{"id":4217,"date":"2020-10-30T13:53:18","date_gmt":"2020-10-30T17:53:18","guid":{"rendered":"https:\/\/blogs.mathworks.com\/steve\/?p=4217"},"modified":"2020-10-30T13:53:18","modified_gmt":"2020-10-30T17:53:18","slug":"how-to-compute-perceptual-color-difference","status":"publish","type":"post","link":"https:\/\/blogs.mathworks.com\/steve\/2020\/10\/30\/how-to-compute-perceptual-color-difference\/","title":{"rendered":"How to Compute Perceptual Color Difference"},"content":{"rendered":"

I wrote previously<\/a> about the new colorChecker<\/tt>, which can detect X-Rite test charts in the R2020b release. Another area of new color-related functionality is computing perceptual color differences. There is the new function deltaE<\/tt>, which computes color differences based on the L*a*b* color space and the CIE76 standard. There is also the imcolordiff<\/tt> function, which can compute color differences based either on the CIE94 or the CIEDE2000 standard.<\/p>

I recently mentioned to someone on the Image Processing Toolbox team that I was planning to write a blog post about these functions, and he posed the following question: which two colors are the furthest apart, perceptually speaking? I decided to give that a try using $\\Delta_E$. I'll formulate the problem this way: which two sRGB colors have the largest $\\Delta_E$ between them?<\/p>

In a related post back in 2015, I demonstrated how to plot the sRGB gamut surface in L*a*b* space, like this:<\/p>

\"\" <\/p>

Looking at this diagram, you might be able to guess the answer. But, let's work through it. The first thing I want to do is to sample the sRGB space.<\/p>

[r,g,b] = ndgrid(linspace(0,1,100));\r\nrgb = [r(:), g(:), b(:)];\r\n<\/pre>
Next, convert the sRGB colors to Lab. (I'm going to stop typing the \"*\" characters.)<\/p>
lab = rgb2lab(rgb);\r\n<\/pre>
Now let's find the colors on the convex hull of the Lab values. The pair of colors most distant from each other must both be on the convex hull.<\/p>
k = convhull(lab);\r\nhull_colors = lab(unique(k(:,1)),:);\r\n<\/pre>
If this were a two-dimensional problem, then I know of an algorithm that can find the most distant pair of points in $O(N)$, where $N$ is the number of points on the convex hull. I don't know of a similar algorithm in three dimensions, so I'm just going to brute force it by computing the distance between every color pair.<\/p>
So, how many colors are we talking about?<\/p>
size(hull_colors)\r\n<\/pre>
\r\nans =\r\n\r\n        3980           3\r\n\r\n<\/pre>
There are 3908 colors, each of which is represented by a 3-element vector in Lab space. The color difference $\\Delta_E$ is the Euclidean distance between two colors in Lab space. I want to compute the distance between every color in that set and every other color. Here is a nifty and compact way to do that using the implicit expansion behavior of MATLAB arithmetic operators.<\/p>
C1 = reshape(hull_colors,[],1,3);\r\nC2 = reshape(hull_colors,1,[],3);\r\nCdiff = C1 - C2;\r\nDE = sqrt(sum(Cdiff.^2,3));\r\nsize(DE)\r\n<\/pre>
\r\nans =\r\n\r\n        3980        3980\r\n\r\n<\/pre>
You can see that DE<\/tt> is an 3980x3980 matrix. It contains the pair-wise distances between all the convex hull colors. (See also pdist2<\/tt>, a function in the Statistics and Machine Learning Toolbox that can compute pair-wise vector distances with much more flexibility, as well as hypot<\/tt>, a MATLAB function that computes the magnitude of a two-element vector in a way that avoids numerical overflow and underflow problems.)<\/p>
Let's find where the maximum distances are in this matrix.<\/p>
[m,n] = find(DE == max(DE(:)))\r\n<\/pre>
\r\nm =\r\n\r\n        3883\r\n          92\r\n\r\n\r\nn =\r\n\r\n          92\r\n        3883\r\n\r\n<\/pre>
What are those colors (in Lab space)?<\/p>
max_colors_lab = hull_colors(m,:)\r\n<\/pre>
\r\nmax_colors_lab =\r\n\r\n   32.2970   79.1875 -107.8602\r\n   87.7347  -86.1827   83.1793\r\n\r\n<\/pre>
And those two colors in sRGB space:<\/p>
max_colors_rgb = lab2rgb(max_colors_lab)\r\n<\/pre>
\r\nmax_colors_rgb =\r\n\r\n    0.0000   -0.0000    1.0000\r\n    0.0000    1.0000    0.0000\r\n\r\n<\/pre>
They are just the sRGB green and blue primaries.<\/p>
colorSwatches(max_colors_rgb)\r\naxis equal<\/span>\r\naxis off<\/span>\r\n<\/pre>\"\" 
(colorSwatches<\/tt> is a DIPUM3E<\/b><\/a> function that is available to you in MATLAB Color Tools<\/i> on the File Exchange<\/a> and on GitHub<\/a>.)<\/p>
These two colors have a $\\Delta_E$ of:<\/p>
DE(m(1),n(1))\r\n<\/pre>
\r\nans =\r\n\r\n  258.6827\r\n\r\n<\/pre>
The deltaE function offers a more convenient way to compute $\\Delta_E$:<\/p>
deltaE([0 1 0],[0 0 1])\r\n<\/pre>
\r\nans =\r\n\r\n  258.6827\r\n\r\n<\/pre>
Some other time, I will write about the other color difference calculations provided by imcolordiff<\/tt>.<\/p>