Compute fuzzy geometric set operations

The spatial plateau set operations plateau intersection, plateau union, and plateau difference implement the respective operations fuzzy geometric intersection, fuzzy geometric union, and fuzzy geometric difference.

Usage

spa_intersection(pgo1, pgo2, itype = "min", as_pcomposition = FALSE)

spa_union(pgo1, pgo2, utype = "max", as_pcomposition = FALSE)

spa_difference(pgo1, pgo2, dtype = "f_diff", as_pcomposition = FALSE)

spa_common_points(pline1, pline2, itype = "min")

Arguments

pgo1

A pgeometry object of any type.

pgo2

A pgeometry object of any type.

itype

A character value that indicates the name of a function implementing a t-norm. The default value is "min", which is the standard operator of the intersection.

as_pcomposition

A logical value; if TRUE, it returns a spatial plateau composition object.

utype

A character value that refers to a t-conorm. The default value is "max", which is the standard operator of the union.

dtype

A character value that indicates the name of a difference operator. The default value is "f_diff", which implements the standard fuzzy difference.

pline1

A pgeometry object of the type PLATEAULINE.

pline2

A pgeometry object of the type PLATEAULINE.

Value

A pgeometry object that is the result of a fuzzy geometric set operation.

Details

They receive two pgeometry objects of the any type as input and yield another pgeometry object as output. The family of fuzzy geometric set operations consists of the following functions:

• spa_intersection() computes the geometric intersection of two spatial plateau objects. The membership degree of common points are calculated by using a t-norm operator given by the parameter itype. Currently, it can assume "min" (default) or "prod".

• spa_union() computes the geometric union of two spatial plateau objects. The membership degree of common points are calculated by using a t-conorm operator given by the parameter utype. Currently, it can assume "max" (default).

• spa_difference() computes the geometric difference of two spatial plateau objects. The membership degree of common points are calculated by using a difference operator given by the parameter dtype. Currently, it can assume "f_diff" (default fuzzy difference), "f_bound_diff" (fuzzy bounded difference), "f_symm_diff" (fuzzy symmetric difference), or "f_abs_diff" (fuzzy absolute difference).

Other t-norms, t-conorms, and difference operators can be implemented and given as values for the parameters itype, utype, and dtype, respectively. For this, the following steps should be performed:

1. Implement your function that accepts two numeric values in [0, 1] as inputs and yields another numeric value in [0, 1] as output. Recall that t-norms and t-conorms must have some specific properties according to the fuzzy set theory.

2. Use the name of your function as the character value of the corresponding parameter itype, utype, or dtype.

An example of operator is the source code of f_bound_diff():

f_bound_diff <- function(x, y) { max(0, (x - y)) }

The spa_common_points() is deprecated. In the past, it computed the common points of two plateau line objects; now, you can use spa_intersection().

References

Carniel, A. C.; Venâncio, P. V. A. B; Schneider, M. fsr: An R package for fuzzy spatial data handling. Transactions in GIS, vol. 27, no. 3, pp. 900-927, 2023.

Underlying concepts and formal definitions of spatial plateau set operations are explained in detail in:

• Carniel, A. C.; Schneider, M. Spatial Plateau Algebra: An Executable Type System for Fuzzy Spatial Data Types. In Proceedings of the 2018 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE 2018), pp. 1-8, 2018.

• Carniel, A. C.; Schneider, M. Spatial Data Types for Heterogeneously Structured Fuzzy Spatial Collections and Compositions. In Proceedings of the 2020 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE 2020), pp. 1-8, 2020.

Examples

library(ggplot2)

# Point components
pcp1 <- create_component("POINT(0 0)", 0.3)
pcp2 <- create_component("MULTIPOINT((2 2), (2 4), (2 0))", 0.5)
pcp3 <- create_component("MULTIPOINT((1 1), (3 1), (1 3), (3 3))", 0.9)

pcp4 <- create_component("MULTIPOINT((2 2), (2 4), (3 2))", 1)
pcp5 <- create_component("MULTIPOINT((0 0), (2 3))", 0.7)
pcp6 <- create_component("MULTIPOINT((0 1), (3 3))", 0.85)
pcp7 <- create_component("MULTIPOINT((1 0), (4 2))", 0.4)
# Line components
lcp1 <- create_component("LINESTRING(0 0, 1 1.5)", 0.2)
lcp2 <- create_component("LINESTRING(1 3, 1 2, 2 0.5)", 0.5)
lcp3 <- create_component("LINESTRING(2 1.2, 3 1.6, 4 4)", 0.7)

lcp4 <- create_component("LINESTRING(1 1.5, 2 1.2)", 1.0)
lcp5 <- create_component("LINESTRING(-1 1, 2 2)", 0.9)
# Polygon components
rcp1 <- create_component("POLYGON((0 0, 1 4, 2 2, 0 0))", 0.4)
rcp2 <- create_component("POLYGON((2 0.5, 4 1, 4 0, 2 0.5))", 0.8)

# Creating plateau point objects
ppoint1 <- create_pgeometry(list(pcp1, pcp2, pcp3), "PLATEAUPOINT")
ppoint2 <- create_pgeometry(list(pcp4, pcp5, pcp6, pcp7), "PLATEAUPOINT")
# Creating plateau line objects
pline1 <- create_pgeometry(list(lcp1, lcp2, lcp3), "PLATEAULINE")
pline2 <- create_pgeometry(list(lcp4, lcp5), "PLATEAULINE")
# Creating a plateau region objects
pregion <- create_pgeometry(list(rcp1, rcp2), "PLATEAUREGION")

# Defining a wrapper to combine plots side by side, for convenience
combine_plots <- function(plot1, plot2, plot3) {
  # setting the same range of coordinates and removing the legend of plot1 and plot2
  plot1 <- plot1 + coord_sf(xlim = c(0, 4), ylim = c(0, 4)) + theme(legend.position = "none")
  plot2 <- plot2 + coord_sf(xlim = c(0, 4), ylim = c(0, 4)) + theme(legend.position = "none")
  plot3 <- plot3 + coord_sf(xlim = c(0, 4), ylim = c(0, 4))
  ggplot() +
    annotation_custom(ggplotGrob(plot1), xmin = 0, xmax = 0.5, ymin = 0.5, ymax = 1) +
    annotation_custom(ggplotGrob(plot2), xmin = 0.5, xmax = 1, ymin = 0.5, ymax = 1) +
    annotation_custom(ggplotGrob(plot3), xmin = 0, xmax = 1, ymin = 0, ymax = 0.5) +
    coord_cartesian(xlim = c(0, 1), ylim = c(0, 1)) +
    theme_void()
}

plot_ppoint1 <- plot(ppoint1) + ggtitle("Plateau point 1")
plot_ppoint2 <- plot(ppoint2) + ggtitle("Plateau point 2")
plot_pline1 <- plot(pline1) + ggtitle("Plateau line 1")
plot_pline2 <- plot(pline2) + ggtitle("Plateau line 2")
plot_pregion <- plot(pregion) + ggtitle("Plateau region")

# Computing the intersection
ppoints_intersec <- spa_intersection(ppoint1, ppoint2)
plot_inter <- plot(ppoints_intersec) + ggtitle("Intersection")
combine_plots(plot_ppoint1, plot_ppoint2, plot_inter)


# \dontrun{
# varying the t-norm 
ppoints_intersec <- spa_intersection(ppoint1, ppoint2, itype = "prod")
plot_inter <- plot(ppoints_intersec) + ggtitle("Intersection (prod)")
combine_plots(plot_ppoint1, plot_ppoint2, plot_inter)


plines_intersec <- spa_intersection(pline1, pline2)
plot_inter <- plot(plines_intersec) + ggtitle("Intersection")
combine_plots(plot_pline1, plot_pline2, plot_inter)


pregion_pline_intersec <- spa_intersection(pline1, pregion)
plot_inter <- plot(pregion_pline_intersec) + ggtitle("Intersection")
combine_plots(plot_pline1, plot_pregion, plot_inter)


# Computing the union
ppoints_union <- spa_union(ppoint1, ppoint2)
plot_union <- plot(ppoints_union) + ggtitle("Union")
combine_plots(plot_ppoint1, plot_ppoint2, plot_union)


plines_union <- spa_union(pline1, pline2)
plot_union <- plot(plines_union) + ggtitle("Union")
combine_plots(plot_pline1, plot_pline2, plot_union)


pregion_pline_union <- spa_union(pline1, pregion)
plot_union <- plot(pregion_pline_union) + ggtitle("Union")
combine_plots(plot_pline1, plot_pregion, plot_union)


# Computing the difference
ppoints_diff <- spa_difference(ppoint1, ppoint2)
plot_diff <- plot(ppoints_diff) + ggtitle("Difference")
combine_plots(plot_ppoint1, plot_ppoint2, plot_diff)


plines_diff <- spa_difference(pline1, pline2)
plot_diff <- plot(plines_diff) + ggtitle("Difference")
combine_plots(plot_pline1, plot_pline2, plot_diff)


pregion_pline_diff <- spa_difference(pline1, pregion)
plot_diff <- plot(pregion_pline_diff) + ggtitle("Difference")
combine_plots(plot_pline1, plot_pregion, plot_diff)

# }