I recently added to my Matlab File Exchange function, Perceptually improved colormaps, a colormap for periodic data like azimuth or phase. I am going to briefly showcase it using data from my degree thesis in geology, which I used before, for example in Visualization tips for geoscientists – Matlab. Figure 1, from that post, shows residual gravity anomalies in milligals.

Figure 1

Often we’re interested in characterizing these anomalies by calculating the direction of maximum dip at each point on the surface, and for that direction display the azimuth, or dip azimuth.  I’ve done this for the surface of residual anomalies from Figure 1 and displayed the azimuth in Figure 2. Azimuth from 0 to 360 degrees are color-coded using Jet, Matlab’s standard colormap (until recently). Typically I do not trust azimuth values when the dip is close to zero because it is often contaminated by noise so I would use shading to de-saturate the colors where dip has the lowest values, but for ease of discussion I haven’t done so in this case.

Figure 2. Azimuth values color-coded with Jet.

There are two problems with Figure 2. First, the well-known problems with the jet colormap. For example, blue is too dark and blue areas appear as bands of constant colour. Yellow is much lighter than any other colour so we see artificial yellow edges that are not really present in the data. But there is an additional issue in Figure 2 because azimuths close in value to 0 and 360 degrees are colored with blue and red, respectively, instead of a single color as they should, causing an additional artificial edge.

In Figure 3 I recolored the map using a colormap that replicates those used in many geophysical software tools to display azimuth or phase data. This is better because it wraps around at 360 degrees but the perceptual issues are unresolved: in this case red, yellow and blue all appear as sharp perceptual edges.

Figure 3. Azimuth values color-coded with generic azimuth colormap.

Figure 4. Azimuth values color-coded with isoluminant azimuth colormap.

In Figure 4 I used my new colormap, called isoAZ (for isoluminant azimuth). This colormap is much better because not only does it wraps around at 360 degrees, but also lightness is held constant for all colors, which eliminates the perceptual anomalies. All the artificial yellow, red, and blue edges are gone, only real edges are left. This can be more easily appreciated in the figure below: if you hover with your mouse over it you are able to switch back and forth between Figure 3 and Figure 4.

From an interpretation point of view, azimuths 180 degrees apart are of opposing colours, which is ideal for dip azimuth data because it allows us to easily recognize folds where dips of opposite direction are juxtaposed at an edge. One example is the sharp edge in the northwest quadrant of Figure 4, where magenta is juxtaposed to green. If you look at Figure 1 you see that there’s a relative high in this area (the edge in Figure 4) with dips of opposite direction on either side (East and West, or 0 and 360 degrees).

The colormap was created in the Lightness-Chroma-Hue color space, a polar transform of the Lab color space, where lightness is the vertical axis and at each value of lightness, chroma is the radial coordinate and hue the polar angle. One limitation of this approach is that due to theirregular  shape of the color gamut section at each lightness value, we can never exceed  chroma values  of about 38-40 (at lightness = 65 in Matlab; in Python, with extensive trial and error, I have not been able to go past 36 using the Scikit-image Color module), which make the resulting colors pale, pastely.

it creates For those that want to experiment with it further, I used just a few lines of code similar to the ones below:

radius = 38; % chroma
theta = linspace(0, 2*pi, 256)'; % hue
a = radius * cos(theta);
b = radius * sin(theta);
L = (ones(1, 256)*65)'; % lightness
Lab = [L, a, b];
RGB=colorspace('RGB<-Lab',Lab(end:-1:1,:));

This code is a modification from an example by Steve Eddins on a post on his Matlab Central blog. In Steve’s example the colormap cycles through the hues as lightness increases monotonically (which by the way is an excellent way to generate a perceptual rainbow). In this case lightness is kept constant and hue cycles through the entire 360 degrees and wraps around. Also, instead of using the Image Processing Toolbox, I used  Colorspace, a free function from Matlab File Exchange, for the color space transformations.

For data like fracture orientation where azimuths 180 degrees apart are equivalent it is better to stack two of these isoluminant colormaps in a row. In this way we place opposing colors 90 degrees apart, whereas color 180 degrees apart are the same. You can do it using Matlab commands repmat or vertcat, as below:

radius = 38; % chroma
theta = linspace(0, 2*pi, 128)'; % hue
a = radius * cos(theta);
b = radius * sin(theta);
L = (ones(1, 128)*65)'; % lightness
Lab = [L, a, b];
rgb=colorspace('RGB<-Lab',Lab(end:-1:1,:));
RGB=vertcat(rgb,rgb);