Object representations at multiple scales from digital elevation models

Abstract

In the last decade landform classification and mapping has developed as one of the most active areas of geomorphometry. However, translation from continuous models of elevation and its derivatives (slope, aspect, and curvatures) to landform divisions (landforms and landform elements) is filtered by two important concepts: scale and object ontology. Although acknowledged as being important, these two issues have received surprisingly little attention.

This contribution provides an overview and prospects of object representation from DEMs as a function of scale. Relationships between object delineation and classification or regionalization are explored, in the context of differences between general and specific geomorphometry. A review of scales issues in geomorphometry—ranging from scale effects to scale optimization techniques—is followed by an analysis of pros and cons of using cells and objects in DEM analysis. Prospects for coupling multi-scale analysis and object delineation are then discussed. Within this context, we propose discrete geomorphometry as a possible approach between general and specific geomorphometry. Discrete geomorphometry would apply to and describe land-surface divisions defined solely by the criteria of homogeneity in respect to a given land-surface parameter or a combination of several parameters. Homogeneity, in its turn, should always be relative to scale.

Highlights

► We provide a critical review on object representation from digital elevation models. ► Scale and object ontology are concepts that filter translation from DEM to landforms. ► Scale optimization techniques remain a priority for future research. ► Footprints of cells have nothing to do with size and shape of landforms. ► We propose discrete geomorphometry as an approach to specific geomorphometry.

1. Introduction

DEMs (Digital Elevation Models) are used for the extraction of land-surface parameters and objects through geomorphometric analysis (Pike, 2000; Pike et al., 2009). As ‘general geomorphometry’ applies to continuous land surface, ‘specific geomorphometry’ applies to discrete landforms (Evans, 1972). Most land-surface parameters and objects vary with spatial scale, which in the digital realm is widely understood as a function of cell size or grid resolution (Wilson and Gallant, 2000). However, grid resolution is not a particularly appropriate representation of scale (Gallant and Hutchinson, 1996). The dependence of land-surface parameters on grid resolution has been described by Evans (1972) as ‘a basic problem in geomorphometry’ (Shary et al., 2002). Since most geomorphometric algorithms work through a fixed neighborhood operation (Pike et al., 2009)—usually within a 3 × 3 kernel—the scale of analysis is tied to the resolution of the input DEM and changes as the resolution changes (Zhu et al., 2008). In the absence of scale optimization techniques (Li, 2008), the geomorphometric analysis is conducted at rather arbitrary scales, which rely on the user's experience. In fact, the scale of analysis often depends on data availability as many users perform what Schmidt and Andrew (2005) called the ‘let's take a DEM… approach’, without much concern for scale effects in analysis. Obviously there is no sound scientific justification for a direct linkage between natural phenomena and data acquisition techniques (Strobl, 2008). Lately, a large amount of literature has developed particularly relevant to applications of geomorphometric analysis (terrain-based environmental modeling), underpinning the impact of scale mismatches between the target variable and the explanatory ones leading to statistical bias. With the advent of increasingly high resolution DEMs, scale is becoming an important issue in geomorphometry (MacMillan et al., 2003).

Particularly when mapping landforms from gridded land-surface models such as DEMs, scale-related shortcomings are partly generated by the implicit assimilation of data model elements (cells or pixels) to geographic objects (Fisher, 1997). Thus, in a somehow counter-intuitive manner, most landform classification systems work through the classification of cells, which are clustered to define the extents of objects—instead of delineating the objects first and then classifying them. The approach of classifying cells directly is limited in several aspects, including the scattered aspect of classification in the so-called ‘salt-and-pepper effect’, tying the scale of analysis to the raster resolution, difficulties in including topological relationships in classification and also in developing hierarchies of landforms. Although delineation techniques have been proposed, quantitative evaluations of their performance have only recently been done (van Niekerk, 2010). Still, fundamental questions pertinent to the nature of objects and their delineation relative to scale have only been touched upon. Minar and Evans (2008) provided a useful review on object ontology, showing the limitations of cell-classification approach in mapping landforms and introducing segmentation of elementary forms as an alternative. Deng (2007) and Hengl and Reuter (2009) reviewed contributions pertinent to scale and object representation from DEMs, but rather in wider contexts. Goodchild (2011) presented an overview of scale in GIS.

This paper provides an overview and prospects for object representation from DEMs as a function of scale. In the next section, relationships between object delineation and classification or regionalization are explored, in the context of differences between general and specific geomorphometry. Rather than being an exhaustive overview, this section will complement Minar and Evans (2008) with a focus on generating objects with the aid of a multi-resolution segmentation (MRS) algorithm. A review of scales issues in geomorphometry—ranging from scale effects to scale optimization techniques—is followed by an analysis of pros and cons of using cells and objects in DEM analysis, in Section 3. This section focuses on methods to establish non-arbitrary scales in the land surface. Prospects for coupling multi-scale analysis and object delineation are then discussed, in Section 4.

2. Object ontology—from cells to landforms

Specific geomorphometry applies to discrete spatial features or geomorphic objects (Evans, 1972). Landforms and landform elements (see MacMillan and Shary, 2009 for a detailed review) are particular cases of geomorphic objects. Extracting landform divisions (landform elements, landforms, etc.) from DEMs usually involves applying an object model on a raster data structure, which is the most popular format for spatial modeling (Pike et al., 2009). This is because raster elements themselves, i.e. cells or pixels, do not have any meaning in reality regardless of their size (Fisher, 1997). Cells as artificial, discrete units exist merely for the purpose of representation (Goodchild et al., 2007). Clearly, footprints of cells have nothing to do with the size and shape of real-world entities such as landforms. Therefore, a model to translate from the continuous land surface to discrete entities is required. This is challenging as the land surface is smoother than other terrain variables, such as vegetation (Hengl and Evans, 2009). Consequently land-surface objects are less obvious on DEMs than land cover patches on satellite images, for instance.

In the absence of comprehensive conceptual data models for land surfaces (Brändli, 1996), extraction of landform divisions has largely been based on classification schemes. A landform is rather represented as a collection of cells that exhibit similar morphometric characteristics (Schmidt and Dikau, 1999). Classification methods have been widely applied to directly assign cells to landform classes. Although this strategy is straightforward two major shortcomings have been identified: Firstly, cells are often treated as spatially independent from each other, and thus adjacent cells are frequently assigned to different classes resulting in a highly scattered spatial representation of landforms (Burrough et al., 2001), the so called ‘salt-and-pepper effect’. Secondly, since landforms are classified on a cell by cell basis after applying statistical rules, they are solely defined thematically, but not spatially (e.g. by location, context or topology) and hence, they do not represent spatially configured objects (Deng, 2007; Minar and Evans, 2008). Often, in a kind of Procrustean bed approach, cells are allocated to classes following pre-defined categories of, or thresholds in, land-surface parameters (e.g. the threshold value of slope to define a flat surface). Landform boundaries are then given by the edges of the aggregated cells (as resulting after some filtering, needed to reduce the ‘salt-and-pepper’ effect). But these boundaries may not coincide with morphologic discontinuities in a given landscape; they are merely conceptual or fiat boundaries (Smith, 1995).

This problem of arbitrary incidence has been exemplified for curvatures by Minar and Evans (2008). The authors observed that ‘isoline boundaries may create artificial areas without sufficient respect to the natural structure of landforms with various types of homogeneity’ (p. 241). We have noticed similar behavior for elevation and slope. Such crude representation of landscapes is part of ‘the conceptual and computational gap between local geometry and meaningful landforms’, which broadens paradoxically with improving quality and resolution of DEMs (Mark, 2009).

Moving on from collections of ‘geomorphometric points’ to ‘geomorphometric objects’ (Schmidt and Dikau, 1999) requires delineating the objects first, then classifying them. Similar to the concept of ‘object–field’ (Cova and Goodchild, 2002), DEMs can be partitioned into discrete, spatially intact land-surface objects, following data-driven approaches rather than pre-defined classification templates. A strategy of clustering similar cells in property space by means of image analysis methods to delimit form types has already been envisioned by Pike (1995) and applied by Irvin et al. (1997). While this method produces less scattered objects, the problem of matching land-surface discontinuities still remains, since clusters are created using global thresholds instead of local contrasts (van Niekerk, 2010). The same applies to classification methods using dynamic but global thresholds (Iwahashi and Pike, 2007).

Identification of spatial discontinuities in land-surface parameters seems to be more appropriate for object delineation. This idea was presented by Minar and Evans (2008) as an axiom: “At a given scale, the land surface may … exhibit discontinuities; these may be recognized as natural boundaries of geomorphic objects”. A manual technique of mapping based on morphological discontinuities was proposed by Savigear (1965). Dymond et al. (1995) described an algorithm for automated mapping of land components through approximation of slope breaks. Recently, image segmentation techniques have increasingly been used to generate objects based on the concept of heterogeneity. The most known algorithm is MRS (Baatz and Schäpe, 2000) as implemented in the eCognition® software. This is a region-merging technique to create objects from pixels through an optimization process that minimizes the internal weighted heterogeneity of each object at a given scale. These objects are then merged or split to create objects at consecutive scales, either higher, created in a bottom-up approach, or lower, created in a top-down one. Therefore, each decision of merging or splitting is based on the attributes of homogeneous structures of a recent scale (Baatz and Schäpe, 2000) and on the user-defined heterogeneity threshold, called scale parameter. Drăguţ and Blaschke (2006) and van Asselen and Seijmonsbergen (2006) introduced this algorithm to the analysis of DEMs. This approach has lately been increasingly used in delineation of landforms or land entities (Drăguţ and Blaschke, 2008; Möller et al., 2008; Schneevoigt et al., 2008; Anders et al., 2009; Blanco et al., 2009; Kringer et al., 2009; Martha et al., 2010). While segmentation relies on local contrasts in drawing meaningful boundaries of objects, classification uses global thresholds to facilitate interpretation of landform classes.

Van Niekerk (2010) recently found that an MRS algorithm is more sensitive to morphological discontinuities than two other alternatives, ALCoM and ISODATA, and he proposed it as the most suitable technique for delineating land components from DEMs. An image segmentation approach should not be confused with landform segmentation as used by Pennock and Corre (2001), which is actually a classification procedure based on pre-defined morphometric categories. Other segmentation algorithms have also been successfully applied for geomorphological mapping or delineation of homogeneous areas from DEMs. For instance, Miliaresis (2001a,b, 2006) used a region-growing algorithm for extraction of bajadas from DEMs and satellite imagery, and for geomorphometric mapping at regional scales; Lucieer and Stein (2005) proposed a region-growing segmentation procedure based on texture to extract landform objects from LiDAR data; Stepinski et al. (2006, 2007), Stepinski and Bagaria (2009), and Ghosh et al. (2010) developed a segmentation approach combined with Artificial Intelligence to automatically map planetary surfaces; Jellema et al. (2009) applied a region-growing algorithm to characterize and evaluate landscapes. A particularly simple and appealing procedure for watershed segmentation of curvature was proposed by Romstad and Etzelmüller (2009) for the purpose of geomorphological mapping.

3. Scale

Due to increasing availability and easier access to DEMs at a broad range of spatial resolutions (from LiDAR at several centimeters up to GTOPO30 at approx. 1 km), multi-scale analysis of the land surface is becoming more feasible. The modeling of scale effects with respect to both changing resolution and varying window size for surface calculations has been identified as a major research topic not only in geomorphometry, but rather in all disciplines dealing with DEMs including hydrology, soil science, and geomorphology. In a recent paper Li (2008) provided a valuable review on the numerous approaches that examine scale dependencies in terrain-based modeling, and has outlined a general strategy for their analysis.

As has been shown parameters such as slope (Deng et al., 2008) and curvatures (Schmidt and Andrew, 2005), do not only change in magnitude, but may even shift their topographic meaning, as for sign of curvature. Surface roughness also varies with scale (Grohmann et al., 2010). Moreover, scale dependencies in terrain analysis are driven by terrain characteristics (e.g. simple or complex, Carter, 1992), and also vary across different landform types (Gao, 1997; Deng et al., 2007). An important development in examining scale effects is that scale has become an integrated part of terrain analysis (Deng, 2007). For example, Wood (1996, 1998) proposed calculating surface parameters at various window sizes for a constant resolution, and for each cell recording the results as a series of values linked with scale information. His open-source software package LandSerf for ‘multi-scale surface characterization’ offers powerful visualization tools for exploring scale effects. Following Wood's approach several researchers focused on exploring the effects of neighborhood size on computed terrain parameters as well as on the application of various window sizes for multi-scale characterization of landforms (Fisher et al., 2004; Schmidt and Hewitt, 2004; Schmidt and Andrew, 2005; Reuter et al., 2006; Deng and Wilson, 2008). Lately, it has been found that parameters are less sensitive to DEM resolution changes than to variations in neighborhood size (Zhu et al., 2008).

Pure modeling of scale effects barely gives clues on how to select non-arbitrary scales for given analyses. Hence, researchers started to examine strategies for scale optimization and scale detection. They conducted either experimental testing in the context of terrain-based environmental modeling, or theoretical analysis in data-driven approaches (for a review see Li, 2008).

In terrain-based environmental modeling it is essential to fit the spatial scale of the terrain data to the scale of the processes or features under investigation in order to obtain valid model results. In doing so, several authors compared results from multi-resolution terrain analysis with reference data such as field measurements of a landscape property (Bian and Walsh, 1993; Florinsky and Kuryakova, 2000; Hengl, 2006; Drăguţ et al., 2009a), or model outputs (Smith et al., 2006; Zhang et al., 2008). Through statistical analysis they were able to identify the grid size or resolution range with the most powerful predictions. However, the optimal grid size might be different for different target variables, so that one can hardly select a single optimal resolution (Hengl, 2006). Especially in soil-landscape modeling, recent efforts have been made towards optimizing neighborhood sizes for the prediction of soil classes (Smith et al., 2006; Zhu et al., 2008; Behrens et al., 2010).

Indeed, it is more desirable to develop methods that work data-driven, and without reference to dependent variables outside the DEMs. Depending on the method one obtains a measure for each scale, and when plotting all these measures or simply the results from terrain analysis against scale, one finally gets a ‘scale signature’, where extremes mark ‘characteristic scales’ (Wood, 1996, 2009). Schmidt and Andrew (2005) introduced a spatially adaptive scale detection technique exemplified for curvatures in order to recognize dominant scale ranges of landforms. Gallant and Dowling (2003) proposed an algorithm to produce a multiresolution index of valley bottoms based on their topographic signatures at multiple scales.

A particularly appealing concept to describe aspects of scale dependency in spatial objects is fractals (Mandelbrot, 1975). However, empirical evidence (Chase, 1992; Evans and McClean, 1995; Perron et al., 2008) suggests that fractal models are not appropriate to the land surface, which has a statistically multidimensional character (Evans, 1998). Deficiencies of unifractal and multifractal models are summarized by Evans (1998). Tate and Wood (2001) provide a comprehensive review on fractals and scale dependencies in the land surface.

Probably the most promising approach for data-driven scale detection is the method of local variance (LV). This method was originally developed in image analysis for the purpose of scale detection (Woodcock and Strahler, 1987). The approach is based on the relationship between the size of objects in the real world and pixel resolution, as expressed by the spatial structure of images. The information on spatial structures of images is coded in the local variance measures. Thus, in a high-resolution scene (Strahler et al., 1986), objects in the real world are represented by multiple pixels, hence spatial autocorrelation is high (Fig. 1, top). Local variance, computed as average value of standard deviation measured in a small neighborhood (3 × 3 moving window), is therefore small. When successive coarser scales are produced from the initial dataset through resampling, local variance increases with the scale levels up to the point where pixels start approximating the representative objects in the scene (Fig. 1, top). At this scale level the maximum value of local variance is recorded as the likelihood of neighbors being similar decreases. At coarser pixel sizes local variance decreases again as a consequence of including more objects within a pixel, hence the spectral difference between neighbor pixels is reduced (Fig. 1, top).

Fig. 1.

Rationales of the method of local variance (LV) as applied on cells (top) and objects (bottom).

Despite its simplicity and usefulness this method was not widely adopted in remote sensing and GIS (Cao and Lam, 1997). Li (2008) suggested the LV method could prove useful as a scale detection technique in DEM analysis. Two recent studies tested the suitability of the LV method for multi-scale pattern analysis in geomorphometry (Drăguţ et al., 2009b, in press). Scale levels were simulated from the same high-resolution datasets through resampling and image segmentation respectively, in a bottom-up approach. The authors found that the LV method performed better when scale levels were created with image segmentation as compared to cell aggregation. Up-scaling through resampling induces an isotropic smoothing that levels out spatial patterns at coarser resolutions. In contrast, image segmentation maintains distinct boundaries of objects (Karl and Maurer, 2010) by steadily adding heterogeneity to objects at each coarser scale level (Fig. 1, bottom). Thus, at each coarser scale level contiguous objects with similar property values are merged into larger ones. The merged object preserves the external boundaries of the previously independent objects.

4. Discrete geomorphometry? Coupling multi-scale pattern analysis and object delineation

Talking about the specific geomorphometry, Mark (1975, p. 165) observed that ‘the specific approach can only be applied once an area has been identified as a drainage basin, an alluvial fan, a drumlin, etc.’. This means that delineation of objects that satisfy the condition of maximizing internal homogeneity and external differences is not a specific approach, as long as the objects do not bear a meaning other than statistical. As Minar and Evans (2008, p. 238–239) pointed out ‘segmentation of the land surface can provide a transition from the field model to the object model, and from general geomorphometry to specific geomorphometry, thus connecting the continuous and atomistic hypotheses’. The main issue is that this transition has not been conceptualized in geomorphometry so far. Therefore, we propose discrete geomorphometry as a possible approach to specific geomorphometry.

Discrete geomorphometry would apply to and describe land-surface divisions defined solely by the criteria of homogeneity in respect to a given land-surface parameter or a combination of several parameters. Homogeneity, in its turn, should always be relative to scale. The main aim of discrete geomorphometry is to produce morphometrically meaningful objects. This approach has a general and objective character as general geomorphometry; however the two approaches are framed into different conceptual models–field vs. object. On the other hand, discrete geomorphometry shares the object model with specific geomorphometry; however, in the specific approach the character of objects is defined before segmentation, while in the discrete the meanings are assigned after or along with segmentation.

Discrete geomorphometry thus centers on objects, defined as homogeneous areas delineated by discontinuities in land-surface parameters, either on individual or combined layers, which reveal the land-surface patterns at a given scale or across scales. Therefore, spatial pattern, which ‘has been missing from most quantitative work’ (Evans, in press) would come into focus. Possible candidates for such objects have been called land components by Dymond et al. (1995), terrain facets by Rowbotham and Dudycha (1998), elementary forms by Minar and Evans (2008), pattern elements by Drăguţ et al. (2009b), and morphometric primitives by Gessler et al. (2009). These objects can be seen as intermediate building blocks in the translation from cells to landform divisions. Once the objects are delineated in the digital realm, further statistical, relational and semantic rules can be applied to map each object to the landform concept to which it comes closest (MacMillan et al., 2004; Minar and Evans, 2008; Bishop, 2009; Eisank, 2010), by incorporating expert knowledge (MacMillan et al., 2005), or to use terrain objects as basic areal divisions for the study of land-surface processes (Rowbotham and Dudycha, 1998). Once the objects are assigned to classes of elementary form, or given interpretations such as ‘terrace’, ‘fan’ or ‘drumlin’, we have moved into specific geomorphometry.

Segmentation is a good candidate as the main method of discrete geomorphometry. In the second section of this paper we presented its advantages over clustering in property space and cell classification (also see Minar and Evans, 2008 for a comprehensive discussion on segmentation techniques). Two main strategies of segmentation can be followed: segmentation into pre-defined types and degrees of homogeneity (Minar and Evans, 2008) and data-driven approach (Drăguţ et al., in press).

In the first strategy, homogeneity is expressed by ‘constant values of altitude or its derived morphometric properties’ (Minar and Evans, 2008, p. 244). These constant values are named the form-defining properties. A unified system of elementary forms was predefined based on variation in the general fitted function. The forms are data-based and assigned to a form class after or along with delimitation. This approach is potentially likely to facilitate the reference of terrain divisions to process or genesis. Although the approach should be in principle independent of scale, the authors acknowledge that many aspects of the land surface are scale dependent (Evans, 2003, 2009, 2010), therefore further investigation is needed before applying it at broader scales (Minar and Evans, 2008).

The data-driven approach was developed with the aid of the multiresolution segmentation algorithm (see Section 2 of this paper). Homogeneity is controlled by a user-defined factor called scale parameter. Selection of the scale parameter was turned into an objective choice, with the aid of the ESP tool (Estimation of Scale Parameters), by using the concept of local variance (Drăguţ et al., 2010). Segmentation of land-surface parameters such as slope gradient (Eisank and Drăguţ, 2010) was performed with the ESP tool in a bottom-up approach, where objects at finer scale were steadily merged into more heterogeneous objects at broader scales (Fig. 1), by changing the scale parameter in a constant increment. Thus, ever coarser object patterns were produced. For each pattern LV was measured by first calculating the standard deviation of objects, and then averaging object values to obtain the pattern mean. Values of LV were plotted against scale parameter; breaks in the LV graph and its rate of change indicated the scales where the probability that delineated land-surface objects match a group of similar-sized real-world forms should be higher than for other scales (Drăguţ et al., in press).

Although the two methods are in incipient stages, they offer good prospects for delineating homogeneous morphometric primitives either independent of scale, or at multiple scales, through multi-scale pattern analysis. However, much work lies ahead before they can be fully operational for what Olaya (2009) called the ‘discrete analysis of the land surface’. Object ontology needs particular attention. For instance, more research on relationships between real land-surface features and segmented objects across scales is required. This is to make sure that we delineate real objects and do not create artificial constructs; arbitrary partitions at various scales would represent a particular case of the Modifiable Area Unit Problem (MAUP) with all its acknowledged shortcomings (Openshaw and Taylor, 1979). A possible strategy for linking object ontologies of segmented objects with concepts of real landforms is semantic modeling (Eisank et al., 2010) as was originally proposed by Dehn et al. (2001).

Another important research topic would be on the nature of boundaries. Since land-surface features show smooth transitions, how can we account for fuzziness on both conceptual and spatial domains? Preliminary results (Drăguţ et al., in press) show that discontinuities in land-surface parameters seem to relate to the quality of object boundaries: sharper contrasts are expressed by smoother lines, while soft transitions give more indented boundaries. If this proves true, the degree of fuzziness could be measured via edge analysis. Models like core vs. transitional areas (Drăguţ and Blaschke, 2008), attractors (Minar and Evans, 2008), or spatial gradation of slope positions (Qin et al., 2009) would be further useful in classification of objects.

Here we presented methods producing non-overlapping objects, potentially amenable to nested hierarchies. However, overlapping areas might be better suited for given applications. Romstad and Etzelmüller (2009) proposed an interesting approach based on overlapping convexities and concavities.

5. Summary

This paper has provided a critical review on object representation from digital elevation models. Translation from continuous models of elevation and its derivatives (slope, aspect, and curvatures) to landform divisions (landforms and landform elements) is filtered by two important concepts: scale and object ontology.

Scale has been acknowledged as a basic problem in geomorphometry. Depending on the DEM resolution and on the size of the analysis window, land-surface parameters have different values at the same location; consequently one landscape can be represented in multiple ways. Thus, scale impacts heavily on the results of geomorphometric analysis. Whereas scale effects on geomorphometric analysis are now relatively well understood, scale optimization techniques remain a priority for future research. Scale becomes more important with increasing DEM resolution (MacMillan et al., 2003): while the number of landscape representations from a low resolution dataset is limited, many more versions of the landscape can be represented from a single very high resolution DEM through up-scaling. Which of those representations are best suited for a given purpose? The answer to this question prompts a reliance either on expert knowledge (Gustavsson and Kolstrup, 2009), or an exploratory attitude in the use of local land-surface parameters (Deng, 2007). An exploratory attitude is unfortunately hindered by poor technical implementation of scale issues in most GIS software packages. LandSerf (Wood, 1996) is a remarkable exception, providing a suite of solutions for scale/multi-scale analysis (Wood, 2009).

Although landform divisions are discrete features by definition (Evans, 1972; MacMillan and Shary, 2009; Pike et al., 2009), most of the procedures to produce them in the digital era have essentially been tributary to a field model: individual cells are allocated to pre-defined classification schemes. Objects emerge then as aggregations of cells through decisions that are not easily applicable to other landscapes. Boundaries of such aggregates are likely to fail in matching land-surface discontinuities. Image segmentation procedures create the technical framework for delineating homogeneous objects delimited by real discontinuities in land-surface parameters. ‘Abstracting complex surfaces as objects is innately human’ (Gessler et al., 2009), so that this technique can bridge the gap between a pixel-based approach and manual mapping based on visual interpretation, while preserving the advantages of speed and objectivity given by computers. Still, the conceptual basis of specific geomorphometry should be improved. Here we proposed discrete geomorphometry as a possible approach between general and specific geomorphometry. Discrete geomorphometry would apply to and describe land-surface divisions defined solely by the criteria of homogeneity in respect to a given land-surface parameter or a combination of several parameters. Homogeneity, in its turn, should always be relative to scale.

The emerging idea of smoothing local variability in land-surface parameters into homogeneous entities looks promising for future developments. Delineation of such objects should produce morphometric patterns that match landscape patterns at given scales. This approach does not incorporate a priori knowledge on a specific landscape, therefore results are transferable. Landform classification can further give meaning to the resulting building blocks (elementary forms, pattern elements or morphometric primitives) by adding semantics. Minar and Evans (2008) set up the stage by creating a unified system of geometric primitives. Coupling multi-scale pattern analysis—with the help of local variance—with object delineation is another promising approach towards hierarchies built on such objects.