Embedding Pyramids into 3D Meshes

cessing and image understanding [11, 13]. The most efficient applications of the pyramid are in the areas of scalespace (or multiresolution) and coarse-to-fine operations. Multiscale or multiresolution image representation is a very powerful tool for analyzing numerous image features at multiple scales [6, 11]. Moreover, pyramid machines are not limited just to image processing tasks. By exploiting the hierarchy inherent in the tree structure of a pyramid, and the parallelism inherent at each level, pyramids can handle various problems in graph theory [9], digital geometry [9], and recursive parallel tasks [5]. The 2D mesh architecture, on the other hand, has had wide availability in the research and commercial community. A natural extension of the 2D mesh is the 3D mesh, which has recently gained marked popularity due to a number of inherent architectural features. These include simple VLSI layout, good scalability, higher bandwidth, and smaller diameter compared to 2D mesh (when they have same link width). In addition, since it has three dimensions, it is capable of modeling many physical world problems more naturally, such as 3D image processing and finite element methods. The advantages and rich topological characteristics of 3D mesh have led to the development of the massively parallel MIT J-Machine [3] and, more recently, the CRAY T3D. With the increasing popularity of the 3D mesh and its potential availability coupled with the suitability of many image processing and computer vision applications on a pyramid, we consider the problem of embedding pyramids into 3D meshes. The rest of the paper is organized as follows. Section 2 presents basic terminology and notations and discusses the measures used for our embedding of the pyramid into 3D mesh. Section 3 presents a simple embedding scheme denoted as natural embedding. Section 4 gives an improved embedding scheme denoted as multiple embedding. Section 5 provides some conclusions.