USING THE DISCRETE 3D VORONOI DIAGRAM FOR THE

MODELLING OF 3D CONTINUOUS INFORMATION IN GEOSCIENCES

GIMA MSc Thesis Midterm Presentation

Tom van der Putte

Supervisors: Hugo Ledoux and Peter van Oosterom

degenerate cases

Contents

• The problem and it’s context

• The research objectives

• The research already performed

• The research still to come

Problem and Context

Modeling continuous fields:Usually represented in raster format

Ledoux (2006) proposed:represent continuous field by creating an exact (vector) Voronoi diagram

The Voronoi Diagram

Point dataset:(seeds)

Voronoi Diagram:

Interpolation method

Why the exact Voronoi diagram?

• It handles anisotropic data well• It can be interpolated more efficient• Very easily manipulated!

BUT:- Struggles with degenerate cases- It can be relatively slow

What is the discrete Voronoi diagram?

N-dimensional regularly tesselated space (raster), in which all tesselations (pixels/voxels) have the value of the closest ‘seed’.

Why the discrete Voronoi diagram?

Expected:

- Same pro’s

BUT:

- No degenerate cases- Expected to be fast (Park et al, 2006)

Research Objectives

Main objective:

To assess the use of the discrete 3D Voronoi diagram for the modelling of 3D continuous information in geosciences.

Research Objectives

Research Questions:

- How to create a discrete 3D Voronoi diagram?- Which GIS can handle discrete 3D data (raster)?- Which data formats are used?- What functionality is needed for modeling ?- What functionality is provided by the GIS?- Discrete vs. Exact: which is best for what?

Already Done

Creating discrete (3D) VD

Numerous ways to create a discrete VD

Poll every pixel/voxel:“Which seed is closest ?”

“Grow” Voronoi Cells(Dilation)

“Growing” Voronoi cells

Through morphological operation: Dilation

Object Structuring element

New Object

“Growing” Voronoi cells

POINT DATA SET

List of points

P1 (x,y,z,a)

P2 (x,y,z,a)

P…. (x,y,z,a)

Pn (x,y,z,a)

“Growing” Voronoi cells

IF neighbour has no value:

assign neighbour current value

ELSE IF distances from pixel to seeds are equal:

choose highest/lowest/random

ELSE:

assign value of closest seed

For each point in a list of points:

Save changed points in new list

Why this way?

- Conceptually very simpel- Relatively efficient- Inserting / removing points very easy and

efficient!

POINT

List of points

P1 (x,y,z,a)……

From discrete 3D VD to GIS

Grass

GIS packages that fully support 3D raster:

(PCRaster)

3D raster vizualisation:

MayaVi

3D raster storage

DISCRETE 3D VD

MayaViGRASS

3D POINT DATA SET

ASCII 3DRASTER

file

VTK file

VTK XML file

Functionality

GRASS MayaViVisualisation -/+ ++

Isosurface - ++

Slicing - ++

Analytical functionality ++ n/a

Multi-D Map Algebra ++ n/a

3D - Reclassification ++ n/a

Interpolation (Point) + n/a

What functionality is offered?

Interpolation- Resampling- Natural Neighbour Interpolation -> Easy because inserting points = quick!

FunctionalityWhat functionality is NOT offered?

Still to do?

• Adding final functionality• Comparisson exact VD and discrete VD:

- Which is better suited for what purpose?- What is the difference in accuracy?

• Summarize problems in current 3D GIS in light of this research

• Put it all on paper