A graphically represented quadtree is a tessellation of a square using differently sized smaller squares. Is there a transformation that deforms the quadtree cells to achieve following 2 contrasting goals?
- equalize the cell sizes by deforming them (cell size is measured by area)
- close cells in the original quadtree shall be close cells in the transformed result
The final result would be a deformed quadtree where all cell sizes are as similiar in area as possible while neighborhood relationship is maintained as good as possible. In ideal case one could weigh what is more important, equal area or neighborhood.
Motivation
I have a 2d point cloud. To each point a value is associated. Geometrically close points have similar values but the gradient varies over the region. There are regions where the point values are quite constant and there are regions where the values of the points differ more with distance. The quadtree groups points of similar values, i.e. the degree of subdivision is high where the values of near points show large fluctuations. However this splitting of the original square results by definition in differently sized cells. For further processing each cell shall be represented with equal size (=equal area). So I want to shrink larger cells and expand smaller cells and keep the neighborhood relationship. This is of course not possible without loosing some other properties. So one way would be to bend the plane in the 3rd dimension or to accept compromisses, i.e. cells are more equal sized than before and some neighborhood relationships are lost. I am looking for such a solution that deals with compromisses.
Centroidal Voronoi tessellation is not applicable. It leads to equally sized cells but doesn't care about neighborhood.
