How do you get the medial axis of a multipolygon using CGAL?

I would like to extract the medial axis of a MultiPolygon using CGAL.

Looking through the CGAL documentation I only see functions to make a straight skeleton.

How can I use CGAL to get the medial axis?

Solution

Overview

If we have a polygon (a planar domain with straight edges, possibly convex) then the edges which form the medial axis of a polygon are a subset of the multipolygon's Voronoi diagram.

Our strategy, then, is to find the polygon's Voronoi diagram and then remove edges from the diagram until we are left with the medial axis. If we have a MultiPolygon we just repeat the process for each of its constituent polygons.

I'll demonstrate the process visually below and then present code to accomplish it. The data for the test polygon I'll use is this WKT:

MULTIPOLYGON(((223.83073496659313 459.9703924976702,256.1247216035629 304.08821449578033,1.1135857461033538 187.63424160514955,468.8195991091309 189.21374898389445,310.69042316258424 318.7630937196408,432.07126948775 451.93407043677666,453.2293986636971 612.1086753947116,32.29398663697134 616.325100949687,223.83073496659313 459.9703924976702)))

The polygon looks like this:

The full Voronoi diagram of the polygon looks like this:

At this point we can consider three types of vertices in the Voronoi diagram: those that are properly within the polygon, those which lie on its border, and those which are properly outside the polygon. It's visually obvious that the medial axis does not contain any edge of the Voronoi diagram which is defined in part by a point which is properly outside the polygon. Therefore, we remove these exterior edges. Leaving us with this:

Not all of the Voronoi edges in the above figure are part of the medial axis. It turns out that those edges which are in part defined by the concave vertices (aka "reflex vertices") of the polygon are not part of the medial axis. The concave vertices of the polygon are those that "point inwards". Computationally, we can find them by using cross-products. (Incidentally, the area of a polygon can be calculated by summing the cross-products of each of its vertices.)

Removing the edges discussed above gives us the final medial axis:

The Code

The following C++ code calculates the medial axis of a multipolygon.

// Compile with: clang++ -DBOOST_ALL_NO_LIB -DCGAL_USE_GMPXX=1 -O3 -g -Wall -Wextra -pedantic -march=native -frounding-math main.cpp -lgmpxx -lmpfr -lgmp
#include <CGAL/Boolean_set_operations_2.h>
#include <CGAL/Exact_predicates_exact_constructions_kernel.h>
#include <CGAL/IO/WKT.h>
#include <CGAL/Polygon_with_holes_2.h>
#include <CGAL/Segment_Delaunay_graph_2.h>
#include <CGAL/Segment_Delaunay_graph_adaptation_policies_2.h>
#include <CGAL/Segment_Delaunay_graph_adaptation_traits_2.h>
#include <CGAL/Segment_Delaunay_graph_traits_2.h>
#include <CGAL/squared_distance_2.h>
#include <CGAL/Voronoi_diagram_2.h>

#include <algorithm>
#include <deque>
#include <fstream>
#include <iomanip>
#include <iostream>
#include <map>
#include <set>
#include <stdexcept>
#include <unordered_set>

typedef CGAL::Exact_predicates_exact_constructions_kernel              K;
typedef CGAL::Segment_Delaunay_graph_traits_2<K>                       Gt;
typedef CGAL::Segment_Delaunay_graph_2<Gt>                             SDG2;
typedef CGAL::Segment_Delaunay_graph_adaptation_traits_2<SDG2>         AT;
typedef CGAL::Segment_Delaunay_graph_degeneracy_removal_policy_2<SDG2> AP;
typedef CGAL::Voronoi_diagram_2<SDG2, AT, AP>      VoronoiDiagram;
typedef AT::Site_2                                 Site_2;
typedef AT::Point_2                                Point_2;
typedef VoronoiDiagram::Locate_result              Locate_result;
typedef VoronoiDiagram::Vertex_handle              Vertex_handle;
typedef VoronoiDiagram::Face_handle                Face_handle;
typedef VoronoiDiagram::Halfedge_handle            Halfedge_handle;
typedef VoronoiDiagram::Ccb_halfedge_circulator    Ccb_halfedge_circulator;
typedef VoronoiDiagram::Bounded_halfedges_iterator BHE_Iter;
typedef VoronoiDiagram::Halfedge                   Halfedge;
typedef VoronoiDiagram::Vertex                     Vertex;
typedef CGAL::Polygon_with_holes_2<K> Polygon;
typedef std::deque<Polygon> MultiPolygon;

/// Creates a hash of a Point_2, used for making O(1) point lookups
// struct Point2Hash {
//   size_t operator()(const Point_2 &pt) const {
//     std::hash<double> hasher;
//     auto seed = hasher(pt.x());
//     // boost::hash_combine from https://stackoverflow.com/q/35985960/752843
//     seed ^= hasher(pt.y()) + 0x9e3779b9 + (seed<<6) + (seed>>2);
//     return seed;
//   }
// };

typedef std::set<Point_2> Point2_Set;
typedef std::map<Vertex_handle, int> VH_Int_Map;


/// Holds a more accessible description of the Voronoi diagram
struct MedialData {
  /// Map of vertices comprising the Voronoi diagram
  VH_Int_Map vertex_handles;
  /// List of edges in the diagram (pairs of the vertices above)
  std::vector<std::pair<int, int>> edges;
  /// Medial axis up governor. 1:1 correspondance with edges above.
  std::vector<VoronoiDiagram::Delaunay_graph::Vertex_handle> ups;
  /// Medial axis down governor. 1:1 correspondance with edges above.
  std::vector<VoronoiDiagram::Delaunay_graph::Vertex_handle> downs;
};


/// Read well-known text from @p filename to obtain shape boundary
MultiPolygon get_wkt_from_file(const std::string& filename){
  std::ifstream fin(filename);
  MultiPolygon mp;
  CGAL::read_multi_polygon_WKT(fin, mp);

  if(mp.empty()){
    throw std::runtime_error("WKT file '" + filename + "' was empty!");
  }
  for(const auto &poly: mp){
    if(poly.outer_boundary().size()==0){
      throw std::runtime_error("WKT file '" + filename + "' contained a polygon without an outer boundary!");
    }
  }

  return mp;
}


/// Converts a MultiPolygon into its corresponding Voronoi diagram
VoronoiDiagram convert_mp_to_voronoi_diagram(const MultiPolygon &mp){
  VoronoiDiagram vd;

  const auto add_segments_to_vd = [&](const auto &poly){
    for(std::size_t i=0;i<poly.size();i++){
      std::cerr<<i<<" "<<std::fixed<<std::setprecision(10)<<poly[i]<<std::endl;
      // Modulus to close the loop
      vd.insert(
        Site_2::construct_site_2(poly[i], poly[(i+1)%poly.size()])
      );
    }
  };

  for(const auto &poly: mp){                    // For each polygon in MultiPolygon
    std::cout<<poly<<std::endl;                 // Print polygon to screen for debugging
    add_segments_to_vd(poly.outer_boundary());  // Add the outer boundary
    for(const auto &hole : poly.holes()){       // And any holes
      add_segments_to_vd(hole);
    }
  }

  if(!vd.is_valid()){
    throw std::runtime_error("Voronoi Diagram was not valid!");
  }

  return vd;
}


/// Find @p item in collection @p c or add it if not present.
/// Returns the index of `item`'s location
int find_or_add(VH_Int_Map &c, const Vertex_handle &item){
  // Map means we can do this in log(N) time
  if(c.count(item) == 0){
    c.emplace(item, c.size());
    return c.size() - 1;
  }

  return c.at(item);
}


/// Convert a map of <T, int> pairs to a vector of `T` ordered by increasing int
std::vector<Vertex_handle> map_to_ordered_vector(const VH_Int_Map &m){
  std::vector<std::pair<Vertex_handle, int>> to_sort(m.begin(), m.end());
  to_sort.reserve(m.size());
  std::sort(to_sort.begin(), to_sort.end(), [](const auto &a, const auto &b){
    return a.second < b.second;
  });

  std::vector<Vertex_handle> ret;
  ret.reserve(to_sort.size());
  std::transform(begin(to_sort), end(to_sort), std::back_inserter(ret),
    [](auto const& pair){ return pair.first; }
  );

  return ret;
}


/// Find vertex handles which are in the interior of the MultiPolygon
std::set<Vertex_handle> identify_vertex_handles_inside_mp(
  const VoronoiDiagram &vd,
  const MultiPolygon &mp
){
  // Used to accelerate interior lookups by avoiding Point-in-Polygon checks for
  // vertices we've already considered
  std::set<Vertex_handle> considered;
  // The set of interior vertices we are building
  std::set<Vertex_handle> interior;

  for (
      auto edge_iter = vd.bounded_halfedges_begin();
      edge_iter != vd.bounded_halfedges_end();
      edge_iter++
  ) {
    // Determine if an orientation implies an interior vertex
    const auto inside = [](const auto &orientation){
      return orientation == CGAL::ON_ORIENTED_BOUNDARY || orientation == CGAL::POSITIVE;
    };

    // Determine if a vertex is in the interior of the multipolygon and, if so,
    // add it to `interior`
    const auto vertex_in_mp_interior = [&](const Vertex_handle& vh){
      // Skip vertices which have already been considered, since a vertex may
      // be connected to multiple halfedges
      if(considered.count(vh)!=0){
        return;
      }
      // Ensure we don't look at a vertex twice
      considered.insert(vh);
      // Determine if the vertex is inside of any polygon of the MultiPolygon
      const auto inside_of_a_poly = std::any_of(
        mp.begin(), mp.end(), [&](const auto &poly) {
          return inside(CGAL::oriented_side(vh->point(), poly));
        }
      );
      // If the vertex was inside the MultiPolygon make a note of it
      if(inside_of_a_poly){
        interior.insert(vh);
      }
    };

    // Check both vertices of the current halfedge of the Voronoi diagram
    vertex_in_mp_interior(edge_iter->source());
    vertex_in_mp_interior(edge_iter->target());
  }

  return interior;
}


/// The medial axis is formed by building a Voronoi diagram and then removing
/// the edges of the diagram which connect to the concave points of the
/// MultiPolygon. Here, we identify those concave points
Point2_Set identify_concave_points_of_mp(const MultiPolygon &mp){
  Point2_Set concave_points;

  // Determine cross-product, given three points. The sign of the cross-product
  // determines whether the point is concave or convex.
  const auto z_cross_product = [](const Point_2 &pt1, const Point_2 &pt2, const Point_2 &pt3){
    const auto dx1 = pt2.x() - pt1.x();
    const auto dy1 = pt2.y() - pt1.y();
    const auto dx2 = pt3.x() - pt2.x();
    const auto dy2 = pt3.y() - pt2.y();
    return dx1 * dy2 - dy1 * dx2;
  };

  // Loop through all the points in a polygon, get their cross products, and
  // add any concave points to the set we're building.
  // `sense` should be `1` for outer boundaries and `-1` for holes, since holes
  // will have points facing outward.
  const auto consider_polygon = [&](const auto &poly, const double sense){
    for(size_t i=0;i<poly.size()+1;i++){
      const auto zcp = z_cross_product(
        poly[(i + 0) % poly.size()],
        poly[(i + 1) % poly.size()],
        poly[(i + 2) % poly.size()]
      );
      if(sense*zcp < 0){
        concave_points.insert(poly[(i + 1) % poly.size()]);
      }
    }
  };

  // Loop over the polygons of the MultiPolygon, as well as their holes
  for(const auto &poly: mp){
    // Outer boundary has positive sense
    consider_polygon(poly.outer_boundary(), 1);
    for(const auto &hole: poly.holes()){
      // Inner boundaries (holes) have negative (opposite) sense
      consider_polygon(hole, -1);
    }
  }

  return concave_points;
}


/// Print the points which collectively comprise the medial axis
void print_medial_axis_points(const MedialData &md, const std::string &filename){
  std::ofstream fout(filename);
  fout<<"x,y"<<std::endl;
  for (const auto &vh : map_to_ordered_vector(md.vertex_handles)) {
    fout << vh->point().x() << "," << vh->point().y() << std::endl;
  }
}


/// Prints the edges of the medial diagram
void print_medial_axis_edges(const MedialData &md, const std::string &filename){
  std::ofstream fout(filename);
  fout<<"SourceIdx,TargetIdx,UpGovernorIsPoint,DownGovernorIsPoint"<<std::endl;
  for (std::size_t i = 0; i < md.edges.size(); i++) {
    fout << md.edges[i].first        << ","
          << md.edges[i].second      << ","
          << md.ups[i]->is_point()   << "," // Is up-governor a point?
          << md.downs[i]->is_point()        // Is down-governor a point?
          << std::endl;
  }
}


MedialData filter_voronoi_diagram_to_medial_axis(
  const VoronoiDiagram &vd,
  const MultiPolygon &mp
){
  MedialData ret;

  const auto interior = identify_vertex_handles_inside_mp(vd, mp);
  const auto concave_points = identify_concave_points_of_mp(mp);

  // Returns true if a point is a concave point of the MultiPolygon
  const auto pconcave = [&](const Point_2 &pt){
    return concave_points.count(pt) != 0;
  };

  // The Voronoi diagram is comprised of a number of vertices connected by lines
  // Here, we go through each edge of the Voronoi diagram and determine which
  // vertices it's incident on. We add these vertices to `ret.vertex_handles`
  // so that they will have unique ids.

  // The `up` and `down` refer to the medial axis governors - that which
  // constrains each edge of the Voronoi diagram
  for (
      auto edge_iter = vd.bounded_halfedges_begin();
      edge_iter != vd.bounded_halfedges_end();
      edge_iter++
  ) {
    const Halfedge& halfedge = *edge_iter;
    const Vertex_handle& v1p = halfedge.source();
    const Vertex_handle& v2p = halfedge.target();

    // Filter Voronoi diagram to only the part in the interior of the
    // MultiPolygon
    if(interior.count(v1p)==0 || interior.count(v2p)==0){
      continue;
    }

    // Drop those edges of the diagram which are not part of the medial axis
    if(pconcave(v1p->point()) || pconcave(v2p->point())){
      continue;
    }

    // Get unique ids for edge vertex handle that's part of the medial axis
    const auto id1 = find_or_add(ret.vertex_handles, v1p);
    const auto id2 = find_or_add(ret.vertex_handles, v2p);
    ret.edges.emplace_back(id1, id2);

    // Keep track of the medial axis governors
    ret.ups.push_back(halfedge.up());
    ret.downs.push_back(halfedge.down());
  }

  return ret;
}


int main(int argc, char** argv) {
  if(argc!=2){
      std::cerr<<"Syntax: "<<argv[0]<<" <Shape Boundary WKT>"<<std::endl;
      return -1;
  }

  CGAL::set_pretty_mode(std::cout);

  const auto mp = get_wkt_from_file(argv[1]);
  const auto voronoi = convert_mp_to_voronoi_diagram(mp);
  const auto ma_data = filter_voronoi_diagram_to_medial_axis(voronoi, mp);

  print_medial_axis_points(ma_data, "voronoi_points.csv");
  print_medial_axis_edges(ma_data, "voronoi_edges.csv");

  return 0;
}

You can plot the resulting medial axis with this Python script:

#!/usr/bin/env python3

import matplotlib.pyplot as plt
import numpy as np
from shapely import wkt

fig, ax = plt.subplots()

# Output file from C++ medial axis code
pts = np.loadtxt("build/voronoi_points.csv", skiprows=1, delimiter=',')

fig4 = wkt.loads(open("fig4_data.wkt").read())
for geom in fig4.geoms:
  xs, ys = geom.exterior.xy
  ax.plot(xs, ys, '-ok', lw=4)

# Output file from C++ medial axis code
ma = np.loadtxt("build/voronoi_edges.csv", dtype=int, skiprows=1, delimiter=',')
for mal in ma:
  print(mal)
  ax.plot((pts[mal[0]][0], pts[mal[1]][0]), (pts[mal[0]][1], pts[mal[1]][1]), '-o')

plt.show()