minkowski_difference

Geometric functions minkowski_difference

syntax

bp_minkowski_difference(polygon_data_item, kernel)
bp_minkowski_difference(polygon_data_item, size, variant)

and the same two forms for bg_minkowski_difference, cgal_minkowski_difference and geos_minkowski_difference.

definition

The Minkowski difference (erosion) of a geometry A by a kernel K is the set of positions at which K fits entirely inside A. It shrinks A by the kernel’s shape, and it is what the old bp_polygon_d4HV family of operators did with twelve fixed kernels.

It is computed as

A (-) K  =  A \ ( (R \ A) (+) -K )

with R a box containing A with room to spare and -K the kernel reflected through the origin: eroding A is dilating everything outside it. All four backends use this identity, on top of their own minkowski_sum.

Since GeoDMS 20.18.0. See minkowski_sum for why the kernel became an argument, and Boost polygon functions for the migration table from the depreciated _d4HV … _dXD operators.

description

Everything in minkowski_sum about the two argument forms, the six variant names, the kernel restrictions and the four backends applies here unchanged, with one difference: this operator erodes, so it accepts the old d-prefixed variant spellings ('d4HV', 'd8D', …) and rejects an i prefix, naming minkowski_sum instead.

Erosion is not the inverse of dilation. Dilating and then eroding by the same kernel is the morphological closing: it fills notches narrower than the kernel and can only add, never remove, so the result always contains the original. Eroding and then dilating is the opening: it removes protrusions thinner than the kernel, and the result is always contained in the original. Neither round-trips back to where it started, which is exactly what makes them useful for cleaning up geometry.

Erosion can empty a feature: anything thinner than the kernel disappears, and a feature that vanishes entirely leaves an empty sequence rather than a null.

applies to

As minkowski_sum.

conditions

As minkowski_sum. In addition, the erosion works within a bounding box grown past the kernel’s reach, so a geometry whose coordinates already sit near the bp_ integer limit (2^25) has less headroom here than under minkowski_sum.

since version

20.18.0

examples

// shrink every parcel by 5 units, with a square kernel
attribute<ipoint> parcel_core (polygon, parcel) := bp_minkowski_difference(parcel/border, 5.0, '4HV');

// morphological opening: drop slivers thinner than 5 units, keep the rest in place
attribute<ipoint> opened (polygon, parcel) :=
    bp_minkowski_sum(bp_minkowski_difference(parcel/border, 5.0, 'd4HV'), 5.0, 'i4HV');

// morphological closing: fill gaps narrower than 5 units
attribute<ipoint> closed (polygon, parcel) :=
    bp_minkowski_difference(bp_minkowski_sum(parcel/border, 5.0, 'i4HV'), 5.0, 'd4HV');