Shape metrics

Background.--The interaction of patch shape and size can influence a number of important ecological processes. Patch shape has been shown to influence inter-patch processes such as small mammal migration (Buechner 1989) and woody plant colonization (Hardt and Forman 1989), and may influence animal foraging strategies (Forman and Godron 1986). However, the primary significance of shape in determining the nature of patches in a landscape seems to be related to the 'edge effect' (see discussion of edge effects for Area and Edge Metrics).

Shape is a difficult parameter to quantify concisely in a metric for the reasons discussed below. Generally speaking, the shape of a geometric object, such as a patch, is a function of its morphology. Thus, one might expect shape metrics to discriminate among patch morphologies. While it is possible to quantitatively distinguish morphological patterns, as is done in the field of computer visions (e.g., face recognition), it is generally deemed unimportant to do so in landscape ecological applications. Instead, the emphasis is on geometric complexity and distinguishing among patches and landscapes on the basis of overall complexity rather than particular morphologies. Conseqently, the shape metrics described below all deal with overall geometric complexity and do not distinguish among distinct morphologies.

FRAGSTATS Metrics.--FRAGSTATS computes several metrics that quantify landscape configuration in terms of the complexity of patch shape at the patch, class, and landscape levels. Most of these shape metrics are based on perimeter-area relationships. Perhaps the simplest shape index is a straightforward Perimeter-area ratio (PARA). A problem with this metric as a shape index is that it varies with the size of the patch. For example, holding shape constant, an increase in patch size will cause a decrease in the perimeter-area ratio. Patton (1975) proposed a diversity index based on shape for quantifying habitat edge for wildlife species and as a means for comparing alternative habitat improvement efforts (e.g., wildlife clearings). This Shape index (SHAPE) measures the complexity of patch shape compared to a standard shape (square) of the same size, and therefore alleviates the size dependency problem of PARA. This shape index is widely applicable in landscape ecological research (Forman and Godron 1986).

Another other basic type of shape index based on perimeter-area relationships is the Fractal dimension index (FRAC). In landscape ecological research, patch shapes are frequently characterized via the fractal dimension of the object (Krummel et al. 1987, Milne 1988, Turner and Ruscher 1988, Iverson 1989, Ripple et al. 1991). The appeal of fractal analysis is that it can be applied to spatial features over a wide variety of scales. Mandelbrot (1977, 1982) introduced the concept of fractal, a geometric form that exhibits structure at all spatial scales, and proposed a perimeter-area method to calculate the fractal dimension of natural planar shapes. The perimeter-area method quantifies the degree of complexity of the planar shapes. The degree of complexity of a polygon is characterized by the fractal dimension (D), such that the perimeter (P) of a patch is related to the area (A) of the same patch by P ≈ √AD (i.e., log P ≈ ½D log A). For simple Euclidean shapes (e.g., circles and rectangles), P ≈ √A and D = 1 (the dimension of a line). As the polygons become more complex, the perimeter becomes increasingly plane-filling and P ≈ A with D -> 2. Although fractal analysis typically has not been used to characterize individual patches in landscape ecological research, we use this relationship to calculate the fractal dimension of each patch separately. Note that the value of the fractal dimension calculated in this manner is dependent upon patch size and/or the units used (Rogers 1993). Thus, varying the cell size of the input image will affect the patch fractal dimension. Therefore, caution should be exercised when using this fractal dimension index as a measure of patch shape complexity.

Fractal analysis usually is applied to the entire landscape mosaic using the perimeter-area relationship A = k P^2/D, where k is a constant (Burrough 1986). If sufficient data are available, the slope of the line obtained by regressing log(P) on log(A) is equal to 2/D (Burrough 1986). Note, fractal dimension computed in this manner is equal to 2 divided by the slope; D is not equal to the slope (Krummel et al. 1987) nor is it equal to 2 times the slope (e.g., O'Neill et al. 1988, Gustafson and Parker 1992). We refer to this index as the perimeter-area fractal dimension (PAFRAC) in FRAGSTATS. Because this index employs regression analysis, it is subject to spurious results when sample sizes are small. In landscapes with only a few patches, it is not unusual to get values that greatly exceed the theoretical limits of this index. Thus, this index is probably only useful if sample sizes are large (e.g., n > 20; although PAFRAC is computed in FRAGSTATS if n = 10). If insufficient data are available, an alternative to the regression approach is to calculate the Mean patch fractal dimension (FRAC_MN) based on the fractal dimension of each patch, or the Area-weighted mean patch fractal dimension (FRAC_AM) at the class and landscape levels by weighting patches according to their size, although these metrics do not have the same interpretation or utility as PAFRAC. In contrast to the fractal dimension of a single patch, which provides an index of shape complexity for that patch, the perimeter-area fractal dimension of a patch mosaic provides an index of patch shape complexity across a wide range of spatial scales (i.e., patch sizes). Specifically, it describes the power relationship between patch area and perimeter, and thus describes how patch perimeter increases per unit increase in patch area. If, for example, small and large patches alike have simple geometric shapes, then PAFRAC will be relatively low, indicating that patch perimeter increases relatively slowly as patch area increases. Conversely, if small and large patches have complex shapes, then PAFRAC will be much higher, indicating that patch perimeter increases more rapidly as patch area increases - reflecting a consistency of complex patch shapes across spatial scales. The fractal dimension of patch shapes, therefore, is suggestive of a common ecological process or anthropogenic influence affecting patches across a wide range of scales, and differences between landscapes can suggest differences in the underlying pattern-generating process (e.g., Krummel 1987).

An alternative method of assessing shape is based on ratio of patch area to the area of the smallest circumscribing circle, known as the related circumscribing circle (CIRCLE)(Baker and Cai 1992). The circumscribing circle provides a measure of overall patch elongation. A highly convoluted but narrow patch will have a low related circumscribing circle index due to the relative compactness of the patch, yet a narrow and elongated patch will have a high related circumscribing square index. This index may be particularly useful for distinguishing patches that are both linear (narrow) and elongated.

A final method of assessing patch shape is based on the spatial connectedness, or contiguity, of cells within a grid-cell patch to provide an index on patch boundary configuration and thus patch shape (LaGro 1991). Contiguity index (CONTIG) is quantified by convolving a 3x3 pixel template with a binary digital image in which the pixels within the patch of interest are assigned a value of 1 and the background pixels (all other patch types) are given a value of zero. A template value of 2 is assigned to quantify horizontal and vertical pixel relationships within the image and a value of 1 is assigned to quantify diagonal relationships. This combination of integer values weights orthogonally contiguous pixels more heavily than diagonally contiguous pixels, yet keeps computations relatively simple. The center pixel in the template is assigned a value of 1 to ensure that a single-pixel patch in the output image has a value of 1, rather than 0. The value of each pixel in the output image, computed when at the center of the moving template, is a function of the number and location of pixels, of the same class, within the nine cell image neighborhood. Specifically, the contiguity value for a pixel in the output image is the sum of the products, of each template value and the corresponding input image pixel value, within the nine cell neighborhood. Thus, large contiguous patches result in larger contiguity index values.

Limitations.--All shape indices based on perimeter-area relationships have important limitations. First, perimeter lengths are biased upward in raster images because of the stair-stepping pattern of line segments, and the magnitude of this bias varies in relation to the grain or resolution of the image. Thus, the computed perimeter-area ratio will be somewhat higher than it actually is in the real-world. Second, as an index of "shape", the perimeter-to-area ratio method is relatively insensitive to differences in patch morphology. Thus, although patches may possess very different shapes, they may have identical areas and perimeters. For this reason, shape indices based on perimeter-area ratios are not useful as measures of patch morphology; they are best considered as measures of overall shape complexity. Alternative indices of shape that are not based on perimeter-area rations are less troubled by these limitations. But these too, generally do not distinguish patch morphology, but instead emphasize one or more aspects of shape complexity (e.g., elongation).