Digital Terrain Modeling: Principles and Methodology - Chapter 5 pps

Figure 5.6 Delaunay triangulation starting from anywhere on the imaginary boundary box the points indicated by triangles are the imaginary points.. The area can then be defined by themin

Generation of Triangular Irregular Networks

InChapter 4,digital terrain surface modeling was discussed It was pointed out thatgrid- and triangle-based modeling approaches are more widely used than point-basedapproaches For grid-based modeling, a grid network needs to be formed through

a random-to-grid interpolation, if the original data are not in grid form The sion of interpolation methods will be conducted inChapter 6.On the other hand, fortriangle-based modeling, a triangular network needs to be formed through a triangula-

discus-tion procedure if the data are not in grid or triangular form The formadiscus-tion of triangular irregular network (TIN) from irregularly distributed data is discussed in this chapter.

FORMATION: PRINCIPLES

There are a number of ways for the construction of a triangulation network from

a given set of randomly (irregularly) distributed data They are based on differentprinciples In this section, these principles will be presented

5.1.1 Approaches for Triangular Irregular Network Formation

To form a TIN, there are two choices for making use of the data points The first is to

consider all the data to form an overall network This is a batch (or static) approach

for the Delaunay triangulation of a set of data points The alternative is to allow theaddition or removal of points during the triangulation process This is a dynamicprocess and thus is called dynamic triangulation, as modifications to the structurecan be made without reconstructing the whole network each time It should be notedthat “dynamic” does not mean that the points are considered to be moving — that isanother property, usually known as kinetic (Guibas et al 1991)

87

Spatial data can be in either vector or raster format Therefore, the triangulationcan be in either vector or raster mode It is possible to convert vector data to rasterand then triangulate in raster mode Alternatively, it is possible to convert raster datainto vector and then triangulate in vector mode.

As will be discussed later in this section, there are many possible criteria forthe construction of triangles, thus leading to many alternative methods The mostwidely used method, as was discussed inChapter 4,is the Delaunay triangulation,which has a dual relationship with the Voronoi diagram It implies that the Delaunaytriangulation network can be formed either directly by algorithm or indirectly throughthe Voronoi diagram However, the triangulation in raster mode is usually achievedvia the Voronoi diagram because in raster space the construction of Voronoi diagrams

is much easier than that of Delaunay triangulation Therefore, the approaches fortriangulation can be summarized as in Figure 5.1

5.1.2 Principles of Triangular Irregular Network Formation

From a set of randomly distributed data, there are alternative ways to form triangularnetworks Figure 5.2 illustrates the three alternative triangular networks generatedfrom the same set of data The question that arises is “which one is the best?”There must be some basic principles to guide the construction of triangular networks.This section discusses these principles

Raster based

Static mode

Vector based

Dynamic mode

Triangular irregular network (TIN)

Voronoi diagram

Figure 5.1 Approaches for triangular irregular network formation.

Figure 5.2 Triangular networks with different shapes constructed from the same data set:

(a) a set of data; (b) result 1; (c) result 2; and (d) result 3.

As mentioned inChapter 4,one of the basic characteristics of Delaunay lation is that no other data points are contained by the circumcircle of a Delaunaytriangle This is one of the basic principles for the generation of a Delaunay triangu-lation from a set of two-dimensional data points, referred to as empty circumcircleprinciple Figure 5.3 illustrates this In Figure 5.3(a), point D is within the circlecircumscribing
ABC if point C is selected to form a triangle with points A and B.

triangu-It means that point D instead of point C should be used to form a triangle with points

A and B Figure 5.3(b) shows this case, where point C does not fall into the circlecircumscribing
ABD

Local equiangularity is another principle suggested by Lawson in 1972 (see Tsai

1993) for Delaunay triangulation It says that the triangular network is optimum

if for every convex quadrilateral formed by two adjacent triangles, the swapping

of diagonals will not cause a decrease in the minimum of the six interior anglesconcerned and at the same time will not cause an increase in the maximum angle

In this way, the minimum angle is maximized and the maximum angle is minimized

for all the triangles This is also called the MAX–MIN angle principle The procedure

for swapping diagonals is called a local optimization procedure (LOP) (Tsai 1993).Figure 5.4 illustrates this principle In Figure 5.4(a), two triangles,
ABC and
ADCare used to form a convex quadrilateral The minimum interior angle is∠CAD andthe maximum interior angle is∠ADC After swapping the diagonal, as shown inFigure 5.4(b), the minimum interior angle then becomes ∠CBD, which is largerthan∠CAD and the maximum interior angle is ∠ADB, which is smaller than ∠ADC.This means that the shape in Figure 5.4(b) is the optimal configuration

Figure 5.3 The empty circumcircle principle for Delaunay triangulation (a) Circumcircle

containing point D (b) Point D is used to form the triangle.

Maximum

Minimum

D

C A

B

Figure 5.4 Illustration of the LOP process for local equiangularity: (a) before swapping the

diagonal; and (b) after swapping the diagonal.

An intuitive principle is the minimum sum-distance, which refers to the sum ofthe distances from a new point to the two end points of a triangle baseline (Yeoli1977) The corresponding algorithms are based on the criterion that the new pointselected to construct a new triangle is the one that has the sum of its distances to theend points of the baseline as the smallest value.

Another simple principle is the minimum circumscribing circle radius (Elfick1979) The corresponding algorithms are based on the criterion that the new pointthat is selected should form a triangle in which its circumscribing circle radius is thesmallest value The minimum distance from the center of the circumscribing circle

to the base line has also been suggested (McLain 1976)

5.2 VECTOR-BASED STATIC DELAUNAY TRIANGULATION

As discussed in the previous section, various principles can be used for the mentation of TINs, leading to different types of algorithms No attempt is made here

imple-to introduce all these algorithms Instead, only the methods for the construction ofDelaunay triangulations will be presented here

5.2.1 Selection of a Starting Point for Delaunay Triangulation

An advantage of Delaunay triangulation is that the resulting triangulation network isindependent of the starting point Therefore, the selection of a starting point is only forthe convenience of algorithm implementation Some choices for the starting point are:

1 the geometric center of the data points (Elfick 1979)

2 the shortest of all possible lines between any two data points (Yeoli 1977)

3 a line segment on the imaginary boundary (McCullagh and Ross 1980)

4 a line segment on the boundary convex hull (Tsai 1993; Gosper 1998)

After choosing the starting point, another point, which is normally the nearestneighbor is selected to form the initial base Then, a third point is searched to formthe first triangle Other triangles can then be formed by using the three sides of theinitial triangle as these bases The search for points to form triangles will be discussed

in the next sub-section

Figure 5.5 shows the triangulation process starting from the geometric center

It is not necessary to have a data point located exactly at the geometric center Thedata point closest to this X and Y average values are selected as the starting point.Point 1 in this Figure 5.5 is closest to the geometric center and thus selected as thestarting point If the shortest of all possible lines between any two data points wasselected as the starting point, then the computation of all distances between two datapoints would be heavy As a consequence, this choice is not very popular

Many triangulations start from anywhere on the boundary of the area to bemodeled; for many applications, the area to be modeled is explicitly defined However,

in many other applications, the boundary is not explicitly defined, then the boundaryneeds to be sorted out first

2 1

3

2 1 3

Figure 5.5 Delaunay triangulation starting from the geometric center: (a) generation of the first

triangle and (b) generation of the second and third triangles.

Figure 5.6 Delaunay triangulation starting from anywhere on the imaginary boundary

box (the points indicated by triangles are the imaginary points) (a) A set of random points (b) Minimum bounding rectangle (c) Imaginary boundary box.

A set of imaginary points can be used to define the boundary box of the area to betriangulated Figure 5.6 illustrates the triangulation from such an imaginary boundarybox Figure 5.6(a) shows a set of random points The area can then be defined by theminimum bounding rectangle as shown in Figure 5.6(b) or by an imaginary boundarybox containing all the data points (Figure 5.6c) Usually, on the imaginary boundarybox, a few imaginary points (e.g., at the four corners and on the four sides) are addedfor convenience of point searching From anywhere on the rectangle or on the box,the triangulation process can start, for example, from the upper/left corner

Often, the convex hull of the data points is used to define the area of interestwhich is the smallest convex polygon containing all data points A convex polygonmeans that a line segment connecting any two points must be completely within it(Tsai 1993) A number of algorithms are available for constructing the convex hull of

a set of points on a 2-D plane such as Graham’s scan, Jarvis’ march (gift wrapping),and Quick hull A detailed discussion of these algorithms can be found elsewhere

(O’Rourke 1993; Gosper 1998) The Gift Wrapping algorithm is simple and popularly

used.Figure 5.7illustrates the working principle of this algorithm The first step is tofind the point with the minimumY coordinate as a starting point; the second step is

Step Step 2 Step 3 Final

Figure 5.8 Search for a point to form a new triangle (a) Only one point inside the circle; (b)

More than one point inside the circle.

to find B where all points lie to the left of line AB by scanning through all the points.That is, B makes the largest right-hand turn from A In a similar way, points A, B, C,

D, E and F could be found to form a convex hull

5.2.2 Searching for a Point to Form a New Triangle

After the starting point (also called rotation point) is determined, the nearest point isselected as the known point to form an initial base Then, a new point is selected as thevertex of a new Delaunay triangle, which is located to the right side (i.e., clockwisefrom the known point to the rotation point) of the base A simple search method is todraw a circle from the middle point of the base with the base as the diameter If there

is only one point inside this circle, then the point will be picked up to form a newtriangle If there is more than one point inside the circle, then the point that has thelargest angle subtended from the base from all possible choices around the startingpoint will be selected If this does not succeed in finding a point, then the search circle

is expanded using the base line as a chord and with progressively larger circles untilthe appropriate neighbors are found Figure 5.8 shows this searching process In thisway, the most likely neighbors are first picked up and are then tested to find one withthe largest angle

.2

.1 10

.3 11

4

8 9

5

.2

.1 10

3 11

Figure 5.9 Triangulation by enclosing from convex hull boundaries (a) First triangle starting

from the boundary (b) Subsequent triangles formed near the boundary.

To increase the efficiency of point searching, data points could be grouped intoblocks beforehand according to their (x or y) coordinates.

5.2.3 The Process of Delaunay Triangulation

Once the starting point is determined and the search principle is specified, the gulation can be carried out smoothly An example of triangulation from the geometriccenter of the data points has already been given inFigure 5.5.Therefore, the examplesgiven in this section are of triangulations starting from the boundaries

trian-The first example starts from the convex hull of the data area and moves alongthe boundary It is illustrated in Figure 5.9(a) The procedure is as follows:

1 The starting point has the point with minimumY coordinate, that is, point 5 in

4 The triangulation proceeds along the convex boundary That is, triangle edge 8, 9

is used as the second base line for the formation of the second triangle

5 The advancing front of the triangulation will be moved forward clockwise andgradually toward the center, until all data points are triangulated

The second example is to start from an imaginary boundary This is illustrated

inFigure 5.10.The lower-left corner is selected as the starting point This time themovement is counterclockwise A shell of triangles is formed when the front hits theimaginary boundary Instead of continuing along the boundary, a new shell will startfrom the imaginary boundary near the first base In this way, triangulation for thewhole data set will be completed shell by shell

Shell 1

Shell 4

Shell 5 Shell 6

Shell 3 Shell 2

Figure 5.10 Delaunay triangulation starting from an imaginary boundary.

5.3 VECTOR-BASED DYNAMIC DELAUNAY TRIANGULATION

In the previous section, Delaunay triangulation in static mode was discussed.The search for points is usually an inefficient process if the amount of data islarge Therefore, triangulation is more often done dynamically by gradually addingnew points into the network This kind of dynamic operation is also referred to

as incremental triangulation There are many incremental triangulation algorithmsavailable in the literature (e.g., Zhu and Chen 1998), but in this section, only thesimplest and most robust one is described as a representative, that is the Bowyer–

Watson algorithm (Bowyer 1981; Watson 1981), which is usually called the simple incremental algorithm.

5.3.1 The Principle of Bowyer–Watson Algorithm for

Dynamic Triangulation

The Bowyer–Watson algorithm (Bowyer 1981; Watson 1981) is regarded as the mostpractical triangulation algorithm The basic idea is to start with coarse triangles andthen to add points sequentially into this coarse triangulation network.Figure 5.11illustrates the refinement of triangles after adding more points

The initial triangulation is usually very simple (e.g., two triangles of the boundingrectangle) enclosing all points in the area of concern The insertion process of thisalgorithm is illustrated inFigure 5.12 When a point p is inserted into a triangle(i.e.,
ABC in Figure 5.12a) this triangle is split into three, with the new pointforming a vertex of each of the three new triangles (Figure 5.12b) Then, each of thethree edges of the old triangle (i.e.,
ABC) is checked to see whether there is a need for

swapping the edge with the alternative diagonal by applying the empty circumcircle principle In this example, edge−→

AB must be swapped with the alternative Dp and

Figure 5.12 Delaunay triangulation by the Bowyer–Watson algorithm: (a) initial triangulation;

(b) splitting the enclosing triangle; and (c) the “swap” operation.

the new triangulation after the insertion of point p is shown in Figure 5.12(c) Morediscussion on a numerical criterion for edge swapping is given in Section 5.3.3

5.3.2 Walk-Through Algorithm for Locating the

Triangle Containing a Point

For a large volume of data, locating the triangle where a point is to be inserted isdone by a walking-through algorithm (Gold et al 1977) to improve efficiency In thisalgorithm, two problems need to be solved The first is to set a numerical criterion,which tells whether a point is within the triangle The second is to give a pointer tothe next triangle to be examined if the current triangle does not contain the point.The directional relationship between a point P and a directed line segment−→

ABcan be determined by the following formula:

Figure 5.13 Local “area coordinates” to determine whether a point is inside a triangle.

This value is in fact twice the area of the triangle formed by points A, B, and P Thedirectional relationship is judged as follows:

> 0, three points are in anticlockwise order,

i.e., P on the leftside of line−→

AB

= 0, three points are in a line

< 0, three points are in clockwise order,

i.e., P on the right side of line−→

AB

(5.2)

By using Equations (5.1) and (5.2), the question of whether point P is within

a triangle can then be answered Figure 5.13 illustrates the principle In this figure,points 1, 2, and 3 are the three vertices For a point P to be checked, three equationscan be established as follows:

of P will be negative

To find a triangle that contains the new point, the algorithm starts from anyarbitrary triangle If all the area coordinates of the point are positive, then the pointfalls within this triangle If not, one crosses the edge that has a negative coordinate andrepeats the same procedure for the new triangle The walk continues until a triangle

is found where all three coordinates of P are positive, which means that p is insidethe triangle.Figure 5.14illustrates the walk operation through the triangulation

5.3.3 Numerical Criterion for Edge Swapping

After the triangle containing P has been found, the point is inserted into the triangle,splitting the old one into three, as inFigure 5.11(a)and Figure 5.11(b) The threeexterior triangle edges now need to be tested, to see if they conform to the Delaunay(empty circumcircle) condition This is computed as follows:

> 0, point D is inside the circumcircle of
ABC

< 0, point D is outside the triangle

If the fourth vertex (obtained from the next triangle outward from the split triangle)

is inside the split triangle’s circumcircle, then their common edge must be exchangedwith the other diagonal, making the inserted point a “neighbor” to the fourth point.The new exterior edges must be stacked for later testing, and this process continuesuntil all exterior edges satisfy the Delaunay condition.Figure 5.15illustrates the edgeswitching Figure 5.15(a) shows the case of vertex D being inside the circumcircle of

ABC, and thus being a neighbor of B, and Figure 5.15(b) shows vertex D outside —with the result that the diagonal BD has been switched and D is no longer a neighbor

of B IfH = 0 the four points are co-circular and either configuration is acceptable.

p

S 1

2 3

Figure 5.14 The walk operation in a dynamic Delaunay triangulation.

C B

C B

D (b)

(a)

Figure 5.15 Empty circumcircle test.

To avoid “infinite loops” a zero value is considered to mean that D is “outside,” andthus will not be switched again

5.3.4 Removal of a Point from the Delaunay Triangulation

It is very useful to allow insertion of points in an existing triangulated network,especially in engineering design It is equally important to allow the removal ofpoints from the network for the consideration of alternative designs The problem ofremoving points from the Delaunay triangulation can be considered as the inverse

of the incremental insertion algorithm (Heller 1990) The potential triangle with thesmallest circumcircle is removed by swapping the edge (the inverse of the insertionalgorithm described previously) to reduce the set of neighbors by one, and the process

is repeated until only three triangles are left Again, as the inverse of the insertionalgorithm, P is removed and the three triangles are merged A triangle considered forremoval is often called a “ear” and consists of adjacent triples of point P.Figure 5.16illustrates an example of point deletion In this figure,
DCB might have the smallestcircumcircle (Figure 5.16a) and thus is removed first (Figure 5.16b) The triple withsmallest circumcircle in the remaining set of neighbors is AED, and this is removedsecond, leaving only three triangles (Figure 5.16c) Point P is removed in the end andthe final result is shown in Figure 5.16(d)

An alternative (Devillers 1999) to the method described earlier is to remove theears in order of the power of P (Aurenhammer 1987) The power of P with respect tothe ear with verticesv1,v2, andv3is simplyH (v1,v2,v3) divided by the triangle area.

Since it is necessary to select the ears for removal in order of their power, a priorityqueue data structure is required This data structure saves ears in the order in whichthey are supplied, and returns them in the order smallest first The ear that has thesmallest power is guaranteed to be Delaunay, and may thus be removed With a similartreatment shown in Figure 5.16, new triangles are formed by swapping the diagonals;powers for the two new triangles are computed and the priority queue updated Thisprocess is repeated until three ears remain in the list Finally, these last three trianglesare collapsed and point P is removed Devillers (1999) shows an example whereHeller’s algorithm does not give correct results Mostafavi et al (2003) give a similaralgorithm where an ear is removed if its circumcircle contains none of the neighboringpoints of P While less efficient in extreme cases, it does not require the calculation of

A

B

C D

E 4 3

E 4

3

2

1 5

P

A

B

C D

E

5

Figure 5.16 Deletion of a point from a triangular network: (a) neighbors of P (to be deleted);

(b) triple with smallest circumcircle deleted; (c) only three triangles left; and (d) point P deleted from the triangulation.

the power of P (which requires a division), which may be an advantage where robustarithmetic is required for degenerate configurations

5.4 CONSTRAINED DELAUNAY TRIANGULATION

As will be discussed later inChapter 8,the accuracy of the resultant DTM surfacewill be much improved if the terrain F-S points and lines are all measured during thedata acquisition This section will discuss the special treatment of these lines duringthe triangulation process

5.4.1 Constraints for Delaunay Triangulation: The Issue

and Solutions

As discussed inChapter 2,the terrain feature lines are special Ridge lines are theconnected lines of local maxima (points) and the ravine lines are the local minima.These lines are so special that they should not be broken by any triangle edges.Figure 5.17shows the consequence of a ravine line being broken by a triangle edge

In other words, special attention should be paid to these lines in the triangulationprocess There are two possible solutions, the first and the simplest is to make thepoints on these lines very dense so that the lines will hopefully not be broken by

628 677

453 531 481 490

628 677

453 531 481 490

461

B

A C

628 677

453 531 481 490

461

A C

Figure 5.17 Consequence of triangulation without consideration of terrain feature lines:

(a) a set of data with a ravine line; (b) a possible profile across ACB; (c) triangulation without considering the ravine; and (d) contouring after considering the ravine.

Figure 5.18 Densification of points on feature lines for triangulation: (a) data with a feature line;

(b) point densification; and (c) triangulation result.

any triangle edges even though they are treated as ordinary points In this method,the critical stage is the point densification process, which in effect transforms theconnected feature lines into ordinary data points Experience shows that the intervalbetween the points along the feature lines has to be equal to or shorter than half

of the average point intervals computed from the whole data set This method canmeet the demand very well in normal circumstances with simplicity, stability, andreliability although the data volume is increased and the original data set is modified

by the densification Figure 5.18 illustrates this treatment A mathematically rigoroussolution for point density along curves is given by Amenta et al (1998) The othermethod is to treat each feature line as a constraint, ensuring that no triangle edge is