(P8) Shape Index

p_ij =                perimeter of patch ij in terms of number of cell surfaces.

min p_ij =         minimum perimeter of patch ij in terms of number of cell surfaces (see below).

Description

SHAPE equals patch perimeter (given in number of cell surfaces) divided by the minimum perimeter (given in number of cell surfaces) possible for a maximally compact patch (in a square raster format) of the corresponding patch area. If a_ij is the area of patch ij (in terms of number of cells) and n is the side of a largest integer square smaller than a_ij, and m = a_ij - n², then the minimum perimeter of patch ij, min-p_ii will take one of the three forms (Milne 1991, Bogaert et al. 2000):

min-p_ii = 4n, when m = 0, or

min-p_ii = 4n + 2, when n² < a_ij ≤ n(1+n), or

min-p_ii = 4n + 4, when a_ij > n(1+n).

Units

None

Range

SHAPE ≥ 1, without limit.

SHAPE = 1 when the patch is maximally compact (i.e., square or almost square) and increases without limit as patch shape becomes more irregular.

Comments

Shape index corrects for the size problem of the perimeter-area ratio index (see previous description) by adjusting for a square (or almost square) standard and, as a result, is the simplest and perhaps most straightforward measure of overall shape complexity. Note, the minimum perimeter for an aggregate of like-valued square pixels (a_ij) is calculated as above. For large patches, say a_ij > 100 pixels, the minimum perimeter asymptotically approaches , the perimeter of an exact square of size a_ij. Previous versions of FRAGSTATS used this large patch approximation in the shape index. Thus, the results will not agree exactly with previous runs, although the differences will be nontrivial only in cases involving very small patches.