Examples

Track an Interface

I use the orthtree to track the boundary of an N-sphere as it moves and dilates. The following operations are performed at each step.

• Compute the distance to the boundary.
• Coarsen cells which are no longer close to the boundary.
• Refine (up to the maximum level) cells which cross the boundary.
• Balance the tree so that adjacent neighbors differ by no more than one level.
• Write the orthtree in VTK format. (This is implemented only for 1-D, 2-D, and 3-D.)

Here are movies showing results for a circle and a sphere.

Image Compression

I use the coarsening capability to perform image compression. Here is a function sampled on a 256 x 256 grid (65,536 nodes).

We compress using the criterion that we can merge children if the variation in their values is less than 0.1. The resulting image has 2,812 nodes. Below we show the compressed image and the image with node boundaries.

Function Sampling

We use the refinement capability to sample a function. Each node in the orthtree hold the values of a function at its corners. If the function values differ by more that a specified threshhold, the node is split. Each child node uses one corner value from the parent and evaluates the function to determine the other values. In the figures below, we show function sampling with maximum allowed variations of 0.2 and 0.1.