常见的平面拟合方法一般是最小二乘法。当误差服从正态分布时，最小二乘方法的拟合效果还是很好的，可以转化成PCA问题。
当观测值的误差大于2倍中误差时，认为误差较大。采用最小二乘拟合时精度降低，不够稳健。
提出了一些稳健的方法：有移动最小二乘法（根据距离残差增加权重）；采用2倍距离残差的协方差剔除离群点；迭代重权重方法(选权迭代法)。
MainWindow中的平面拟合方法，调用了ccPlane的Fit方法。
1 void MainWindow::doActionFitPlane() 2 { 3 doComputePlaneOrientation(false); 4 } 5 6 void MainWindow::doActionFitFacet() 7 { 8 doComputePlaneOrientation(true); 9 } 10 11 static double s_polygonMaxEdgeLength = 0; 12 void MainWindow::doComputePlaneOrientation(bool fitFacet) 13 { 14 ccHObject::Container selectedEntities = m_selectedEntities; 15 size_t selNum = selectedEntities.size(); 16 if (selNum < 1) 17 return; 18 19 double maxEdgeLength = 0; 20 if (fitFacet) 21 { 22 bool ok = true; 23 maxEdgeLength = QInputDialog::getDouble(this,"Fit facet", "Max edge length (0 = no limit)", s_polygonMaxEdgeLength, 0, 1.0e9, 8, &ok); 24 if (!ok) 25 return; 26 s_polygonMaxEdgeLength = maxEdgeLength; 27 } 28 29 for (size_t i=0; i<selNum; ++i) 30 { 31 ccHObject* ent = selectedEntities[i]; 32 ccShiftedObject* shifted = 0; 33 CCLib::GenericIndexedCloudPersist* cloud = 0; 34 35 if (ent->isKindOf(CC_TYPES::POLY_LINE)) 36 { 37 ccPolyline* poly = ccHObjectCaster::ToPolyline(ent); 38 cloud = static_cast<CCLib::GenericIndexedCloudPersist*>(poly); 39 shifted = poly; 40 } 41 else 42 { 43 ccGenericPointCloud* gencloud = ccHObjectCaster::ToGenericPointCloud(ent); 44 if (gencloud) 45 { 46 cloud = static_cast<CCLib::GenericIndexedCloudPersist*>(gencloud); 47 shifted = gencloud; 48 } 49 } 50 51 if (cloud) 52 { 53 double rms = 0.0; 54 CCVector3 C,N; 55 56 ccHObject* plane = 0; 57 if (fitFacet) 58 { 59 ccFacet* facet = ccFacet::Create(cloud, static_cast<PointCoordinateType>(maxEdgeLength)); 60 if (facet) 61 { 62 plane = static_cast<ccHObject*>(facet); 63 N = facet->getNormal(); 64 C = facet->getCenter(); 65 rms = facet->getRMS(); 66 67 //manually copy shift & scale info! 68 if (shifted) 69 { 70 ccPolyline* contour = facet->getContour(); 71 if (contour) 72 { 73 contour->setGlobalScale(shifted->getGlobalScale()); 74 contour->setGlobalShift(shifted->getGlobalShift()); 75 } 76 } 77 } 78 } 79 else 80 { 81 ccPlane* pPlane = ccPlane::Fit(cloud, &rms); 82 if (pPlane) 83 { 84 plane = static_cast<ccHObject*>(pPlane); 85 N = pPlane->getNormal(); 86 C = *CCLib::Neighbourhood(cloud).getGravityCenter(); 87 pPlane->enableStippling(true); 88 } 89 } 90 91 //as all information appears in Console... 92 forceConsoleDisplay(); 93 94 if (plane) 95 { 96 ccConsole::Print(QString("[Orientation] Entity '%1'").arg(ent->getName())); 97 ccConsole::Print(" - plane fitting RMS: %f",rms); 98 99 //We always consider the normal with a positive 'Z' by default! 100 if (N.z < 0.0) 101 N *= -1.0; 102 ccConsole::Print(" - normal: (%f,%f,%f)",N.x,N.y,N.z); 103 104 //we compute strike & dip by the way 105 PointCoordinateType dip = 0, dipDir = 0; 106 ccNormalVectors::ConvertNormalToDipAndDipDir(N,dip,dipDir); 107 QString dipAndDipDirStr = ccNormalVectors::ConvertDipAndDipDirToString(dip,dipDir); 108 ccConsole::Print(QString(" - %1").arg(dipAndDipDirStr)); 109 110 //hack: output the transformation matrix that would make this normal points towards +Z 111 ccGLMatrix makeZPosMatrix = ccGLMatrix::FromToRotation(N,CCVector3(0,0,PC_ONE)); 112 CCVector3 Gt = C; 113 makeZPosMatrix.applyRotation(Gt); 114 makeZPosMatrix.setTranslation(C-Gt); 115 ccConsole::Print("[Orientation] A matrix that would make this plane horizontal (normal towards Z+) is:"); 116 ccConsole::Print(makeZPosMatrix.toString(12,' ')); //full precision 117 ccConsole::Print("[Orientation] You can copy this matrix values (CTRL+C) and paste them in the 'Apply transformation tool' dialog"); 118 119 plane->setName(dipAndDipDirStr); 120 plane->applyGLTransformation_recursive(); //not yet in DB 121 plane->setVisible(true); 122 plane->setSelectionBehavior(ccHObject::SELECTION_FIT_BBOX); 123 124 ent->addChild(plane); 125 plane->setDisplay(ent->getDisplay()); 126 plane->prepareDisplayForRefresh_recursive(); 127 addToDB(plane); 128 } 129 else 130 { 131 ccConsole::Warning(QString("Failed to fit a plane/facet on entity '%1'").arg(ent->getName())); 132 } 133 } 134 } 135 136 refreshAll(); 137 updateUI(); 138 }
ccPlane的fit方法：
ccPlane* ccPlane::Fit(CCLib::GenericIndexedCloudPersist *cloud, double* rms/*=0*/)
{
//number of points
unsigned count = cloud->size();
if (count < 3)
{
ccLog::Warning("[ccPlane::Fit] Not enough points in input cloud to fit a plane!");
return 0;
}
CCLib::Neighbourhood Yk(cloud);
//plane equation
const PointCoordinateType* theLSPlane = Yk.getLSPlane();
if (!theLSPlane)
{
ccLog::Warning("[ccPlane::Fit] Not enough points to fit a plane!");
return 0;
}
//get the centroid
const CCVector3* G = Yk.getGravityCenter();
assert(G);
//and a local base
CCVector3 N(theLSPlane);
const CCVector3* X = Yk.getLSPlaneX(); //main direction
assert(X);
CCVector3 Y = N * (*X);
//compute bounding box in 2D plane
CCVector2 minXY(0,0), maxXY(0,0);
cloud->placeIteratorAtBegining();
for (unsigned k=0; k<count; ++k)
{
//projection into local 2D plane ref.
CCVector3 P = *(cloud->getNextPoint()) - *G;
CCVector2 P2D( P.dot(*X), P.dot(Y) );
if (k != 0)
{
if (minXY.x > P2D.x)
minXY.x = P2D.x;
else if (maxXY.x < P2D.x)
maxXY.x = P2D.x;
if (minXY.y > P2D.y)
minXY.y = P2D.y;
else if (maxXY.y < P2D.y)
maxXY.y = P2D.y;
}
else
{
minXY = maxXY = P2D;
}
}
//we recenter the plane
PointCoordinateType dX = maxXY.x-minXY.x;
PointCoordinateType dY = maxXY.y-minXY.y;
CCVector3 Gt = *G + *X * (minXY.x + dX / 2) + Y * (minXY.y + dY / 2);
ccGLMatrix glMat(*X,Y,N,Gt);
ccPlane* plane = new ccPlane(dX, dY, &glMat);
//compute least-square fitting RMS if requested
if (rms)
{
*rms = CCLib::DistanceComputationTools::computeCloud2PlaneDistanceRMS(cloud, theLSPlane);
plane->setMetaData(QString("RMS"),QVariant(*rms));
}
return plane;
}
Efficient Ransac shape extract插件调用的模板类Plane实现，可以看到使用的Jacobi特征值分解的方法实现。
template< class PointT >
template< class PointsForwardIt, class WeightsForwardIt >
bool Plane< PointT >::Fit(const PointType &origin, PointsForwardIt begin,PointsForwardIt end, WeightsForwardIt weights)
{
MatrixXX< PointType::Dim, PointType::Dim, ScalarType > c, v;
CovarianceMatrix(origin, begin, end, weights, &c);
VectorXD< PointType::Dim, ScalarType > d;
if(!Jacobi(c, &d, &v))
{
//std::cout << "Jacobi failed:" << std::endl;
//std::cout << "origin = " << origin[0] << "," << origin[1] << "," << origin[2] << std::endl
// << "cov:" << std::endl
// << c[0][0] << c[1][0] << c[2][0] << std::endl
// << c[0][1] << c[1][1] << c[2][1] << std::endl
// << c[0][2] << c[1][2] << c[2][2] << std::endl;
//std::cout << "recomp origin:" << std::endl;
//PointT com;
//Mean(begin, end, weights, &com);
//std::cout << "origin = " << origin[0] << "," << origin[1] << "," << origin[2] << std::endl;
//std::cout << "recomp covariance:" << std::endl;
//CovarianceMatrix(com, begin, end, weights, &c);
//std::cout << "cov:" << std::endl
//<< c[0][0] << c[1][0] << c[2][0] << std::endl
//<< c[0][1] << c[1][1] << c[2][1] << std::endl
//<< c[0][2] << c[1][2] << c[2][2] << std::endl;
//std::cout << "weights and points:" << std::endl;
//WeightsForwardIt w = weights;
//for(PointsForwardIt i = begin; i != end; ++i, ++w)
// std::cout << (*i)[0] << "," << (*i)[1] << "," << (*i)[2]
// << " weight=" << (*w) << std::endl;
return false;
}
for(unsigned int i = 0; i < PointType::Dim; ++i)
d[i] = Math< ScalarType >::Abs(d[i]);
EigSortDesc(&d, &v);
_normal = PointType(v[PointType::Dim - 1]);
_d = -(_normal * origin);
return true;
}