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Copy of paper presented at the International Cartographic Associatation Annual Conference, Ottawa 1999.
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Full paper (Word 97 doc.) | Full paper (Zipped Word 97 doc.) | LandSerf software |
This paper examines some of the techniques available for the visualisation of surface models of terrain. In particular it concentrates on those visualisation processes that involve examining and understanding scale dependency within a landscape. Scale dependency is defined in this context as the characteristics of a point on a terrain model that vary when measured over different spatial extents. Scale dependency is an important property of a surface as it allows us to identify critical points on a landscape and critical scales of analysis. Identifying scale dependency is an important part of surface generalisation, which in turn is necessary for efficient surface visualisation.
Two methods of surface visualisation are implemented using the Java programming language and the OpenGL graphics library. The first involves creating a '3-dimensional' perspective rendering of the terrain model which can be explored by 'flying' over the surface in real time. This process exploits our cognitive ability to assimilate visual information at a variety of scales simultaneously. Near, foreground, large scale detail can be represented at the same time as far, background, small scale generalisation. By 'flying' over a landscape, the regions of the terrain represented at large and small scales can be changed continuously. By allowing an observer to change viewpoint interactively, the spatial organisation of scale dependency can be explored.
The second method of visualisation involves calculating morphometric measurements over a range of grid resolutions. Users of the system can interactively interrogate parts of the landscape and are presented with graphical summaries of how these measurements vary with grid resolution. Measurements vary from simple surface properties such as slope and curvature, to higher order structural partitions of the landscape such as drainage basins and ridge networks.
The approach described here is new in that it makes the visualisation process central to terrain analysis, and considers analysis at scales other than that imposed by the grid resolution of the underlying elevation model. The approach is accessible since it implements the visualisation on low-cost PC platforms using freely available software.
The visual representation of landscape form has long history dating at least as far back as the images of mountains scratched onto earthenware in Mesopotamia over 4000 years ago [Imhoff, 1982]. The subsequent development of cartographic terrain representation can be seen as a struggle to symbolise multiple perspectives of 3-dimensional surface form in a (usually) static 2-dimensional medium. Solutions have included early mimetic symbolisations of relief (e.g. Abraham Ortelius' 1585 map of Iceland); the use of hachuring to symbolise lines of steepest decent (e.g. Ordnance Survey of Great Britain's first series topographic maps); the widespread use of contour lines to represent slope normals, and the more recent use of automated shaded relief calculation from Digital Elevation Models e.g. [Yoeli, 1983]. A problem faced when interpreting all of these representations is that surface form varies with both scale and perspective. The apparent shape of a patch of land will look different depending on the relative distance and direction of the observer.
Several solutions have been adopted to address the inherent subjectivity of relief representation. The historical transition from mimetic to abstract symbolisation (from pictures of hills, to hachures to contour lines) can be seen as a move from the personally subjective to the more universally objective. Other solutions include incorporation of multiple perspectives within the same map through the use of insets or non-linear projections (e.g. Wainwright's guides to the Lakeland Fells [Wainwright, 1992] show mountain paths from both a bird's-eye and oblique perspective view simultaneously). Alternatively, the scope and perspective of a map's relief representation may be made specific and explicit, such as the cartographic conventions adopted in geomorphological mapping e.g. [Evans, 1990].
Yet all of these approaches are limited by the inevitable fact that they represent surface form at a relatively fixed level of detail. A cartographic representation of a steep slope may symbolise that slope over a distance of 100m on the ground, but it is unlikely to also represent the slope over a range of 10cm or 10km. This is of course, a specific case of a problem faced by all cartographic representations. It could be argued that by stating the scale (ratio) of a map, the reader is able to infer the implied level of detail. However, the representation of relief using computers presents particular problems since the notion of a scale ratio is somewhat ambiguous in a dynamic on-screen environment. This is compounded further by the fact that surface features tend not to have clearly demarcated scales at which they are expressed.
The argument presented in this paper therefore is that effective representation of relief must allow scale-dependencies in the landscape to be visualised. If the character of the landscape varies at different levels of detail, its visualisation must reflect such change.
We shall define scale dependency in this context as the notion that the characteristics of a point on a surface vary when measured over different spatial extents or different levels of detail. We will also consider the more neutral term scale tendency which describes the effects that may or may not present when measuring a property at different scales [Goodchild and Quattrochi, 1997]. Although related to the idea of map scale, scale dependent properties are a function of two separate characteristics - measurement resolution and measurement extent. We define resolution as the smallest spatial extent on the ground that may be measured and modelled as part of a visualisation, and extent as the area on the ground used to measure a surface property. The idea of scale dependency is an important one as it allows us to identify critical scales of analysis and representation (such as form-process relationships). It is also an important part of the surface generalisation processes such as resampling, aggregation and TIN vertex selection which, is in turn necessary for efficient surface visualisation.
Within geomorphology there is plenty of evidence that surface form exhibits scale dependency. Many of the empirically derived quantitative relationships identified in the 1950s and 1960s were in the form of non-linear scalings of surface form (e.g. Drainage Area-Mainstream length [Hack, 1957], Alluvial fan Area - Drainage basin area [Bull, 1964]). Such relations have been termed allometric relations [Bull, 1975] and have been placed in a wider quantitative framework by Church and Mark [Church and Mark, 1980].
The scale dependencies observed above are largely a response to spatial extent rather than spatial resolution. The development of the theory fractal geometry [e.g. Mandelbrot, 1977, Goodchild, 1980] suggest that resolution has an influence over measured properties, even those generated by scale independent process. The more recent development of multifractals (e.g. [Schertzer and Lovejoy, 1994]) suggest that even the fractal dimension, which predicts how measurement changes with resolution, itself varies with scale [Pecknold et al, 1997]. This is supported by geomorphological investigation that suggests that a unifractal model of most landscapes is inappropriate (e.g. [Evans and McClean, 1995]).
Further evidence of the importance of considering scale dependency is provided by those who have examined the effect of resolution on computer generated measurements of surfaces. Hodgeson [Hodgeson, 1995] shows that slope and aspect measured from a raster Digital Elevation Model actually describe the (different) slope and aspect properties at a resolution 1.6 - 2.0 times the DEM resolution. Chang and Tsai [Chang and Tsai, 1991] demonstrated that where the underlying shape of a surface varied at a resolution significantly finer than a DEM, results measured from the DEM would show statistical bias in favour of lower slope values. The nature and quality of surface features measured from DEMs have also been shown to be dependent on the underlying DEM resolution [Wood, 1998; Martz and Garbrecht, 1998].
Together, the evidence presented suggests that we ignore scale-dependency at our peril. This applies as much to the visualisation of surfaces as it does to their analysis. This is particularly the case when visualisation is used to generate ideas for hypothesis testing [McKim, 1972]. Two solutions are presented here to the problem of visualising surface properties at multiple resolutions. The first considers how multiple 'immersive' viewpoints may be generated using interactive perspective rendering. The second considers the idea of dynamically linked windows to show scale tendency explicitly.
The historical transition from mimetic to symbolic representation of landscape in maps can be seen as a move towards a more universal objective representation. This has obvious advantages in terms of the applicability of the visualisation in a variety of contexts. For example, a contour map effectively provides a 'reference source' of implied spot heights on a landscape The information content is much higher than an equivalent hill-shaded representation. However, in losing the subjective viewpoint we lose the sense of immersion within a landscape that can be vital in the generation of ideas. Importantly, we also lose much of the multi-scale representation inherent in subjective symbolisation.
Consider the problem of visually representing the slope at an arbitrary point on a landscape. The case is illustrated with a (random) fractal landscape in Figures 1 and 2. The slope may be inferred in only the most general terms from Figure 1a. The colour of any point gives an indication of elevation, and the rate of change in colour over the image gives some idea of slope. We might infer from this image that the slope surrounding the peak towards the right of the image is somewhat steeper than that of the (green) peninsula towards the centre. An alternative view of the same surface is given in Figure 1b, which combines the elevation coded hue with shaded relief using simple Lambertian lighting model [Foley et al, 1993]. Here shading is calculated based on the surface normal calculated by comparing each raster cell with the elevation of its 8 neighbours. We now get a better indication of locally steep slope at the cost of the broader-scale picture. No longer is it clear that the slope surrounding the peak to right is any steeper than that of the peninsular. Indeed, it would appear as though the peninsula is the roughest part of the surface exhibiting the steepest slope. Finally, Figure 1c shows slope calculated analytically and coloured as indicated by the key. As with the shaded relief, slope is calculated locally on a cell-by-cell basis resulting in a picture of local surface variation at the expense of a more regional view.
Figure 1 Three two dimensional representations of a fractal surface (see text for discussion).
Figure 2 Three three-dimensional perspective representations of a fractal surface (see text for discussion).
Figure 2 shows the same surface as represented in Figure 1c but using a 3-dimensional perspective projection. The images show snapshots of an interactive 'fly-through' over the surface where the viewer is immersed within the viewing space itself. Here it is possible to gain both a detailed 'large-scale' view of the surface in the foreground simultaneously with a generalised 'small-scale' view of the background. By allowing the user to control the imaginary camera position interactively, the relationship between cell-by-cell measures (as indicated by the coloured surface) and the more regional morphometry of the surface may be investigated. While by its very definition, this style of interactive visualisation is subjective, it offers advantages over static perspective views in that multiple viewpoints can be explored with ease. For example, horizon profiles can be particularly effective at representing scale tendency as shown in the final frame of Figure 2. By rotating viewing direction, different parts of the surface may be viewed in such a manner.
The visualisations shown here are implemented in the software LandSerf written in Java and OpenGL available from www.landserf.org. Java is used for the 2-dimensional visualisation and analysis, OpenGL for the perspective rendering. Implementation in a low-level graphics language such as OpenGL allows full control over viewing parameters while enabling fast interactive rendering of images. The dynamic perspective images shown in Figures 2 and 3 were rendered at about 20 frames per second on a Pentium II 266 Mhz with a FireGL 1000 graphics card costing about £100.
Figure 3 The effect of viewing parameters on perspective projection. (a) 25 degree field of view; (b) 55 degree field of view; (c) 110 degree field of view; (d) 80 degree field of view with local shaded relief and fog depth cueing.
Controlling the various projection parameters required to produce a perspective view allows the viewer to explore the relationship between large- and small-scale features. After viewing position, the parameter that has most visual effect on this relationship is the image's field of view (FOV). Figure 3 shows the same fractal surface shown in Figures 1 and 2 with a chequered control surface draped over it in order to assess the scaling effects of the projection. Figures 3a-3c show the effect of increasing the FOV combined with movement of the viewing position such that the hills in the background are rendered at approximately the same scale in all four images. Increasing the FOV tends to exaggerate the differences in scale between near and far features, the effect of which is to increase the sense of immersion within the landscape. Low FOVs tend to imply a more objective rendering that diminishes any scale dependencies within the surface. Finally Figure 3d further increases the distinction between near and far views by adding a distance-dependent fog effect and local shaded relief representation. The danger in adopting the style of visualisation shown in Figure 3d is that the amplified subjectivity of the rendering results in very different views of the same spatial location as perspective changes. The morphometric character of the landscape appears to have a low 'map stability' [Muehrcke, 1990]. Yet when combined with interactive exploration, this variation becomes its very strength for Dibiase's private 'visual thinking' rather than public 'visual communication' [DiBiase, 1990].
One of the problems of visualisation with a low 'map stability' is that there is no guarantee that the viewer will be presented with a sufficiently wide range of views to allow a representative picture of the landscape to be inferred. As MacEachren suggests [MacEachren, 1995], the only way we can assess the confidence of what we view is to compare multiple views. The multiple views afforded by the interactive movement described above are not sufficient alone in providing a thorough picture of scale tendency within a landscape. More objective statistical measures can be identified by measuring some surface property over a range of resolutions and/or spatial extents. This section considers how the scale tendency of the morphometric measures such as slope, aspect, and curvature may be assessed within a visual exploratory environment.
The identification of slope, aspect and curvature at a point on a surface requires some form of continuous surface representation. Using (discrete) DEMs, this is usually achieved by some form of interpolation between adjacent DEM cells within a local region. The method adopted here is a generalised version of the method described by Travis [Travis, 1975] and Evans [Evans, 1980]. The method described by these authors involved fitting a bivariate quadratic surface through a local 3x3 kernel centred on the point of interest. This function can then be differentiated in order to measure the first derivatives (slope and aspect), and the second derivatives (profile and plan curvature). The main problem with this method is that the measures represent form over a fixed spatial extent of 9 DEM cell units. If we wish to consider measures over different spatial extents, the DEM has to be resampled. The method described here and implemented in the LandSerf software uses least squares regression to fit a quadratic surface through any arbitrary set of points with a user-defined distance weighting function. The method [Wood, 1996a; Wood 1996b] allows surface measures to be taken for the same location over a range of spatial extents.
Figure 4 shows an example of the variation in cross-sectional curvature for Crater Lake, Oregon. The six images show the distribution of ridges and channels at scales of approximately 100m to 2km. The finer scales (4a) tend to be dominated by errors in the DEM (manifesting itself in left-right orientated lineations) and fluvial channels sloping inward towards the crater. At coarser scales, the larger ridge and channel networks flowing away from the crater dominate. Certain features appear to dominate at characteristic scales, most notably the mid-crater island, which is expressed most strongly at the 300m scale.
Figure 4 Cross-sectional curvature for Crater Lake, Oregon. Darker blue indicates increasing surface concavity, darker red surface convexity. Curvature calculated using a kernel of (a) 90m; (b) 210m; (c) 330m; (d) 630m; (e) 1050m; (e) 2070m.
Scale tendency can thus be characterised by a 'scale-signature' - the set of surface measurements centred at a given point over a range of local neighbourhoods. This presents a problem for visualisation as it immediately increases the dimensionality of the data we wish to view. For a given attribute, every point on the surface has 2 spatial dimensions and a scale dimension. Thus, to visualise the 3-dimensional pattern of scale dependency we can think of the problem as one of projection pursuit, that is the collapsing of one or more dimensions in order to project onto a 2-dimensional medium. Figure 4 shows one solution suitable for visual presentation, namely the use of 'small multiples' [Tufte, 1990] to present snapshots through the scale dimension. An alternative for visual exploration is to hold the spatial dimensions constant and plot the change in measure over scale. Figure 5 shows some examples of this process, whereby the viewer selects a point on the surface with a mouse in order to produce a graph of how cross-sectional curvature changes with scale. With repeated selection of points, the viewer can explore those parts of the surface which exhibit greatest scale dependency. Points of inflexion on the subsequent reflect 'characteristic scales' at which a particular measurement is most prominent. By maintaining a dynamic link between the different representations (windows) of the surface, the multiple viewpoints recommended by MacEachren [MacEachren, 1995) can be explored.
Figure 5 Interactive query of cross-sectional curvature at two points on the Crater Lake DEM. Each have different characteristic scales at which they are most convex (300m for point a, 1.2km for point b).
Finally, an alternative projection pursuit can be investigated by collapsing the scale variation dimension by summarising the behaviour of the scale signature for each point on the surface. An simple way of doing this is to display the mean of the surface characteristic over multiple scales to represent its central tendency, and the standard deviation to represent its dispersion. These two measures are shown in Figure 6 for the Crater Lake DEM. The standard deviation (Figure 6b) is an explicit visualisation of the scale-tendency of curvature in the landscape. Darker areas are scale dependent in that their character varies depending on the scale of analysis, while lighter area are scale independent, having similar characteristics over a range of spatial scales.
Figure 6 (a) Mean, (b) Standard deviation of curvature for the Crater Lake DEM over kernel sizes from 90m-750m. (c) Hue-intensity combined image of mean and standard deviation.
The cartographic history of terrain representation shows a trend from the pictorial representation of the subjective towards a more objective form of representation (e.g. in the form of contours). It has been suggested here that advances in computer visualisation technology, particularly the perspective projection of surfaces, allow us to return to a more subjective form of representation. However, unlike the earlier static representations, we are now able to produce multiple subjective viewpoints suitable for exploratory visualisation. This type of surface visualisation is particularly appropriate for the investigation of scale tendencies within the landscape, as it tends to differentiate between the representation of 'near' large-scale features and 'far' smaller scale features.
There is sound geomorphological evidence that scale tendency is an important property of real terrain. It follows therefore that as cartographers, we must have appropriate techniques for visualising scale dependency. Several methods have been presented here that allow us to both explore and present scale tendency within the landscape. Three-dimensional perspective views allow us to explore subjective and immersive views of landform. They allow us to exploit our intuitive experience of judging many scales simultaneously within the three-dimensional world. By navigating within an implied three-dimensional space, we can explore multiple viewpoints over a range of spatial resolutions and extents. Measuring surface properties over a range of local neighbourhoods and visualising the result can generate further viewpoints. When implemented in a dynamically linked visual environment, the scale tendency of such measures can be explored interactively. Statistical summaries of morphometric measures are perhaps more appropriate for presentation rather than exploration as they are more computationally intensive to calculate.
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