{"id":4436,"date":"2021-04-27T17:05:39","date_gmt":"2021-04-27T21:05:39","guid":{"rendered":"https:\/\/blogs.mathworks.com\/steve\/?p=4436"},"modified":"2021-05-11T11:37:24","modified_gmt":"2021-05-11T15:37:24","slug":"plotting-a-and-b-colors","status":"publish","type":"post","link":"https:\/\/blogs.mathworks.com\/steve\/2021\/04\/27\/plotting-a-and-b-colors\/","title":{"rendered":"Plotting a* and b* colors"},"content":{"rendered":"
Today's blog post comes from planning one topic, but then taking a sharp left turn and doing something else completely. I was thinking about writing something related to <\/span>meshgrid<\/span>, and so I was looking at some old blog posts in which <\/span>meshgrid<\/span> was used. For example, <\/span>meshgrid<\/span> was used in this <\/span>30-Dec-2010 post<\/span><\/a> about the a* and b* component in the Lab color space.<\/span><\/div>
In reading over that old post, however, I realized that I made a rather egregious conceptual error in it. I plotted colors over the domain <\/span>$ -100 \\leq a^* \\leq 100 $<\/span>, <\/span>$ -100 \\leq b^* \\leq 100 $<\/span>, using <\/span>$ L^* = 90 $<\/span>, without realizing or explaining that most of those <\/span>$ (L^*,a^*,b^*) $<\/span> combinations are far out of the sRGB gamut. In other words, they are not really displayable (even on a wide-gamut monitor). Also, the functions I used back then have since been superseded by new functions that are not only easier to use, but are also more helpful at looking at in-gamut vs. out-of-gamut questions.<\/span><\/div>
I decided, therefore, to update and improve that old post.<\/span><\/div>

Showing <\/span>$ (a^*,b^*) $<\/span> colors for a fixed <\/span>$ L^* $<\/span><\/h2>
As I did last time, I'll start with <\/span>$ L^*=90 $<\/span>, which is at the bright end of the scale. For that choice of <\/span>$ L^* $<\/span>, let's see what the colors in the <\/span>$ (a^*,b^*) $<\/span> plane look like, taking into account colors that might be out of gamut.<\/span><\/div>
clear<\/span><\/span><\/div><\/div>
close <\/span>all<\/span><\/span><\/div><\/div>
a = -110:0.4:110;<\/span><\/span><\/div><\/div>
b = -110:0.4:110;<\/span><\/span><\/div><\/div>
[aa,bb] = meshgrid(a,b);<\/span><\/span><\/div><\/div>
L = 90;<\/span><\/span><\/div><\/div>
LL = L*ones(size(aa));<\/span><\/span><\/div><\/div>
lab = cat(3,LL,aa,bb);<\/span><\/span><\/div><\/div><\/div>
In the old post, I used <\/span>makecform<\/span> and <\/span>applycform<\/span> to convert Lab values to sRGB values. This time, I'll use <\/span>lab2rgb<\/span>.<\/span><\/div>
rgb = lab2rgb(lab);<\/span><\/span><\/div><\/div><\/div>
When using <\/span>lab2rgb<\/span>, out-of-gamut values are indicated by output values that are greater than 1 or less than 0.<\/span><\/div>
out_gamut_mask = any((rgb > 1) | (rgb < 0),3);<\/span><\/span><\/div><\/div><\/div>
Let's replace the out-of-gamut colors with gray and display the result. Set the <\/span>y<\/span>-direction of the image display to <\/span>'normal'<\/span> so that positive <\/span>$ b^* $<\/span> is at the top, which is the usual convention for these kinds of plots. Also, display the axes ticks and labels.<\/span><\/div>
rgb(repmat(out_gamut_mask,1,1,3)) = 0.6;<\/span><\/span><\/div><\/div>
imshow(rgb,<\/span>'XData'<\/span>,b,<\/span>'YData'<\/span>,a)<\/span><\/span><\/div><\/div>
axis <\/span>xy<\/span><\/span><\/div><\/div>
axis <\/span>on<\/span><\/span><\/div><\/div>
xlabel(<\/span>'$a^*$'<\/span>,<\/span>'Interpreter'<\/span>,<\/span>'latex'<\/span>)<\/span><\/span><\/div><\/div>
ylabel(<\/span>'$b^*$'<\/span>,<\/span>'Interpreter'<\/span>,<\/span>'latex'<\/span>)<\/span><\/span><\/div><\/div>
title(<\/span>'$L^*=90$'<\/span>,<\/span>'Interpreter'<\/span>,<\/span>'latex'<\/span>)<\/span><\/span><\/div>
<\/div><\/div><\/div><\/div><\/div>
As you can see, with <\/span>$ L^*=90 $<\/span>, only a relatively small portion of the <\/span>$ (a^*,b^*) $<\/span> plane is in gamut.<\/span><\/div>

Yellow is the brightest color<\/span><\/h2>
Let's repeat that process with an even brighter <\/span>$ L^* $<\/span> value, almost all the way to white.<\/span><\/div>
L = 96;<\/span><\/span><\/div><\/div>
LL = L*ones(size(aa));<\/span><\/span><\/div><\/div>
lab = cat(3,LL,aa,bb);<\/span><\/span><\/div><\/div>
rgb = lab2rgb(lab);<\/span><\/span><\/div><\/div>
out_gamut_mask = any((rgb > 1) | (rgb < 0),3);<\/span><\/span><\/div><\/div>
rgb(repmat(out_gamut_mask,1,1,3)) = 0.6;<\/span><\/span><\/div><\/div>
imshow(rgb,<\/span>'XData'<\/span>,b,<\/span>'YData'<\/span>,a)<\/span><\/span><\/div><\/div>
axis <\/span>xy<\/span><\/span><\/div><\/div>
axis <\/span>on<\/span><\/span><\/div><\/div>
xlabel(<\/span>'$a^*$'<\/span>,<\/span>'Interpreter'<\/span>,<\/span>'latex'<\/span>)<\/span><\/span><\/div><\/div>
ylabel(<\/span>'$b^*$'<\/span>,<\/span>'Interpreter'<\/span>,<\/span>'latex'<\/span>)<\/span><\/span><\/div><\/div>
title(<\/span>'$L^*=96$'<\/span>,<\/span>'Interpreter'<\/span>,<\/span>'latex'<\/span>)<\/span><\/span><\/div>
<\/div><\/div><\/div><\/div><\/div>
The plot above demonstrates that yellow is the only color that can be displayed almost as bright as white.<\/span><\/div>

Blue is the darkest color<\/span><\/h2>
Now let's try something dark.<\/span><\/div>
L = 25;<\/span><\/span><\/div><\/div>
LL = L*ones(size(aa));<\/span><\/span><\/div><\/div>
lab = cat(3,LL,aa,bb);<\/span><\/span><\/div><\/div>
rgb = lab2rgb(lab);<\/span><\/span><\/div><\/div>
out_gamut_mask = any((rgb > 1) | (rgb < 0),3);<\/span><\/span><\/div><\/div>
rgb(repmat(out_gamut_mask,1,1,3)) = 0.6;<\/span><\/span><\/div><\/div>
imshow(rgb,<\/span>'XData'<\/span>,b,<\/span>'YData'<\/span>,a)<\/span><\/span><\/div><\/div>
axis <\/span>xy<\/span><\/span><\/div><\/div>
axis <\/span>on<\/span><\/span><\/div><\/div>
xlabel(<\/span>'$a^*$'<\/span>,<\/span>'Interpreter'<\/span>,<\/span>'latex'<\/span>)<\/span><\/span><\/div><\/div>
ylabel(<\/span>'$b^*$'<\/span>,<\/span>'Interpreter'<\/span>,<\/span>'latex'<\/span>)<\/span><\/span><\/div><\/div>
title(<\/span>'$L^*=25$'<\/span>,<\/span>'Interpreter'<\/span>,<\/span>'latex'<\/span>)<\/span><\/span><\/div>
<\/div><\/div><\/div><\/div><\/div>
Although you can make out some relatively unsaturated green and red regions above, blue is the only fully saturated color.<\/span><\/div>

What is the in-gamut <\/span>$ (a^*,b^*) $<\/span> region for every <\/span>$ L^* $<\/span>?<\/span><\/h2>
Now, I want to make multidimensional cubes of <\/span>$ L^* $<\/span>, <\/span>$ a^* $<\/span>, and <\/span>$ b^* $<\/span> values over their entire ranges and calculate the in-gamut colors all at once.<\/span><\/div>
Find <\/span>$ L^* $<\/span>, <\/span>$ a^* $<\/span>, <\/span>$ b^* $<\/span> values filling a three-dimensional region. Convert the triples to sRGB.<\/span><\/div>
<\/div><\/div>
L = 0:100;<\/span><\/span><\/div><\/div>
[LL,bb,aa] = ndgrid(L,b,a);<\/span><\/span><\/div><\/div>
rgb = lab2rgb([LL(:) aa(:) bb(:)]);<\/span><\/span><\/div><\/div><\/div>
Reshape so that the color component is in the 4th dimension. Make a three-dimensional out-of-gamut mask. Replace out-of-gamut values with gray.<\/span><\/div>
rgb = reshape(rgb,[size(LL) 3]);<\/span><\/span><\/div><\/div>
mask = any((rgb > 1) | (rgb < 0),4);<\/span><\/span><\/div><\/div>
rgb(repmat(mask,1,1,3)) = 0.6;<\/span><\/span><\/div><\/div><\/div>
Display the set of colors for <\/span>$ L^*=50 $<\/span> (corresponding to the index 51 in the first dimension). After indexing with 51 in the first dimension, use <\/span>squeeze<\/span> to eliminate that dimension before displaying as an image.<\/span><\/div>
rgb_50 = squeeze(rgb(51,:,:,:));<\/span><\/span><\/div><\/div>
imshow(rgb_50,<\/span>'XData'<\/span>,b,<\/span>'YData'<\/span>,a)<\/span><\/span><\/div><\/div>
axis <\/span>xy<\/span><\/span><\/div><\/div>
axis <\/span>on<\/span><\/span><\/div><\/div>
xlabel(<\/span>'$a^*$'<\/span>,<\/span>'Interpreter'<\/span>,<\/span>'latex'<\/span>)<\/span><\/span><\/div><\/div>
ylabel(<\/span>'$b^*$'<\/span>,<\/span>'Interpreter'<\/span>,<\/span>'latex'<\/span>)<\/span><\/span><\/div><\/div>
title(<\/span>'$L^*=50$'<\/span>,<\/span>'Interpreter'<\/span>,<\/span>'latex'<\/span>)<\/span><\/span><\/div>
<\/div><\/div><\/div><\/div><\/div>
Here's an animation, creating using <\/span>imwrite<\/span>, showing the in-gamut colors for the values of <\/span>$ L^* $<\/span> ranging from 0 to 100.<\/span><\/div>
\"varying-L.gif\"<\/img><\/div>

What is the in-gamut <\/span>$ L^* $<\/span> range everywhere in the <\/span>$ (a^*,b^*) $<\/span> plane?<\/span><\/h2>
These experiments caused me to wonder how to visualize the range of colors more broadly, while keeping the visualization in the <\/span>$ (a^*,b^*) $<\/span> plane. Here's a relevant question: given particular <\/span>$ a^* $<\/span> and <\/span>$ b^* $<\/span> values, what is the lowest <\/span>$ L^* $<\/span> value that is in gamut? What is the highest value? Further, can we tell if <\/span>no<\/span> <\/span>$ L^* $<\/span> values are in gamut for a particular <\/span>$ a^* $<\/span> and <\/span>$ b^* $<\/span>?<\/span><\/div>
I worked this out based on the multidimensional arrays computed above.<\/span><\/div>
Find the minimum in-gamut values in the <\/span>LL<\/span> array along the first dimension, taking advantage of the fact that the <\/span>min<\/span> function ignores <\/span>NaN<\/span> values by default. Use the <\/span>squeeze<\/span> function to eliminate the first dimension, which has a size of 1 following the <\/span>min<\/span> operation.<\/span><\/div>
LL_min = LL;<\/span><\/span><\/div><\/div>
LL_min(mask) = NaN;<\/span><\/span><\/div><\/div>
LL_min = squeeze(min(LL_min,[],1));<\/span><\/span><\/div><\/div><\/div>
Following a similar procedure, find the maximum in-gamut values in the <\/span>LL<\/span> array along the first dimension.<\/span><\/div>
LL_max = LL;<\/span><\/span><\/div><\/div>
LL_max(mask) = NaN;<\/span><\/span><\/div><\/div>
LL_max = squeeze(max(LL_max,[],1));<\/span><\/span><\/div><\/div><\/div>
Here are the darkest and brightest in-gamut <\/span>$ L^* $<\/span> values on the <\/span>$ (a^*,b^*) $<\/span> plane, displayed with a grayscale colormap, along with the associated colors.<\/span><\/div>
tiledlayout(2,2)<\/span><\/span><\/div><\/div>
nexttile(1)<\/span><\/span><\/div><\/div>
imshow(LL_min,[],<\/span>'XData'<\/span>,a,<\/span>'YData'<\/span>,b)<\/span><\/span><\/div><\/div>
axis <\/span>xy<\/span><\/span><\/div><\/div>
axis <\/span>on<\/span><\/span><\/div><\/div>
title(<\/span>'Minimum in-gamut $L^*$'<\/span>,<\/span>'Interpreter'<\/span>,<\/span>'latex'<\/span>)<\/span><\/span><\/div><\/div>
xlabel(<\/span>'$a^*$'<\/span>,<\/span>'Interpreter'<\/span>,<\/span>'latex'<\/span>)<\/span><\/span><\/div><\/div>
ylabel(<\/span>'$b^*$'<\/span>,<\/span>'Interpreter'<\/span>,<\/span>'latex'<\/span>)<\/span><\/span><\/div><\/div>
<\/div><\/div>
nexttile(2)<\/span><\/span><\/div><\/div>
imshow(LL_max,[],<\/span>'XData'<\/span>,a,<\/span>'YData'<\/span>,b)<\/span><\/span><\/div><\/div>
axis <\/span>xy<\/span><\/span><\/div><\/div>
axis <\/span>on<\/span><\/span><\/div><\/div>
title(<\/span>'Maximum in-gamut $L^*$'<\/span>,<\/span>'Interpreter'<\/span>,<\/span>'latex'<\/span>)<\/span><\/span><\/div><\/div>
xlabel(<\/span>'$a^*$'<\/span>,<\/span>'Interpreter'<\/span>,<\/span>'latex'<\/span>)<\/span><\/span><\/div><\/div>
ylabel(<\/span>'$b^*$'<\/span>,<\/span>'Interpreter'<\/span>,<\/span>'latex'<\/span>)<\/span><\/span><\/div><\/div>
<\/div><\/div>
nexttile(3)<\/span><\/span><\/div><\/div>
[aa,bb] = meshgrid(a,b);<\/span><\/span><\/div><\/div>
rgb_min = lab2rgb(cat(3,LL_min,aa,bb));<\/span><\/span><\/div><\/div>
imshow(rgb_min,<\/span>'XData'<\/span>,a,<\/span>'YData'<\/span>,b)<\/span><\/span><\/div><\/div>
axis <\/span>xy<\/span><\/span><\/div><\/div>
axis <\/span>on<\/span><\/span><\/div><\/div>
xlabel(<\/span>'$a^*$'<\/span>,<\/span>'Interpreter'<\/span>,<\/span>'latex'<\/span>)<\/span><\/span><\/div><\/div>
ylabel(<\/span>'$b^*$'<\/span>,<\/span>'Interpreter'<\/span>,<\/span>'latex'<\/span>)<\/span><\/span><\/div><\/div>
<\/div><\/div>
nexttile(4)<\/span><\/span><\/div><\/div>
rgb_max = lab2rgb(cat(3,LL_max,aa,bb));<\/span><\/span><\/div><\/div>
imshow(rgb_max,<\/span>'XData'<\/span>,a,<\/span>'YData'<\/span>,b)<\/span><\/span><\/div><\/div>
axis <\/span>xy<\/span><\/span><\/div><\/div>
axis <\/span>on<\/span><\/span><\/div><\/div>
xlabel(<\/span>'$a^*$'<\/span>,<\/span>'Interpreter'<\/span>,<\/span>'latex'<\/span>)<\/span><\/span><\/div><\/div>
ylabel(<\/span>'$b^*$'<\/span>,<\/span>'Interpreter'<\/span>,<\/span>'latex'<\/span>)<\/span><\/span><\/div>
<\/div><\/div><\/div><\/div><\/div>
Finally, I want to make an animation showing how the colors in the above plot change as they go from the darkest available color to the lightest, starting with another four-dimensional cube of RGB images:<\/span><\/div>
N = 96;<\/span><\/span><\/div><\/div>
[bb2,aa2] = ndgrid(b,a);<\/span><\/span><\/div><\/div>
rgb_scan = zeros([size(bb2) 3 (N+1)]);<\/span><\/span><\/div><\/div>
LL_diff = LL_max - LL_min;<\/span><\/span><\/div><\/div>
for <\/span>k = 0:N<\/span><\/span><\/div><\/div>
L_k = LL_min + (LL_diff * (k\/N));<\/span><\/span><\/div><\/div>
rgb_scan(:,:,:,k+1) = lab2rgb(cat(3,L_k,aa2,bb2));<\/span><\/span><\/div><\/div>
end<\/span><\/span><\/div><\/div><\/div>
Here's the resulting animation, also created using <\/span>imwrite<\/span>. (Note that quality of the animation suffers because of the GIF limitation of no more than 256 distinct colors in each frame.)<\/span><\/div>
\"Lmin-to-Lmax.gif\"<\/img><\/div>\r\n<\/div>