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I have a data frame in r with x and y coordinates and each point has a circumference (tree_crown) around it. I'm wanting to build a raster file that each cell would have a value equal to the percent of the area being covered by the points (and the area of the circles around them). The x and y range is from 0 to 5147.815, and I'm wanting the resolution to be 1x1

My data looks like this: enter image description here

Here is the idea behind the output:

enter image description here

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You can construct polygons like the ones shown in your image by converting your points into an sf object and then using st_buffer. To compute the fraction of each grid cell that is covered by the polygons, you can use exactextractr::coverage_fraction. It's important to first dissolve the buffered points into a single polygon using st_union, so that areas covered by more than one tree aren't double-counted. Here's an example using constructed data:

library(sf)
library(exactextractr)
library(raster)

set.seed(123)

n_trees <- 50

trees <- data.frame(x_cor=runif(n_trees, min=0, max=50),
                    y_cor=runif(n_trees, min=0, max=50),
                    tree_crown=runif(n_trees, min=0.5, max=5))

grid <- raster(xmn=0, xmx=50, ymn=0, ymx=50, res=10, crs=NA)

canopy <- st_as_sf(trees, coords=c('x_cor', 'y_cor')) %>%
  st_buffer(dist=.$tree_crown) %>%
  st_union()

pct_cov <- coverage_fraction(grid, canopy)[[1]]

plot(pct_cov)
plot(st_geometry(canopy), add=TRUE)

enter image description here

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