Voronoi mosaics

While looking for ideas to implement voronoi in sage, I stumbled upon a beautiful paper written by a group of japanese computer graphic professionals from the universities of Hokkaido and Tokyo: A Method for Creating Mosaic Images Using Voronoi Diagrams. The first step of their algorithm is simple yet brilliant: Start with any given image, and superimpose an hexagonal tiling of the plane. By a clever approximation scheme, modify the tiling to become a voronoi diagram that adaptively minimizes some approximation error. As a consequence, the resulting voronoi diagram is somehow adapted to the desired contours of the original image.

(Fig. 1)	(Fig. 2)	(Fig. 3)	(Fig. 4)

In a second step, they manually adjust the Voronoi image interactively by moving, adding, or deleting sites. They also take the liberty of adding visual effects by hand: emphasizing the outlines and color variations in each Voronoi region, so they look like actual pieces of stained glass (Fig. 4).

Image Processing with numpy, scipy and matplotlibs in sage

In this post, I would like to show how to use  a few different features of numpy, scipy and matplotlibs to accomplish a few basic image processing tasks: some trivial image manipulation, segmentation, obtaining of structural information, etc.  An excellent way to show a good set of these techniques is by working through a complex project.  In this case, I have chosen the following:

Given a HAADF-STEM micrograph of a bronze-type Niobium Tungsten oxide (left), find a script that constructs a good approximation to its structural model (right).

Courtesy of ETH Zurich

For pedagogical purposes, I took the following approach to solving this problem:

1. Segmentation of the atoms by thresholding and morphological operations.
2. Connected component labeling to extract each single atom for posterior examination.
3. Computation of the centers of mass of each label identified as an atom. This presents us with a lattice of points in the plane that shows a first insight in the structural model of the oxide.
4. Computation of Delaunay triangulation and Voronoi diagram of the previous lattice of points. The combination of information from these two graphs will lead us to a decent (approximation to the actual) structural model of our sample.

Let us proceed in this direction:

Triangulations

As part of a project developed by Professor Bradley J. Lucier, to code a PDE solver written in scheme, I worked in some algorithms to perform “good triangulations” of polygons with holes (“good triangulations” meaning here, those where all the triangles have their three angles as close to 60º as possible). I obtained the necessary theoretical background and coding strategies from the following references:

• Mark de Berg et al., “Computational Geometry by Example.“
• Francis Chin and Cao An Wang, “Finding the Constrained Delaunay Triangulation and Constrained Voronoi Diagram of a simple Polygon in Linear Time.“
• Joseph O’Rourke, “Computational Geometry in C.“
• Jim Ruppert, “A Delaunay Refinement Algorithm for Quality 2-dimensional Mesh Generation.“