Post on 10-Apr-2018
transcript
8/8/2019 DTM and TIN
1/82
Source : Dr. Michela Bertolotto
TerrainTerrain modelingmodeling and TINand TIN
8/8/2019 DTM and TIN
2/82
Terrain dataTerrain data
Terrain dataTerrain data relates to the 3D configuration of the surface ofrelates to the 3D configuration of the surface of
the Earththe Earth
On the other hand,On the other hand, map datamap data refers to data located on therefers to data located on thesurface of the Earth (2D)surface of the Earth (2D)
The geometry of a terrain is modeled as a 2The geometry of a terrain is modeled as a 2 --dimensionaldimensional
surface, i.e., a surface in 3D space described by a bivariatesurface, i.e., a surface in 3D space described by a bivariatefunctionfunction
8/8/2019 DTM and TIN
3/82
Mathematical terrain modelsMathematical terrain models
AA topographic surfacetopographic surface oror terrainterrain can be mathematicallycan be mathematically
modeled by the image of a real bivariate functionmodeled by the image of a real bivariate function
z =z = JJ(x,y)(x,y)
defined over a domaindefined over a domain DD such that Dsuch that D 22
The pairThe pair TT=(=(DD,, JJ)) is called ais called a mathematical terrain modelmathematical terrain model
UnidimensionalUnidimensional
profile of aprofile of a
mathematicalmathematical
terrain modelterrain model
D
J
8/8/2019 DTM and TIN
4/82
Digital Terrain Models (DTM)Digital Terrain Models (DTM)
AA digitaldigital terrain modelterrain modelis a model providing ais a model providing a
representation of a terrain on the basis of a finite set ofrepresentation of a terrain on the basis of a finite set of
sampled datasampled data
Elevation dataElevation data refers to measures of elevation at a set ofrefers to measures of elevation at a set of
pointspoints VVof the domain plus possibly a setof the domain plus possibly a setEEof nonof non--
crossing line segments with endpoints incrossing line segments with endpoints in VV
D
J
8/8/2019 DTM and TIN
5/82
Elevation data acquisitionElevation data acquisition
Elevation data can be acquired through:Elevation data can be acquired through:
sampling technologiessampling technologies (by means of on(by means of on--site measurementssite measurements
or of remote sensing techniques)or of remote sensing techniques)
digitisationdigitisation of existing contour mapsof existing contour maps
Elevation data can be scattered (irregularly distributed)Elevation data can be scattered (irregularly distributed)
or form a regular gridor form a regular grid
The set of nonThe set of non--crossing lines can form a collection ofcrossing lines can form a collection ofpolygonal chainspolygonal chains
8/8/2019 DTM and TIN
6/82
ContoursContours
Given aGiven a terrain modelterrain model
TT= (= (DD,, JJ))
and a real valueand a real value v,v,the set ofthe set ofcontourscontours ofofTTat heightat height vvisis
{ (x,y){ (x,y)D,D,JJ(x,y) = v(x,y) = v}}
This is a set of simple lines (non selfThis is a set of simple lines (non self--intersecting)intersecting)
D
Planez = v
8/8/2019 DTM and TIN
7/82
DTMsDTMs
Digital terrain models represent an approximation ofDigital terrain models represent an approximation of
mathematical terrain modelsmathematical terrain models
Sampled modelSampled model Digital terrain modelDigital terrain model
8/8/2019 DTM and TIN
8/82
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 theThe 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 differentDifferent distributions might be required by different
configurations of the terrain reliefconfigurations of the terrain relief
8/8/2019 DTM and TIN
9/82
Sampled data distributionSampled data distribution
Sometimes it can be useful to haveSometimes it can be useful to have irregularly distributedirregularly distributedsets ofsets of
datadata
For example, only a few sampled points where the terrain isFor example, only a few sampled points where the terrain is
quite flat and more values where the surface presents specificquite flat and more values where the surface presents specific
features such as peaks etc.features such as peaks etc.
8/8/2019 DTM and TIN
10/82
Sampled data distribution (cont.d)Sampled data distribution (cont.d)
Regular samplingRegular samplingis good in areas where the terrainis good in areas where the terrain
elevation is more or less constantelevation is more or less constant
8/8/2019 DTM and TIN
11/82
DTMsDTMs
In general, a largerIn general, a larger
number of samplednumber of sampledpoints allows for apoints allows for a
better representation:better representation:
8/8/2019 DTM and TIN
12/82
Terrain modelsTerrain models
GlobalGlobalterrain models: defined by means of a singleterrain models: defined by means of a single
function interpolating all datafunction interpolating all data
L
ocalL
ocalterrain models: piecewise defined on a partitionterrain models: piecewise defined on a partitionof the domain into patches (regions)of the domain into patches (regions)
In other words, they represent the terrain by means of aIn 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 isin which the domain is
subdividedsubdivided
In general it is very difficult to find a single functionIn general it is very difficult to find a single function
that interpolates all available data, so usually localthat interpolates all available data, so usually local
models are usedmodels are used
8/8/2019 DTM and TIN
13/82
Types of DTMsTypes of DTMs
Polyhedral terrain modelsPolyhedral terrain models
Gridded elevation modelsGridded elevation models
Contour mapsContour maps
8/8/2019 DTM and TIN
14/82
Polyhedral terrain models: definitionPolyhedral terrain models: definition
AApolyhedral terrain modelpolyhedral terrain modelfor a set of sampled pointsfor a set of sampled points VV
can be defined on the basis of:can be defined on the basis of:
1.1. aa partitionpartition of the domainof the domain DDinto polygonal regionsinto polygonal regionshaving their vertices at points inhaving their vertices at points in VV
2.2. aa functionfunctionffthat isthat is linearlinearover each region of theover each region of the
partition (i.e., the image ofpartition (i.e., the image offfover each polygonalover each polygonalregion is aregion is aplanar patchplanar patch this will guaranteethis will guarantee
continuity of the surface along the common edges)continuity of the surface along the common edges)
((ffis also called ais also called apiecewise linearpiecewise linearfunction). Imagefunction). Image
analysis techniques calledanalysis techniques calledfacet modelsfacet models have somehave some
similarity with these methods.similarity with these methods.
8/8/2019 DTM and TIN
15/82
Polyhedral terrain models: propertiesPolyhedral terrain models: properties
-- They can be used for any type of sampled pointsetThey can be used for any type of sampled pointset
(regularly and irregularly distributed)(regularly and irregularly distributed)
-- They can adapt to the irregularity of terrainsThey can adapt to the irregularity of terrains
-- They represent continuous surfacesThey represent continuous surfaces
8/8/2019 DTM and TIN
16/82
Triangulated Irregular NetworksTriangulated Irregular Networks
The most commonly used polyhedral terrain modelsThe most commonly used polyhedral terrain modelsareare 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
8/8/2019 DTM and TIN
17/82
TINsTINs
Example of a TIN based on irregularly distributedExample of a TIN based on irregularly distributed
datadata
8/8/2019 DTM and TIN
18/82
TINs for regular dataTINs for regular data
Regular sampling is enough in areas where theRegular sampling is enough in areas where the
terrain elevation is more or less constantterrain elevation is more or less constant
8/8/2019 DTM and TIN
19/82
TINs: important propertiesTINs: important properties
They guarantee the existence of a planar patch forThey guarantee the existence of a planar patch for
each region (triangle) of the domain subdivisioneach 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 areThe most commonly used triangulations are
Delaunay triangulationsDelaunay triangulations
Triangular Irregular Network (TIN), whichrepresents a surface as a set of non-overlapping
contiguous triangular facets, of irregular size and
shape.
8/8/2019 DTM and TIN
20/82
Why Delaunay TriangulationsWhy Delaunay Triangulations
They generate the most equiangular triangles in theThey generate the most equiangular triangles in the
domain subdivision (thus minimising numericaldomain subdivision (thus minimising numerical
problems: e.g.,problems: e.g.,point locationpoint location))
Their Dual is a Voronoi diagram. Therefore, someTheir Dual is a Voronoi diagram. Therefore, some
proximity queries can be solved efficientlyproximity queries can be solved efficiently
8/8/2019 DTM and TIN
21/82
Delaunay TriangulationsDelaunay Triangulations
Intuitively: given a set V of points, among all the triangulationsIntuitively: given a set V of points, among all the triangulations
that can be generated with the points of V, the Delaunaythat can be generated with the points of V, the Delaunay
triangulation is the one in which triangles are as muchtriangulation is the one in which triangles are as much
equiangular as possibleequiangular as possible
In other words, Delaunay triangulations tend to avoid long andIn other words, Delaunay triangulations tend to avoid long and
thin triangles: important for numerical problemsthin triangles: important for numerical problems
t P
DoesDoes P lie inside t or on its boundary?lie inside t or on its boundary?
8/8/2019 DTM and TIN
22/82
Voronoi DiagramsVoronoi Diagrams
Given a set V of points in the plane, theGiven a set V of points in the plane, the VoronoiVoronoi Diagram for V is theDiagram for V is the
partition of the plane into polygons such that each polygon contains onepartition of the plane into polygons such that each polygon contains one
pointpointpp of V and is composed of all points in the plane that are closer toof V and is composed of all points in the plane that are closer topp
than to any other point of Vthan to any other point of V
8/8/2019 DTM and TIN
23/82
Voronoi Diagrams (cont.d)Voronoi Diagrams (cont.d)
Property: the straightProperty: the straight--lineline dualdual of theof the VoronoiVoronoi diagram of V is adiagram of V is a
Delaunay triangulation of VDelaunay triangulation of V
Dual:Dual: obtained by replacing each polygon with a point and each pointobtained by replacing each polygon with a point and each point
with a polygon.Connect all pairs of points contained inwith a polygon. Connect all pairs of points contained in VoronoiVoronoi cells thatcells thatshare an edgeshare an edge
8/8/2019 DTM and TIN
24/82
Voronoi Diagrams (cont.d)Voronoi Diagrams (cont.d)
VoronoiVoronoi diagrams are used as underlying structures to solvediagrams are used as underlying structures to solveproximityproximity
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?)
KK--nearest neighbours (what are the k points of V nearest to P?)nearest neighbours (what are the k points of V nearest to P?)
Etc.Etc.
P
8/8/2019 DTM and TIN
25/82
Why Delaunay Triangulations (cont.d)Why Delaunay Triangulations (cont.d)
It has been proven that they generate the bestIt has been proven that they generate the bestsurface approximation (in terms of roughness).surface approximation (in terms of roughness).
There are several efficient algorithms to calculateThere are several efficient algorithms to calculate
themthem
8/8/2019 DTM and TIN
26/82
Gridded modelsGridded models
AAGridded Elevation ModelGridded Elevation Modelis defined on the basis of ais defined on the basis of a
domain partition into regular polygonsdomain partition into regular polygons
8/8/2019 DTM and TIN
27/82
RSGsRSGs
The most commonly used gridded elevation models areThe most commonly used gridded elevation models are
Regular Square Grids (RSGs)Regular Square Grids (RSGs)where each polygon in thewhere each polygon in the
domain partition is a squaredomain partition is a square
The function defined on each square can be a bilinearThe function defined on each square can be a bilinear
function interpolating all four elevation pointsfunction interpolating all four elevation points
corresponding to the vertices of the squarecorresponding to the vertices of the square
8/8/2019 DTM and TIN
28/82
RSG: an exampleRSG: an example
8/8/2019 DTM and TIN
29/82
RSG (cont.d)RSG (cont.d)
Alternatively, a constant function can be associatedAlternatively, a constant function can be associated
with each square (i.e., a constant elevation value). Thiswith each square (i.e., a constant elevation value). This
is called ais 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
UnidimensionalUnidimensional
profile of aprofile of a steppedstepped
modelmodel
8/8/2019 DTM and TIN
30/82
TINs & RSGsTINs & RSGs
Both models support automated terrain analysisBoth models support automated terrain analysisoperationsoperations
RSGs are based on regular data distributionRSGs are based on regular data distribution
TINs can be based both on regular and irregularTINs can be based both on regular and irregular
data distributiondata distribution
Irregular data distribution allows to adapt to theIrregular data distribution allows to adapt to thevariability of the terrain relief: more appropriatevariability of the terrain relief: more appropriate
and flexible representation of the topographicand flexible representation of the topographic
surfacesurface
8/8/2019 DTM and TIN
31/82
Calculations from a Gridded
DEM
8/8/2019 DTM and TIN
32/82
Slope and Aspecta b c
d p e
f g h
Consider a 3x3 neighbourhood at every pixel for computation
Slope = Rate of change of terrain property at a given point
Aspect = Direction of Change
Both are expressed as angles
8/8/2019 DTM and TIN
33/82
Slope and Aspect
Slope = S: tan S =
1
22 2
( ) ( )z z
x y
x x
x x
Where z is the altitude and x and y are the coordinate
axes
Aspect: tan A = /
/
z y
z x
x x
x x
Let dxx = (a + 2d + f) (c + 2e + h)
dyy = (a + 2b + c) (f + 2g + h)
8/8/2019 DTM and TIN
34/82
Slope
z
x
x !x
dxx/(8*dx) where dx = X-resolution
z
y
x
xdyy/(8*dy) where dy = Y-resolution
Slope
S = arctan
(To convert S into degrees, multiply by 57.29578)
1
22 2
( ) ( )z z
x y
x x
x x
8/8/2019 DTM and TIN
35/82
Aspect
a) If and are positive,
Aspect = 90o + 57.3*tan-1[abs()]
b) If > 0 and < 0,
Aspect = 90 - 57.3*tan-1[abs()]
z
x
x
x
z
y
x
x
z
x
x
x
z
y
x
x
Aspect: tan A =/
/
z y
z x
x x
x x
8/8/2019 DTM and TIN
36/82
Aspect
c) If < 0 and > 0,
Aspect = 270 - 57.3*tan-1[abs( )]
d) If < 0 and < 0,
Aspect = 180 - 57.3*tan-1[abs( )]
In all cases, if Aspect < 0, Aspect = Aspect + 360
/
/
z y
z x
x x
x x
z
x
x
x
z
y
x
x
z
x
x
x
z
y
x
x
/
/z yz x
x x
x x
8/8/2019 DTM and TIN
37/82
Digital Contour MapsDigital Contour Maps
Given a sequence {Given a sequence { vv00, ,v, ,vnn } of real values, a} of real values, a digitaldigital
contour mapcontour map of a mathematical terrain model (of a mathematical terrain model (DD,, JJ) is) is
an approximation of the set of contour linesan approximation of the set of contour lines
{ ({ (x,yx,y))D,D,JJ(x,y) = v(x,y) = vii}} i = 0, , ni = 0, , n
A set of contourA set of contour
lineslines
8/8/2019 DTM and TIN
38/82
Digital Contour MapsDigital Contour Maps
A line interpolating points of a contour can be obtainedA line interpolating points of a contour can be obtained
in different waysin different ways
ExamplesExamples:: polygonal chainspolygonal chains, or lines described by, or lines described by
higher order equationshigher order equations
8/8/2019 DTM and TIN
39/82
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 terrainThey are not good for complex automated terrain
analysisanalysis
8/8/2019 DTM and TIN
40/82
DTMs: accuracyDTMs: accuracy
In general, the more data is available, the better theIn general, the more data is available, the better the
representationrepresentation
Using very large datasets requires large storage spaceUsing very large datasets requires large storage spaceand processing timeand processing time
Use of generalisation techniques: select a subset of dataUse of generalisation techniques: select a subset of data
that still maintains acceptable accuracy in thethat still maintains acceptable accuracy in therepresentationrepresentation
8/8/2019 DTM and TIN
41/82
DTMs: accuracy (cont.d)DTMs: accuracy (cont.d)
AnAn approximate terrainapproximate terrain model is a model that uses amodel is a model that uses a
subset of the data availablesubset of the data available
The approximationThe approximation errorerror EEis calculated with respect tois calculated with respect toa reference model built using the whole dataseta reference model built using the whole dataset
For exampleFor example: the error could be the maximum: the error could be the maximum
difference between the elevation value at a point anddifference between the elevation value at a point andthe interpolated value in the approximated modelthe interpolated value in the approximated model
8/8/2019 DTM and TIN
42/82
DTMs: accuracy (cont.d)DTMs: accuracy (cont.d)
Example:Example:
Model builtModel built
using the wholeusing the whole
data set:data set:
reference modelreference model
Model builtModel built
using a subset ofusing a subset of
the data:the data:
approximatedapproximated
modelmodel
8/8/2019 DTM and TIN
43/82
DTMs: approximation errorDTMs: approximation error
Maximum difference: approximationMaximum difference: approximation errorerror
Calculate all differences betweenCalculate all differences between
elevation data and interpolatedelevation data and interpolated
datadata
8/8/2019 DTM and TIN
44/82
DTMs: accuracy (DTMs: accuracy (cont.dcont.d))
TheThe accuracyaccuracy of the model is inversely proportional toof the model is inversely proportional to
the errorthe errorEEassociated with the model and is defined as:associated with the model and is defined as:
Some applications might require to build anSome applications might require to build an
approximate model with accuracy (error) within aapproximate model with accuracy (error) within a
given thresholdgiven threshold
E11
8/8/2019 DTM and TIN
45/82
How to pick points
Given a set of points with known values or given a set of
digitized contours, how should points be selected so thatthe surface is accurately represented?
8/8/2019 DTM and TIN
46/82
VIP Algorithm
Each point has 8 neighbors, forming 4 diametricallyopposite pairs, i.e. up and down, right and left, upperleft and lower right, and upper right and lower left
For each point, examine each of these pairs ofneighbors in turn
Connect the two neighbors by a straight line, andcompute the perpendicular distance of the central
point from this line
8/8/2019 DTM and TIN
47/82
VIP Algorithm
Average the four distances to obtain a measure of
"significance" for the point
Delete points from the DEM in order of increasingsignificance, deleting the least significant first . This continues
until one of two conditions is met: The number of points
reaches a predetermined limit . The significance reaches a
predetermined limit
8/8/2019 DTM and TIN
48/82
8/8/2019 DTM and TIN
49/82
8/8/2019 DTM and TIN
50/82
Comments
Due to its local nature, this method is best when the
proportion of points deleted is low.
Due to its emphasis on straight lines, and the TIN's useof planes, it is less satisfactory on curved surfaces.
8/8/2019 DTM and TIN
51/82
Given a set of points, calculate a triangulationGiven a set of points, calculate a triangulation
Particular properties, e.g., equiParticular properties, e.g., equi--angularity (Delaunayangularity (Delaunaytriangulation)triangulation)
TriangulationCalculationTriangulationCalculation
8/8/2019 DTM and TIN
52/82
Given a set of points, calculate a triangulationGiven a set of points, calculate a triangulation
Particular properties, e.g., equiangularity (DelaunayParticular properties, e.g., equiangularity (Delaunaytriangulation)triangulation)
TriangulationCalculationTriangulationCalculation
8/8/2019 DTM and TIN
53/82
Given a setGiven a set VVof points, calculate aof points, calculate a DelaunayDelaunaytriangulationtriangulation with vertices at points ofwith vertices at points ofVV
Watsons algorithm is one of the soWatsons algorithm is one of the so--calledcalled onon--lineline
methodsmethods: based on the modification of an existing: based on the modification of an existingDelaunay triangulation when a new point is insertedDelaunay triangulation when a new point is inserted
In onIn on--line methodsline methods, the first step consists of building a, the first step consists of building aDelaunay triangulation of the domain (containing allDelaunay triangulation of the domain (containing all
data points)data points)
Then all points ofThen all points ofVVare added incrementallyare added incrementally
Watsons algorithm (1981)Watsons algorithm (1981)
8/8/2019 DTM and TIN
54/82
InWatsons algorithm, the initial triangulation of theIn Watsons algorithm, the initial triangulation of the
domain is built by considering a fictitious triangledomain is built by considering a fictitious triangle
containing all points ofcontaining all points ofVVin its interiorin its interior
Watson: initial stepWatson: initial step
Dataset VDataset V
8/8/2019 DTM and TIN
55/82
After building the initial triangle, all points ofAfter building the initial triangle, all points ofVVareare
added one at a timeadded one at a time
Finally the initial triangle and all edges incident at itsFinally the initial triangle and all edges incident at its
vertices are deletedvertices are deleted
The main step in this algorithm is the insertion of aThe main step in this algorithm is the insertion of a
new point in the current Delaunay triangulationnew point in the current Delaunay triangulation
Watson: processWatson: process
8/8/2019 DTM and TIN
56/82
We call theWe call the influence polygoninfluence polygon RRPP of a pointof a point PP in ain a
triagulationtriagulation TTthe union of all triangles ofthe union of all triangles ofTTwhosewhose
circumscribing circle containcircumscribing circle contain PP
After insertingAfter inserting PP inin TT, we update, we update TTby deleting allby deleting all
edges internal toedges internal to RRPP and by joiningand by joining PPwith all thewith all thevertices ofvertices ofRR
PP
Watson: insertion of a new pointWatson: insertion of a new point
P
RP
8/8/2019 DTM and TIN
57/82
We call theWe call the influence polygoninfluence polygon RRPP of a pointof a point PP in ain a
triagulationtriagulation TTthe union of all triangles ofthe union of all triangles ofTTwhosewhose
circumscribing circle containscircumscribing circle contains PP
After insertingAfter inserting PP inin TT, we update, we update TTby deleting allby deleting all
edges internal toedges internal to RRPP and by joiningand by joining PPwith all thewith all thevertices ofvertices ofRR
PP
Watson: insertion of a new pointWatson: insertion of a new point
P
RP
8/8/2019 DTM and TIN
58/82
Points inPoints in VVare added one at a time in the currentare added one at a time in the current
Delaunay triangulationDelaunay triangulation
Watson: insertion stepWatson: insertion step
8/8/2019 DTM and TIN
59/82
First vertexFirst vertex PP11: the fictitious triangle is its influence: the fictitious triangle is its influencepolygonpolygon RRP1P1; join; join PP11 with its three verticeswith its three vertices
Watson: insertion step (cont.d)Watson: insertion step (cont.d)
8/8/2019 DTM and TIN
60/82
Inserting the second vertexInserting the second vertexPP
22: calculation of: calculation ofRRP1P1
Watson: insertion step (cont.d)Watson: insertion step (cont.d)
8/8/2019 DTM and TIN
61/82
Inserting the second vertexInserting the second vertexPP
22: updating the current: updating the currenttriangulationtriangulation
Watson: insertion step (cont.d)Watson: insertion step (cont.d)
8/8/2019 DTM and TIN
62/82
Inserting new verticesInserting new vertices
Watson: insertion step (cont.d)Watson: insertion step (cont.d)
8/8/2019 DTM and TIN
63/82
Inserting new verticesInserting new vertices
Watson: insertion step (cont.d)Watson: insertion step (cont.d)
8/8/2019 DTM and TIN
64/82
Inserting new verticesInserting new vertices
Watson: insertion step (cont.d)Watson: insertion step (cont.d)
8/8/2019 DTM and TIN
65/82
Inserting new verticesInserting new vertices
Watson: insertion step (cont.d)Watson: insertion step (cont.d)
8/8/2019 DTM and TIN
66/82
Inserting new verticesInserting new vertices
Watson: insertion step (cont.d)Watson: insertion step (cont.d)
8/8/2019 DTM and TIN
67/82
Inserting new verticesInserting new vertices
Watson: insertion step (cont.d)Watson: insertion step (cont.d)
8/8/2019 DTM and TIN
68/82
Inserting new verticesInserting new vertices
Watson: insertion step (cont.d)Watson: insertion step (cont.d)
8/8/2019 DTM and TIN
69/82
Inserting new verticesInserting new vertices
Watson: insertion step (cont.d)Watson: insertion step (cont.d)
8/8/2019 DTM and TIN
70/82
Inserting new verticesInserting new vertices
Watson: insertion step (cont.d)Watson: insertion step (cont.d)
8/8/2019 DTM and TIN
71/82
Inserting new verticesInserting new vertices
Watson: insertion step (cont.d)Watson: insertion step (cont.d)
8/8/2019 DTM and TIN
72/82
Inserting new verticesInserting new vertices
Watson: insertion step (cont.d)Watson: insertion step (cont.d)
8/8/2019 DTM and TIN
73/82
Delete the fictitious triangle and all edges incident atDelete the fictitious triangle and all edges incident at
its verticesits vertices
Watson: final stepWatson: final step
8/8/2019 DTM and TIN
74/82
Delete the fictitious triangle and all edges incident atDelete the fictitious triangle and all edges incident at
its verticesits vertices
Watson: final stepWatson: final step
8/8/2019 DTM and TIN
75/82
GivenGiven VV, Watsons algorithm calculates the Delaunay, Watsons algorithm calculates the Delaunay
triangulation with vertices at points of oftriangulation with vertices at points of ofVV
Watson: final stepWatson: final step
8/8/2019 DTM and TIN
76/82
The time complexity ofWatsons algorithm is O(The time complexity ofWatsons algorithm is O(nn22))
wherewhere nn is the number of input points (worst case)is the number of input points (worst case)
This depends on the fact that the insertion of a newThis depends on the fact that the insertion of a new
point requires changing all triangles of the influencepoint requires changing all triangles of the influence
polygon. In the worst case, the influence polygonpolygon. In the worst case, the influence polygon
includes all triangles of the current triangulationincludes all triangles of the current triangulation
NOTE: remember that the number of triangles in aNOTE: remember that the number of triangles in a
triangulation is O(triangulation is O(vv), with v the number of vertices in), with v the number of vertices inthe triangulationthe triangulation
Watson: complexityWatson: complexity
8/8/2019 DTM and TIN
77/82
Although the worst case time complexity for WatsonsAlthough the worst case time complexity for Watsonsalgorithm is O(algorithm is O(nn22), it has been shown that in the), it has been shown that in the
average case the number of triangles in the influenceaverage case the number of triangles in the influence
polygon is equal to 6polygon is equal to 6
Therefore, the average case time complexity is linearTherefore, the average case time complexity is linear
in the number of input pointsin the number of input points
Watson: complexity (cont.d)Watson: complexity (cont.d)
8/8/2019 DTM and TIN
78/82
Storage of Triangle Data
8/8/2019 DTM and TIN
79/82
Storage of Triangle Data
8/8/2019 DTM and TIN
80/82
The structure of a TIN. The TIN is a topological data model. The data are stored in
a set of tables that retain the coordinate values as well as the spatial relations of
the facets.
TIN S
8/8/2019 DTM and TIN
81/82
TIN Storage
Performance of any structure based operation, e.g.,
algorithm development depends on the way the dataare stored.
In TIN based DTM, data can be stored as
Triangle-based (for slope, aspect, volume
computations) Node-based
Side-based (for contouring or any traversal procedure)
Combination of the above (for visibility analysis)
TIN structure is a case of establishing a good andefficient triangle topology
TIN A li ti
8/8/2019 DTM and TIN
82/82
TIN Applications
Slope and aspect Contouring
Finding drainage networks
Creating cross-sections
Visualization of a 3-dimensional surface