DGtal  0.9.3
testVoronoiMap.cpp
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1 
30 #include <iostream>
32 #include <array>
33 #include <algorithm>
34 
35 #include "DGtal/base/Common.h"
36 #include "DGtal/helpers/StdDefs.h"
37 #include "DGtal/images/CConstImage.h"
38 #include "DGtal/geometry/volumes/distance/VoronoiMap.h"
39 #include "DGtal/geometry/volumes/distance/ExactPredicateLpSeparableMetric.h"
40 #include "DGtal/geometry/volumes/distance/InexactPredicateLpSeparableMetric.h"
41 #include "DGtal/geometry/volumes/distance/DistanceTransformation.h"
42 #include "DGtal/kernel/BasicPointPredicates.h"
43 #include "DGtal/io/boards/Board2D.h"
44 #include "DGtal/io/colormaps/HueShadeColorMap.h"
46 
47 using namespace std;
48 using namespace DGtal;
50 // Functions for testing class VoronoiMap.
52 
53 
54 template<typename Point>
55 double mynorm(const Point &point, const double p)
56 {
57  double res=0.0;
58  for(unsigned int i=0; i< Point::dimension; i++)
59  res += std::pow ( (double)abs(point[i]) , p);
60 
61  return std::pow(res, 1.0/p);
62 }
63 
64 template <typename VoroMap>
65 void saveVoroMap(const std::string &filename,const VoroMap &output,const double p)
66 {
67  typedef HueShadeColorMap<double,2> Hue;
68 
69  const double maxdt = mynorm( output.domain().upperBound() - output.domain().lowerBound(), p );
70 
71  Board2D board;
72  Hue hue(0, maxdt);
73 
74  for(typename VoroMap::Domain::ConstIterator it = output.domain().begin(),
75  itend = output.domain().end();
76  it != itend; ++it)
77  {
78  typename VoroMap::Value point = output(*it);
79  board << CustomStyle( (*it).className(), new CustomColors( hue(mynorm(point- (*it),p)),
80  hue(mynorm(point- (*it),p))))
81  << (*it);
82  }
83 
84  board.saveSVG(filename.c_str());
85 }
86 
95 template < typename Point, typename Domain >
97 {
98  auto const & lowerBound = aDomain.lowerBound();
99  auto const & upperBound = aDomain.upperBound();
100 
101  for ( std::size_t i = 0; i < Domain::dimension; ++i )
102  aPoint[i] = ( aPoint[i] - 2*lowerBound[i] + upperBound[i] + 1 ) % ( upperBound[i] - lowerBound[i] + 1 ) + lowerBound[i];
103 
104  return aPoint;
105 }
106 
117 template < std::size_t N >
118 std::array<bool, N> getPeriodicityFromInteger( std::size_t anInteger )
119 {
120  std::array<bool, N> periodicity;
121  for ( std::size_t i = 0, mask = 1 ; i < N ; ++i, mask *= 2 )
122  periodicity[i] = anInteger & mask;
123 
124  return periodicity;
125 }
126 
133 template < std::size_t N >
134 std::string formatPeriodicity( std::array<bool, N> const& aPeriodicity )
135 {
136  std::string str;
137  for ( std::size_t i = 0; i < N; ++i )
138  str += aPeriodicity[i] ? '1' : '0';
139 
140  return str;
141 }
142 
145 template < typename Set, typename Voro>
146 bool checkVoronoi(const Set &aSet, const Voro & voro)
147 {
148  typedef typename Voro::Point Point;
149 
150  // Domain extent
151  Point const extent = voro.domain().upperBound() - voro.domain().lowerBound() + Point::diagonal();
152 
153  // Site shifting depending on the periodicity.
154  std::vector< typename Voro::PeriodicitySpec > periodicShift;
155  for ( std::size_t i = 0; i < ( 1u << Voro::Space::dimension ); ++i )
156  {
157  const auto periodicity = getPeriodicityFromInteger< Voro::Space::dimension >( i );
158 
159  // Checking if this periodicity possibility is valid.
160  bool isValid = true;
161  for ( std::size_t j = 0; j < periodicity.size(); ++j )
162  if ( periodicity[j] && ! voro.isPeriodic(j) )
163  {
164  isValid = false;
165  break;
166  }
167 
168  if ( isValid )
169  periodicShift.push_back( periodicity );
170  }
171 
172  // Checking site for all domain points.
173  for ( auto const& pt : voro.domain() )
174  {
175  // Calculating reference (raw-)distance.
176  const Point psite = voro(pt);
177  const auto dist = voro.metric()->rawDistance( pt, psite );
178 
179  // Checking that the reference site is actually a given site.
180  // Also testing projectPoint method.
181  if ( std::find( aSet.begin(), aSet.end(), voro.projectPoint( psite ) ) == aSet.end() )
182  {
183  trace.error() << "The Voro point " << psite
184  << " projected to " << voro.projectPoint( psite )
185  << " is not a valid site." << std::endl;
186  return false;
187  }
188 
189  // Checking if we can found a better site.
190  for ( auto site : aSet )
191  {
192  // Trying all shifting possibilities of the site depending on the domain periodicity.
193  for ( auto const & periodicity : periodicShift )
194  {
195  auto currSite = site;
196 
197  // Shifting site.
198  for ( std::size_t dim = 0; dim < Voro::Space::dimension ; ++dim )
199  if ( periodicity[dim] )
200  currSite[dim] += ( pt[dim] < currSite[dim] ? -1 : 1 ) * extent[dim];
201 
202  // Checking raw-distance.
203  const auto dbis = voro.metric()->rawDistance( pt, currSite );
204  if ( dbis < dist )
205  {
206  trace.error() << "DT Error at " << pt
207  << " Voro:" << psite << " (" << dist << ")"
208  << " from set:" << site << "(" << dbis << ")"
209  << " projected from " << currSite << "." << std::endl;
210  return false;
211  }
212  }
213  }
214  }
215 
216  return true;
217 }
218 
219 
221 {
224 
225  return true;
226 }
227 
232 bool testVoronoiMap( std::array<bool, 2> const& periodicity = { {false, false} } )
233 {
234  unsigned int nbok = 0;
235  unsigned int nb = 0;
236 
237  trace.beginBlock ( "Testing VoronoiMap2D with periodicity " + formatPeriodicity(periodicity) );
238 
239  Z2i::Point a(-10,-10);
240  Z2i::Point b(10,10);
241  Z2i::Domain domain(a,b);
242 
243  Z2i::DigitalSet mySet(domain);
244 
245  for(Z2i::Domain::ConstIterator it = domain.begin(), itend = domain.end();
246  it != itend;
247  ++it)
248  mySet.insertNew( *it );
249 
250 
251  Z2i::DigitalSet sites(domain);
252 
253  sites.insertNew( Z2i::Point(3,-6));
254  sites.insertNew( Z2i::Point(9,0));
255  sites.insertNew( Z2i::Point(-2,0));
256 
257  for(Z2i::DigitalSet::ConstIterator it = sites.begin(), itend = sites.end();
258  it != itend; ++it)
259  mySet.erase (*it);
260 
261 
262 
265  L2Metric l2;
266  Voro2 voro(&domain, &mySet, &l2, periodicity);
267 
268  for(int j=-10; j <= 10; j++)
269  {
270  for(int i=-10; i<=10; i++)
271  trace.info() << "("<<voro( Z2i::Point(i,j))[0]<<","<< voro( Z2i::Point(i,j))[1]<<") ";
272  trace.info()<<std::endl;
273  }
274 
275  trace.info()<<"Exporting to SVG"<<std::endl;
276 
277  Board2D board;
278  for(Voro2::OutputImage::Domain::ConstIterator it = voro.domain().begin(),
279  itend = voro.domain().end();
280  it != itend; ++it)
281  {
282  const auto p = calcPointModuloDomain( voro(*it), voro.domain() );
283  const unsigned char c = ( p[1]*13 + p[0] * 7) % 256;
284  board << CustomStyle( (*it).className(), new CustomColors(Color(c,c,c),Color(c,c,c)))
285  << (*it);
286  }
287 
288  board.saveSVG( ( "Voromap." + formatPeriodicity(periodicity) + ".svg" ).c_str() );
289 
290  trace.beginBlock("Validating the Voronoi Map");
291  nbok += checkVoronoi(sites,voro) ? 1 : 0;
292  nb++;
293  trace.endBlock();
294 
295  trace.endBlock();
296 
297  return nbok == nb;
298 }
299 
300 
301 
306 template<typename Set>
307 bool testVoronoiMapFromSites2D( const Set &aSet,
308  const std::string &name,
309  std::array<bool, 2> const& periodicity = { {false, false} } )
310 {
311  unsigned int nbok = 0;
312  unsigned int nb = 0;
313 
314  Set mySet(aSet.domain());
315 
316  for(typename Set::Domain::ConstIterator it = aSet.domain().begin(), itend = aSet.domain().end();
317  it != itend;
318  ++it)
319  mySet.insertNew( *it );
320 
321 
322  for(typename Set::ConstIterator it = aSet.begin(), itend = aSet.end();
323  it != itend; ++it)
324  mySet.erase (*it);
325 
326 
327  trace.beginBlock(" Voro computation");
330  L2Metric l2;
331  Voro2 voro(aSet.domain(), mySet, l2, periodicity );
332 
333  trace.endBlock();
334 
335  trace.beginBlock(" Voronoi computation l_6");
337  L6Metric l6;
339  Voro6 voro6( aSet.domain(), mySet, l6, periodicity );
340  trace.endBlock();
341 
342 
343 
344  trace.beginBlock(" DT computation");
346  DT dt( aSet.domain(), mySet, l2, periodicity );
347  trace.endBlock();
348 
349 
350  if ( (aSet.domain().upperBound()[1] - aSet.domain().lowerBound()[1]) <20)
351  {
352  for(int j= aSet.domain().lowerBound()[1]; j <= aSet.domain().upperBound()[1]; j++)
353  {
354  for(int i=aSet.domain().lowerBound()[0]; i<=aSet.domain().upperBound()[0]; i++)
355  if ( aSet.find( Z2i::Point(i,j) ) != aSet.end() )
356  std::cout <<"X ";
357  else
358  std::cout<<"0 ";
359  trace.info()<<std::endl;
360  }
361 
362  trace.info() << std::endl;
363 
364  for(int j= aSet.domain().lowerBound()[1]; j <= aSet.domain().upperBound()[1]; j++)
365  {
366  for(int i=aSet.domain().lowerBound()[0]; i<=aSet.domain().upperBound()[0]; i++)
367  trace.info() << "("<<voro( Z2i::Point(i,j))[0]<<","<< voro( Z2i::Point(i,j))[1]<<") ";
368  trace.info()<<std::endl;
369  }
370  }
371 
372  Board2D board;
373  board << voro.domain();
374  for(typename Voro2::OutputImage::Domain::ConstIterator it = voro.domain().begin(), itend = voro.domain().end();
375  it != itend; ++it)
376  {
377  if (!mySet(*it))
378  board << (*it);
379  }
380  std::string filename = "Voromap-" + name + "-orig." + formatPeriodicity(periodicity) + ".svg";
381  board.saveSVG(filename.c_str());
382 
383  board.clear();
384  board << voro.domain();
385  board.setPenColor(Color(0,0,0));
386  for(typename Voro2::OutputImage::Domain::ConstIterator it = voro.domain().begin(), itend = voro.domain().end();
387  it != itend; ++it)
388  {
389  Z2i::Point p = voro(*it);
390  if ((p != (*it)) && (p != voro.domain().upperBound() + Z2i::Point::diagonal(1))
391  && (p != voro.domain().lowerBound()))
392  Display2DFactory::draw( board, p - (*it), (*it));
393  }
394 
395  filename = "Voromap-" + name + "-diag." + formatPeriodicity(periodicity) + ".svg";
396  board.saveSVG(filename.c_str());
397 
398 
399  board.clear();
400  board << voro.domain();
401  for(typename Voro2::OutputImage::Domain::ConstIterator it = voro.domain().begin(), itend = voro.domain().end();
402  it != itend; ++it)
403  {
404  const auto p = calcPointModuloDomain( voro(*it), voro.domain() );
405  unsigned char c = (p[1]*13 + p[0] * 7) % 256;
406  board << CustomStyle( (*it).className(), new CustomColors(Color(c,c,c),Color(c,c,c)))
407  << (*it);
408  }
409 
410  filename = "Voromap-" + name + "." + formatPeriodicity(periodicity) + ".svg";
411  board.saveSVG(filename.c_str());
412 
413  filename = "Voromap-" + name + "-hue." + formatPeriodicity(periodicity) + ".svg";
414  saveVoroMap(filename.c_str(),voro,2);
415 
416 
417  board.clear();
418  for(typename Voro6::OutputImage::Domain::ConstIterator it = voro6.domain().begin(),
419  itend = voro6.domain().end();
420  it != itend; ++it)
421  {
422  Z2i::Point p = voro6(*it);
423  if (p != (*it))
424  Display2DFactory::draw( board, p - (*it), (*it));
425  }
426 
427  filename = "Voromap-diag-l6-" + name + "." + formatPeriodicity(periodicity) + ".svg";
428  board.saveSVG(filename.c_str());
429 
430  board.clear();
431  for(typename Voro6::OutputImage::Domain::ConstIterator it = voro6.domain().begin(), itend = voro6.domain().end();
432  it != itend; ++it)
433  {
434  const auto p = calcPointModuloDomain( voro6(*it), voro6.domain() );
435  unsigned char c = (p[1]*13 + p[0] * 7) % 256;
436  board << CustomStyle( (*it).className(), new CustomColors(Color(c,c,c),Color(c,c,c)))
437  << (*it);;
438  }
439 
440  filename = "Voromap-l6" + name + "." + formatPeriodicity(periodicity) + ".svg";
441  board.saveSVG(filename.c_str());
442 
443  filename = "Voromap-hue-l6-" + name + "." + formatPeriodicity(periodicity) + ".svg";
444  saveVoroMap(filename.c_str(),voro6,3);
445 
446 
447  trace.beginBlock("Validating the Voronoi Map");
448  nbok += checkVoronoi(aSet, voro) ? 1 : 0;
449  nb++;
450  trace.endBlock();
451 
452  trace.beginBlock("Validating the Voronoi Map l_6");
453  nbok += checkVoronoi(aSet, voro6) ? 1 : 0;
454  nb++;
455  trace.endBlock();
456 
457  return nbok == nb;
458 }
459 
464 template<typename Set>
465 bool testVoronoiMapFromSites( const Set &aSet )
466 {
467  std::array<bool, Set::dimension> periodicity;
468  periodicity.fill( false );
469  return testVoronoiMapFromSites( aSet, periodicity );
470 }
471 
472 template<typename Set>
473 bool testVoronoiMapFromSites( const Set &aSet, std::array<bool, Set::Space::dimension> const & periodicity )
474 {
475  unsigned int nbok = 0;
476  unsigned int nb = 0;
477 
478  Set mySet(aSet.domain());
479 
480  for(typename Set::Domain::ConstIterator it = aSet.domain().begin(),
481  itend = aSet.domain().end();
482  it != itend;
483  ++it)
484  mySet.insertNew( *it );
485 
486 
487  for(typename Set::ConstIterator it = aSet.begin(), itend = aSet.end();
488  it != itend; ++it)
489  mySet.erase (*it);
490 
491 
492 
493  trace.beginBlock(" Voronoi computation l_2");
496  L2Metric l2;
497  Voro2 voro(aSet.domain(), mySet, l2, periodicity);
498  trace.endBlock();
499 
500  trace.beginBlock("Validating the Voronoi Map");
501  nbok += checkVoronoi(aSet, voro) ? 1 : 0;
502  nb++;
503  trace.endBlock();
504 
505  trace.beginBlock(" Voronoi computation l_3");
508  L3Metric l3;
509  Voro3 voro3(aSet.domain(), mySet, l3, periodicity);
510  trace.endBlock();
511 
512  trace.beginBlock("Validating the Voronoi Map l_3");
513  nbok += checkVoronoi(aSet, voro3) ? 1 : 0;
514  nb++;
515  trace.endBlock();
516 
517  trace.beginBlock(" DT computation");
519  DT dt(aSet.domain(), mySet, l2, periodicity);
520  trace.endBlock();
521 
522  return nbok == nb;
523 }
524 
525 
527 {
528 
529  Z2i::Point a(-10,-10);
530  Z2i::Point b(10,10);
531  Z2i::Domain domain(a,b);
532 
533  Z2i::DigitalSet sites(domain);
534  bool ok = true;
535 
536  sites.insertNew( Z2i::Point(3,-6));
537  sites.insertNew( Z2i::Point(9,0));
538  sites.insertNew( Z2i::Point(-3,0));
539 
540  for ( std::size_t i = 0; i < 4; ++i )
541  {
542  auto const periodicity = getPeriodicityFromInteger<2>(i);
543  trace.beginBlock( "Simple2D with periodicity " + formatPeriodicity(periodicity) );
544  ok = ok && testVoronoiMapFromSites2D( sites, "simple", periodicity );
545  trace.endBlock();
546  }
547 
548  return ok;
549 
550 }
551 
553 {
554  unsigned size=16;
555  Z2i::Point a(0,0);
556  Z2i::Point b(size,size);
557  Z2i::Domain domain(a,b);
558 
559  Z2i::DigitalSet sites(domain);
560  bool ok = true;
561 
562  for(unsigned int i = 0 ; i < size; ++i)
563  {
564  Z2i::Point p( rand() % (b[0]) - a[0], rand() % (b[1]) + a[1] );
565  sites.insert( p );
566  }
567 
568  for ( std::size_t i = 0; i < 4; ++i )
569  {
570  auto const periodicity = getPeriodicityFromInteger<2>(i);
571  trace.beginBlock( "Random 2D with periodicity " + formatPeriodicity(periodicity) );
572  ok = ok && testVoronoiMapFromSites2D( sites, "random", periodicity );
573  trace.endBlock();
574  }
575 
576  for(unsigned int i = 0 ; i < size*size-size; ++i)
577  {
578  Z2i::Point p( rand() % (b[0]) - a[0], rand() % (b[1]) + a[1] );
579  sites.insert( p );
580  }
581 
582  for ( std::size_t i = 0; i < 4; ++i )
583  {
584  auto const periodicity = getPeriodicityFromInteger<2>(i);
585  trace.beginBlock( "Random 2D (dense) with periodicity " + formatPeriodicity(periodicity) );
586  ok = ok && testVoronoiMapFromSites2D( sites, "random-dense", periodicity );
587  trace.endBlock();
588  }
589 
590  return ok;
591 }
592 
593 
595 {
596 
597  Z3i::Point a(-10,-10,-10);
598  Z3i::Point b(10,10,10);
599  Z3i::Domain domain(a,b);
600 
601  Z3i::DigitalSet sites(domain);
602  bool ok = true;
603 
604  sites.insertNew( Z3i::Point(0,0,-6));
605  sites.insertNew( Z3i::Point(6,0,0));
606  sites.insertNew( Z3i::Point(-6,0,3));
607 
608  for ( std::size_t i = 0; i < 8; ++i )
609  {
610  auto const periodicity = getPeriodicityFromInteger<3>(i);
611  trace.beginBlock( "Simple3D with periodicity " + formatPeriodicity(periodicity) );
612  ok = ok && testVoronoiMapFromSites( sites, periodicity );
613  trace.endBlock();
614  }
615 
616  return ok;
617 }
618 
620 {
621  std::size_t const N = 16;
622 
623  Z3i::Point a(0, 0, 0);
624  Z3i::Point b(N, N, N);
625  Z3i::Domain domain(a,b);
626 
627  Z3i::DigitalSet sites(domain);
628  bool ok = true;
629 
630  for(unsigned int i = 0 ; i < N; ++i)
631  {
632  Z3i::Point p( rand() % (b[0]) - a[0],
633  rand() % (b[1]) + a[1],
634  rand() % (b[2]) + a[2] );
635  sites.insert( p );
636  }
637 
638  for ( std::size_t i = 0; i < 8; ++i )
639  {
640  auto const periodicity = getPeriodicityFromInteger<3>(i);
641  trace.beginBlock( "Random 3D with periodicity " + formatPeriodicity(periodicity) );
642  ok = ok && testVoronoiMapFromSites( sites, periodicity );
643  trace.endBlock();
644  }
645 
646  return ok;
647 }
648 
649 
650 
652 {
653 
654  typedef SpaceND<4> Space4;
655  Space4::Point a(0,0,0,0);
656  Space4::Point b(5,5,5,5);
658 
660  bool ok = true;
661 
662  sites.insertNew( Space4::Point(1,4,1,1));
663  sites.insertNew( Space4::Point(3,1,3,1));
664  sites.insertNew( Space4::Point(0,0,0,0));
665 
666  for ( std::size_t i = 0; i < 16; ++i )
667  {
668  auto const periodicity = getPeriodicityFromInteger<4>(i);
669  trace.beginBlock( "Simple4D with periodicity " + formatPeriodicity(periodicity) );
670  ok = ok && testVoronoiMapFromSites( sites, periodicity );
671  trace.endBlock();
672  }
673 
674  return ok;
675 }
676 
677 
679 // Standard services - public :
680 
681 int main( int argc, char** argv )
682 {
683  trace.beginBlock ( "Testing class VoronoiMap" );
684  trace.info() << "Args:";
685  for ( int i = 0; i < argc; ++i )
686  trace.info() << " " << argv[ i ];
687  trace.info() << endl;
688 
689  bool res = testCheckConcept()
690  && testVoronoiMap()
691  && testVoronoiMap( { {true, false} } )
692  && testVoronoiMap( { {false, true} } )
693  && testVoronoiMap( { {true, true} } )
694  && testSimple2D()
695  && testSimpleRandom2D()
696  && testSimple3D()
697  && testSimpleRandom3D()
698  && testSimple4D()
699  ; // && ... other tests
700 
701  trace.emphase() << ( res ? "Passed." : "Error." ) << endl;
702  trace.endBlock();
703  return res ? 0 : 1;
704 }
705 // //
void beginBlock(const std::string &keyword="")
const Point aPoint(3, 4)
Aim: Implementation of the linear in time Voronoi map construction.
Definition: VoronoiMap.h:126
const ConstIterator & end() const
void insertNew(const Point &p)
Aim: Implementation of the linear in time distance transformation for separable metrics.
const Domain domain(Point(1, 2), Point(6, 5))
MyDigitalSurface::ConstIterator ConstIterator
double mynorm(const Point &point, const double p)
Trace trace
Definition: Common.h:137
Aim: SpaceND is a utility class that defines the fundamental structure of a Digital Space in ND...
Definition: SpaceND.h:95
bool testSimpleRandom3D()
const Point & upperBound() const
STL namespace.
double endBlock()
bool testCheckConcept()
Custom style class redefining the pen color and the fill color. You may use Board2D::Color::None for ...
Definition: Board2D.h:278
Aim: This class template may be used to (linearly) convert scalar values in a given range into a colo...
std::array< bool, N > getPeriodicityFromInteger(std::size_t anInteger)
bool checkVoronoi(const Set &aSet, const Voro &voro)
void saveSVG(const char *filename, PageSize size=Board::BoundingBox, double margin=10.0) const
Definition: Board.cpp:1012
Aim: implements separable l_p metrics with exact predicates.
bool testSimple2D()
std::ostream & emphase()
void clear(const DGtal::Color &color=DGtal::Color::None)
Definition: Board.cpp:152
const ConstIterator & begin() const
bool testSimpleRandom2D()
bool testSimple4D()
bool testVoronoiMapFromSites2D(const Set &aSet, const std::string &name, std::array< bool, 2 > const &periodicity={ {false, false} })
DGtal is the top-level namespace which contains all DGtal functions and types.
MyPointD Point
Definition: testClone2.cpp:383
Aim: A wrapper class around a STL associative container for storing sets of digital points within som...
unsigned int dim(const Vector &z)
std::ostream & info()
bool testVoronoiMap(std::array< bool, 2 > const &periodicity={ {false, false} })
void saveVoroMap(const std::string &filename, const VoroMap &output, const double p)
const Point & lowerBound() const
Point calcPointModuloDomain(Point aPoint, Domain const &aDomain)
bool testSimple3D()
Structure representing an RGB triple with alpha component.
Definition: Color.h:66
Board & setPenColor(const DGtal::Color &color)
Definition: Board.cpp:298
Aim: Defines the concept describing a read-only image, which is a refinement of CPointFunctor.
Definition: CConstImage.h:94
std::ostream & error()
Container::const_iterator ConstIterator
ConstIterator type of the container;.
std::string formatPeriodicity(std::array< bool, N > const &aPeriodicity)
int main(int argc, char **argv)
Aim: A container class for storing sets of digital points within some given domain.
void draw(const Iterator &itb, const Iterator &ite, Board &aBoard)
Aim: This class specializes a &#39;Board&#39; class so as to display DGtal objects more naturally (with <<)...
Definition: Board2D.h:70
bool testVoronoiMapFromSites(const Set &aSet)