173991,-0.00126245,-0.00325203,-0.000486578,
0.00271479,0.00422385,0.00120929,-0.00248099,-0.00107542,
0.0013427,0.00223839,-0.00306751,0.00309787,0.00376061,0.}

We already know that the fundamental is the same whether we compute it using the dual basis or using the projection operator. Let me illustrate that the fundamental computed using RGB is the same as the fundamental computed using XYZ.

I have plotted points using the previous fundamental, and I have plotted a piecewise linear curve using the new fundamental:

I should probably emphasize that although this drawing looks like one from the previous post, it illustrates something else. The apparently identical drawing in the previous post showed the equivalence of the fundamental computed two ways using XYZ; this drawing shows the equivalence of the fundamental from the previous post with the fundamental from this post.

And the null? Close, too. We have

old n minus new n =

{0.00362232,-0.000221807,-0.00281357,0.00101889,0.00138667,
0.00346164,0.00173991,0.00126245,0.00325203,0.000486578,
-0.00271479,-0.00422385,-0.00120929,0.00248099,0.00107542,
-0.0013427,-0.00223839,0.00306751,-0.00309787,-0.00376061,0.}

We can check that n is in the nullspace of $A^T\$ by computing $A^T\ n\$:

{-6.77791*10^-14, 6.72509*10^-15, -3.50254*10^-14}.

We should also check that the fundamental has the same 3 components as the entire reflected spectrum. That is, we had

$A^T\ S = \{70.5889,\ 61.249,\ 47.8412\}\$.

for $A^T\$ applied to the reflected spectrum. If we apply $A^T\$ to the fundamental, we get

$A^T\ f = \{70.5889,\ 61.249,\ 47.8412\}$.

The same. As they should be.

You know what? There’s another pair of calculations I can do, to make a point.

What are the components of the fundamental wrt the orthonormal basis v1? We just dot f into the v1 matrix, because its columns are orthonormal basis vectors:

$f \cdot v1 =\$
{-129.491,141.916,-79.7421,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}.

That confirms that no part of the fundamental lies in the nullspace. That was the whole point of the projection operator.

And what are the components of the residual? I hope you’ve got it: exactly the first three are zero:

$n \cdot v1 =\$
{0,0,0,-24.6623,-15.3786,17.8955,33.3365,29.6755,19.4414,
-0.412828,-18.9652,-15.4907,8.59491,37.8598,57.8821,63.3301,
60.1642,52.0835,59.2778,33.0245,39.625}.

The decomposition looks the same as in the previous post. The reflected spectrum is in yellow, the fundamental in white, and their difference n, the residual, is in black.

NOTE that although reflected and fundamental are non-negative, the residual is not. The good news is that, in principle, we could generate the fundamental spectrum, as well as the reflected. But I can’t imagine how to generate a spectrum with negative values.

We face this problem not just for the residual, but for just about every basis vector we’ve computed!

(The middle vector of the orthonormal basis, i.e. the middle column of v1, is the only exception so far; it is everywhere non-negative. But that’s only for the orthonormal basis for the RGB color matching functions; for the previous post, every column of v1 has some negative values.)

Because the basis vectors are, frankly, weird spectra, don’t expect anything nice if you try to find their RGB, XYZ, or xyz values. Most of them lie outside the CIE chromaticity chart! Incidentally, I have computed Cohen’s F1 and F2 bases for this case, and some of them are still negative, too. Well, they have to be: things have to cancel out so that the dot products among the orthonormal vectors are actually zero.

But while we’re looking at the linear algebra, let’s take a look at the orthonormal basis for the null space.

## the null space basis

Now for something new.

We have 18 basis vectors (metameric black spectra) for the nullspace. It’s rather confusing to look at all of them… it’s not much better to look at a few of them close together. In the top left, we have the 1st 5th, 9th, and 13th (1,5,9,13)… the top right is 2,6,10,14… bottom left is 3,7,11,15… bottom right is 4,8,12,16:

There are two more to show (17 & 18) and I thought I show a few more adjacent pairs, too. I have added 1 & 2 and 9 & 10:

While I’m playing around…. What about four spectra centered around 500? And add them up, to get another metameric black.

Let me do one last thing.

## Graphs at 5 nm

What would the bases look like if I used the full 5 nm tables? The fundamental depended on the reflected spctrum, but the bases depend only on my choice of color matching functions. So, let me show them for the 1931 CIE rgb bar tables at 5nm intervals.

Here are the orthonormal basis vectors “v1” — at 20 nm on the left, 5 nm on the right:

And the dual basis vectors “E”:

And the dual basis “E” juxtaposed with the orthonormal basis “v1”:

I might as well do this for the XYZ color matching functions, too. The more detailed curves are from the 1931 xyz bar tables, at 5 nm intervals.

Here are the orthonormal basis vectors “v1”:

The dual basis vectors “E”:

And the dual basis “E” juxtaposed with the orthonormal basis “v1”:

### 3 Responses to “Color: decomposing the RGB A transpose matrix”

1. rip Says:

I was just leafing thru the end of Cohen, and I see that he graphed the 3 fundamentals for the 1964 CIE; that is, the 3 dual basis vectors E.

His drawing on p 163. looks just like my drawing of the 1931 CIE basis vectors E. We’ve seen before that the differences between the 1931 and the 1964 are pretty small.