1. The Etree Library: A System for Manipulating Large Octrees on Disk David R. O’Hallaron Associate Professor of CS and ECE Carnegie Mellon University (joint work with Tiankai Tu and Julio Lopez) www.cs.cmu.edu/~euclid

2. Geometric and physical properties and max freq Problem definition Source definition CVM 4D Output Input mesh Solver Solver specific code and data Viz Validate Verify Seismo extraction Publish Inversion Std Output Convert Pathway 2 Work Flow

3. Motivation Problem: –Need interoperability between AWM and analysis codes. General Solution Approach: –Represent input CVM and AWM data sets in a standard format that can be queried efficiently. Specific Solution Approach: –Represent CVM and AWM data sets as hierarchical spatial data structures called octrees »Octrees are compact ways to represent equivalent regions of space i.e., Regions of the CVM with similar density –Store the octrees in indexed databases (etrees) for fast retrieval and querying. –Provide an interface for C programs (etree library)

4. Geometric and physical properties and max freq Problem definition Source definition CVM (3D etree) 4D Output Input mesh Solver Solver specific code and data Viz Validate Verify Seismo extraction Publish Inversion 4D etree Convert Pathway 2 Work Flow

5. An Octree Domain Pixels 0123456871091211131415 x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 y

6. Equivalent Octree Representations 0123456871091211131415 x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 y b c e g h i j k l m Domain Representation a c dm e f k l g h j i b Level 0 Level 1 Level 2 Level 3 : Interior octant: Leaf octant Tree Representation

7. 10010000_010 Append level of f to obtain locational code Morton code: Maps n-dimensional points to one-dimensional scalars Locational code: Appends an octant’s level to the Morton code of its left-lower corner Computing Addresses of Octants b c e g h i j k l m 0100 10010000 1000 Interleave the bits to obtain Morton code f’s left lower corner (4, 8) Binary form (0100, 1000)

8. Linear Octrees abcdefghijkl m B-tree index B-tree Pages b c e g h i j k l m a c d m e f k l g h j i b Octants are stored on disk in locational code order –Interior nodes are optional Cool fact: the following are equivalent: –Sorted order on disk –Space filling Z-order –Preorder tree traversal

9. Finding a Leaf Octant Without Knowing its Exact Locational Code b c e g h i j k l m Query pixel q a c dm e f k l g h j i b q Another cool (and powerful) fact: In the linear ordering on disk h < q < i

10. Representing Point Sets with Sparse Octrees 001011 00011011 01

11. The Etree Library C library with 22 functions for manipulating large linear octree files (called etrees) on disk –Opening and closing etrees –Inserting and deleting octants and subtrees –Searching for octants –Traversing octants in Z-order (preorder traversal) –Loading and storing application metadata Supports 3D or 4D spatial data sets Portable file format –Standard header characterizes byte ordering and data sizes –Text trailer contains arbitrary application-level metadata Code and documentation available on the Web –www.cs.cmu.edu/~euclid

12. Current Pathway 2 Etree Application Codes Discrete Community Velocity Model –3D etree created by sampling Magistrale/Day CVM program. –Distributed with etree library CVM Web Service –Web server that queries the discrete CVM etree. –Returns 2D GIF slices or 3D FD and FEM meshes. –Distributed with etree library Finite Element Mesh Generation [Tu, 2002] –Converts discrete CVM etree to FEM mesh. –Mesh represented as an element etree and a node etree. Finite Difference Mesh Generation –Generates mesh for Kim Olsen’s AWM –Current part of CVM Web service

13. Future Pathway 2 Etree Application Codes Conversion from flat AWM files to etree files. Visualization of 3D discrete CVM etree files. Visualization of 4D etrees.