So now let me explain our method in more detail.
There's just about one kind of colormap where we know exactly how to control luminance: gray scales [showing the fixed gray and grayscale ramp next to "Luminance pattern"]. We know, obviously, how to keep a shade of gray fixed in luminance. We also know how to make it increase monitonically, perhaps taking into account the the gamma of the device and that non-linear relationship to brightness.
Now suppose you have a set of RGB colors that you want as control points in your colormap [top of slide; "Input control points"], but you want to make them have the same pattern of luminance variation as the gray scale. So, you go through the control points, and match the luminance of the color, to the luminance of the corresponding gray ["Output control points"]. How you match the luminance is the major contribution of our paper, and I'll talk about it shortly.
You adjust the luminance of the color by moving it up and down the "lightness" axis of the HLS double-hexcone color space. Its a simple device-dependent colorspace with a well-known transform to RGB space. This means that we loose saturation as soon as we have to increase the lightness past 0.5, here in the middle, but I don't see how else to create colors with the same hue at a high luminance. Finally, you interpolate between the luminance adjusted control points, in a way that itself controls luminance, and that's something else I'll talk about.
The result of interpolating between the new control points is a perceptual colormap. The steps shown are basically the method described in the paper. So there are now these two aspects of the method which I'll describe in a little more detail: the luminance matching, and the color interpolation.