Computation of Lacunarity from Covariance of Spatial Binary Maps

Abstract

We consider a spatial binary coverage map (binary pixel image) which might represent the spatial pattern of the presence and absence of vegetation in a landscape. ‘Lacunarity’ is a generic term for the nature of gaps in the pattern: a popular choice of summary statistic is the ‘gliding-box lacunarity’ (GBL) curve. GBL is potentially useful for quantifying changes in vegetation patterns, but its application is hampered by a lack of interpretability and practical difficulties with missing data. In this paper we find a mathematical relationship between GBL and spatial covariance. This leads to new estimators of GBL that tolerate irregular spatial domains and missing data, thus overcoming major weaknesses of the traditional estimator. The relationship gives an explicit formula for GBL of models with known spatial covariance and enables us to predict the effect of changes in the pattern on GBL. Using variance reduction methods for spatial data, we obtain statistically efficient estimators of GBL. The techniques are demonstrated on simulated binary coverage maps and remotely sensed maps of local-scale disturbance and meso-scale fragmentation in Australian forests. Results show in some cases a fourfold reduction in mean integrated squared error and a twentyfold reduction in sensitivity to missing data.

Supplementary materials accompanying the paper appear online and include a software implementation in the R language.

Appendix

Proof of (22)

The volume $|X \cap (B\oplus \mathbf {y})|$ can be written as an integral of indicator functions

$$\begin{aligned} |X \cap (B\oplus \mathbf {y})|&=\int _{\mathbb {R}^d} \mathbf {1}_{X \cap W}(\mathbf {x}) \mathbf {1}_{B\oplus \mathbf {y}} (\mathbf {x}) \, \mathrm{d}\mathbf {x} = \int _{\mathbb {R}^d} \mathbf {1}_{W}(\mathbf {x}) \mathbf {1}_{X}(\mathbf {x}) \mathbf {1}_{B} (\mathbf {x}-\mathbf {y}) \, \mathrm{d}\mathbf {x}. \end{aligned}$$

so the first moment is (using the Fubini-Tonelli theorem)

$$\begin{aligned} \frac{1}{|W|}\int _{\mathbb {R}^d} |X \cap (B\oplus \mathbf {y})|\, \mathrm{d}\mathbf {y}&= \frac{1}{|W|}\int _{\mathbb {R}^d} \int _{\mathbb {R}^d}\mathbf {1}_{W}(\mathbf {x}) \mathbf {1}_{X}(\mathbf {x}) \mathbf {1}_{B} (\mathbf {x}-\mathbf {y}) \, \mathrm{d}\mathbf {x} \, \mathrm{d}\mathbf {y} \nonumber \\&= \frac{1}{|W|} \int _{\mathbb {R}^d} \mathbf {1}_{W}(\mathbf {x}) \mathbf {1}_{X}(\mathbf {x}) \int _{\mathbb {R}^d}\mathbf {1}_{B} (\mathbf {x}-\mathbf {y})\, \mathrm{d}\mathbf {y} \, \mathrm{d}\mathbf {x} \nonumber \\&= \frac{|X \cap W|}{|W|} |B| = \hat{p} |B|. \end{aligned}$$

(23)

With similar arguments the second moment is

$$\begin{aligned} \frac{1}{|W|}\int _{\mathbb {R}^d}&|X \cap (B\oplus \mathbf {y})|^2 \, \mathrm{d}\mathbf {y} \nonumber \\&= \frac{1}{|W|} \int _{\mathbb {R}^d}\, \int _{\mathbb {R}^d} \mathbf {1}_{W}(\mathbf {x}) \mathbf {1}_{X}(\mathbf {x}) \mathbf {1}_{B} (\mathbf {x}-\mathbf {y}) \, \mathrm{d}\mathbf {x} \int _{\mathbb {R}^d} \mathbf {1}_{W}(\mathbf {z}) \mathbf {1}_{X}(\mathbf {z}) \mathbf {1}_{B} (\mathbf {z}-\mathbf {y}) \, \mathrm{d}\mathbf {z}\, \, \mathrm{d}\mathbf {y} \nonumber \\&= \frac{1}{|W|} \int _{\mathbb {R}^d}\, \int _{\mathbb {R}^d} \mathbf {1}_{W}(\mathbf {x}) \mathbf {1}_{X}(\mathbf {x}) \mathbf {1}_{W}(\mathbf {z}) \mathbf {1}_{X}(\mathbf {z}) \int _{\mathbb {R}^d} \mathbf {1}_{B} (\mathbf {x}-\mathbf {y}) \mathbf {1}_{B} (\mathbf {z}-\mathbf {y}) \, \mathrm{d}\mathbf {y}\, \, \mathrm{d}\mathbf {x}\, \, \mathrm{d}\mathbf {z} \nonumber \\&= \frac{1}{|W|} \int _{\mathbb {R}^d} \int _{\mathbb {R}^d} \mathbf {1}_{W}(\mathbf {x}) \mathbf {1}_{X}(\mathbf {x}) \mathbf {1}_{W}(\mathbf {z}) \mathbf {1}_{X}(\mathbf {z}) \gamma _B(\mathbf {z}-\mathbf {x}) \, \mathrm{d}\mathbf {x}\, \, \mathrm{d}\mathbf {z} \nonumber \\&= \frac{1}{|W|} \int _{\mathbb {R}^d} |((X \cap W) \oplus \mathbf {v}) \cap (X \cap W)| \gamma _B(\mathbf {v}) \, \mathrm{d}\mathbf {v} \nonumber \\&= \frac{1}{|W|} \int _{\mathbb {R}^d} \gamma _{X \cap W}(\mathbf {v}) \gamma _B(\mathbf {v}) \, \mathrm{d}\mathbf {v} = \int _{\mathbb {R}^d} \frac{\gamma _{X \cap W}(\mathbf {v})}{|W|} \gamma _B(\mathbf {v}) \, \mathrm{d}\mathbf {v}. \end{aligned}$$

(24)

Substitution of (23) and (24) into (21) proves statement (22). $\square$