Manual of Digital Earth pp 279-324 | Cite as

# Transformation in Scale for Continuous Zooming

## Abstract

This chapter summarizes the theories and methods in continuous zooming for Digital Earth. It introduces the basic concepts of and issues in continuous zooming and transformation in scale (or multiscale transformation). It presents the theories of transformation in scale, including the concepts of multiscale versus variable scale, transformation in the Euclidean space versus the geographical space, and the theoretical foundation for transformation in scale, the Natural Principle. It addresses models for transformations in scale, including space-primary hierarchical models, feature-primary hierarchical models, models of transformation in scale for irregular triangulation networks, and the models for geometric transformation of map data. It also discusses the mathematical solutions to transformations in scale (including upscaling and downscaling) for both raster (numerical and categorical data) and vector (point set data, line data set and area data) data. In addition, some concluding remarks are provided.

## Keywords

Continuous zooming Transformation in scale Natural principle Multiscale Variable scale## 8.1 Continuous Zooming and Transformation in Scale: An Introduction

### 8.1.1 Continuous Zooming: Foundation of the Digital Earth

Imagine, for example, a young child going to a Digital Earth exhibit at a local museum. After donning a head-mounted display, she sees Earth as it appears from space. Using a data glove, she zooms in, using higher and higher levels of resolution, to see continents, then regions, countries, cities, and finally individual houses, trees, and other natural and man-made objects.

The cascade scene seen by the young child is a result of continuous zooming. Such zooming can be realized by continuously displaying a series of Earth images taken at a given position and changing the focal length of the camera lens continuously or displaying images taken at different heights continuously but with at a fixed camera focal length.

In theory, to make the display visually smooth, the differences between two images should be sufficiently small, thus the number of images in such a series is very large, which demands huge data storage. Thus, it is a very difficult, if not impossible, problem.

### 8.1.2 Transformation in Scale: Foundation of Continuous Zooming

*multiscale representation*. Figure 8.1 shows a series of satellite images covering Hong Kong Polytechnic University at six different scales, extracted from Google Maps. If such images at discrete scales are displayed in sequence, there will be a visual jump between two images. The obviousness of the visual jump is dependent on the magnitude of the scale difference. The smaller the difference between the two scales is, the less apparent the visual jump will be.

To minimize the effect of such visual jumps, some techniques are required to smooth the transformations from one scale to another scale to make the display appear like continuous zooming. This transformation in scale is the foundation of continuous zooming. Thus, transformation in scale, also called multiscale transformation, is the topic of this chapter.

### 8.1.3 Transformation in Scale: A Fundamental Issue in Disciplines Related to Digital Earth

Transformation in scale is one of the most important but unsolved issues in various disciplines related to Digital Earth, such as mapping, geography, geomorphology, oceanography, soil science, social sciences, hydrology, environmental sciences and urban studies. Typical examples are map generalization and the modifiable areal unit problem (MAUP). Although transformation in scale is a traditional topic, it has been a critical issue in this digital era.

Transformation in scale has attracted attention from disciplines related to Digital Earth since the 1980s because a few important publications on the scale issue in that period awakened researchers in relevant areas. Openshaw (1984) revisited the *MAUP*. Abler (1987) reported that multiscale representation was identified as one of the initiatives of the National Center for Geographic Information and Analysis (NCGIA), and noted that zooming and overlay are the two most exciting functions in a geographical information system. Since then, the scale issue has been included in many research agendas (e.g., Rhind 1988; UCGIS 2006) and has become popular in the geo-information community.

The first paper on the scale issue in remote sensing was also published in 1987 (Woodcock and Strahler 1987). Later, in 1993, the issue of scaling from point to regional- or global-scale estimates of the surface energy fluxes attracted great attention at the Workshop on Thermal Remote Sensing held at La Londe les Maures, France from September 20–24. Scale became a hot topic in remote sensing as well.

As a result, many papers on the scale issue have been published in academic journals and at conferences related to Digital Earth. Other papers have been p in the form of edited books, such as *Scaling Up in Hydrology Using Remote Sensing* edited by Stewart et al. (1996), *Scale in Remote Sensing and GIS* edited by Quattrochi and Goodchild (1997), *Scale Dependence and Scale Invariance in Hydrology* edited by Sposito (1998), *Modelling Scale in Geographical Information Science* edited by Tate and Atkinson (2001), *Scale and Geographic Inquiry: Nature, Society and Method* edited by Sheppard and McMaster (2004), *Generalisation of Geographic Information: Cartographic Modelling and Applications* edited by Mackaness et al. (2007), and *Scale Issues in Remote sensing* edited by Weng (2014). Authored research monographs have also been published by researchers, e.g., *Algorithmic Foundation of Multi*-*Scale Spatial Representation* by Li (2007) and *Integrating Scale in Remote Sensing and GIS* by Zhang et al. (2017).

## 8.2 Theories of Transformation in Scale

Transformation in scale is the modeling of spatial data or spatial representations from one scale to another by employing mathematical models and/or algorithms developed based on certain scaling theories and/or principles. This section describes such scaling theories and/or principles.

### 8.2.1 Transformation in Scale: Multiscale Versus Variable Scale

*cartographic ratio*. Similarly, image data and digital elevation models (DEMs) are also produced and stored at discrete scales. In these two cases, the scale is normally indicated by

*resolution*.

### 8.2.2 Transformations in Scale: Euclidean Versus Geographical Space

*Euclidean space*, an increase in scale will commonly cause an increase in length, area and volume; and a decrease in scale will cause a decrease in length, area and volume, accordingly. Figure 8.4 shows an example of scale reduction and increase in a 2D Euclidean space. In such a transformation in scale, the absolute complexity of a feature or features remains unchanged. That is, the transformations are reversible.

However, the geographical space *is fractal*. If one measures a coastal line using different measurement units, then different lengths will be obtained. The smaller the measurement unit is, the longer the length obtained. Similarly, different length values will be obtained when measuring a coastal line represented on maps at different scales using identical measurement units at map scale. That is, the transformation in scale in fractal geographical space is quite different from that in Euclidean space.

The transformation in scale is also termed *scaling*. The process of making the resolution coarser (or making the map scale smaller) is called *upscaling*. In contrast, the transformation process to make the resolution finer (or map scale larger) is called *downscaling*.

### 8.2.3 Theoretical Foundation for Transformation in Scale: The Natural Principle

One question that arises is “does such a transformation follow any principle or law?” The answer is “yes”. Li and Openshaw (1993) formulated the *Natural Principle* for such a transformation in scale in fractal geographical space.

*Natural Principle*, as follows:

When one views the terrain surface from the Moon, all terrain variations disappear, and one can only see a blue ball;

When one views the terrain surface from a satellite, then the terrain surface becomes visible, but the terrain surface looks very smooth;

When one views the terrain surface from an airplane, the main characteristics of the terrain variations become very clear, but small details do not appear; and

When one views the terrain surface from a position on ground, the main characteristics of the terrain variations become lost, and one sees small details.

*Natural Principle*as termed by Li and Openshaw (1993). It can be stated as follows:

For a given scale of interest, all details about the spatial variations of geographical objects (features) beyond a certain limitation cannot be presented and can thus be neglected.

By using a criterion similar to the limitation of human eyes’ resolution, and, neglecting all the information about the spatial variation of spatial objects (features) beyond this limitation, zooming (or generalization) effects can be achieved.

*k*is the SVS value in terms of map distance at the target scale and

*K*is the SVS value in terms of ground distance at the target scale. Through intensive experimental testing, Li and Openshaw (1992) recommend a

*k*value between 0.5 and 0.7 mm, i.e.,

## 8.3 Models for Transformations in Scale

To realize a transformation in scale, some transformation models must be adopted and algorithms and/or mathematical functions for these models are applied. The former is the topic of this section and the latter are described in Sect. 8.4.

### 8.3.1 Data Models for Feature Representation: Space-Primary Versus Feature-Primary

To record features in geographical space, two different viewpoints can be taken: feature-primary and space-primary (Lee et al. 2000).

*vector data model*. Figure 8.8a–c show the representation of points, a line and an area using a vector model.

In a space-primary view, the geographical space is considered as being tessellated by space cells. In such a tessellation (partitioning), square raster cells are popularly employed, leading to the popular term *raster data model*. In each raster cell, there could be a feature or there might be no features. A point is represented by a pixel (picture element); a line is represented by a string of connected pixels and an area is formed by a set of connected pixels, as shown in Fig. 8.8d–f. The cells can be in any form, regular or irregular. Irregular triangular networks are another popular tessellation.

As the natures of the raster and vector data models are quite different, the model for transformation in scale in these two data models might also differ. Thus, separate subsections are devoted to these topics.

### 8.3.2 Space-Primary Hierarchical Models for Transformation in Scale

A more general form of transformation to create a hierarchical representation is to transform N × N pixels into M × M pixels, e.g., a 5 × 5 into a 2 × 2 or a 3 × 3 into a 2 × 2. In such cases, a resampling process (instead of simple aggregation) is required.

For hierarchical representation on a spherical surface, the Open Geospatial Consortium (OGC) approved a new standard called the Discrete Global Grid System (DGGS) (OGC 2019) The hierarchical representation of QTM as shown in Fig. 8.9b is an example of such a DGGS.

### 8.3.3 Feature-Primary Hierarchical Models for Transformation in Scale

### 8.3.4 Models of Transformation in Scale for Irregular Triangulation Networks

*Vertex removal*: A vertex in the triangular network is removed and new triangles are formed.*Triangle removal*: A complete triangle with three vertices is removed and new triangles are formed.*Edge collapse*: An edge with two vertices is collapsed to a point and new triangles are formed.*Triangle collapse*: A complete triangle with three vertices is collapsed to a point and new triangles are formed.

### 8.3.5 Models for Geometric Transformation of Map Data in Scale

Models for geometric transformations in scale of individual point features (Li 2007)

Transformation model | Large-scale | Photo-reduced | Small-scale |
---|---|---|---|

(move because it is too close to another feature) | |||

(too small to represent, thus removed) | |||

(enlarged due to importance) |

Models for geometric transformations in scale of a set of point features (Li 2007)

Transformation model | Large-scale | Photo-reduced | Small-scale |
---|---|---|---|

(group points and make a new one) | |||

(delineate a boundary outlined by points and make a new area feature) | |||

(retain more important points and omit less important ones) | |||

(cluster complexity; the main structure is retained) | |||

(typical pattern kept while points removed for clarity) |

Models for geometric transformations in scale of individual line features (Li 2007)

Transformation model | Large-scale | Photo-reduced | Small-scale | |
---|---|---|---|---|

(to move a line away from the position because it is too close to another feature) | ||||

(to remove the line because it is too minor to be included) | ||||

(main structure suitable at target scale retained but small details removed) | ||||

(to modify the shape of a segment within a line) | ||||

(to reduce the number of points by removing less important points) | ||||

(to make the data appear smoother) |
(to fit a curve through a set of points) | |||

(to filter out the high-frequency components or small details of a line) | ||||

(typical patterns of the line bends retained while removing some of them) |

Models for geometric transformations in scale of a set of line features (Li 2007)

Transformation model | Large-scale | Photo-reduced | Small-scale | |
---|---|---|---|---|

(to select more important points and remove less important points) | ||||

(to reduce the dimension) | Ring-to-point | |||

Double-to-single | ||||

(to keep the characteristics clear) | ||||

(to combine to two or more close lines together) | ||||

(to move one away from others or both away from each other) |

Models for geometric transformations in scale of individual area features (Li 2007)

Transformation model | Large-scale | Photo-reduced | Small-scale | |
---|---|---|---|---|

(to reduce the dimension of features) | Area-to-point | |||

Area-to-line | ||||

Partial | ||||

(to move the area to a slightly different position to solve the conflict problem) | ||||

(to enlarge one or two dimensions of a small area) |
(to enlarge an area feature in a direction) | |||

(to uniformly magnify in all directions) | ||||

(to widen the bottleneck of an area feature) | ||||

(to eliminate data that is too small to represent | ||||

(to reduce the complexity of a boundary) | ||||

(to split an area into two because the connection between them is too narrow) |

Models for geometric transformations in scale of a set of area features (Li 2007)

Transformation model | Large-scale | Photo-reduced | Small-scale |
---|---|---|---|

(to combine area features, e.g., buildings separated by open space) | |||

(to make area features bounded by thin area features into adjacent area features) | |||

(to combine area features, e.g., buildings separated by another feature such as roads) | |||

(to split a small area into pieces and merge these pieces into adjacent areas) | |||

(to combine two adjacent areas into one) | |||

(to move more than one feature around to solve the crowding problem) | |||

(to retain the structure of area patches by selecting important ones) | |||

(to retain the typical pattern, e.g., a group of areas aligned in rows and columns) |

### 8.3.6 Models for Transformation in Scale of 3D City Representations

LOD 0—regional, landscape

LOD 1—city, region

LOD 2—city districts, projects

LOD 3—architectural models (outside), landmarks

LOD 4—architectural models (interior)

Models for transformation in scale of 3D features

Transformation model | At large scale | Photo-reduced | At small scale | |

Elimination | Geometric elimination | |||

Thematic elimination | ||||

Exaggeration | Thematic exaggeration | |||

Geometric exaggeration | ||||

Simplification | Vertical simplification | |||

Flattening | ||||

Squaring | ||||

Thematic simplification | ||||

Displacement | ||||

Typification |

## 8.4 Mathematical Solutions for Transformations in Scale

In the previous section, several sets of models for the transformation in scale were described. These models express what is achieved in such transformations, e.g., the shape is simplified, important points retained, and/or the main structure is preserved. To make these transformations work, mathematical solutions (e.g., algorithms and mathematical functions) must be developed for each of these transformations. A selection of these solutions is presented in this section.

### 8.4.1 Mathematical Solutions for Upscaling Raster Data: Numerical and Categorical

**raster-based numerical data**such as images and digital terrain models (DTMs), aggregation is widely used to generate hierarchical models. In recent years, wavelet transform (e.g., Mallat 1989), Laplacian transform (Burt and Adelson 1983) and other more advanced mathematical solutions have also been employed. The commonly used aggregation methods are by mode, by median, by average, and by Nth cell (i.e., Nth cell in both the row and column). Figure 8.22 shows a “3 × 3 to 1 × 1” aggregation with these four methods. The 6 × 6 grid is then aggregated into a 2 × 2 grid.

Once the coefficients \( a_{0} \text{,}\,a_{1} \text{,}\;a_{2} ,\;a_{3} \) are computed, the height \( {\text{Z}}_{P} \) of any point *P* with a given set of coordinates \( ({\text{x}}_{P} ,y_{P)} \) can be obtained by substituting \( ({\text{x}}_{P} ,y_{P)} \) into Eq. (8.1).

*d*is the distance from a reference point to the interpolation point. In the case of interpolating the height of P in Fig. 8.23b, the four distances from the four (old) cell centers to point P will be used. Figure 8.23b also shows that the distance of each cell center to the interpolation point P is directly related to the size of the area contributed by each (old) cell to the new cell. If the area size is denoted as A, the weighting function is

**raster-based categorical data**, the averaging and median are no longer applicable. The mode (also called the majority in some literature) is still valid and widely used. Figure 8.24b shows such a result. However, the value for the upper right cell is difficult to determine as there is no mode (majority) in the 3 × 3 window at the upper right corner of the original data (Fig. 8.24a). Notably, some priority rules or orders are in practical use. For example, a river feature is usually given a priority because thin rivers are likely to be broken after aggregation. Figure 8.25 shows the improvement in the connectivity of river pixels with water as the priority. Figure 8.24c-e show the results with different options, e.g., random selection and central pixel. It is also possible to consider the statistical distribution of the original data (e.g., A = 8, T = 10, W = 6, S = 11) to try to maintain the distribution as much as possible.

*Natural Principle*(Li and Openshaw 1993) described in Sect. 8.2.3. Mathematically,

*K*is the SVS value in terms of ground distance at the target scale computed by Eq. (8.1), and W is the size of the window’s side in terms of pixel numbers (of input data).

### 8.4.2 Mathematical Solutions for Downscaling Raster Data

Downscaling produces a finer spatial resolution raster data than that of the input data through prediction. It is possible to use simple resampling (as described in Sect. 8.4.1) to achieve downscaling. However, methods based on spatial statistical analysis are more theoretically grounded and have become popular (Atkinson 2008, 2013), particularly area-to-point prediction (ATPP). Double dictionary learning has also been used (Xu and Huang 2014).

Area-to-point kriging (ATP Kriging or ATPK) (Kyriakidis 2004) is the typical method. ATP Kriging can ensure the coherence of predictions, such as by ensuring that the sum of the downscaled predictions within any given area are equal to the original aggregated count. Some variants of ATP Kriging have also been developed, e.g., ATP Poisson Kriging (Goovaerts 2008, 2009, 2010), indicator cokriging (Boucher and Kyriakidis 2006) and ATP regression Kriging (Wang et al. 2015). In this section, the base version of ATP Kriging is described.

*d*apart, which can be expressed as follows:

*A.*

### 8.4.3 Mathematical Solutions for Transformation (in Scale) of Point Set Data

As discussed in Sect. 8.3.5, a number of transformations are possible, such as regionalization, aggregation, selective omission, structural simplification, and typification. In both aggregation and regionalization, the clustering plays a central role. In aggregation, a cluster is represented by a point; in regionalization, a cluster is represented by an area. Thus, clustering is discussed here.

*K*clusters (i.e., the objective function) as follows:

- (1)
arbitrarily select K points from data set (X) as initial cluster centroids;

- (2)
assign each point in X to the cluster whose centroid is closest to the point;

- (3)
compute the new cluster centroid for each cluster; and

- (4)
repeat Steps (2) and (3) until no change can be made.

*f*is the clustering model; A is the analysis scale (the size of clusters or the degree of homogeneity within clusters); and D is the data scale (e.g., resolution and extent).

- (1)
Control the data scale: Determine the SVS (smallest visible size) based on input and output data scales and following the Natural Principle, and ignore all the points within an SVS in the calculation of point data density.

- (2)
Identify high-density points: The probability density function (PDF) of the dataset is estimated with adaptive analysis scales. The PDF are statistically tested against a null distribution. Points with a significantly higher density are then identified.

- (3)
Group the high-density points into clusters: Clusters with different densities are formed by adaptively breaking the long edges in the triangulation of high-density points. The significance of clusters obtained at multiscales can be statistically evaluated.

### 8.4.4 Mathematical Solution for Transformation (in Scale) of Individual Lines

As discussed in Sect. 8.3.5, there are eight different types of transformation for individual lines and the algorithms/mathematical solutions for the transformation models are discussed in detail by Li (2007). In this section, two classic algorithms are described in detail, i.e., the Douglas–Peucker algorithm (Douglas and Peucker 1973) and the Li–Openshaw algorithm (Li and Openshaw 1992).

The algorithm first selects two end points (i.e., the first and last points). It then searches for the point that has the largest deviation from the straight-line segment connecting these two end points, i.e., at point 2 in Fig. 8.27. If the deviation is larger than \( \varepsilon \), then this point is selected; otherwise, all other points can be ignored. In this example, point 2 is selected and it splits the line into two pieces. The search is then carried out for both pieces. Then, points (3, 1) and (3, 2) are selected. These two points split the whole line into four pieces, and the search will be carried out for these four pieces. The process continues until all the deviations are smaller than \( \varepsilon \).

*effective area*of a point as the threshold, as illustrated in Fig. 8.28. For example, the effective area of point 2 is the area covered by the triangle formed by points 1, 2 and 3. The basic idea of this algorithm is to progressively eliminate the point with smallest effective area from the list, and the effective areas of the two points adjoining the recently deleted point should be immediately updated. In this example, point 11 is first eliminated and point 13 is removed. The points are ranked from least to most important according to the sequence of elimination.

*Natural Principle*(Li and Openshaw 1993) described in Sect. 8.2.3, i.e., to neglect all spatial variations within the SVS that is computed by using input and output scales. The SVS is mimicked by a cell or pixel although other geometric elements are also possible (e.g., hexagon by Raposo in 2013). The cells can be organized in the form of a none overlapped tessellation or with overlaps. If there is no overlap, it becomes a pure raster template. Figure 8.30 shows the generalization (transformation) process with a raster template. In this example, each SVS is represented by a raster pixel and the result is represented by pixels, as shown in Fig. 8.30b, or by its geometric center.

Similar to the algorithm in raster mode, overlap between SVSs can also be adopted, although it is not too critical. Notably, it is not necessary to take the average to represent a cell. It does not matter what point within the cell is used, as the cell itself is an SVS. Thus, it is also possible to take an original point, which is considered a critical point to represent the cell.

### 8.4.5 Mathematical Solutions for Transformation (in Scale) of Line Networks

In geographical space, three types of line networks are commonly used, contour line networks, hydrological networks and transportation networks. Some hierarchical models were presented in Sect. 8.3.3. The mathematical solutions for the transformation in scale of these networks are discussed in detail by Li (2007). Here, only the construction of a hierarchy for transportation networks is described.

*importance of roads*. As road networks are stored in segments and intersections in a database, two steps are required, to build strokes and to order strokes, as illustrated in Fig. 8.33. To build strokes means to concatenate continuous and smooth network segments (see Fig. 8.33a) into a whole (see Fig. 8.33b). To order strokes means to rank the strokes in a descending order based on their importance from high to low (see Fig. 8.33b). The importance of each stroke can be calculated according to various properties, i.e., geometric properties such as length (Chaudhry and Mackaness 2005), topological properties such as degree, closeness and/or betweenness (Jiang and Claramunt 2004), and thematic properties such as road class. A comparative analysis of the methodology for building strokes was carried by Zhou and Li (2012). With each stroke, given an importance, a stroke-based hierarchy of a line network can be built.

*p*

_{ij}) from node

*i*to any of its immediate neighbor nodes can be defined as the reciprocal of the degree of connectivity (

*k)*of node

*i*. Mathematically,

For instance, in Fig. 8.34a, the ego is connected to both alter1 and alter2, so its degree of connectivity is 2; thus, the strengths of links from this ego to alter1 and to alter2 are both 1/2 = 0.5. The strengths of other links are also indicated in Fig. 8.34.

*i*and node

*j*are not directly linked but are linked via another node

*q*in the neighbor (

*ne*), the strength of the link from node

*i*to node

*j (i.e., p*

_{ij}) is defined as:

*C*

_{ij}) from node

*i*to node

*j*is defined as the square of the sum of the direct link strength and the indirect link strength from node

*i*to node

*j*. Mathematically,

The *C*_{ij} value reveals the constraint of *i* by j. The larger the C value is, the larger the constraint over *i*, and the smaller the opportunity for *i*.

The ego-network is a feasible and effective solution for the formation of hierarchies for road networks. However, Zhang and Li (2011) identified two significant limitations, the deviation of the link intensity definition from reality and the so-called ‘degree 1 effect’. They subsequently developed a weighted ego-network analysis method.

*mesh density*-

*based approach*proposed by Chen et al. (2009). The so-called mesh is a closed region surrounded by several road segments. In this approach, the density of each mesh in the road network is computed according to the following formula:

Generally, a road network is often a hybrid of linear and areal patterns, thus Li and Zhou (2012) proposed the construction of hybrid hierarchies, i.e., an integration of a line hierarchy and an area hierarchy.

### 8.4.6 Mathematical Solutions for Transformation of a Class of Area Features

Li (1994) argued that the transformation in scale should be better performed in raster space (because a scale reduction causes a space reduction and the raster format takes care of space) and proposed the use of techniques in mathematical morphology for transformation in scale. Li et al. have developed a complete set of algorithms for such transformations based on mathematical morphology.

_{1}and B

_{2}are the two structuring elements.

_{1}and B

_{2}. Su et al. (1997) suggest that the size of B

_{1}and B

_{2}should be determined by the input and output scales, following the Natural Principle described in Sect. 8.2.3. Figure 8.39 shows the combination of buildings using this model for two different scales: one for a scale reduction by 7 times and the other by 10 times. The results are also compared with those using simple photoreduction. The combined results are very reasonable. However, the combined results are very irregular and the simplification of boundaries could be discussed. A detailed description of such a simplification is omitted here but can be found in the work of Su et al. (1997) and the book by Li (2007). The result is shown in Fig. 8.40.

### 8.4.7 Mathematical Solutions for Transformation (in Scale) of Spherical and 3D Features

In the previous sections, mathematical solutions for transformation of 2D features have been presented. Mathematical solutions for transformation of spherical (e.g., Dutton 1999) and 3D features (e.g., Anders 2005) have also been researched, although the body of literature is much smaller than that for map generalization. In recent years, there have been more papers on the generalization of buildings-based CityGML (e.g., Fan and Meng 2012, Uyar and Ulugtekin 2017); details on such methodologies are omitted here due to page limitations.

## 8.5 Transformation in Scale: Final Remarks

The beginning of this chapter emphasized that continuous zooming is at the core of Digital Earth as initiated by Al Gore. Continuous zooming is a kind of transformation of spatial representation in scale. In this chapter, the theoretical foundation for transformations in scale was presented in Sect. 8.2. Then, models for such transformations were described in Sect. 8.3 for raster and vector data, images, digital terrain models and map data. A selection of algorithms and/or mathematical functions for achieving these transformations was presented in Sect. 8.4.

Notably, the content of this chapter was concentrated on the theories and methodology to achieve continuous zooming and some important issues related to transformation in scale have been omitted, such as temporal scale, scale effect and optimum scale selection. For the content of the models for transformation in scale, emphasis was on the representations. Thus, other models such as geographical and environmental processes were excluded. However, these aspects are important but were omitted due to page limitations.

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