Home Articles Extraction of mountains from digital elevation models using mathematical morphology

Extraction of mountains from digital elevation models using mathematical morphology


Dinesh Sathyamoorthy
Science and Technology Research Institute for Defence (STRIDE),
Ministry of Defence, Malaysia.
[email protected]

Abstract
In this paper, a mathematical morphological based algorithm to extract mountains from digital elevation models (DEMs) is proposed. First, ultimate erosion is used to extract the peaks of the DEM. Conditional dilation is performed on the extracted peaks to obtain the mountain regions. The effectiveness of the proposed algorithm is tested by implementing it on the Global Digital Elevation Model (GTOPO30) of Great Basin, Nevada, USA. Connected component labelling is used to identify the individual mountains objects. Each mountain object is described based on their size, perimeter length, maximum elevation, mean gradient, local relief and relative massiveness

1 Introduction

Mountains are the portions a terrain that are sufficiently elevated above the surrounding land (greater than 300 to 600m) and have comparatively steep sides. In a mountain, two parts are distinctive:

  1. The summit, the highest point (the peak) or the highest ridges
  2. The mountainside, the part of a mountain between the summit and the foot (Bates and Jackson, 1987).

The mapping of mountains is generally performed manually through fieldwork and visual interpretation of topographic maps, which is a time consuming and labor intensive activity. In recent times, extraction techniques have evolved from manual through computer assisted to automated methods; with digital elevation models (DEMs) as the input data. In seeking the efficient extraction of mountains from DEMs, various algorithms have been proposed (Graff and Usery, 1993; Miliaresis and Argialas, 1999; Miliaresis, 2000)

In this paper, a mathematical morphological based algorithm to perform the extraction of mountains from digital elevation models (DEMs) is developed. Mathematical morphology is a branch of image processing that deals with the extraction of image components that are useful for representational and descriptional purposes (Gonzalex and Wood, 1993). In mathematical morphology, the grey level of a greyscale image is taken to represent height above a base plane, so that the greyscale image represents a topographic surface in 3D Eucledian space. Hence, a DEM can be easily represented as a greyscale image since it is enough to associate each elevation with a grey level proportional to the considered elevation. A DEM is therefore thought of as a greyscale function defined over a subset of 2D digital space, with the grey level at any point being the altitude at this point (Soille and Ansoult, 1990). The fundamental morphological operators are discussed in Matheron (1975), Serra (1982) and Soille (2003). Morphological operators generally require two inputs; the input image A, which can be in binary or grayscale form, and the kernel B, which is used to determine the precise effect of the operator (Serra, 1982).

In the Section 2, the proposed mathematical morphological based mountain extraction algorithm is discussed. In Section 3, the effectiveness of the proposed algorithm is tested by implementing on the Global Digital Elevation Model (GTOPO30) of Great Basin, Nevada, USA. Concluding remarks regarding the scope of the study is provided in the final section.

Table 1: Numerical description of the extracted mountain objects

Object ID Area(pixels) Perimeter(pixels) Maximum elevation(grey level) Mean gradient(°)
1 1227 219 178 11.94
2 10422 2277 191 10.61
3 1353 161 149 8.28
4 298 88 123 10.08
5 14232 3391 240 10.22
6 432 113 143 10.87
7 6444 999 255 9.56
8 311 162 112 10.66
9 1119 353 167 8.03
10 219 124 130 6.48
11 3574 754 237 9.18
12 3058 651 231 13.20
13 494 172 170 9.55
14 261 112 170 8.01

2 The Proposed Mountain Extraction Algorithm
Mountains have a gradient range of 6º and above. The gradient values of a terrain are usually minimized in the pits and peaks, in contrast to the usually steep valley sides or cliff sides. Hence, physiographic segmentation cannot be performed through gradient thresholding of the DEM. The proposed mountain extraction algorithm is as follows:


An example of the ultimate erosion operation. Ultimate erosion is implemented through the iterative erosion of the image until all objects vanish (images Xi), and the reconstruction of each eroded image using the eroded image E(Xi) as the mask and the erosion of smaller size as the marker. The reconstructed images (images Yi) are subtracted from the corresponding eroded images to form the eroded sets (images Ui). The final resultant image, known as the ultimate eroded set, contains only the objects’ pseudo-centres. (Source: Duchane and Lewis, 1996).
 

2.1 Peak extraction
The peaks of a terrain refer to the highest points of the mountains of the terrain. In DEMs, peaks are connected components that are completely surrounded by pixels of lower elevation. The detection of peaks is the first step in most techniques used to perform DEM characterization and to describe the general geomorphometry of a surface. Dilation sets the pixel values within the kernel to the maximum value of the pixel neighbourhood. The dilation operation is expressed as:

Erosion sets the pixels values within the kernel to the minimum value of the kernel. Erosion is the dual operator of dilation:

where Ac denotes the complement of A, and B is symmetric with respect to reflection about the origin. Greyscale erosion can be used to remove bright areas in greyscale images/malaysia/2006/oct-dec. It causes small peaks in the image to disappear. However, it also causes valley widening which results in formation of larger peaks (Serra, 1982). Morphological reconstruction allows for the isolation of certain features within an image based on the manipulation of a mask image, X and a marker image, Y. It is founded on the concept of geodesic transformations, where dilations or erosion of a marker image are performed until stability is achieved (represented by a mask image) (Vincent, 1993). The geodesic dilation dG used in the reconstruction process is performed through iteration of elementary geodesic dilations d (1) until stability is achieved.

The elementary dilation process is performed using standard dilation of size one followed by an intersection.

The operation in equation 4 is used for elementary dilation in binary reconstruction. In greyscale reconstruction, the intersection in the equation is replaced with a pointwise minimum (Vincent, 1993). Morphological reconstruction can be used to maintain the peak removal effect of erosion while avoiding its the valley enlargement effect (Vincent, 1993). The peaks removed by erosion can be obtained by subtracting the reconstructed eroded image from the original image.

In order to extract the peaks of a DEM, ultimate erosion is performed on the DEM. Ultimate erosion is implemented by successively eroding an image until all particles vanish and performing morphological greyscale reconstruction on each eroded image into the erosion of smaller size (Duchane and Lewis, 1996). Figure 1 demonstrates the operation of ultimate erosion. The generated ultimate eroded set of the DEM forms the peaks of the DEM.

2.2 Mountains extraction
Step 1: Conditional dilation of the peaks of the DEM The peaks of the DEM are dilated with a square kernel of size 3. The boundary pixels of the dilated peaks that have gradient less than 6° are deleted. The conditional dilation of the peaks is repeated until no further changes are produced. In the image produced from this step, the pixels with value 1 (white pixels) are mountain pixels while pixels with value 0 (black pixels) are non-mountain pixels.

Step 2: Removal of small islands of non-mountain pixels observed on mountaintops These pixels are flat to gently sloping regions, so the gradient was less than 6°. These pixels were not classified as peaks and Step 1 did not flag them as mountain pixels due to their gradient being less than 6°. However, these pixels have the geometric proximity to be mountain pixels. These erroneous non-mountain pixels are removed by assigning them as mountain pixels.

Step 3: Removal of small islands of mountain pixels observed in flat areas In flat areas of DEMs, the noise (mean error in elevation) to signal (elevation) ratio is high, causing the formation of spurious peaks. These spurious peaks do not form larger mountain regions as there are small gradient values in their neighborhood. These erroneous mountain pixels are removed by converting these erroneous mountain regions into non-mountain pixels.

3 Case Study
The DEM in Figure 2 shows the area of Great Basin, Nevada, USA. The area is bounded by latitude 38° 15’ to 42° N and longitude 118° 30’ to 115° 30’W. The DEM was rectified and resampled to 925m in both x and y directions. The DEM is a Global Digital Elevation Model (GTOPO30 DEM) and was downloaded from the USGS GTOPO30 website


The GTOPO30 DEM of Great Basin. The elevation values of the terrain (minimum 1005 meters and maximum 3651 meters) are rescaled to the interval of 0 to 255 (the brightest pixel has the highest elevation). The scale is approximately 1:3,900,00.

). GTOPO30 DEMs are available at a global scale, providing a digital representation of the Earth’s surface at a 30 arc-seconds sampling interval. The land data used to derive GTOPO30 DEMs are obtained from digital terrain elevation data (DTED), the 1-degree DEM for USA and the digital chart of the world (DCW). The accuracy of GTOPO30 DEMs varies by location according to the source data. The DTED and the 1-degree dataset have a vertical accuracy of + 30m while the absolute accuracy of the DCW vector dataset is +2000m horizontal error and +650 vertical error (Miliaresis and Argialas, 2002). Tensional forces on the terrain’s crust and thins by normal faulting have caused the formation an array of tipped mountain blocks that are separated from broad plain basins, producing a basin-and-range physiography (Howell, 1995).

The DEM of Great Basin has a mean gradient of 4.94. Figure 3(a) shows the pixels of the DEM in the gradient range of 0 to 57.12 rescaled to the interval 0-254. The DEM contains 34,248 pixels (37.46%) with gradient higher than 6°. As shown in Figure 3(b), the gradient thresholding of the DEM is an invalid mountain extraction method as it fails to classify the peaks and mountaintops of the DEM as mountain pixels.


Gradient analysis of the DEM of Great Basin. (a) The pixels of the DEM (in the gradient range of 0° to 57.12°) rescaled to the interval of 0 to 255 (The brightest pixel has the highest gradient. (b) Gradient thresholding of the DEM. The pixels in white have gradient higher than 6°.
 

Ultimate erosion is performed on the depressionless DEM to extract the peaks of the DEM (Figure 4). A total of 1,315 peaks are extracted from the DEM. A total 6,010 pixels (6.60%) are classified as peak pixels.

The conditional dilation process is repeated on the extracted peaks until convergence (Figure 5(a)). The small islands of non-mountain pixels enclosed by mountain pixels are assigned as mountain pixels (Figure 5(b)). The mountain regions with size less than 180 pixels are removed by converting these pixels to non-mountain pixels (Figure 5(c)). A total of 42,168 pixels (46.13%) are classified as mountain class pixels. These pixels form 14 distinct mountain objects (Figure 5(d)) which are identified using the connected component labelling proposed in Pitas (1993). Each mountain object is described based on their size, perimeter length, maximum elevation and mean gradient (Table 1).

In Miliaresis and Argialas (1999), mountain extraction is performed by first extracting the seed ridge pixels using runoff simulation. The mountain regions are obtained by performing region growing on the seed ridge pixels. The results obtained using this algorithm is shown in Figure 6. The algorithm resulted in 40,419 pixels (43.50%) being classified as mountain pixels. The mountain pixels form 36 distinct mountain regions.

A good match was evident between the results obtained using the proposed mathematical morphological based algorithm and the results obtained in Miliaresis and Argialas (1999), although some differences exist. The mountain objects in Figure 5(c) are wider than the corresponding mountain objects in Figure 6. A number of single mountain objects in Figure 5(c) appeared broken in Figure 6, resulting in Figure 5(b) having fewer distinct mountain objects the Figure 6. This difference occurs because the seed ridge pixel image does not contain a number of the peaks of the terrain despite containing most of the highest points of the mountain regions. Hence, region growing on the seed ridge pixel image is unable to extract all of the mountain regions of the DEM, particularly the mountaintop regions.

Runoff simulation is unable to operate effectively on flat areas in DEMs, causing errors in the extracted seed ridge pixels, and hence errors in the extracted mountain regions. The proposed mountain extraction algorithm does not rely on flow directional based algorithms, and is hence able to operate effectively on flat areas in DEMs. The differences observed were also due to the errors generated during the rescaling of the elevations (ranging between 1005 meters to 3561 meters) of the DEM to the interval of 0 to 255.


A total of 1,315 peaks are extracted from the DEM of Great Basin.


Mountain extraction. (a) The mountain pixels (the pixels in white) of the DEM. The black pixels are non-mountain pixels. (b) The mountain pixels after the removal of erroneous non-mountain regions enclosed by mountain pixels. (c) The mountain pixels after removal of erroneous mountain pixels. (d) The identification of the individual mountain objects.


The mountains extracted using the algorithm proposed in Miliaresis and Argialas (1999)

4 Conclusion
In this paper, a mathematical morphological based mountain extraction algorithm was proposed. First, ultimate erosion is performed on the DEM to extract the peaks of the DEM. Conditional dilation is performed on the extracted peaks to extract the mountains regions of the DEM. The proposed mountain extraction algorithm performs well in areas where a set of mountain features is developed in between basins standing at different baselevels, and is able to operate effectively on the flat areas in the DEM.

It can be seen in Figure 5(c) that the extracted mountains have varying shapes and sizes. Experiments are currently being carried out to characterize the shape-size complexity of mountains extracted from DEMs. This will be performed using the pattern spectrum procedure proposed in Maragos (1989).

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