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. 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. (http://edcwww.cr.usgs.gov/landdaac/gtopo30/gtopo30.html). 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°. |