Digital Terrain Modeling: Principles and Methodology - Chapter 9 pptx

It would be desirable to maintain and update the DTM data at the largest scale only and to derive DTM data at any smaller scale by a generalization process, that is, DTM generalization..

Multi-Scale Representations

of Digital Terrain Models

The previous two chapters addressed the issues on the quality of DTMs The repre-sentations of DTM data will be discussed in the next three chapters This chapter will look at some issues on multi-scale representation

9.1 MULTI-SCALE REPRESENTATIONS OF DTM: AN OVERVIEW

Before multi-scale representations of DTM data can be discussed, it is essential to clarify some concepts related to scale, to understand what is to be addressed in this chapter

9.1.1 Scale as an Important Issue in Digital Terrain Modeling

“Scale is a confusing concept, often misunderstood, and meaning different things depending on the context and disciplinary perspective” (Quattrochi and Goodchild

1997, p 395) Scale is an old issue in geosciences such as cartography and geography

In cartography, maps are produced at different scales, for example, 1:10,000 and 1:100,000 A ground area with a fixed size will be mapped into a bigger map space at a larger scale For example, a ground area of 10 km× 10 km will be a map area of 1 m× 1 m at a scale of 1:10,000 However, at 1:100,000, it is an area

of only 0.1 m× 0.1 m on map Due to the reduction in size (from 1 m × 1 m to 0.1 m× 0.1 m), the same level of detail (LOD) as represented on the larger-scale map cannot be represented on the smaller-scale map This means that the representations

of the same feature in the same area will be different when the maps are at different scales Therefore, there is a multi-scale issue in cartography, that is, how to derive small-scale maps from large-scale maps through operations such as simplification

and aggregation This issue is called generalization.

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In geography, there is a similar issue Normally, geographical data are sampled from enumeration units The size of the enumeration unit is a scale indicator In some geographical applications, a larger unit is needed as the basis for analysis The sampled data need to be aggregated into a larger enumeration unit from the original unit However, the statistical results may be different when the analysis is carried out based on different sizes (i.e., different scales) of the enumeration unit Therefore, the issue is how to aggregate data from small enumeration units to larger units for processing This issue is called the “modifiable areal unit” issue (Openshaw 1994) Indeed, there are similar issues in all geosciences, such as geomorphology, oceano-graphy, soil science, biology, biophysics, social sciences, hydrology, environmental sciences, and so on

In digital terrain modeling, there is also a similar issue Currently, DTM data at

a national level are produced at various scales, for example, 1:10,000, 1:50,000, and 1:100,000, for different applications They are mainly derived from contour maps at the corresponding scales It would be desirable to maintain and update the DTM data

at the largest scale only and to derive DTM data at any smaller scale by a generalization process, that is, DTM generalization

In computer graphics and games, landscape is often used as a background in various situations such as driving and flight simulations In such cases, the LOD

of the terrain surface may appear to be different if viewed from different distances Therefore, in a scene, some parts may be represented in more detail and other parts in less detail This is also a multi-scale issue and is simply termed LOD, which is also

an important issue in DTM visualization

As will be discussed later, LOD is an issue quite different from generalization

In this chapter, the emphasis is on generalization although the basic ideas on LOD will also be outlined

9.1.2 Transformation in Scale: An Irreversible Process in

Geographical Space

Here, the term geographical space means the real world However, the term real world

is still confusing because different disciplines study different aspects of it Nuclear physics studies particles at the sub-molecular level in units of nanometers This is an extreme at a micro-scale At the other extreme, astro-physics studies the planets at an intergalactic level in units of light-years (the distance travelled by light in the period

of a year) Such studies are at a macro-scale Between these two extremes, many scientific disciplines study the planet earth, such as geology, geography, geomatics,

geomorphology, geophysics, which are collectively called geosciences Here, real world refers to the world studied by the geosciences Such studies are at a scale called geo-scale By an analogy to electromagnetic (EM) spectrum, the scale range, from micro-scale to geo-scale to macro-scale, is termed the scale spectrum(Figure 9.1) Like the visible light band in the EM spectrum, geo-scale is also a small band in the scale spectrum (Li 2003) Digital terrain modeling is a branch of geoscience The transformation in the scale of geographical space is much more complicated than that in Euclidean space In Euclidean space, any object has an integer dimension, that is, 0 dimension for a point, 1 for a line, 2 for a plane, and 3 for a volume

Geo-scale Macro-scale

Micro-scale

Astro-physics

Geosciences Meteorology

Biology Nuclear physics Space science

Figure 9.1 Geo-scale in the scale spectrum.

Scale 3 Scale 2

Scale 1

Figure 9.2 Scale change in 2-D Euclidean space: a reversible process.

An increase (or decrease) in scale will cause an increase or (or decrease) in length in

a 2-D space and in volume in a 3-D space However, the shape of the object remains unchanged Figure 9.2 illustrates the scale reduction in a 2-D Euclidean space Scale 2

is a two-time reduction of scale 1 and scale 3 is a four-time reduction of scale 1 In such

a transformation process, the length of the perimeter is reduced by two and four times, respectively, and the area of the object is reduced by 22and 42times, respectively When the object at scale 3 is increased four times, the shape of the object is identical

to the original one shown at scale 1 That is, the transformations are reversible However, in geographical space, the dimensions are not integers The concept of fractal dimensions was introduced by Mandelbrot (1967) In such a space, a value between 1.0 and 2.0 is the dimension of a line and a value between 2.0 and 3.0 is the dimension of a surface In fractal geographical space, it was discovered long ago that different lengths will be obtained for a coastline represented on maps at different scales The length measured from smaller-scale maps will be shorter if the same unit size (at map scale) is used for measurement This is because different levels of reality (i.e., the Earth’s surface with different degrees of abstraction) have been measured Indeed, on maps at a smaller scale, the level of complexity of an object is reduced

to suit the representation at that scale But when the representation at a smaller scale

is enlarged back to the original size, the level of complexity cannot be recovered

Scale 1 Scale 2 Scale 3

Figure 9.3 Scale change in 2-D geographical space: an irreversible process.

Figure 9.3 illustrates that such a transformation is not reversible Li (1996, 1999) regarded such kind of transformation as “transformation in scale dimension.” This means that the generalization of a DTM from a large-scale to a smaller-scale representation is an irreversible process It will be shown in Section 9.3 that this process follows a natural principle (Li and Openshaw 1993)

9.1.3 Scale, Resolution and Simplification of Representations

The size of the basic unit for measurement or representation is referred to as resolution.

If the data are in raster format, the size of the raster of pixels is referred to as the resolution The larger the pixel size, the lower the resolution In the case of grid DTM, the grid interval is usually regarded as the resolution

Normally, the scale and resolution of spatial data are tightly packed With the resolution of one’s eyes fixed, when one views an object more closely, the images formed in one’s eyes are larger, thus the images have a higher resolution or are at a larger scale That is, in normal cases, the resolution is also good indicator of scale

for DTM data This is because resolution means the level of detail and scale means the level of abstraction However, scale is not equal to resolution Scale could refer

to the ratio of distances as well as the relative size of interest.Figure 9.4shows four images at the same scale but with four different resolutions Similarly, digital maps can be plotted at any scale one wishes, but the resolution is fixed

With the introduction of resolution, it is easier to explain the difference between scale reduction in Euclidean and geographical spaces In Euclidean space, reduction in the size of an object does not cause a change in its complexity This can be understood with the following line of thought When the scale is reduced, the size of the object is also reduced, but at the same time, the basic resolution of the observation instrument

is also refined by the same magnification This is implied in Euclidean geometry For example, if the scale is reduced by two times, the size is reduced by two times and the resolution is two times finer (i.e., higher) On the other hand, in geographical space, when the scale is reduced, the size of an object is also reduced, but the basic resolution of the observation instrument remains unchanged That is, this change of complexity is achieved by changing the relationship between the size of the object and the resolution of observation

(a) (b)

Figure 9.4 Four images of Hong Kong with the same scale but different resolutions The

color plate can be viewed at http://www.crcpress.com/e_products/downloads/ download.asp?cat_no = TF1732.

Table 9.1 The Cause and Effect of Scale Reduction in Euclidean and Geographical Space

Relative Absolute Instrumentation Observer’s

Geographical space Unchanged Decreased Unchanged Unchanged

There are ways in which to achieve this result (Table 9.1) The first is to change the size of the object but, at the same time, to retain the basic resolution of the observational instrument The second is (a) to retain the size of the object but change the basic resolution of the observation instrument and then (b) to change the size of the observed objects by simple reduction in Euclidean space

9.1.4 Approaches for Multi-Scale Representations

It becomes clear that there are two different types of multi-scale representations The first one is map-like, emphasizing the metric quality, and is thus useful for measurement The multi-scale issue on the DTM is related to how to automatically

(a) (b)

Figure 9.5 Steps and linear slope compared to discrete and continuous transformations:

(a) discrete (steps) and (b) continuous (linear slope).

derive DTM data suitable for any smaller-scale representation from the DTM at the

largest scale, which is updated continuously Such a process is called generaliza-tion and it is applied uniformly across the whole area covered by the DTM data so

that all data points within the area will have a uniform accuracy This is referred

to as metric multi-scale representation here The other type, for visual impression only (e.g., for computer graphics and games), is called visual multi-scale repre-sentation On the same image, the scale of the image pixels produced from DTM

data is not the same over the whole image and is a function of viewing distance

In other words, the LOD represented on an image varies from place to place This

kind of approach is simply called LOD in computer graphics In some literature, it is also called view-dependent LOD and, in contrast, metric multi-scale representation

is called view-independent LOD.

There are also two types of transformations in scale, discrete and continu-ous transformations “Discrete” means that there are only a few scales available,

for example, 1:10,000, 1:100,000, and 1:1,000,000 The transformation jumps from 1:10,000 to 1:100,000, then to 1:1,000,000 Discrete transformation is like fixed steps

in a staircase (Figure 9.5a) “Continuous” means that transformation can be to any scale, for example, 1:50,000 or 1:56,999, although in practice some scales are not used (e.g., 1:56,999) Continuous transformation is like a linear slope (Figure 9.5b) From the above discussions, the approaches for multi-scale representation

of DTM data can be summarized as inFigure 9.6

9.2 HIERARCHICAL REPRESENTATION OF DTM

AT DISCRETE SCALES

The hierarchical representation seems to be popular for the representation of DTM data at discrete scales, particularly in computer graphics and games and terrain visu-alization, in order to speed up the data processing Both grid and triangular networks could be represented in hierarchy (de Berg and Dobrindt 1998; de Floriani 1989)

9.2.1 Pyramidal Structure for Hierarchical Representation

The pyramid is the most commonly used hierarchical representation of DTM data Figure 9.7shows three-level pyramid structures of square grid and triangular grids That is, four squares (or triangles) at the third level form a larger square quadrilateral

Isotropic

Continuous

View-dependent

Visual multi-scale representation (View-dependent LOD)

Metric multi-scale representation (Generalization, or isotropic LOD

or view-independent LOD)

Figure 9.6 Alternative approaches for multi-scale representation of DTM data.

Figure 9.7 Pyramid representations of grid and triangular networks.

(or triangle) at the second level Similarly, four squares (or triangles) at the second level form the largest square (or triangle) at the first level The number of squares (triangles) at thenth level is 4 n−1.

The sizes of the squares at the same level of the pyramid structure are identical Figure 9.8is an example of the grid pyramid representation of DTM data, where the original DTM is represented in three hierarchical levels In the four-to-one aggregation process, simple averaging is adopted to compute the height value of the new grids For example, if the heights of the four grid nodes at the fourth level are 5, 6, 4.5, and 5.5 m, then the average height value of these, that is, (5+6+4.5+5.5) m/4 = 5.25 m,

is used as the height for the new grid node at the third level

The hierarchical concept of a simple pyramid emphasizes the level of grid sizes, that is, different resolutions, to represent the terrain surface at different scales This

is also the simplest LOD for visualization of DTM data However, it does not take into consideration certain terrain features and thus usually produces relatively obvious visual distortions due to the loss of important surface characteristics and discontinuity

at the boundaries between grid cells

(a) (b)

Figure 9.8 Pyramid representation of grid DEM: (a) original DEM at 1:20,000; (b) the second

level; (c) the third level; and (d) the fourth level.

9.2.2 Quadtree Structure for Hierarchical Representation

A major shortcoming of the simple pyramid structure is that the grid intervals among the grid cells are identical at any hierarchical level irrespective of whether the ter-rain surface is complicated or simple This may cause problems if some parts of an area are complicated while other parts are simple In this case, a hierarchical struc-ture with varying grid sizes would be more desirable The complicated parts could

be represented with grid cells with finer resolution (i.e., smaller grid interval) and the simpler parts could be represented by grid cells with coarser resolution that is,

(larger grid interval) Area quadtree, or simply quadtree, is such a kind of grid in

common use.Figure 9.9is an illustration of quadtree structure Similarly, triangular cells also can be represented by quadtree.Figure 9.10is an example of triangular quadtree

The aggregation of four cells into one is similar to the simple pyramid The only difference is that in a quadtree, some criteria must be set so as to determine whether aggregation should take place for the four given cells For example, if the height differences are larger than the threshold, then no aggregation should take place, otherwise they are aggregated into one.Figure 9.11is an example of a hierarchical representation by a quadtree

In visualization with such a representation, when a more detailed level is desired, the next level with smaller grid intervals will be displayed, only in those parts with complicated terrain variations To speed up the visualization process, one would like

to generate an LOD for in as many levels as possible However, in normal practice, only three to five LODs are produced and stored Further levels are generated in real time by algorithms

Figure 9.9 Hierarchical representation of grid by quadtree structure.

Figure 9.10 Hierarchical description of terrain triangular network by use of quadtree.

Figure 9.11 Quadtree representation of DTM (Cheng 2000).

9.3 METRIC MULTI-SCALE REPRESENTATION OF DTM

AT CONTINUOUS SCALES: GENERALIZATION 9.3.1 Requirements for Metric Multi-Scale Representation of DTM

Since the later 1970s, metric multi-scale representation of DTM data has been a research topic A list of six criteria were proposed by Weibel (1987) for the evaluation

of the methodology used:

1 to run as automatically as possible

2 to perform a broad range of scale changes

3 to be adaptable to the given relief characters

4 to work directly on the basis of the DTM

5 to enable an analysis of the results

6 to provide the opportunity for feature displacement based on the recognition of the major topographic features and individual landforms (major scale reduction)

Three approaches have been proposed for metric multi-scale representation of DTM data

1 filtering methods (e.g., Loon 1978)

2 generalization of structure lines (e.g., Wu 1997)

3 a hybrid (Weibel 1987) of the above two

However, these methods do not directly relate to the filtering process to scales and, therefore, are not true generalization methods

If the set of criteria proposed by Weibel (1987) are used to evaluate these pyramidal representations, the results are not too good The most serious shortcom-ing is that only a fixed number of scales are produced, at least for hierarchical grid networks, that is, 2-time, 4-time, 8-time, 16-time, scale reduction Therefore,

this approach is only convenient for data structures with which visualization of DTMs could be speeded up, but it cannot be used to produce smaller-scale DTMs from larger-scale ones In this section, an approach based on a natural principle will be introduced

9.3.2 A Natural Principle for DTM Generalization

The natural principle formalized by Li and Openshaw (1993) mimics the general-ization process of the Nature The example they used is the Earth’s surface viewed from different heights If one views the Earth from the Moon, all terrain variations disappear and the Earth appears like a blue ball If one views it from a satellite, then the terrain surface becomes very smooth These phenomena can easily be checked

by forming a stereo model from a pair of satellite images such as SPOT images with 10-m resolution or Spacelab Metric Camera photography Such a stereo model

is at a very small scale When one views the terrain surface from an airplane, small details are still not visible but the main characteristics of the terrain variations become clear It is a commonplace to photogrammetrists that stereo models formed from