Linear Color: Applying the Forward Matrix

Now that we know how to create a 3×3 linear matrix to convert white balanced and demosaiced raw data into XYZ_{D50}  connection space – and where to obtain the 3×3 linear matrix to then convert it to a standard output color space like sRGB – we can take a closer look at the matrices and apply them to a real world capture chosen for its wide range of chromaticities.

Figure 1. Image with color converted using the forward linear matrix discussed in the article.

The Forward Matrix Analyzed

I captured a ColorChecker Passport Photo 24 patch target shortly after squeezing the trigger on the scene depicted in Figure 1, with a Nikon D610 and Nikkor 24-120mm f/4 at base ISO.  The time was about 1:15pm of a clear, late summer day at an altitude of 1600m, so I assumed the illuminant to be fairly close to D50 ‘daylight’.  It turns out that Correlated Color Temperature was actually 5350K.

I then followed the procedure outlined in the last article to obtain the linear matrix needed to convert white balanced and minimally demosaiced raw data[2] – indicated as RGB_{rwd} in this article – into CIE XYZ_{D50} Profile Connection Space with the given illuminant:

Figure 2. Forward Matrix and White Balanced Raw->sRGB Matrix for a Nikon D610+24-120mm f/4 computed for a ‘daylight’ illuminant  near D50 as described in the previous article.

The Forward Matrix is applied to every RGB_{rwd} pixel by matrix multiplication as follows:

    \[ \left[ \begin{array}{c} X_{D50} \\ Y_{D50} \\ Z_{D50} \end{array} \right] = \begin{bmatrix} 0.7502 & 0.2138 & -0.0027 \\ 0.2833 & 0.9803 & -0.2635 \\ 0.0321 & -0.2477 & 1.0349\end{bmatrix} \left[ \begin{array}{c} R_{\text{rwd}} \\ G_{\text{rwd}} \\ B_{\text{rwd}} \end{array} \right] \]

Note that the matrix contains some negative coefficients, which are larger in the Y and Z rows.   However, RGB values from the camera can only be positive and XYZ  primaries were chosen specifically so that visible colors would only have positive values within the space.  This means that after multiplication by the Forward Matrix some values captured in the raw data could  turn out to be negative, falling outside of the visible realm.

The sum of the coefficients of each row shown at the bottom of Figure 2 are the XYZ values that will be obtained when the RGB_{rwd} input is [1 1 1], that is from a properly white balanced neutral white patch of demosaiced raw data.  The result should therefore represent the coordinates of D50 white in XYZ space.  A quick check with Bruce Lindbloom’s calculator[3] shows indeed a correlated color temperature of 5003.8K at the  shown [0.9613  1.0000  0.8193] XYZ coordinates. The matrix is doing its job.

The second row of the Forward Matrix produces the Y values , which are supposed to be proportional  to photometric Luminance.  Note that its coefficients add to 1.000 by design.  The constant K in Figure 2 is the factor that, multiplied by the matrix, makes that happen.  It represents how close the capture was to an ‘ideal’ exposure based on ColorChecker gray patch readings assuming maximum diffuse white at L* = 100.  In this case it is 1.0313, which means that I ‘underexposed’ by about 3% or 0.04 stops.

Applying the Forward Matrix

The capture in Figure 1 was exposed ‘properly’ by my standards, with RawDigger reporting less than 2k pixels blown in the R and B raw channels combined and just 7k in the G channels. That’s as captured.  However the white balance multipliers obtained from ColorChecker’s second gray patch from the left were 1.9615 for R and 1.3336 for B, so after white balancing the situation may worsen. Below you can see an animated GIF alternating every 2s of the white balanced raw data before clipping.  You can open it full size in a new tab by clicking on it.

Figure 3. R and B Pixels clipped after white balancing the raw data. It’s an animation, so if you do not see it alternating every 2s click on it to open it in its own tab.  The color indicates the channel being clipped.

Other than for a couple of irrelevant objects, blown pixels relate mainly to specular reflection flecks in the stone, logs and flowers – but the subject of the image proper, including most of the flowers, fits pretty well within XYZ bounds.   Since this is a wholly linear exercise I did not perform any highlight reconstruction but instead simply clipped every value to the same level as G’s maximum at this point,  per the procedure outlined in the article on rendering.

After multiplication of the white balanced and demosaiced raw data by the Forward Matrix the situation in XYZ is shown below.  We now have a few values being clipped (immaterial rounding errors, first frame of the animation below) and many being sent below zero (material, second frame):

Figure 4. Negative and Clipped values after transformation from white balanced raw to XYZ. The animation shows the image, clipped and negative frames alternating every 2s.  The color shows the channel being clipped/blocked with X in red, Y in green and Z in blue.

The story here is all about the yellow flowers: there is a  little bit of clipping in the Y channel  (green in the animation) and a lot of negative Z values (blue in the animation).  There are also some negative values in the green foliage on the log.  The rest of the image, including the other flowers and the sky, pretty well all fit within XYZ space.

Clipping is a sign of ‘overexposure’, while negative values indicate raw data captured by the camera outside of the visible range, hence ‘imaginary colors’.

xy Chromaticities of the Capture

XYZ is considered a color space with direct correspondence to human perception of color.  In other words if two tones have the same coordinates in XYZ they should appear to have the same color to a human observer.  Therefore while in XYZ we can take a look at the data superimposed onto the classic chromaticity Helmholtz horseshoe, which is supposed to represent the limits of color vision[4].

First I am going to show the chromaticity diagram of the white balanced ColorChecker target used to determine the color matrix of Figure 2, transformed into XYZ by the matrix we derived.

Figure 5. xy chromaticity diagram of the ColorChecker Passport Photo 24 patch target used to obtain the relative Forward Matrix, 1 dot per patch. sRGB, Adobe RGB and ProPhoto RGB shown.

The 24 dots represent the chromaticities of the 24 patches.  The six gray patches in the bottom row of the ColorChecker are bunched up near the center of the horseshoe where D50 resides, as they should.  A couple of dots in the yellow-orange region appear to be just outside of sRGB (white solid line).  They actually could be – but recall from the previous article that the Forward Matrix is a compromise, with some noticeable errors.  Shown below are \Delta E_{00} deviations from BabelColor 30 database reference data[5].  Note that orange and yellow are indeed two of the worst matches.

Figure 6. CIEDE2000 color difference between the ColorChecker Passport Photo 24 patch target as captured by a Nikon D610+24-120mm/4 around 1.30pm of a clear mountain day and the average of BabelColor’s database. 1 unit represents a just noticeable difference.

If every raw pixel of the image in Figure 1 is white balanced, demosaiced, converted to XYZ and plotted as in Figure 5 we get the chromaticities below left.  Too many dots, but it shows the gamut of chromaticities at the scene captured by a D610 and 24-120mm f/4 under the ‘daylight’ 5350K mountain sun.  Some values are indeed negative and some values fall outside of Helmholtz’ horseshoe, hence outside the limits of human vision.

Figure 7. 7a Left: xy chromaticity diagram of every pixel of the image in Figure 1 white balanced, demosaiced and converted with the derived Forward Matrix.  7b Right: The same data as left but shown as a histogram, log10 scale. The sRGB (yellow) Adobe RGB (red) and ProPhoto (black) color spaces are shown for reference.

Since it’s hard to determine from Figure 7a how many pixels correspond to each chromaticity,  the diagram on the right (7b)  shows instead a histogram of the number of pixels within the relative .01x.01 xy chromaticity square.  The counts are in log10 units and color coded, so dark blue means that there are less than 10  pixels in such a square, light blue less than 100, cyan less than 1000 … all the way to yellow with more than 100k.   We can then easily see where we are losing color information in what numbers by referencing the standard color gamuts (yellow, red and black triangles for sRGB, Adobe RGB and ProPhoto resp.).

For instance it would be a pity to lose chromaticities by the hundreds or thousands (indicated by cyan and green in the histograms of figures 7b and 8). Note that even Adobe RGB would not be quite enough for this scene and that the hardware is capturing colors that we cannot see (e.g. outside of the NW to SE going line of the ProPhoto color space, which coincides with the edge of the horseshoe).

It actually makes more sense to look at this data in the perceptually more uniform CIELUV color space:

Figure 8.  Same information as in Figure 7b but shown as u’v’ chromaticities of the perceptually more uniform CIELUV color space.

The weighted histogram shows that image ‘colors’ fit better into the visible range.  The yellow dots, representing the most numerous chromaticities, obviously relate to the blue sky.

Taking It to sRGB

Now that we understand the limitations of the data in XYZ_{D50} we can multiply it by the standard matrix  that will take it to our chosen colorimetric ‘output’ color space, in this case sRGB_{D65}, as discussed earlier.  I did not clip values below zero or above max in order to maintain linearity until the end. From Bruce Lindbloom’s site[3]:

You can see that the coefficients in this matrix are much larger than those of the Forward Matrix shown in Figure 2, so they are much more aggressive on the data.  Indeed this is the step in the color rendering process that leaves the most information on the raw conversion floor, both clipping (first frame below, flowers etc.) and blocking (negative values, second frame, flowers and foliage).  Gray tones made it through to sRGB, colored items did not.

Figure 9. Lost color information during XYZD50->sRGB transform. Animation: first frame = clipped values; second frame + negative values. This is really a cumulative display of clipped/blocked pixels from the two matrix multiplications, as I did not clip/block the XYZ data before transforming it to sRGB.

All those tones above full scale and below zero need to be clipped/blocked to fit into the sRGB color space and are therefore corrupted.   Interesting how much of the foliage did not make it through unscathed.

WB Raw to sRGB in One Matrix

Alternatively in this case the full linear trip from white balanced and demosaiced raw data to sRGB could be accomplished with the following single matrix as discussed in the previous article

Ideally we would like the sum of the row coefficients to be all ones so that a neutral tone in the raw data (say [0.5 0.5 0.5]) would map to the same neutral tone in sRGB, but this is close enough considering the fact that a Forward Matrix is by definition a compromise.  If we forced the sums of all row coefficients to be equal to one, as most matrix generators do, then that would be accomplished at the expense of some other tones.   This being a landscape where whites are not critical I prefer to leave the matrix which yields the best overall color compromises as-is.

How Do You Like’m Linear Blues?

The final step in obtaining the image of Figure 1 is to apply sRGB gamma.  Here it is again below, followed by a few ‘flat’ renditions by real raw converters, after I tried to turn all corrections/curves off. All conversions were white balanced off the same gray ColorChecker patch but Figure 10 (the one we’ve been working on) was the only one that had the benefit of having the applied Forward Matrix designed from a ColorChecker Passport Photo under the correct illuminant at the time of capture.  If you click on one of them it opens up in a new tab where it can be zoomed to 1:1 if desired[G].

Figure 10 . The image rendered by Matlab/Octave as discussed in this article.  It uses a Forward Matrix derived from a ColorChecker Passport Photo, per the procedure described in the previous article.
Figure 11. RawTherapee 4.2.175 with Neutral Processing Profile, Camera Standard Input Profile and everything else I could find turned off. I assume it uses interpolated matrices as available in Adobe DNG converter.
Figure 12.  Nikon Capture NX-D with the ‘Flat’ Picture Control, -1.2EC and everything else I could find turned off.
Figure 13. Adobe Camera Raw CC with Camera Neutral v2 profile, -1EC and everything else I could find turned off.  I assume it uses the same matrices as those  provided by DNG converter.  It did not have a ‘Camera Flat’ profile so note the higher contrast built into the Neutral one.

Figure 10 and Figure 11 (mine and RawTherapee’s) are the purest linear renditions and look very similar, even though their Forward Matrices most likely are not.  I assume RT uses interpolated matrices from Adobe DNG.

I was not able to defeat all corrections in Capture NX-D and Adobe Camera Raw CC, hence the differences in color, contrast and chromaticity.  I know that ACR’s colors, and I am pretty sure CNX-D’s, are fine tuned non linearly during conversion.   In addition I believe they both apply a positive ‘Baseline Exposure’ correction, counting on nonlinear highlight correction algorithms in order not to clip then.  I had to apply -1.2 and -1.0 stops of EC to CNX-D and ACR respectively in order to keep their respective brightnesses roughly comparable to the other two.

Bonus: Nonlinear Blues

This article is about linear color, so no attempt was made to make the reference image more accurate through an ‘HSV map’ or more pleasing by applying ‘look’ and ‘tone curve’ fine tuning aimed at a specific class of output media like monitors.  The quoted terms refer to advanced color corrections applied in commercial raw converters.  Some of these corrections are applied while in XYZ space via round trips to ProPhotoRGB and HSV and back; others can just as well be applied via an editor like PhotoShop after rendering.

To give you an idea  of how the linearly converted image would look after such corrections, here is a version with a very mild custom profile generated by DcamProf with Anders Torger’s neutral+ operator[6], as rendered by RawTherapee.  It is meant to be shown in Adobe RGB on a wide gamut monitor but I converted it to sRGB for wider consumption.  To see its real colors you should probably click on the image to open it in its own tab, save it and open it in a color managed editor like PhotoShop.

Figure 14.  The image rendered by RawTherapee with Anders Torger’s neutral+ custom look profile produced by DcamProf.

Beauty is in the eye of the beholder and lots of adjustments could be made in post to improve these images for the intended display medium and purpose.  But in this case nothing at all was done to them other than white balance, demosaic by averaging the green channels in each quartet and apply color.  This concludes this short series on how (linear) color is rendered during raw conversion.

The same five final images are assembled in a gallery below.  If you click on one of them they open up full screen and you can navigate between them using the superimposed left and right arrows.  The bottom left hand corner indicates the image you are viewing (sRGB is mine, the others are self explanatory).


Notes and References

1. Lots of provisos and simplifications for clarity as always.  I am not a color scientist, so if you spot any mistakes please let me know.
2. Minimally demosaiced means that for each quartet the Red and Blue values were taken as-is from the raw data, while Green was the average of the G1 and G2 raw values, equivalent to dcraw -h.
3. This is the link to Bruce Lindbloom’s site.
4. Helmholtz’s horseshoe was drawn by excellent matlab plugin optprop.
5. The ColorChecker pages at can be found here. Careful that there was a change in formulations in November 2014.
6. Anders Torger’s site and his excellent profile maker DcamProf can be found here.