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spatLac: Spatial Lacunarity

DOI

The spatLac R package helps researchers compute Lacunarity for binary and continuous Spatial Raster objects. spatLac uses fast C++ code, allowing for low memory usage and multithreading.

Installation

You can install the development version from GitHub with:

# install.packages("devtools")
devtools::install_github("STBrinkmann/spatLac")

Lacunarity

“Lacunarity” literally refers to the gappiness or heterogeneity of a fractal or non-fractal image. As a scale dependent measure of heterogeneity, Dong (2000) introduced lacunarity for spatial heterogeneity measurements in GIS. Based on the research of Plotnick et al. (1993) he implemented a Lacunarity algorithm for raster images. Hoechstetter et al. (2011) further improved Lacunarity measurements for continuous images. In a recent paper Labib et al. (2020) applied Lacunarity on binary (Land Use) and continuous (NDVI and LAI) raster images, to derive scale sensitive weights used in a buffer analysis to build multiple-exposure metrics.

To compute Lacunarity we use a sliding box algorithm, the box can be either square or round as demonstrated in the animation below.

Lacunarity - Square Lacunarity - Round

Lacunarity calculation algorithms

Lacunarity for binary raster images is calculated using Eq. 1-5, for continuous images only function 1 and 6 are being used.

Lacunarity \Lambda(r) (“lambda”) for box diameter r of an image with width W and length L is beeing calculated as follows:

First the total number of boxes N[r] can be described as (Eq. 1):

\begin{align*} N[r] = (W - r + 1)(L - r + 1) && \text{(1)} \end{align*}

Next, the box mass S for each box of size r is beeing calculated by taking the box sum, or the range of all box-values, for binary or continuous raster, respectively.

For binary images, the number of boxes of size r containing box mass S are counted as n[S,r], and converted into a probability distribution Q(S,r) by dividing by N[r] (Eq. 2):

\begin{align*} Q(S,r) = \cfrac{n[S,r]}{N[r]} && \text{(2)} \end{align*}

Next, the first and second moments of this probability distribution are estimated using Eq. (3), (4) respectively:

\begin{align*} Z(1) = \sum{S(r)\times Q(S,r)} && \text{ (3)} \end{align*}

\begin{align*} Z(2) = \sum{S^2(r)\times Q(S,r)} && \text{(4)} \end{align*}

Lacunarity \Lambda(r) can now be computed as (Eq. 5):

\begin{align*} \Lambda(r) = \cfrac{Z(2)}{Z^2(1)} && \text{(5)} \end{align*}

If the raster is continuous the first and second moment can be expressed as the mean E[S(r)] and variance Var[S(r)] of the box mass values, respectively. Lacunarity can now be derived as (Eg. 6):

\begin{align*} \Lambda(r) = 1 + \cfrac{Var[S(r)]}{E^2[S(r)]} && \text{(6)} \end{align*}

r should not be greater than half of the shorter dimension of the input raster. If r is not provided by the user, it will be automatically generated as the function r_p = 2^p + 1, where p is a whole number commencing at 1 until r_p reaches half of the shorter dimension.

Examples

For demonstration purposes I replicated the examples provided by Hoechstetter et al. (2011).

library(spatLac)
library(terra)

sample_rast <- rast("data_raw/hoechstetter.tif")

sample_lac <- lacunarity(x = sample_rast, r_vec = 3:50, plot = TRUE)

To compute Lacunarity for a larger study, it can be computed for all raster objects (ending with .tif) in a folder. Furthermore, individual and summary plots can be automatically created, too.

lacunarity(x = "folder_path_with_TIF_files",
           save_plot = "folder_path_to_save_plots")

About

Citation

Run this command to get info on how to cite this package.

citation("spatLac")
#> 
#> To cite spatLac in publications use:
#> 
#>   Brinkmann, S.T. (2021). spatLac: R package for computing Lacunarity
#>   for Spatial Raster. doi: 10.5281/zenodo.5786547.
#> 
#> A BibTeX entry for LaTeX users is
#> 
#>   @Manual{,
#>     title = {spatLac: R package for computing Lacunarity for Spatial Raster},
#>     author = {Brinkmann S.T.},
#>     year = {2021},
#>     url = {https://doi.org/10.5281/zenodo.5786547},
#>   }

Package contributors

Brinkmann, Sebastian (Creator and author) e-mail: sebastian.brinkmann@fau.de

Dr. S.M. Labib (Contributor) e-mail: sml80@medschl.cam.ac.uk

Bibliography

Dong P. (2000): Lacunarity for Spatial Heterogeneity Measurement in GIS. Geographic Information Sciences 6:1, pages 20-26. DOI: 10.1080/10824000009480530.

Hoechstetter S., Walz U. and Thinh N.X. (2011): Adapting lacunarity techniques for gradient-based analyses of landscape surfaces. Ecological Complexity 8:3, pages 229-238. DOI: 10.1016/j.ecocom.2011.01.001.

Labib S.M., Lindley S. and Huck J.J. (2020): Scale effects in remotely sensed greenspace metrics and how to mitigate them for environmental health exposure assessment. Computers, Environment and Urban Systems 82. DOI: 10.1016/j.compenvurbsys.2020.101501.

Plotnick R.E., Gardner R.H. and O’Neill R.V. (1993): Lacunarity indices as measures of landscape texture. Landscape Ecology 8, pages 201–211. DOI: 10.1007/BF00125351