A hierarchical triangle-based model
for
terrain description
Leila De Floriani
Dipartimento di Informatica e Scienze dell'Informazione
Universitk di Genova
Viale Benedetto XV, 3 - 16132 Genova- ITALY
Enrico Puppo
Istituto per la Matematica Applicata
Consiglio Nazionale delle Ricerche
Via L.B. Alberti, 4 - 16132 Genova - ITALY
Abstract
This article describes a new hierarchical model for representing a terrain. The model, called a Hierarchical Triangulated Irregular Network
(HTIN), is a method for compression of spatial data and representation
of a topographic surface at successively finer levels of detail. A HTIN
is a hierarchy of triangle-based surface approximations, where each node,
except for the root, is a triangulated irregular network refining a triangle
face belonging to its parent in the hierarchy. In this paper we present
an encoding structure for a HTIN and we describe an algorithm for its
construction.
1
Introduction
In the last few years hierarchical surface models have been developed to describe
terrains at different levels of resolution. Variable-resolution surface models provide a d a t a compression m e c h a n i s m which allows a reduction of the number of
points needed to describe a surface: fewer points, for instance, should be used
to represent large surface regions of almost constant slope. Such models are
also used for robot navigation on a terrain, since a coarse m a p of the terrain is
necessary to plan an initial path for the robot, while a more detailed description
is required to plan local motions.
237
Hierachical models are used in a wide range of applications for representing
point data, planar regions, surfaces, and 3D objects [Sam90]. Such models allow
the manipulation of a geometric entity at different levels of resolution. Usually, they are classified into space-dependent representations (like the quadtree,
the octree and their variants), which are based on a recursive decomposition
of the space occupied by an object, and object-dependent representations (like
different hierarchical triangulations or the Delaunay pyramid), which provide a
description of the object in an object-centered coordinate frame.
A terrain can be described by a mathematical model, expressed as a pair
A4 = (D, f ) , where z = f ( x , y) is a function (with suitable continuity properties)
defined over a domain D in the x-y plane. In practice, a terrain is known
through elevation values given at a discrete set of points in the x-y plane. Thus,
we are interested in a discrete model of the terrain based on such data. A
Digital Elevation Model (DEM) can be defined as a pair 9r = (~, F), where
E is a partition of the domain D into regions { R 1 , . . . , R,~} and F is a family
of continuous functions z = fi(x, y), each defined on a region Ri of E, which
provide a piecewise continuous approximation of the terrain.
Digital elevation models are usually classified into Regular Grids (RGs) and
Triangulated Irregular Networks (TINs) depending on the characteristics of the
domain subdivision E: The great advantage of TINs lies in their capability of
adapting to the changes in the roughness of terrain. Also, surface specific points
(peaks, pits, passes) and lines (ridges or valleys), which characterize the surface
independently of the data sampling, can be included in a TIN as vertices and
edges. Hence, in spite of the simplicity of regular grids, TINs provide a more
appropriate and flexible means for describing a terrain. An arbitrary triangulation of the domain does not usually represent an acceptable basis for building
a TIN, because of numerical interpolation problems. Thus, TINs are generally
based on a Delaunay triangulation of the projections of the data points on the
x-y plane. Delaunay triangulation has some important properties, like local definition, equiangularity (i.e., its triangles are as much equiangular as possible),
minimal roughness (i.e., the surface described by a Delaunay-based TIN is as less
rough as possible, independently of the elevation values) [Rip90]. In a Delaunaybased TIN, the domain subdivision depends only on the point distribution on
the x-y plane. In [Dyn90] some domain triangulations that depend also on the
elevation values are discussed, and corresponding TINs are compared with a
Delaunay-based TIN.
In this paper, we present a new terrain model, which is based on an irregular
domain triangulation and is hierarchical as well, thus providing a triangle-based
representation at increasingly higher levels of resolution. The basic idea in such
model, called a hierarchical TIN, is to combine the advantages of a TIN with
the benefits of a hierachical description (for instance, local refinement in areas of interest). At a higher abstraction level, a hierarchical TIN consists of a
Delaunay-based TIN, while each other node is a TIN refining a triangle belong-
238
ing to its parent in the hierarchy. The refinement of a triangle is performed by
inserting new points in its interior and/or on its edges and computing a Delaunay triangulation of such points. Thus, the underlying triangulation is locally a
Delaunay one.
The remainder of the paper is organized as follows. Section 2 briefly reviews
hierarchical models for terrain description both for regularly and irregularly
spaced data. Section 3 introduces the hierarchical model we propose by giving
the basic definitions and aome examples. Section 4 describes a data structure for
encoding a hierachical TIN, while Section 5 presents an algorithm for building
such a model.
2
Hierarchical terrain models: an overview
We call Hierarchical Digital Elevation Model (HDEM) any DEM which provides
a multiresolution description of a terrain. As standard DEMs, HDEMs can be
classified depending o n t h e shape of the underlying subdivision into: quadtreebased models and hierarchical triangulated models.
Quadtree-based models require regularly spaced data: they are based on
a subdivision of the domain defined by the recursive partition of a rectangle
enclosing the projections of the data points into a set of nested rectangles having
vertices at such projections. Each rectangle is split into four subrectangles by
joining each internal point to its projections on the four sides of the recatngle.
Chen and Tobler evaluate different interpolation techniques for approximating
a surface defined by a quadtree in terms of accuracy, computational speed and
storage cost [Che86]. The problem with such interpolants is the difficulty in
preserving the continuity of the surface approximation along the sides of the
subdivision. Von Iterzen and Barr propose a method for avoiding discontinuities
[Von88]. Such method uses a modified version of the quadtree (that they call
a restricted qnadtree), and triangulates the leaves in the quadtree to achieve
continuity at the borders of adjacent regions.
Hierachical triangulated models can be further classified into strictly hierarchical triangulations, which are described by a domain partition tree, and
multiresolution triangulations, which are basically sequences of TINs. The first
hierarchical triangulated models appeared in the literature are the ternary and
the quaternary hierarchical triangulations [Gom79,Bar84,DeF84,Fek84,Pon87].
These models are based on the recursive subdivision of an initial triangle (with
vertices at data points and containing all the other points inside) into a set
of nested subtriangles with vertices at the data points. In a ternary triangulation, a subdivision of a triangle ~ consists of joining an internal point P to
the three vertices of t, while, in a quaternary triangulation, each triangle is
subdivided into four triangles formed by joining three points, each lying on a
239
different triangle side. The major problem with a ternary triangulation lies in
the elongated shape of its triangles, which leads to inaccuracies in numerical
intel;polation. On the other hand, a quaternary triangulation is only well suited
when the data are regularly distributed and suffers of the same discontinuity
problems at the boundaries of adjacent triangles as quadtree-based models. A
quaternary triangulation, called triarcon, has been applied by Goodchild and
Shiren as a hierarchical representation of the globe in a geographic information
system [Goo92].
More recently, Scarlatos and Pavlidis have proposed a hierarchical triangulated model, which generalizes the ternary triangulation, by allowing splitting
of the triangles along the edges as well [Sca90]. The idea of the method is to
define alternative ways of refining a triangle either by adding internal points or
by splitting its edges. The objective is to insert into the model significant terrain characteristics hpproximating critical lines and points at different levels of
detail. The authors show experimentally that the number of edge and triangle
splits necessary to achieve a certain degree of approximation is less than the
number of refinements required in a ternary triangulation.
The Delaunay pyramid described in [DeF89] is a multiresolution hierarchical
model consisting of an ordered sequence of Delaunay based TINs, each of which
contains an increasing number of points and provides a more accurate surface
description. Any two consecutive TINs are connected by a set of links which join
the triangles modified between the two levels. The Delaunay pyramid has been
extended to include also a set of segments as edges in the model at increasing
resolution, thus producing a constrained Delaunay pyramid [Jon91]. This allows
the explicit representation of surface-specific lines as well as their refinement
when the resolution increases. A constrained Delaunay pyramid can be built by
using the incremental method described in [DeF92].
A Delaunay pyramid cannot be described by a tree, since a triangle belonging
to the triangulation at a given level i can intersect a portion of the domain
covered by several triangles in the triangulation at level i - 1. A Delaunay
pyramid is encoded as a sequence of TINs plus a set of links which describe the
connections among the triangles. One drawback of such model is thus its high
storage cost. Moreover, it is difficult to reconstruct a surface approximation in
which the approximation error is different in different regions of the domain.
3
Hierarchical Triangulated Irregular Network
In this section we introduce the definitions of Hierarchical Triangulation (HT)
and of Triangulated Irregular Network (TIN), and we combine such concepts to
define our new hierarchical triangle-based surface model cMled the Hierarchical
Triangulated Irregular Network (HTIN).
240
Figure 1: Two adjacent triangles and their refinement
All the structures we define are based on the triangulation of a finite set of
points V in the Euclidean plane, defined as follows:
a triangulation T(V) of set V is a maximal Planar Straight Line
Graph having V as set of vertices [Pre85].
T(V) is a planar subdivision formed by triangular faces and covering the convex
hull of V in IR2; in the following, the domain of a triangulation T will be denoted
D(T). We also say that an edge e of a triangulation T is a boundary edge for T
if it lies on the boundary of D(T).
3.1
Hierarchical Triangulation
A Hierarchical Triangulation (117') is obtained by applying the concept of recursive refinement to the triangulation model. Intuitively, given a triangulation,
any of its triangles is seen as an individual entity. We can expand any such
triangle at a higher level of detail into another triangulation, whose domain covers the triangle. Different triangles can be refined independently; nevertheless,
the refinement of one triangle can produce splitting of its edges into chains of
edges. As edges are entities shared by adjacent triangles, we enforce adjacent
triangles to be expanded into triangulations that "match" along their common
241
edge; this requirement will avoid discontinuities along the boundaries of each
new triangulation (see Figure 1).
A I-IT is described by a tree 7-I = (T,E), where T = {T0,...,T2v} is a
family of triangulations with triangular domains (possibly except D(To)), and
s = {(T, T ~) I T, T ~ E T} is a set of directed arcs linking pairs of elements of
T. In order to define the properties that characterize a HT, we introduce the
following definitions:
9 SONS(T)={Tj I (T, Tj) E E};
9 F A T H E R ( T ) = ~ such that (Ti, T) e ~;
9 ANCESTORS(T)={Tj t there exists a path in ~/from Tj to T};
9 D E S C E N D A N T S ( T ) = { ~ ! there exists a path in ~ from T to Tj};
A Hierarchical Triangulation satisfies the following properties:
9 h i e r a r c h y rule: for every Tj E SONS(T) there exists exactly one triangle tj in T such that tj = D(Tj); in other words, each triangulation~
except for the root, is the refinement of some triangle of another (coarser)
triangulation in the hierarchy; we say that Tj refines tj and we denote:
- tj = A B S T R A C T I O N ( T j )
- ~ =REFINEMENT(tj)
9 m a t c h i n g rule: let T ~ and T ~/be two triangulations of T and let V ~ and
V I~ be the sets of vertices of T ~ and T" respectively; we say that T ~ and T ~
are adjacent along a straight line I segment if their domains intersect only
along/; we say that T ~ and T '~ are matching along l if they are adjacent
along l and l M V ~ = l M V"; then, if T~,T ~ E T are adjacent along l, one
of the following conditions must hold:
(a) T I and T" are matching along l;
(b) 1 M V' C l M V" and there exists T* EDESCENDANTS(T') that is
matching with T" along I;
(c) l M V' D l M V" and there exists T** EDESCENDANTS(T *') that is
matching with T ~ along I.
In practice, the hierarchy rule guarantees the consistency of the hierarchical
relations between triangulations in ~/, while the matching rule ensures that any
edge of any triangulation T of T is always refined consistently in the subtree
rooted at T.
242
Figure 2: An approximation of terrain represented by a TIN
3.2
Triangulated
Irregular
Network
Let z = f(x, y) be a function of two variables describing a terrain. A Triangulated Irregular Network (TIN) approximating such terrain is a continuous
function interpolating f at a discrete set of points V = {vt, ..., yr,} in the plane,
defined over a two-dimensional triangulation T(V) of the points of V. A TIN is
defined as a pair f = (T(V), F), where triangulation T(V) is formed by triangles
{ t t , . . . , tn} and F = { f l , . . . , fn} is a family of functions, where each function
fi is defined over ti and interpolates f at the vertices of ti.
Here, we restrict to piecewise linear TINs, i.e., we impose that F is a set of
linear functions. In practice, for every triangle ~i of T, a corresponding triangular
planar patch ~ in ]R3 approximates f(x, y), for all (x, y) lying inside ti.
The precision of .T in approximating function f is measured by some distance E(f, F) between functions f and the linear interpolants over each triangle. Here, we define the precision of function fi E F over triangle ti as
Ei(f, fi) = maxt, l f ( z ) - f,(z)l, and we define E ( f , : ) = max~=lE,(f,f, ). We
will call E the error function.
In Figure 2 a perspective view of a terrain represented by a TIN is shown.
243
3.3
Hierarchical
Triangulated
Irregular
Network
In order to define a hierachical model of terrain surface, we combine the concept
of TIN with the one of hierarchical triangulation. Given a function f defined as
above, a Hierarchical Triangulated Irregular Nelwork (HTIN) approximating f
is a hierarchy of n TINs such that:
9 for every i-- 1,..., n, TIN 5ri = (T/(~), Fi) approximates f in D(Ti);
| triangulations T/(V/), i = 1 , . . . , n, form a Hierarchical Triangulation 7/.
In the following, we will refer interchangeably to a TIN ~" and to the triangulation T describing it, whenever no ambiguity arises. Similarly, we will refer to
the precision of a triangulation or of a triangle by meaning the approximation
error of the corresponding TIN or triangular patch, respectively. For simplicity,
we will also denote with E(T) the error of a TIN described by triangulation T,
and with E(t) the error of a triangular patch corresponding to triangle t.
A Hierarchical Delaunay TIN (HDT) is a HTIN with the further property
that any triangulation in the hierarchy is a Delaunay triangulation.
The above definitions do not yield any particular criterion of refinement. For
instance, one could build the HTIN by imposing that each triangle is refined into
a triangulation containing no more than a maximum number nrnax of triangles
(such an approach could be useful, for instance, to speedup point location). In
the next Section, we will show how a HTIN (a HDT) can be built based on a
finite sequence of decreasing tolerance values, in such a way that each refinement
causes an increase in the precision of the approximation of a portion of terrain.
4
A d a t a s t r u c t u r e for e n c o d i n g a hierarchical
TIN
In this Section, we describe a data structure for encoding a hierarchical TIN.
The information stored in the data structure allow fast traversal of the model,
as required by algorithms that build an "expanded" TIN at a specified level of
accuracy, or more generally, which navigate the model at different degrees of
resolution. Navigation algorithms are the basis, for instance, of algorithms for
contour extraction or for visibility computation.
The internal encoding structure of a hierarchical TIN is a combination of
individual data structures representing the various TINs composing the HTIN,
and information describing the hierarchy as well as adjacency relations between
triangulations sharing common boundaries.
244
Several data structures have been proposed in the literature for encoding a
triangulation [Gui83,Pre85,Woo85]. Here, we use a variant of the symmetric
structure developed by Woo [Woo85] for describing the triangulations forming
the hierarchy. For each trianglulation T E T, we encode its three basic entities,
namely vertices, edges and triangles, together with four incidence relations. Each
edge e of T is described by its two extreme vertices (Edge-Vertex relation), and
by the two triangles sharing it (Edge-Triangle relation). Each triangle t is related
to its bounding edges (Triangle-Edge relation). Finally, we store with each vertex
v one of the edges of T incident on v (partial Vertex-Edge relation). It can be
shown that the data structure is optimal with respect to both storage cost and
time complexity of the structure accessing algorithms operating on it.
The following information are stored to describe the hierarchical organizaition
of a HTIN. For every node T in the HTIN, we store:
o a link to its parent triangulation in the hierarchy FATHER(T) together
with an indirect link to triangle ABSTRACTION(T) in FATHER(T);
9 for every triangle t in T that is refined in the hierarchy, a link to triangulation REFINEMENT(t);
| three sequences of boundary edges, corresponding to the refinements of the
three edges of ABSTRACTION(T) respectively;
9 for each boundary edge e of T, the sequence of boundary edges containing
e and the position of e in such a sequence.
Information on boundary edges allow maintaining links towards the exterior of
triangulation T in an implicit form.- Such links allow navigation of the hierarchical structure at a given level of detail.
Let e be a boundary edge of T. We denote, for analogy, ABSTRACTION(e)
the edge of ABSTRACTION(T) that is expanded in T into a sequence s containing e. Let us suppose for now that ABSTRACTION(e) is not a boundary edge
of FATHER(T). Then, there must exist a triangle t ~ belonging to FATHER(T)
and adjacent to ABSTRACTION(T) along ABSTRACTION(e); in the general
case, ~ is expanded in the hierarchy into a triangulation T ~ that matches with
T aIong s. Then the position of e in s can be used to locate its corresponding
edge in the corresponding boundary chain of T ~. If also ABSTRACTION(e) is
a boundary edge in FATHER(T), then the position of e is pushed onto a stack
and upper levels in the hierarchy are searched following the same rule, until a
non-boundary edge e* is found. Then, the hierarchy is descended from e* using
the stack of positions, until the edge corresponding to e at the same level of
detail is found (see Figure 3).
245
/
/
/
/
I
t
1
/
Figure 3: Navigation of a HTIN
246
5
Building a hierarchical T I N
We are interested in building a hierarchical terrain model at various levels of
resolution. We assume to have an "exact" representation of a terrain (typically,
an interpolation function based on a dense regular grid of sampled data), and
to build our model by extracting points from it.
Techniques to obtain a TIN at a fixed level of resolution from a dense set
of data have been described in [Fow79,DeF85,Sca90,Pup92]. In [DeF85] an algorithm was proposed that refines an existing Delaunay-based TIN by inserting
one point at a time and updating the Delaunay triangulation until the required
precision is met. The point inserted at each cycle is the one that causes the maximum error in the approximation; this criterion is named the Delaunay selector.
In [Pup92] a parallel algorithm based on the Delaunay selector is proposed, that
performs refinement by inserting many points at a time (namely, one per existing
triangle).
Let {c0,..., ok} be a given decreasing sequence of positive real values, called
the tolerance values. We build first the root of the HTIN To, which is a TIN at
level of precision c0 (i.e., E(To) < c0). Then each triangle ti of To refined into
a triangulation T~ such that E ( ~ ) <_ cl. This process is repeated on the new
triangulations with decreasing tolerance values, until each portion of the initial
domain has been refined at precision r
In order to build our HTIN, we produce first triangulation To by applying
a Delaunay selector algorithm with precision r to the initial data set. Then,
an iterative refinement procedure is applied, that refines at each cycle all triangulations corresponding to the leaves of the current hierarchy tree whose error
exceeds the current tolerance level.
The basic step of the refinement procedure consists in taking a triangulation
T = { t l , . . . ,t,~} such that r
< E(T) < r and computing a set of triangulations {T1, ... ,Tn}, where Vj = 1,... ,n, ~ approximates with precision ci+l
the portion of terrain corresponding to triangle tj E T. In order to satisfy the
matching rule, this task is performed in three steps (see Figure 4):
(i) edge refinement: each edge e of T is refined by inserting new data points
on it; then, a chain of segments is obtained that approximates the terrain
profile along e with precision ci+i;
(ii) first expansion: for all j, a triangulation Tj is computed, that contains
the vertices of tj plus all vertices added to its edges;
(iii) t r i a n g u l a t i o n refinement: for all j, a Delaunay selector algorithm with
precision r
is applied to Tj.
247
Figure 4: Steps of refinement algorithm.
248
Figure 5: Refinement of an edge approximating a terrain profile
It is possible that some triangle tj in T satisfies precision ei+l. In this case,
tj will be refined into a trivial triangulation formed by a single triangle. In
order to reduce the storage complexity of the hierarchy, such a triangulation is
eliminated at the next step (at precision ~i+2) and replaced in the hierarchy by
the refinement of the single triangle forming it. Trivial triangulations remaining
as leaves of the final HTIN (at precision ~k) are pruned. Thus, if we consider any
triangle t belonging to any triangulation in the final hierarchy, either t satisfies
precision ek and is not refined further, or there exists an index i < k such
that Eiq-1 <~ E(t) <_ ei and t is expanded into a triangulation satisfying at least
precision ei+l.
Edge refinement can be produced independently for all triangulations and
for all edges in each triangulation. Given an edge e, its approximation error
that we denote E(e), is the maximum difference between the terrain profile
along e and its corresponding interpolated function (i.e., a straight-line segment).
A refinement of e into a chain of edges to improve the approximation can be
obtained through a well-known iterative algorithm that splits e by inserting at
each cycle the point that causes the maximum error (see Figure 5) [Bal82].
After edge splitting has been performed, the first expansion can by computed
for each triangle t independently, by applying any Delaunay triangulation algorithm to the set of vertices corresponding to the chains that refine the edges
of t. The triangulation refinement can be performed on each new triangulation independently through a Delaunay selector algorithm (either sequential or
parallel).
249
6
Concluding remarks
We have defined a hierarchical triangle-based model for terrains. This model,
called a hierarchical TIN, consists of a hierarchy of triangulated surface approximations, each obtained as a refinement of a triangular facet in its parent.
The HTIN is a method for compressing spatial data according to a criterion
based on the accuracy of the approximation. It represents a surface at increasingly finer levels of detail. Being a surface-dependent representation, the HTIN
is invariant through geometric transformations. Compared with quadtree-based
surface models, a hierarchical TIN has the advantage of dealing with arbitrarily
distributed data. Unlike other hierarchical triangle-based models, a HTIN locally satisfies the equiangularity property and guarantees the global continuity
of the approximating surface. Compared with the Delaunay pyramid, a HTIN
has a lower storage complexity and, above all, allows local refinements in area
of interests.
Further developments of this work will involve: experimenting the model with
different underlying hierarchical triangulations, like those obtained by applying data dependent optimization criteria [Dyn90]; designing and implementing
neighbor finding algorithms to navigate the model and to extract global surface
approximations at different levels of precision; using HTIN as the basic data
model for designing a hierarchical geographic database.
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