One reason the images I referenced in my previous post caught my eye was that I've been playing around with a similar technique for a couple of months now. I dusted off the code and improved it to support Delaunay images as well as to do shading of the triangles or polygons.
Image 1 below shows a Delaunay image constructed from the Mona Lisa. The triangles in the first image are coloured evenly and the shade is the average colour of the three vertices. Image 2 is the same except I'm colouring the triangle pixels based on a function of how far they are from the various vertices and the colours at those vertices. It gives a much more realistic image.
I've removed the triangle edges in image 3 and image 4 is the original for reference. I like this technique because you can easily control where the resulting image is more detailed by just using more control points in that region or by shading the triangles differently.
There is a related type of diagram that is based on Voronoi cells. This next image is the Voronoi diagram using the same control points as above. The regions are polygons of arbitrary number of sides rather than triangles. The last image uses more control points to get more details from the underlying base image.
I created these images with custom software written in Processing that relies heavily on the Mesh library by Lee Byron. I also used the Mesh demo created by Marius Watz as a starting point for my code. Thanks!