Octree

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An octree node, depicted as dividing a colour space.

An octree is a tree data structure in which each internal node has up to eight children. Octrees are most often used to partition a three dimensional space by recursively subdividing it into eight octants.

Some common uses of octrees are:

Spatial indexing
Efficient collision detection in three dimensions
View frustum culling

Octrees are the three-dimensional analog of quadtrees. The name is formed from oct + tree, and normally written octree not octtree.

Each node in an octree subdivides the space it represents into eight subspaces (octants). In a point region (PR) octree, the node stores an explicit 3-dimensional point, which is the "center" of the subdivision for that node; the point defines one of the corners for each of the eight children. In an MX octree, the subdivision point is implicitly the center of the space the node represents. The root node of a PR quadtree can represent infinite space; the root node of an MX octree must represent a finite bounded space so that the implicit centers are well-defined.

[edit] Application to Colour Quantization

The octree colour quantization algorithm, invented by Gervautz and Purgathofer in 1988, encodes image colour data as an octree up to nine levels deep. Octrees are used because $23 = 8$ and there are three colour components in the RGB system. The node index to branch out from at the top level is determined by a formula that uses the most significant bits of the red, green, and blue colour components, e.g. 4r + 2g + b. The next lower level uses the next bit significance, and so on. Less significant bits are sometimes ignored to reduce the tree size.

The algorithm is highly memory efficient because the tree's size can be limited. The bottom level of the octree consists of leaf nodes that accrue colour data not represented in the tree; these nodes initially contain single bits. If much more than the desired number of palette colours are entered into the octree, its size can be continually reduced by seeking out a bottom-level node and averaging its bit data up into a leaf node, pruning part of the tree. Once sampling is complete, exploring all routes in the tree down to the leaf nodes, taking note of the bits along the way, will yield approximately the required number of colours.