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A Landform-based Approach for the Representation of
Terrain Silhouettes
Yann Chevriaux
Eric Saux
Christophe Claramunt
Naval Academy Research Institute
Lanvéoc-Poulmic
29240 Brest Naval, France
33 2 98 23 37 50
Naval Academy Research Institute
Lanvéoc-Poulmic
29240 Brest Naval, France
Naval Academy Research Institute
Lanvéoc-Poulmic
29240 Brest Naval, France
[email protected]
[email protected]
[email protected]
ABSTRACT
The research proposed in this paper introduces a qualitative
approach whose objective is to identify a computational
representation of terrain features that might help to reconcile
people’s perspective with common GIS models based on a
quantitative representation. We consider a plausible modelling
alternative where the observer is part of the environment. The
model developed is derived from the horizon and the forms that
materialise the frontier between Earth and the sky. This frontier is
represented as a silhouette and modelled as a sequence of
constituent landforms. These constituent landforms give a
complete set of orthogonal and sound primitives that can be used
and computed to represent silhouettes at different levels of
abstraction. This flexibility is adapted to geographical contexts
where terrain entities are interpreted and made of indeterminate
boundaries. A prototype implementation illustrates the potential
of the approach by an interface that allows for the integration of
sample silhouettes and computation of landform sequences. Precategorisation of the set of landform constituents favours the
generation of landform sequences adapted to user requirements.
Such a model and prototype should be of interest for the
description of terrain features perceived by an observer located on
the ground. This will allow a twofold application: giving a
description of a physical environment according to an observer
perspective and according to complementary levels of abstraction.
Categories and Subject Descriptors
I.2.10 [Artificial Intelligence]: Vision and Scene Understanding
– perceptual reasoning, shape.
General Terms
Algorithms, Theory, Human Factors
Keywords
GIS, terrain modelling, landforms, qualitative shape analysis
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ACM GIS’05, November 4-5, 2005, Bremen, Germany
Copyright 2005 ACM …$5.00.
1. INTRODUCTION
Over the past years, Geographical Information Systems (GIS)
have been successfully developed using quantitative models and
geometrical representations mainly inherited from the
cartographical view of the world. Despite the fact that these maporiented models and their computational capabilities are adapted
to many application domains, there are still many cases where the
cartographical view does not reflect the way people conceptualize
their surrounding world [10][14]. This leads to the search for
alternative modelling approaches more closely linked to spatial
cognition where the role of perception and communication is reemphasised [13][15][17]. One common principle of these
cognitive models of space relies in the way people perceive and
conceptualise space, and the actions they perform on it [19]. In
particular, it has been suggested that people perceive objects with
qualitative attributes and incomplete geometrical properties rather
than fields with accurate physical properties [3].
This duality of representations is illustrated by the example of a
terrain modelled using either quantitative or qualitative
approaches. In the former a usual representation is a field of
elevation points upon which topographic maps or digital terrain
models are derived, and where the point of view is a global one.
In the latter, landforms exist as mental representations of the
perceived environment, not always well-defined, and where the
point of view is the one of an observer located on the ground.
The objective of the research presented in this paper is to explore
and develop a qualitative representation of space where the
observer is part of the environment, and where the space to model
is the one derived from the horizon and the distinguishable
landforms that constitute silhouettes between Earth and the sky.
While a silhouette can be modelled by a polyline using a
quantitative representation, our objective is to follow a qualitative
approach where a silhouette is modelled as a sequence of
landforms. Each landform is considered as a primitive associated
to a given category. This approach supports derivation of
sequences of landforms, these being categorized using a shapebased analysis of the landforms geometry.
The remainder of the paper is organized as follows. Section 2
introduces basic cognitive properties of space, landforms and
categorisations. Section 3 develops the principles of the
qualitative shape analysis used for the representation of
landforms. Section 4 describes the main principles of the
landform-based approach. The prototype implementation and
results are discussed in section 5. Finally, section 6 summarizes
the research, preliminary results and outlines further work.
detail. However, these shape representations are mainly supported
by geometrical descriptions, and do not consider the semantic
components of the forms that emerge from the geometry.
2. LANDFORMS AND CATEGORIES
In qualitative spatial analysis, the outline of a shape is commonly
modelled using geometrical primitives. A formal grammar has
been developed by Galton and Meathrel [9]. But their intention
was to propose a general theory of shape’s representation [18]
letting the semantics of forms apart. In a recent terrain language
introduced by Kulik and Egenhofer [11], a category is modelled
by a sequence of oriented primitive lines. The assumption is that a
landform is an aggregation of well-defined geometric primitives,
but the approach is limited by a constraint of many landforms
which are often seen as objects of undefined limits [22].
The Galileo-Newtonian perspective has long claimed that the
Nature has an objective and inherent structure which is
independent from human observation, and that space is a primary
and universal framework. However, cognitive studies have shown
that the way people represent and reason in space is mostly
qualitative [4]. When perceiving landforms, human beings
organise them using categories. The importance of categories in
human reasoning has been raised by cognitive scientists and
Experientialism [12]. Experientialism (also called experiential
realism or embodied realism) underlines the way human beings
physically interact with the World, and particularly with
geographical systems. Influence of experientialism in geography
and spatial cognition has already been discussed (e.g., [2][16][8]).
It is claimed that the human perception of space lays on categories
which are derived from the physical nature of the human
organism, individual and collective experiences, and physical
interaction with the world, rather than a direct representation of
the physical nature of space.
Applied to the representation of landforms, it appears that the
modelling of a given silhouette, i.e., the way landforms are
perceived at the horizon, should be logically based on categories,
and inferred from end-user knowledge. More precisely, people
looking at a silhouette on the horizon do not perceive a sequence
of elevation points, but rather a visual entity characterized by the
forms it contains, and associated to some primitive categories. In
fact, and although usual terrain geometrical properties such as
summits or saddle points are often perceived, people associate
landforms to them although those are not always well-delimited
[22]. A landform is a subjective individuation of a part of the
Earth’s surface. A difficult modelling issue is to find where, for
example, a mountain begins and ends.
Our research approach follows this view as we consider that a
given silhouette, that represents landforms at the horizon, is
modelled by a sequence of landform primitives which are
semantically categorised, achieving thus a description at a higher
level of abstraction. In other words, the geometric properties of a
given silhouette are encapsulated into the semantic properties of
constituent landforms. Considering different levels of abstraction
also gives a relative flexibility to the model as it favours the
modelling of silhouettes at the level required by the user.
3. QUALITATIVE SHAPE ANALYSIS
Qualitative shape analysis has been long applied to the qualitative
description of geometrical forms in many areas, such as robotic
navigation [5] and computer vision [6]. Several models based on a
syntactic approach have been developed to capture salient
features within the outline of a 2D shape. In computer vision,
Richards and Hoffman [20] modelled the outline of a 2D shape by
a sequence of segments characterized by basic geometrical
properties such as the curvature. Since each segment is associated
to a symbol, a sequence of segments can be considered as a
qualitative description of the form, although the method is mainly
quantitative. Richards and Hoffman’s approach has been extended
by Cinque and Lombardi [1] towards a hierarchical representation
that allows for the modelling of shapes at different levels of
Landforms perceived on the horizon are 2D-shapes characterised
by their outlines, i.e. the limit between the sky and Earth. A
qualitative representation of a 2D-shape is influenced by the level
of resolution that determines the geometric primitives of salient
features. For instance, let us assume that the category ‘mountain’
is modelled by a sequence of a right-ascending straight line
followed by a right-descending straight line. According to this
definition, the high-resolution landform drawn for illustration
purpose in Figure 1 (right) is a sequence of very short segments
and, therefore, not perceived as a mountain, whereas a lower
resolution representation of the same landform can be considered
as a mountain (Figure 1, left). This leads to a close relationship
between the level of resolution and the landforms identified, and
emphasizes a limitation of many approaches where a simplified
level of generalisation is considered only. Several alternative
methods can be used to promote multi-resolution representations
of landforms such as hierarchical modelling (e.g., [21], [1]) or
generalisation (e.g., [11]).
Another problem to deal with in qualitative shape analysis is
related to the fact that many landforms have indeterminate
boundaries. Figure 2 illustrates this dilemma in evaluating the
limits of a landform. In this example, each contour leads to the
detection of a same mountain. The problem to address there is to
maintain the larger possible extent of each landform identified.
Figure 1 – Low resolution vs. high resolution landforms.
The two examples above show that the respective influence of the
level of resolution, and the one of the undefined boundaries
should be reflected in a qualitative landform model. This leads us
to retain a modelling approach where the influence of the level of
resolution, and the one of the undetermined boundaries are taken
into account as follows:
Constraint 1: A landform is detected independently of the
quantity of details used to represent it (Figure 1).
Constraint 2: A landform primitive is associated to one and only
one category.
Constraint 3: A landform can be detected through several
polyline representations which all refer to a same category (cf.
Figure 2).
location in the bounding box of a given salience sk. Therefore, a
landform L is associated to a sequence of salience locations
denoted S(L) and given as follows:
S(L )= (Loc(s1),Loc(s2),…, Loc(sn))
An example of landform sequence is given in Figure 5.
Figure 2 – On the difficulty to evaluate a landform limits.
4. LANDFORM-BASED APPROACH
The landform-based approach is achieved by a characterisation of
the salient features of the landforms. These salient features can be
equivalently described by either characteristic points or
characteristic segments. However, characteristic points are less
dependent than segments on the level of resolution, e.g., the
summit of a mountain is still the highest point while the slope
complexity increases with the level of resolution. As one of the
properties of our model, represented by Constraint 1, is to
maintain several levels of resolution, we retain characteristic
points as salient features for the modelling of a given landform.
4.1 Landforms modelling
In order to characterise and differentiate primitive landforms, we
retain an approach where the saliencies of a given landform are
characterised by their intersections with the bounding box of that
landform (Figure 3, one can remark that this property is another
illustration of the persistence of saliencies over different
resolutions). The bounding box and the intersections with the
saliencies are characterised using eight locations that materialise
the four borders and four corners. This is sufficient enough to
qualitatively model the directional vectors often used in twodimensional modelling as suggested in [7].
Figure 5 – A mountain modelled as sequence (LD,T,RD).
Constraint 3 implies that each landform L is associated to a set of
polylines Λ(L) ={Pi} that forms its spatial extent, denoted
Extent(L), and given as follows:
Extent(L)=
U
Pi ∈Λ ( L )
Pi
Since a polyline is defined as a set of points, the extent of a
landform is also a set of points (this will support the manipulation
of landform extents using set operators). We make the difference
between boundary saliencies located at the limit, that is, the
starting and ending vertices of the landform, and the others
denoted as interior saliencies. Boundary saliencies delimitate the
landform (e.g., the feet of a mountain) while interior saliencies
are between the limits of the landform (e.g., the summit of a
mountain). More formally:
Boundary(L) =(s1,sn)
Interior(L) = Salience(L) – Boundary(L)
Figure 3 – Landforms, bounding boxes and saliencies.
Therefore, the reference bounding box Β of a landform is defined
by a sequence of eight characteristic vertices successively
connected and denoted as B={LT, T, RT, R, RD, D, LD, L} for
respectively the locations Left, Right, Top and Down: Left-Top,
Top, Right-Top, Right, Right-Down, Down, Left-Down and Left.
The distinction between boundary and interior saliencies permits
to check whether a landform duplicates another. We say that two
landforms are duplicated when they share the same interior and
are characterized by the same sequence of vertices, more
formally:
Definition 1: A landform L duplicates a, landform L if and only if
Interior(L1) = Interior(L2) and S(L1) = S(L2).
Figure 6 - A mountain included into another.
Figure 4 - Reference bounding box.
A landform L is modelled by a sequence of saliencies
Salience(L)=(s1,s2,…). Each salience sk of a landform L is located
at one of the characteristic vertices, denoted vk, of the bounding
box with vk ∈ B. Let us denote Loc(sk) the function that returns the
Two computational steps are required to capture the landforms of
a given silhouette represented by an ordered set of points, that is,
extraction of soundness landforms, and removal of duplicate
landforms.
Table 1 - Examples of common categories.
captureLandforms(S: silhouette, SC: set of categories)
// extract soundness landforms
Ω = ∅
for each point pi in S do {
for each point pj in S with pj•pi do {
let L be form represented by points pi to pj
// is that form a categorized landform?
if S(L)∈SC then Ω Ω ∪ {L}
}
}
// remove duplicates
for each landform Li in Ω do {
Category
Schematised
representation
Sequence
mountain
(LD,T,RT)
valley
(LT,D,RT)
horn
(LD,L,LT,RD)
for each landform Lj in Ω with Lj•Li {
if (S(Li)=S(Lj) ∧ Interior(Li)=Interior(Lj)) then
{
Λ(Li)Λ(Li)∪ Λ(Lj)
Ω Ω - {Lj}
}
(LT,T,RT,R,RD)
}
}
Step
The application of the algorithm CaptureLandform gives a set Ω
that associates each landform L to its spatial extension Extent(L).
The algorithm detects the landforms within a silhouette,
independently of their respective extents. The landforms which
are constituent of a larger one should be ignored in order to
achieve the description at the highest possible level of abstraction
(Figure 6). This leads us to characterise the notion of constituent
landform as follows:
Definition 2: A landform Lj is a constituent of a landform Li if
and only if the extent of Lj is included into the extent of Li . The set
of landforms constituent of Li is denoted CNT(Li).
CNT(Li) = {Lj | Extent(Lj) ⊂ Extent(Li)}
4.2 Categories modelling
Since landforms are characterised by vertex locations, landform
categories are derived from vertex locations. Constraint 2 implies
that a sequence cannot model more that one category, but a
category might be modelled by several sequences. For instance,
landforms that belong to the category ravine are modelled either
by (LD,L,LT,RD,R,RT) or by (LT,L,LD,RT,R,RD).
A category C is modelled by a set of sequences as follows:
C:={S1, S2, …}
Table 1 shows some examples of identified categories, and
associated sequences of vertex locations and schematised
representations.
(LD,L,LT,T,RT)
(LD,L,LT,RD,R,RT)
Ravine
(LT,L,LD,RT,R,RD)
From the possible combination of landform sequences, we
characterise the ones that model a physically and sound landform
using the notion of regular sequence.
Definition 3: A regular sequence denotes a physically observable
landform. A regular sequence fulfils the following properties:
1.
A polyline that represents a regular sequence has no
auto-intersection (Figure 7 – right). This is due to the
fact that a silhouette represents a limit with no cycles
between the sky and the Earth.
2.
In order to be consistent with the reference bounding
box, a regular sequence must include at least one vertex
in each of the four borders (Figure 7 – left).
The number of regular sequences is determined by a recursive
algorithm that identifies the ones that fulfil the first property
above, and then removal of the ones that do not comply with the
second property above.
Let us construct a sequence made of the vertices of the reference
bounding box. The choice of the first vertex restricts the choice of
the second vertex amongst one of the remaining seven vertices
(Figure 8 – left). For instance, a sequence whose first vertex is LT
restricts the choice of the next one to one of the vertices T, RT, R,
RD, D, LD or L. The choice of the second vertex implies to
consider, for the choice of the third vertex, two different subsets
of connected and ordered vertices denoted P2 and Q2 (Figure 8 –
right).
Table 2 - µ n .
Figure 7 - Irregular sequences.
In order to fulfil the non-intersecting constraint of the sequence,
the choice of the third vertex in either P2 or Q2 also restricts the
choice of the fourth vertex in P2 or Q2, respectively. This
restriction is applied recursively for the choices of the remaining
vertices. Therefore, the number of sequences that can be
constructed, once the second vertex has been fixed, is equal to the
number of sequences that can be constructed if the third vertex is
chosen in P2 plus the number of sequences that can be constructed
if the third vertex is chosen in Q2.
n
0
1
2
3
4
5
6
7
µn
0
1
4
13
40
121
364
1093
The first vertex is chosen amongst the eight vertices of the
reference bounding box, then the second is chosen among a
connected and ordered subset of seven vertices. It follows, that
one can build 8.(1+µ7) = 8752 non-intersecting sequences.
Among the non-intersecting sequences that have been checked,
752 do not reach the four borders, and therefore are not consistent
with the reference bounding box. Therefore, we deduce that 8,000
regular sequences constitute an orthogonal and complete set of
landform primitives used to describe silhouettes.
The set of identified primitives is relatively large, but it has the
advantage of representing extensively the complexity of terrain
landforms. The modelling approach is completed by an interface
manipulation level where the user can define and interact with his
own categories, the system being responsible for mining
relationships between landform primitive and categories. These
components and the interface manipulation level are described
and exemplified in the next section.
5. IMPLEMENTATION
Figure 8 – Choosing the first and second vertices.
Let us consider the case where the third vertex is chosen in P2.
Since P2 is ordered, its vertices are enumerated from v1 to vn,
where n is the cardinality of P2. Let denote k the rank of the
chosen third vertex within P2. Thus, P2 is split into two new
connected and ordered subsets P3 and Q3 whose respective
cardinalities are (k-1) and (n-k) with 1≤k≤n. Figure 9 illustrates an
example with n =6 and k=3.
The principles of our modelling approach and the manipulation
levels are illustrated by a prototype implementation developed in
Java (SDK 1.4.1). The prototype includes an interface
manipulation level where the user can import some real-world
silhouettes, define its own landform categories and trigger a
landform analysis. A crucial part of the prototype is related to the
user-definition of categories, designed as an exploratory process.
5.1 Mining categories
The first step of a prototype analysis relies in the definition of
landform categories. Figure 10 shows some examples of
categories and the associated sequences represented by a schema.
Figure 9 – Derivation of two new connected and ordered set
with n=6 and k=3.
Therefore, the number denoted µn of new non-intersecting
sequences one can build, once the ith vertex has been chosen
amongst n vertices, is equal to n sequences of i vertices plus the
number of sequences one can build once the (i+1)th vertex has
been chosen amongst (k-1) vertices, plus the number of sequences
one can build once the (i+1)th vertex has been chosen amongst (nk) vertices, k varying from 1 to n. Applied recursively, this gives:
n
µ n = n + ∑ ( µ k −1 + µ n− k )
k =1
With µ0=0, µn is iteratively obtained for any 0<=n<=7 (Table 2).
Figure 10 – Prototype - Categories
In order to allow the user to deal with a limited set of primitives,
the set of regular sequences can be further filtered. For instance,
the user can exclude overhangs and generate a set of 753
sequences which is about 90%. According to these principles,
several predefined filters have been implemented, some examples
are given below:
allows for a qualitative characterisation of some specific sorts of
terrain silhouettes.
– “No overhang”: Removal of sequences that contain overhangs,
that is, segments from the right to the left.
5.2 Silhouette description
– “Overhangs only”: Selection of sequences made of left to right
segments.
– “Start with”: Selection of sequences that start with a given
vertex.
– “End with”: Removal of sequences that end with a given
vertex.
We illustrate the potential of the prototype using an analysis of
the silhouette extracted from [11] (Figure 13) in which we seek
for mountains, valleys, horns, steps and ravines as figured in
Table 1. This particular silhouette has been chosen because it is
linearized and encapsulates a representative variety of landforms.
However, in order to increase the expressiveness of the example,
lines are over segmented.
– “Contain”: Selection of sequences that contain a given vertex.
– “Exclude”. Removal of sequences that contain a given vertex.
– “Number of vertices”: Selection of sequences whose number
of vertices is within a given range.
In order to illustrate the role of these filters, let us consider the
case of a user that defines some elevation type categories such as
mountain or horn. The corresponding sequences do not include
overhangs, start with Left-Down vertex, and end with RightDown vertex. The schematized sequences presented in Figure 11
show some elevations of various degrees of complexity, including
the sequences modelling mountains and horns in Table 1. The
user can interact with these sequences by dispatching the
sequences over existing categories, creating new categories, or
considering that some sequences have no particular meaning from
his/her point of view.
Figure 13 – Constituent landforms of a silhouette.
From the initial 57 points, 1596 polylines are explored. Amongst
them, 30 polylines are associated to one of the primitive
categories. Table 3 presents the 12 remaining landforms after
removal of the duplicates. The second column indicates the
included points, while the last indicates the constituent landforms,
if any. The prototype provides, at the highest abstraction level
available, a descriptive sequence: (mountain, valley, mountain,
valley, mountain, valley, mountain).
Table 3 – Derived landforms.
Li
Range
Sequence
Category
L1
(2, 9)
(LD,T,RD)
mountain
L2
(6, 15)
(LT,D,RT)
valley
L3
(10, 18)
(LD,T,RD)
mountain
L4
(11, 13)
(LD,T,RD)
mountain
Figure 11 - Filtering “elevations” among the 8,000 regular
sequences.
L5
(15, 21)
(LT,D,RT)
valley
L6
(18, 33)
(LD,T,RD)
mountain
The user may also add a sequence by selecting a prototype among
the regular sequences, possibly after a filter has been applied or
by editing a schematised representation (Figure 12).
L7
(24, 41)
(LT,D,RT)
valley
L8
(37, 57)
(LD,T,RD)
mountain
Figure 12 - Editing a sequence.
These examples show how end-users, likely geomorphology
experts, can interact with the prototype and the underlying model
properties. Categorising landforms in such an exploratory model
L9
(47, 50)
(LD,L,LT,T,RT)
step
L10
(48, 51)
(LT,T,RT,R,RD)
step
L11
(51, 53)
(LT,T,RT,R,RD)
step
L12
(53, 55)
(LT,T,RT,R,RD)
step
CNT(Li)
L4
L9,L10,L11,L12
The prototype offers the possibility to browse through different
levels of abstraction. For instance, the mountain L8 might be
described with more details as a mountain whose right slope is
ended by a series of steps. The prototype provides, for each
landform at the highest level of abstraction, the list of constituent
landforms, and for each landform the corresponding polylines at
any level of abstraction.
The prototype also illustrates two properties of our qualitative
modelling approach. Firstly, a landform can be compared with the
representation of a category whatever the granularity of the
landform’s representation. Secondly, a user can define a category
by sketching or choosing some sequences at a high level of
abstraction without manipulating all the complexity of the
underlying geometrical primitives. Further developments of the
prototype should favour comparison of different terrains and
similarity analysis.
6. CONCLUSION
This paper introduces a qualitative approach for the modelling of
silhouettes. By contrast to conventional cartographic
representation of space, we consider a qualitative representation
and the particular point of view of an observer located on the
ground, and perceiving constituent landforms and silhouettes at
the horizon. Primitive landforms are categorised and modelled
using a qualitative approach. The model is flexible enough to
describe landforms in the horizon at different levels of
abstraction. Preliminaries prototyping results presented in section
5 illustrate the potential and the flexibility of the approach, and
the silhouette descriptions that can be derived. The model still
deserves further developments and extensions such as the
development of similarity analysis, and further consideration of
the undefined component of landform boundaries. Overall this
work is intended to present a modelling and computational
solution to observers located on the ground, and whose objective
is to give a description of a physical environment, and conversely
a modelling language that will allow the description of a
numerical terrain representation to an observer located on the
ground.
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