Rotation and translation of objects represented by octrees

This paper describes an algorithm for performing arbitrary translations and rotations of objects represented by octrees. Given an octree in a standard position and a transformation, the algorithm builds a new tree in a top down fshion, visiting each node in the new tree only once, and constructing only those nodes that appear in the final tree. It works by projecting the transformed space over the original tree, and labeling the new nodes according to the labels of the nodes in the underlying untransformed tree. 1. Introduction Octrees are finding applications in a number of are-of computer graphics and robot vision because of the ease with which certain operations can be performed (for example, rendering and hidden surface removal in graphics 13, 5, 81, and collision avoidance and path planning in robotics(2, 6, 101). The advantages of the octree arise because it is a t r u e volumetric representation, and yet can be stored in a compact form. An octree 17, 8) is a recursive decomposition of a cubic space Into subcubcs. Initially, the whole space is represented by a single node in the tree, called the root node. If the cubic volume is homogeneous (full or empty), then the root is not decomposed at all, and comprises the complete description of the space. Otherwise , it is split into eight equal subcubes (octants), which become the children of the root. This process continua until all the nodes are homogeneous, or until some resolution limit is reached. Nodes corresponding to cubic regions that are completely full are called object (or full) leaf nodes. Nodes corresponding to empty regions are called background (or empty) leaf nodes, and nodes corresponding to mixed regions (non-leaf nodes) are called non-terminal (or gray) nodes. The octree has advantages over other volumetric representations because of the speed with which volume elements can be located and their relative poeitions established. Octrees, however, suffer from two major disadvantages. First, they do not provide an exact representation for objects, and, Second, the sire of the tree representing an object depends critically on the position of the object in the space. Thus, translating or rotating an object can dramatically increase OT decrease the size of the tree. This paper provides an algwithm for computing such a new octree resulting from an arbitrary translation and rotation of an initial tree (ar tmcs), which attempts to minimize For static objects, a …