Spatial Information Systems (SIS) COMP 30110 Terrain modeling (2)

Spatial Information Systems (SIS)Spatial Information Systems (SIS)

COMP 30110COMP 30110

Terrain modeling (2)Terrain modeling (2)

Digital Terrain Models (DTM)Digital Terrain Models (DTM)

A A digital terrain model digital terrain model is a model providing a is a model providing a representation of a terrain relief on the basis of a finite representation of a terrain relief on the basis of a finite set of sampled dataset of sampled data

Terrain dataTerrain data refers to measures of elevation at a set of refers to measures of elevation at a set of points points V V of the domain plus possibly a set of the domain plus possibly a set EE of non- of non-crossing line segments with endpoints in crossing line segments with endpoints in VV

D

Sampled data distributionSampled data distribution

Sampled data can be scattered (irregularly distributed) Sampled data can be scattered (irregularly distributed) or form a regular grid on the domainor form a regular grid on the domain

The distribution of the sampled data can depend on the The distribution of the sampled data can depend on the acquisition technique or on the specific applicationacquisition technique or on the specific application

Different distributions might be required by different Different distributions might be required by different configurations of the terrain reliefconfigurations of the terrain relief

Sampled data distributionSampled data distribution

Sometimes it can be useful to have Sometimes it can be useful to have irregularly distributedirregularly distributed sets sets of dataof data

For example, only a few sampled points where the terrain is For example, only a few sampled points where the terrain is quite flat and more values where the surface presents specific quite flat and more values where the surface presents specific features such as peaks etc.features such as peaks etc.

Sampled data distribution (cont.d)Sampled data distribution (cont.d)

Regular samplingRegular sampling is good in areas where the terrain is good in areas where the terrain elevation is more or less constantelevation is more or less constant

DTMsDTMs

In general, a larger In general, a larger number of sampled number of sampled points allows for a points allows for a better representation: better representation: multiresolution terrain multiresolution terrain modelsmodels (more on this (more on this later)later)

Terrain modelsTerrain models

• GlobalGlobal terrain models: defined by means of a single terrain models: defined by means of a single function interpolating all datafunction interpolating all data

• LocalLocal terrain models: piecewise defined on a partition terrain models: piecewise defined on a partition of the domain into “patches” (regions)of the domain into “patches” (regions)

In other words, they represent the terrain by means of a In other words, they represent the terrain by means of a different function on each of the regionsdifferent function on each of the regions in which the domain in which the domain is subdividedis subdivided

• In general it is very difficult to find a single function In general it is very difficult to find a single function that interpolates all available data, so usually local that interpolates all available data, so usually local models are usedmodels are used

Types of DTMsTypes of DTMs

• Polyhedral terrain modelsPolyhedral terrain models

• Gridded elevation modelsGridded elevation models

• Contour mapsContour maps

Polyhedral terrain models: definitionPolyhedral terrain models: definition

A A polyhedral terrain modelpolyhedral terrain model for a set of sampled pointsfor a set of sampled points V V can be defined on the basis of:can be defined on the basis of:

1.1. a a partitionpartition of the domain of the domain D D into polygonal regions into polygonal regions having their vertices at points in having their vertices at points in V V

2.2. a a functionfunction f f that is that is linearlinear over each region of the over each region of the partition (i.e., the image of partition (i.e., the image of f f over each polygonal over each polygonal region is a region is a planar patchplanar patch – this will guarantee – this will guarantee continuity of the surface along the common edges)continuity of the surface along the common edges)

((ff is also called a is also called a piecewise linearpiecewise linear function) function)

Polyhedral terrain models: propertiesPolyhedral terrain models: properties

- They can be used for any type of sampled pointset - They can be used for any type of sampled pointset (regularly and irregularly distributed)(regularly and irregularly distributed)

- They can adapt to the irregularity of terrains- They can adapt to the irregularity of terrains

- They represent continuous surfaces- They represent continuous surfaces

Triangulated Irregular NetworksTriangulated Irregular Networks

The most commonly used polyhedral terrain models The most commonly used polyhedral terrain models are are Triangulated Irregular NetworksTriangulated Irregular Networks (TINs), where (TINs), where each polygon of the domain partition is a triangleeach polygon of the domain partition is a triangle

TINsTINs

Example of a TIN based on irregularly distributed Example of a TIN based on irregularly distributed datadata

TINs for regular dataTINs for regular data

Regular sampling is enough in areas where the Regular sampling is enough in areas where the terrain elevation is more or less constantterrain elevation is more or less constant

TINs: important propertiesTINs: important properties

They guarantee the existence of a planar patch for They guarantee the existence of a planar patch for each region (triangle) of the domain subdivision each region (triangle) of the domain subdivision (three points define a plane): the resulting surface (three points define a plane): the resulting surface interpolates all elevation datainterpolates all elevation data

The most commonly used triangulations are The most commonly used triangulations are Delaunay triangulationsDelaunay triangulations

Why Delaunay TriangulationsWhy Delaunay Triangulations

They generate the most equiangular triangles in the They generate the most equiangular triangles in the domain subdivision (thus minimising numerical domain subdivision (thus minimising numerical problems: e.g., problems: e.g., point locationpoint location))

Their Dual is a Voronoi diagram. Therefore, some Their Dual is a Voronoi diagram. Therefore, some proximity queries can be solved efficientlyproximity queries can be solved efficiently

Delaunay TriangulationsDelaunay Triangulations

• Intuitively: given a set V of points, among all the triangulations Intuitively: given a set V of points, among all the triangulations

that can be generated with the points of V, the Delaunay that can be generated with the points of V, the Delaunay

triangulation is the one in which triangles are as much triangulation is the one in which triangles are as much

equiangular as possible equiangular as possible

• In other words, Delaunay triangulations tend to In other words, Delaunay triangulations tend to avoid long and avoid long and

thin triangles: thin triangles: important for numerical problemsimportant for numerical problems

t P

Does Does P lie inside lie inside tt or on its boundary? or on its boundary?

Voronoi DiagramsVoronoi Diagrams

• Given a set V of points in the plane, the Given a set V of points in the plane, the Voronoi DiagramVoronoi Diagram for V is the for V is the

partition of the plane into polygons such that each polygon contains one partition of the plane into polygons such that each polygon contains one

point point pp of V and is composed of all points in the plane that are closer to of V and is composed of all points in the plane that are closer to p p

than to any other point of Vthan to any other point of V

Voronoi Diagrams (cont.d)Voronoi Diagrams (cont.d)

• Property: the straight-line Property: the straight-line dualdual of the Voronoi diagram of V is a of the Voronoi diagram of V is a

Delaunay triangulation of VDelaunay triangulation of V

• Dual:Dual: obtained by replacing each polygon with a point and each point obtained by replacing each polygon with a point and each point

with a polygon. Connect all pairs of points contained in Voronoi cells that with a polygon. Connect all pairs of points contained in Voronoi cells that

share an edgeshare an edge

Voronoi Diagrams (cont.d)Voronoi Diagrams (cont.d)

• Voronoi diagrams are used as underlying structures to solve Voronoi diagrams are used as underlying structures to solve proximityproximity

problems (queries):problems (queries):

• Nearest neighbour (what is the point of V nearest to P?)Nearest neighbour (what is the point of V nearest to P?)

• K-nearest neighbours (what are the k points of V nearest to P?)K-nearest neighbours (what are the k points of V nearest to P?)

• Etc.Etc.

P

Why Delaunay Triangulations (cont.d)Why Delaunay Triangulations (cont.d)

It has been proven that they generate the best It has been proven that they generate the best surface approximation (in terms of roughness) surface approximation (in terms of roughness) independently of the independently of the zz values (Rippa, 1990) values (Rippa, 1990)

There are several efficient algorithms to calculate There are several efficient algorithms to calculate them (Watson, 1981)them (Watson, 1981)

Gridded modelsGridded models

A A Gridded Elevation ModelGridded Elevation Model is defined on the basis of a is defined on the basis of a domain partition into regular polygonsdomain partition into regular polygons

RSGsRSGs

The most commonly used gridded elevation models are The most commonly used gridded elevation models are Regular Square Grids (RSGs) Regular Square Grids (RSGs) where each polygon in the where each polygon in the domain partition is a square domain partition is a square

The function defined on each square can be a bilinear The function defined on each square can be a bilinear function interpolating all four elevation points function interpolating all four elevation points corresponding to the vertices of the squarecorresponding to the vertices of the square

RSG: an exampleRSG: an example

RSG (cont.d)RSG (cont.d)

Alternatively, a constant function can be associated Alternatively, a constant function can be associated with each square (i.e., a constant elevation value). This with each square (i.e., a constant elevation value). This is called a is called a stepped modelstepped model (it presents discontinuity steps (it presents discontinuity steps along the edges of the squares)along the edges of the squares)

D

Unidimensional Unidimensional profile of a profile of a stepped stepped modelmodel

TINs & RSGsTINs & RSGs

Both models support automated terrain analysis Both models support automated terrain analysis operationsoperations

RSGs are based on regular data distributionRSGs are based on regular data distribution

TINs can be based both on regular and irregular TINs can be based both on regular and irregular data distributiondata distribution

Irregular data distribution allows to adapt to the Irregular data distribution allows to adapt to the “variability” of the terrain relief: more appropriate “variability” of the terrain relief: more appropriate and flexible representation of the topographic and flexible representation of the topographic surfacesurface

Digital Contour MapsDigital Contour Maps

Given a sequence { Given a sequence { vv0 0 , …, v, …, vn n } of real values, a } of real values, a digital digital

contour mapcontour map of a mathematical terrain model ( of a mathematical terrain model (DD, , ) is ) is an approximation of the set of contour linesan approximation of the set of contour lines

{ ({ (x,yx,y))D, D, (x,y) = v(x,y) = vii } } i = 0, …, ni = 0, …, n

A set of contour A set of contour lineslines

Digital Contour MapsDigital Contour Maps

Contours are usually available as sequences of pointsContours are usually available as sequences of points

A line interporlating points of a contour can be A line interporlating points of a contour can be obtained in different waysobtained in different ways

ExamplesExamples: polygonal chains, or lines described by : polygonal chains, or lines described by higher order equationshigher order equations

Digital Contour MapsDigital Contour Maps

A line interporlating points of a contour can be A line interporlating points of a contour can be obtained in different waysobtained in different ways

ExamplesExamples: : polygonal chainspolygonal chains, or lines described by , or lines described by higher order equationshigher order equations

Digital Contour Maps: propertiesDigital Contour Maps: properties

They are easily drawn on paperThey are easily drawn on paper

They are very intuitive for humansThey are very intuitive for humans

They are not good for complex automated terrain They are not good for complex automated terrain analysisanalysis