SpaDaGIS 2003 1

Efficiency Issues in Multi-resolution Terrain Modeling

Leila De FlorianiLeila De Floriani * Paola MagilloPaola MagilloDepartment of Computer Science

University of Genova, Genova (Italy)

* currently at the University of Maryland, College Park, MD

SpaDaGIS 2003 2

Terrain Models

Terrain dataTerrain data

• points in the plane

• height values

Terrain modelTerrain model• triangle mesh connecting the points

• linear interpolation of heights

SpaDaGIS 2003 3

Multi-Resolution

Large-size data sets high storage space and processing time

multi-resolutionmulti-resolution

Dynamically adapt resolution to user needs tradeoff accuracy / size

SpaDaGIS 2003 4

Regular and Irregular Multi-Resolution Models

Data on a grid / scattered data Regular / irregular multi-resolution models Both are instances of a Multi-Triangulation Compare efficiency of data structures and of queries

SpaDaGIS 2003 5

Changing the Resolution of a Mesh Modification:Modification: two alternative sets of triangles covering a region at lower / higher resolution

Can adapt resolution by playing with modifications

SpaDaGIS 2003 6

The Multi-Triangulation (MT)

A base meshbase mesh A set of modificationsmodifications A partial orderpartial order

(dependency relation)

M2 depends on M1

iff M2 changes some triangles changed by M1

SpaDaGIS 2003 7

Irregular MT: Vertex-Based MT

DataData: scattered ModificationModification: vertex insertion

Built while refining a mesh through vertex insertion (VI)OR

Built while decimating a mesh through vertex removal

• single vertex (VR)

• set of independent vertices (IVR)

SpaDaGIS 2003 8

Regular MT: Hierarchy of Right Triangles (HRTHRT)

DataData: on a regular grid ModificationModification: simultaneous bisection of two adjacent

right triangles

SpaDaGIS 2003 9

Data Structure for Vertex-Based MT

Partial orderPartial order• As a directed acyclic graph

ModificationsModifications• modification M = two triangle meshes (M-,M+)

• triangles of M+ uniquely defined

• triangles of M- must be encoded

CoordinatesCoordinates and heightheight valuesvalues of vertices ApproximationApproximation errors of triangles

SpaDaGIS 2003 10

Data Structure for Vertex-Based MT

Encode the triangles of M-Encode the triangles of M-• anchor edge

• bit stream (depth-first traversal of a tree of triangles)

10 00 11 11

SpaDaGIS 2003 11

Data Structure for HRT

Each triangle uniquely identified by a location codelocation code

Partial order and modifications are retrieved from location codes and not stored

Height valuesHeight values of vertices ApproximationApproximation errorserrors of triangles

SpaDaGIS 2003 12

Comparison:Storage Costs of the Data Structures

n = number of data points

Full-resolution meshFull-resolution mesh = 54n bytes

Vertex-based MTVertex-based MT• in theory = 33n bytes• in practice depends on

construction process (VI, VR, IVR)

HRT HRT = 6n bytes

SpaDaGIS 2003 13

Comparison: Queries to Extract a Mesh

Variable resolutionVariable resolution focused in a window

error

trianglesWorse (more triangles)

Uniform resolutionUniform resolution on the whole domain

Better (fewer triangles)Plot:

SpaDaGIS 2003 14

Comparison: Uniform Resolution

Best = VI Motivation: error-driven construction strategy

VRIVRHRTVI

Mount Marcy Devil Peak

HRTVRIVRVI

SpaDaGIS 2003 15

Comparison: Uniform Resolution

HRT 22045 triangles VI 16208 triangles

SpaDaGIS 2003 16

Comparison: Uniform Resolution

HRT 3648 triangles VI 1951 triangles

SpaDaGIS 2003 17

Comparison: Variable Resolution

Best = HRT Worst = VR Motivation: smaller modifications, fewer dependency links

Mount Marcy

VRVI

IVRHRT

VRVI

IVRHRT

Devil Peak

SpaDaGIS 2003 18

Comparison: Variable Resolution

HRT 1614 triangles VI 2072 triangles

SpaDaGIS 2003 19

Summary

Data distribution

Space required wrt the mesh at the maximum resolution

LOD Queries

Vertex-based

any about half better at a uniform resolution

Right triangles

on a grid about 1/9 better at a variable resolution

SpaDaGIS 2003 20

The End