Digital Terrain Modeling: Principles and Methodology - Chapter 15 (end) ppsx

In the work by Tse and Gold 2002, the standardTIN model is extended by merging some aspects of terrain modeling TINs, com-putational geometry the Quad-Edge data structure, and computer a

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Beyond Digital Terrain Modeling

Chapter 14 discusses the more traditional development and applications of terrainmodels Here, we look at some extensions of these models for specific problems

CONSTRUCTION 15.1.1 Manual Addition of Constructions on Terrain Surface

For simplicity it is usually assumed that terrain models are monotonic in X and

Y — there is only one possible Z for each XY location This is often true in the

real world, but not always — occasionally there are caves, tunnels, overhangingcliffs, bridges, and overpasses In the work by Tse and Gold (2002), the standardTIN model is extended by merging some aspects of terrain modeling (TINs), com-putational geometry (the Quad-Edge data structure), and computer aided design orCAD (Euler operators, which guarantee to preserve the connectivity of the surfaceafter they are applied) They found it easy to combine them to give the usual oper-ations on a 2D triangulation — as well as add an operator that generates a holebetween any two nonadjacent triangles (which is really the same thing as adding

a bridge or handle to the surface) Figure 15.1, Figure 15.2, andFigure 15.3givesimple examples, and Figure 15.4 and Figure 15.5 show part of a Hong Kongcity model

Thus, a simple modification of the basic triangulation algorithm allows one

to interactively modify the terrain model to add complex features that are wise unavailable Because one is still forming a connected surface, a variety oftopological operations, such as neighborhood selection and flow modeling, may

other-be performed Clearly another, even higher, layer of operations would permit one

to add predesigned features such as buildings, dams, tunnels, etc to our terrainmodel

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Figure 15.3 TIN model of a ground surface with buildings and bridge: (a) a building on a TIN

model and (b) a bridge connecting two buildings.

15.1.2 Semiautomated Modification of the Terrain Surface

Section 15.1.1 showed simple terrain modification based on modifying the TIN byadding and deleting individual points with (X, Y , Z) coordinates This is effective

but slow to do by hand An alternative approach is to “cut” the triangulated surfacewith a “knife” in order to sculpt it to the form desired One first sets the knife size,location, and orientation and then performs the cut (or intersect) operation One may

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Figure 15.4 A partial view of Hong Kong harbor.

Figure 15.5 Part of the Hong Kong city model.

either lower the terrain surface to the knife position, for example, cutting into theside of a hill, or else raise the surface to the knife position, creating an embankment

or dam More points are added to the triangulation to form the intersection linesbetween the knife and the original terrain, and one assumes a maximum slope (lessthan vertical) for the edges of the cut or embankment.Figure 15.6(a)shows a simpleTIN model with the knife in place Figure 15.6(b) shows the result after the surface

is lowered to the knife (with a 45◦ embankment specified) Figure 15.6(c) showsthe knife positioned across a valley, and Figure 15.6(d) shows the result of raising theterrain surface to the knife, forming a dam structure across the valley

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Figure 15.6 Terrain modification by “cutting” the triangulated surface with a “knife.” (a) The knife

positioned on the terrain (b) Modified terrain after lowering the surface to the knife blade (c) The knife positioned across a valley (d) The dam created after raising the surface to the knife blade.

With the introduction of the concept digital earth, global modeling of the Earth’s

surface (Gold and Mostafavi 2000) has become a hot topic The digital terrainmodeling techniques described in this book can also be extended to sphericalterrain modeling

15.2.1 Generation of TIN and Voronoi Diagram on Sphere

In planimetric terrain modeling, as discussed in Chapter 4, grids and TINs havebeen widely used to tessellate the terrain On the sphere, a similar tessellation modelneeds to be used The concept of spherical surface tessellation was presented byFuller, a German cartographer, for map projection in the 1940s (Dutton 1996) Sincethen, many researchers have approached this problem to project, analyze, and index

global data Many methods are based on inscribed polyhedrons, such as the

tetra-hedron, the cube (Snyder 1992), the octahedron (Dutton 1989, 1996; Goodchild et al.

1991; Goodchild and Yang 1992; Otoo and Zhu 1993; Clarke and Mulcahy 1995),

the dodecahedron (Wickman and Elvers 1974), and the icosahedron (Fekete 1990;

White et al 1992; Lee and Samet 2000), as shown inFigure 15.7.The edges of thepolyhedron are projected to the spherical surface and form the edges of sphericaltriangles

The octahedron-based tessellation is a regular triangular mesh on the sphere,called the octahedral quaternary triangular mesh (O-QTM) Figure 15.8shows an

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(a)

(b)

Figure 15.7 Spherical surface tessellation based on inscribed polyhedra (Reprinted with

per-mission from White et al 1992): (a) five polyhedra and (b) projected to the spherical surface.

Figure 15.8 Hierarchical tessellation of the spherical facet based on octahedron (Dutton 1996):

(a) level 1; (b) level 2; and (c) level 3.

example of O-QTM at three difference levels (Dutton 1996) Terrain modeling canthen be applied to the QTM

The QTM can also be used as a coordinate system on the sphere, just like theregular grid or triangular network on a 2D plane In the QTM, a point is represented

by a triangle, an arc is represented by a series of neighbor triangles, and a region

is represented by a series of neighbor triangles on and within its boundary trace.From the QTM, a TIN can then be constructed Alternatively, spherical TINs canalso be derived from spherical Voronoi diagrams (Augenbaum 1985; Robert 1997;Chen et al 2003).Figure 15.9shows an example of spherical Voronoi diagram andits dual — the spherical TIN

15.2.2 Voronoi Diagram for Modeling Changes in Sea Level on Sphere

Mostafavi and Gold (2004) used the dynamic Voronoi diagram on the sphere to modelthe continually changing height of the sea, rather than of terrain Figure 15.10(a)shows an initial set of cells, each representing a fixed mass of water, and uses the freeLagrange method to simulate flow under lunar gravitational influence, and hence thesea height Coastlines were modeled by a double line of fixed Voronoi cell generators.Figure 15.10(b) shows the result after simulation started: high water (HW) is indicated

by smaller, and therefore higher, cells, while low water (LW) is shown by larger,

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Figure 15.10 Dynamic Voronoi diagram on sphere to model the continually changing heights

of sea: (a) initial configuration of Voronoi cells and (b) Voronoi cell configuration indicating lunar tides.

lower cells.Figure 15.11is a Mercator projection showing the flow directions andvelocities of each cell

Two dimensions are required in terrain modeling to generate the underlyingtriangulation or grid Once elevations are added as an attribute, the result is usuallyknown as “2.5D” modeling, although the data structures remain 2D Once the topol-ogy (or connectedness) can no longer be represented on the plane (as in 3D objects

in CAD or games), a surface representation, composed usually of triangles, is oftenused, as in Section 15.1

However, for some applications a surface model is inappropriate, and a full 3Dvolumetric model is needed Examples include geological, atmospheric, and oceano-graphic models, where attributes need to be assigned to arbitrary locations in 3Dspace In some cases a 3D grid may be used, or an octree where nodes repre-sent volumes A more flexible approach is to replace the 2D triangulation structure

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Figure 15.11 Mercator projection showing cell velocities and directions.

Figure 15.12 Delaunay and Voronoi cells in three dimensions.

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Various interpolation techniques, such as the equivalent of the 2D Sibson ornatural neighbor interpolation, may be used, and these behave well for the aniso-tropic distributions of data that are often found in three dimensions For visualizationpurposes the individual tetrahedral may be sliced, based on the values at the cornervertices, to give the 3D equivalent of 2D contours Figure 15.13 shows a single 3Disosurface constructed in this way.

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Epilogue

It was natural that we felt relieved and excited somehow after having completed thefinal draft of this book and having uploaded the materials onto the ftp site of thepublisher However, soon we started to feel obliged to write this epilogue becausethere are a few issues confronting us

We thought it is really a pity that no authored book in this discipline had been

made after over 40 years of development although there are two edited works, Terrain

Modelling in Surveying and Civil Engineering by Petrie and Kennie (1990) and Digital Elevation Model Techniques and Applications: The DEM User Manual by Maune

(2001) Our aim was to write a book systematically covering a wide range of topics

in digital terrain modeling so as to fill in the gap in this area While writing, we werefaced with a number of challenges

The first challenge was related to the selective omission of materials It wasdifficult to make decisions This is because the term “digital terrain modeling” wouldmean different things to different groups of terrain specialists and practitioners To theproducers (including photogrammetrists and surveyors), data acquisition and terrainsurface modeling are of most concern; to geographers, terrain analysis and appli-cations are the most important; to geologists, interpolation techniques seem to becritical; It is really hard to satisfy all these groups In the end, we decided that

those topics are simplified if they have rich bodies of literature available For example,

we did not include many algorithms and techniques for interpolation and triangulation

as there is a huge body of literature (e.g., Su 1989; Chin 1995; Sakhnovich 1997;

de Berg 2000; Phillips 2003) in these areas covering the techniques developed incomputational geometry and geosciences Contouring is a traditional topic in digitalterrain modeling but is only briefly discussed in this book because a book authored

by Watson (1992) has been dedicated to this topic Similarly, DTM-based terrainanalysis is briefly discussed because of a recent book edited by Wilson and Gallant(2000)

The second challenge was related to the depth of discussion We may disappointthose readers who are interested in mathematics because we present neither mathe-matical proofs nor technical details Indeed, it is the main aim of this book to present

a systematic accounting of stories in digital terrain modeling at the level of principlesand methodology, as the title of the book suggests

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The third challenge was related to the boundary of the discipline Because theterrain surface is of concern to all geosciences and a huge body of related model-ing methodology and applications is available, we have to cut down the contentssomewhere Therefore, we did not cover much on Voronoi diagrams (e.g., Davies2000; Okabe et al 2000) although we are very interested in this topic Similarly, wedid not cover much on geostatistics (e.g., Olea 1999) and even omitted the famousKriging technique We only simply mentioned the surface modeling on sphere andwith construction inChapter 15

All in all, we are pleased with the compilation of some materials presented toyou, but also feel guilty about the imperfection Your comments are appreciated sothat we could make improvements in another edition, if possible

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