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openMVS/vcglib/vcg/complex/algorithms/update/curvature.h

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/****************************************************************************
* VCGLib o o *
* Visual and Computer Graphics Library o o *
* _ O _ *
* Copyright(C) 2004-2016 \/)\/ *
* Visual Computing Lab /\/| *
* ISTI - Italian National Research Council | *
* \ *
* All rights reserved. *
* *
* This program is free software; you can redistribute it and/or modify *
* it under the terms of the GNU General Public License as published by *
* the Free Software Foundation; either version 2 of the License, or *
* (at your option) any later version. *
* *
* This program is distributed in the hope that it will be useful, *
* but WITHOUT ANY WARRANTY; without even the implied warranty of *
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the *
* GNU General Public License (http://www.gnu.org/licenses/gpl.txt) *
* for more details. *
* *
****************************************************************************/
#ifndef VCGLIB_UPDATE_CURVATURE_
#define VCGLIB_UPDATE_CURVATURE_
#include <vcg/space/index/grid_static_ptr.h>
#include <vcg/simplex/face/jumping_pos.h>
#include <vcg/complex/algorithms/update/normal.h>
#include <vcg/complex/algorithms/point_sampling.h>
#include <vcg/complex/algorithms/intersection.h>
#include <vcg/complex/algorithms/inertia.h>
#include <Eigen/Core>
namespace vcg {
namespace tri {
/// \ingroup trimesh
/// \headerfile curvature.h vcg/complex/algorithms/update/curvature.h
/// \brief Management, updating and computation of per-vertex and per-face normals.
/**
This class is used to compute or update the normals that can be stored in the vertex or face component of a mesh.
*/
template <class MeshType>
class UpdateCurvature
{
public:
typedef typename MeshType::FaceType FaceType;
typedef typename MeshType::FacePointer FacePointer;
typedef typename MeshType::FaceIterator FaceIterator;
typedef typename MeshType::VertexIterator VertexIterator;
typedef typename MeshType::VertContainer VertContainer;
typedef typename MeshType::VertexType VertexType;
typedef typename MeshType::VertexPointer VertexPointer;
typedef vcg::face::VFIterator<FaceType> VFIteratorType;
typedef typename MeshType::CoordType CoordType;
typedef typename CoordType::ScalarType ScalarType;
typedef typename MeshType::VertexType::CurScalarType CurScalarType;
typedef typename MeshType::VertexType::CurVecType CurVecType;
private:
// aux data struct used by PrincipalDirections
struct AdjVertex {
VertexType * vert;
float doubleArea;
bool isBorder;
};
public:
/// \brief Compute principal direction and magnitudo of curvature.
/*
Compute principal direction and magniuto of curvature as describe in the paper:
@InProceedings{bb33922,
author = "G. Taubin",
title = "Estimating the Tensor of Curvature of a Surface from a
Polyhedral Approximation",
booktitle = "International Conference on Computer Vision",
year = "1995",
pages = "902--907",
URL = "http://dx.doi.org/10.1109/ICCV.1995.466840",
bibsource = "http://www.visionbib.com/bibliography/describe440.html#TT32253",
*/
static void PrincipalDirections(MeshType &m)
{
tri::RequireVFAdjacency(m);
vcg::tri::UpdateNormal<MeshType>::PerVertexAngleWeighted(m);
vcg::tri::UpdateNormal<MeshType>::NormalizePerVertex(m);
for (VertexIterator vi =m.vert.begin(); vi !=m.vert.end(); ++vi) {
if ( ! (*vi).IsD() && (*vi).VFp() != NULL) {
VertexType * central_vertex = &(*vi);
std::vector<float> weights;
std::vector<AdjVertex> vertices;
vcg::face::JumpingPos<FaceType> pos((*vi).VFp(), central_vertex);
// firstV is the first vertex of the 1ring neighboorhood
VertexType* firstV = pos.VFlip();
VertexType* tempV;
float totalDoubleAreaSize = 0.0f;
// compute the area of each triangle around the central vertex as well as their total area
do
{
// this bring the pos to the next triangle counterclock-wise
pos.FlipF();
pos.FlipE();
// tempV takes the next vertex in the 1ring neighborhood
tempV = pos.VFlip();
assert(tempV!=central_vertex);
AdjVertex v;
v.isBorder = pos.IsBorder();
v.vert = tempV;
v.doubleArea = vcg::DoubleArea(*pos.F());
totalDoubleAreaSize += v.doubleArea;
vertices.push_back(v);
}
while(tempV != firstV);
// compute the weights for the formula computing matrix M
for (size_t i = 0; i < vertices.size(); ++i) {
if (vertices[i].isBorder) {
weights.push_back(vertices[i].doubleArea / totalDoubleAreaSize);
} else {
weights.push_back(0.5f * (vertices[i].doubleArea + vertices[(i-1)%vertices.size()].doubleArea) / totalDoubleAreaSize);
}
assert(weights.back() < 1.0f);
}
// compute I-NN^t to be used for computing the T_i's
Matrix33<ScalarType> Tp;
for (int i = 0; i < 3; ++i)
Tp[i][i] = 1.0f - powf(central_vertex->cN()[i],2);
Tp[0][1] = Tp[1][0] = -1.0f * (central_vertex->N()[0] * central_vertex->cN()[1]);
Tp[1][2] = Tp[2][1] = -1.0f * (central_vertex->cN()[1] * central_vertex->cN()[2]);
Tp[0][2] = Tp[2][0] = -1.0f * (central_vertex->cN()[0] * central_vertex->cN()[2]);
// for all neighbors vi compute the directional curvatures k_i and the T_i
// compute M by summing all w_i k_i T_i T_i^t
Matrix33<ScalarType> tempMatrix;
Matrix33<ScalarType> M;
M.SetZero();
for (size_t i = 0; i < vertices.size(); ++i) {
CoordType edge = (central_vertex->cP() - vertices[i].vert->cP());
float curvature = (2.0f * (central_vertex->cN().dot(edge)) ) / edge.SquaredNorm();
CoordType T = (Tp*edge).normalized();
tempMatrix.ExternalProduct(T,T);
M += tempMatrix * weights[i] * curvature ;
}
// compute vector W for the Householder matrix
CoordType W;
CoordType e1(1.0f,0.0f,0.0f);
if ((e1 - central_vertex->cN()).SquaredNorm() > (e1 + central_vertex->cN()).SquaredNorm())
W = e1 - central_vertex->cN();
else
W = e1 + central_vertex->cN();
W.Normalize();
// compute the Householder matrix I - 2WW^t
Matrix33<ScalarType> Q;
Q.SetIdentity();
tempMatrix.ExternalProduct(W,W);
Q -= tempMatrix * 2.0f;
// compute matrix Q^t M Q
Matrix33<ScalarType> QtMQ = (Q.transpose() * M * Q);
// CoordType bl = Q.GetColumn(0);
CoordType T1 = Q.GetColumn(1);
CoordType T2 = Q.GetColumn(2);
// find sin and cos for the Givens rotation
float s,c;
// Gabriel Taubin hint and Valentino Fiorin impementation
float alpha = QtMQ[1][1]-QtMQ[2][2];
float beta = QtMQ[2][1];
float h[2];
float delta = sqrtf(4.0f*powf(alpha, 2) +16.0f*powf(beta, 2));
h[0] = (2.0f*alpha + delta) / (2.0f*beta);
h[1] = (2.0f*alpha - delta) / (2.0f*beta);
float t[2];
float best_c, best_s;
float min_error = std::numeric_limits<ScalarType>::infinity();
for (int i=0; i<2; i++)
{
delta = sqrtf(powf(h[i], 2) + 4.0f);
t[0] = (h[i]+delta) / 2.0f;
t[1] = (h[i]-delta) / 2.0f;
for (int j=0; j<2; j++)
{
float squared_t = powf(t[j], 2);
float denominator = 1.0f + squared_t;
s = (2.0f*t[j]) / denominator;
c = (1-squared_t) / denominator;
float approximation = c*s*alpha + (powf(c, 2) - powf(s, 2))*beta;
float angle_similarity = fabs(acosf(c)/asinf(s));
float error = fabs(1.0f-angle_similarity)+fabs(approximation);
if (error<min_error)
{
min_error = error;
best_c = c;
best_s = s;
}
}
}
c = best_c;
s = best_s;
Eigen::Matrix2f minor2x2;
Eigen::Matrix2f S;
// diagonalize M
minor2x2(0,0) = QtMQ[1][1];
minor2x2(0,1) = QtMQ[1][2];
minor2x2(1,0) = QtMQ[2][1];
minor2x2(1,1) = QtMQ[2][2];
S(0,0) = S(1,1) = c;
S(0,1) = s;
S(1,0) = -1.0f * s;
Eigen::Matrix2f StMS = S.transpose() * minor2x2 * S;
// compute curvatures and curvature directions
float Principal_Curvature1 = (3.0f * StMS(0,0)) - StMS(1,1);
float Principal_Curvature2 = (3.0f * StMS(1,1)) - StMS(0,0);
CoordType Principal_Direction1 = T1 * c - T2 * s;
CoordType Principal_Direction2 = T1 * s + T2 * c;
(*vi).PD1().Import(Principal_Direction1);
(*vi).PD2().Import(Principal_Direction2);
(*vi).K1() = Principal_Curvature1;
(*vi).K2() = Principal_Curvature2;
}
}
}
class AreaData
{
public:
float A;
};
/** Curvature meseaure as described in the paper:
Robust principal curvatures on Multiple Scales, Yong-Liang Yang, Yu-Kun Lai, Shi-Min Hu Helmut Pottmann
SGP 2004
If pointVSfaceInt==true the covariance is computed by montecarlo sampling on the mesh (faster)
If pointVSfaceInt==false the covariance is computed by (analytic)integration over the surface (slower)
*/
typedef vcg::GridStaticPtr <FaceType, ScalarType > MeshGridType;
typedef vcg::GridStaticPtr <VertexType, ScalarType > PointsGridType;
static void PrincipalDirectionsPCA(MeshType &m, ScalarType r, bool pointVSfaceInt = true,vcg::CallBackPos * cb = NULL)
{
std::vector<VertexType*> closests;
std::vector<ScalarType> distances;
std::vector<CoordType> points;
VertexIterator vi;
ScalarType area;
MeshType tmpM;
typename std::vector<CoordType>::iterator ii;
vcg::tri::TrivialSampler<MeshType> vs;
tri::UpdateNormal<MeshType>::PerVertexAngleWeighted(m);
tri::UpdateNormal<MeshType>::NormalizePerVertex(m);
MeshGridType mGrid;
PointsGridType pGrid;
// Fill the grid used
if(pointVSfaceInt)
{
area = Stat<MeshType>::ComputeMeshArea(m);
vcg::tri::SurfaceSampling<MeshType,vcg::tri::TrivialSampler<MeshType> >::Montecarlo(m,vs,1000 * area / (2*M_PI*r*r ));
vi = vcg::tri::Allocator<MeshType>::AddVertices(tmpM,m.vert.size());
for(size_t y = 0; y < m.vert.size(); ++y,++vi) (*vi).P() = m.vert[y].P();
pGrid.Set(tmpM.vert.begin(),tmpM.vert.end());
}
else
{
mGrid.Set(m.face.begin(),m.face.end());
}
int jj = 0;
for(vi = m.vert.begin(); vi != m.vert.end(); ++vi)
{
vcg::Matrix33<ScalarType> A, eigenvectors;
vcg::Point3<ScalarType> bp, eigenvalues;
// int nrot;
// sample the neighborhood
if(pointVSfaceInt)
{
vcg::tri::GetInSphereVertex<
MeshType,
PointsGridType,std::vector<VertexType*>,
std::vector<ScalarType>,
std::vector<CoordType> >(tmpM,pGrid, (*vi).cP(),r ,closests,distances,points);
A.Covariance(points,bp);
A*=area*area/1000;
}
else{
IntersectionBallMesh<MeshType,ScalarType>( m ,vcg::Sphere3<ScalarType>((*vi).cP(),r),tmpM );
vcg::Point3<ScalarType> _bary;
vcg::tri::Inertia<MeshType>::Covariance(tmpM,_bary,A);
}
// Eigen::Matrix3f AA; AA=A;
// Eigen::SelfAdjointEigenSolver<Eigen::Matrix3f> eig(AA);
// Eigen::Vector3f c_val = eig.eigenvalues();
// Eigen::Matrix3f c_vec = eig.eigenvectors();
// Jacobi(A, eigenvalues , eigenvectors, nrot);
Eigen::Matrix3d AA;
A.ToEigenMatrix(AA);
Eigen::SelfAdjointEigenSolver<Eigen::Matrix3d> eig(AA);
Eigen::Vector3d c_val = eig.eigenvalues();
Eigen::Matrix3d c_vec = eig.eigenvectors(); // eigenvector are stored as columns.
eigenvectors.FromEigenMatrix(c_vec);
eigenvalues.FromEigenVector(c_val);
// EV.transposeInPlace();
// ev.FromEigenVector(c_val);
// get the estimate of curvatures from eigenvalues and eigenvectors
// find the 2 most tangent eigenvectors (by finding the one closest to the normal)
int best = 0; ScalarType bestv = fabs( (*vi).cN().dot(eigenvectors.GetColumn(0).normalized()) );
for(int i = 1 ; i < 3; ++i){
ScalarType prod = fabs((*vi).cN().dot(eigenvectors.GetColumn(i).normalized()));
if( prod > bestv){bestv = prod; best = i;}
}
(*vi).PD1().Import(eigenvectors.GetColumn( (best+1)%3).normalized());
(*vi).PD2().Import(eigenvectors.GetColumn( (best+2)%3).normalized());
// project them to the plane identified by the normal
vcg::Matrix33<CurScalarType> rot;
CurVecType NN = CurVecType::Construct((*vi).N());
CurScalarType angle;
angle = acos((*vi).PD1().dot(NN));
rot.SetRotateRad( - (M_PI*0.5 - angle),(*vi).PD1()^NN);
(*vi).PD1() = rot*(*vi).PD1();
angle = acos((*vi).PD2().dot(NN));
rot.SetRotateRad( - (M_PI*0.5 - angle),(*vi).PD2()^NN);
(*vi).PD2() = rot*(*vi).PD2();
// copmutes the curvature values
const ScalarType r5 = r*r*r*r*r;
const ScalarType r6 = r*r5;
(*vi).K1() = (2.0/5.0) * (4.0*M_PI*r5 + 15*eigenvalues[(best+2)%3]-45.0*eigenvalues[(best+1)%3])/(M_PI*r6);
(*vi).K2() = (2.0/5.0) * (4.0*M_PI*r5 + 15*eigenvalues[(best+1)%3]-45.0*eigenvalues[(best+2)%3])/(M_PI*r6);
if((*vi).K1() < (*vi).K2()) { std::swap((*vi).K1(),(*vi).K2());
std::swap((*vi).PD1(),(*vi).PD2());
if (cb)
{
(*cb)(int(100.0f * (float)jj / (float)m.vn),"Vertices Analysis");
++jj;
} }
}
}
/// \brief Computes the discrete mean gaussian curvature.
/**
The algorithm used is the one Desbrun et al. that is based on a discrete analysis of the angles of the faces around a vertex.
It requires FaceFace Adjacency;
For further details, please, refer to: \n
<b>Discrete Differential-Geometry Operators for Triangulated 2-Manifolds </b><br>
<i>Mark Meyer, Mathieu Desbrun, Peter Schroder, Alan H. Barr</i><br>
VisMath '02, Berlin
*/
static void MeanAndGaussian(MeshType & m)
{
tri::RequireFFAdjacency(m);
float area0, area1, area2, angle0, angle1, angle2;
FaceIterator fi;
VertexIterator vi;
typename MeshType::CoordType e01v ,e12v ,e20v;
SimpleTempData<VertContainer, AreaData> TDAreaPtr(m.vert);
SimpleTempData<VertContainer, typename MeshType::CoordType> TDContr(m.vert);
vcg::tri::UpdateNormal<MeshType>::PerVertexNormalized(m);
auto KH = vcg::tri::Allocator<MeshType>:: template GetPerVertexAttribute<ScalarType> (m, std::string("KH"));
auto KG = vcg::tri::Allocator<MeshType>:: template GetPerVertexAttribute<ScalarType> (m, std::string("KG"));
//Compute AreaMix in H (vale anche per K)
for(vi=m.vert.begin(); vi!=m.vert.end(); ++vi) if(!(*vi).IsD())
{
(TDAreaPtr)[*vi].A = 0.0;
(TDContr)[*vi] =typename MeshType::CoordType(0.0,0.0,0.0);
KH[*vi] = 0.0;
KG[*vi] = (ScalarType)(2.0 * M_PI);
}
for(fi=m.face.begin();fi!=m.face.end();++fi) if( !(*fi).IsD())
{
// angles
angle0 = math::Abs(Angle( (*fi).P(1)-(*fi).P(0),(*fi).P(2)-(*fi).P(0) ));
angle1 = math::Abs(Angle( (*fi).P(0)-(*fi).P(1),(*fi).P(2)-(*fi).P(1) ));
angle2 = M_PI-(angle0+angle1);
if((angle0 < M_PI/2) && (angle1 < M_PI/2) && (angle2 < M_PI/2)) // triangolo non ottuso
{
float e01 = SquaredDistance( (*fi).V(1)->cP() , (*fi).V(0)->cP() );
float e12 = SquaredDistance( (*fi).V(2)->cP() , (*fi).V(1)->cP() );
float e20 = SquaredDistance( (*fi).V(0)->cP() , (*fi).V(2)->cP() );
area0 = ( e20*(1.0/tan(angle1)) + e01*(1.0/tan(angle2)) ) / 8.0;
area1 = ( e01*(1.0/tan(angle2)) + e12*(1.0/tan(angle0)) ) / 8.0;
area2 = ( e12*(1.0/tan(angle0)) + e20*(1.0/tan(angle1)) ) / 8.0;
(TDAreaPtr)[(*fi).V(0)].A += area0;
(TDAreaPtr)[(*fi).V(1)].A += area1;
(TDAreaPtr)[(*fi).V(2)].A += area2;
}
else // obtuse
{
if(angle0 >= M_PI/2)
{
(TDAreaPtr)[(*fi).V(0)].A += vcg::DoubleArea<typename MeshType::FaceType>((*fi)) / 4.0;
(TDAreaPtr)[(*fi).V(1)].A += vcg::DoubleArea<typename MeshType::FaceType>((*fi)) / 8.0;
(TDAreaPtr)[(*fi).V(2)].A += vcg::DoubleArea<typename MeshType::FaceType>((*fi)) / 8.0;
}
else if(angle1 >= M_PI/2)
{
(TDAreaPtr)[(*fi).V(0)].A += vcg::DoubleArea<typename MeshType::FaceType>((*fi)) / 8.0;
(TDAreaPtr)[(*fi).V(1)].A += vcg::DoubleArea<typename MeshType::FaceType>((*fi)) / 4.0;
(TDAreaPtr)[(*fi).V(2)].A += vcg::DoubleArea<typename MeshType::FaceType>((*fi)) / 8.0;
}
else
{
(TDAreaPtr)[(*fi).V(0)].A += vcg::DoubleArea<typename MeshType::FaceType>((*fi)) / 8.0;
(TDAreaPtr)[(*fi).V(1)].A += vcg::DoubleArea<typename MeshType::FaceType>((*fi)) / 8.0;
(TDAreaPtr)[(*fi).V(2)].A += vcg::DoubleArea<typename MeshType::FaceType>((*fi)) / 4.0;
}
}
}
for(fi=m.face.begin();fi!=m.face.end();++fi) if( !(*fi).IsD() )
{
angle0 = math::Abs(Angle( (*fi).P(1)-(*fi).P(0),(*fi).P(2)-(*fi).P(0) ));
angle1 = math::Abs(Angle( (*fi).P(0)-(*fi).P(1),(*fi).P(2)-(*fi).P(1) ));
angle2 = M_PI-(angle0+angle1);
// Skip degenerate triangles.
if(angle0==0 || angle1==0 || angle2==0) continue;
e01v = ( (*fi).V(1)->cP() - (*fi).V(0)->cP() ) ;
e12v = ( (*fi).V(2)->cP() - (*fi).V(1)->cP() ) ;
e20v = ( (*fi).V(0)->cP() - (*fi).V(2)->cP() ) ;
TDContr[(*fi).V(0)] += ( e20v * (1.0/tan(angle1)) - e01v * (1.0/tan(angle2)) ) / 4.0;
TDContr[(*fi).V(1)] += ( e01v * (1.0/tan(angle2)) - e12v * (1.0/tan(angle0)) ) / 4.0;
TDContr[(*fi).V(2)] += ( e12v * (1.0/tan(angle0)) - e20v * (1.0/tan(angle1)) ) / 4.0;
KG[(*fi).V(0)] -= angle0;
KG[(*fi).V(1)] -= angle1;
KG[(*fi).V(2)] -= angle2;
for(int i=0;i<3;i++)
{
if(vcg::face::IsBorder((*fi), i))
{
CoordType e1,e2;
vcg::face::Pos<FaceType> hp(&*fi, i, (*fi).V(i));
vcg::face::Pos<FaceType> hp1=hp;
hp1.FlipV();
e1=hp1.v->cP() - hp.v->cP();
hp1.FlipV();
hp1.NextB();
e2=hp1.v->cP() - hp.v->cP();
KG[(*fi).V(i)] -= math::Abs(Angle(e1,e2));
}
}
}
for(vi=m.vert.begin(); vi!=m.vert.end(); ++vi) if(!(*vi).IsD() /*&& !(*vi).IsB()*/)
{
if((TDAreaPtr)[*vi].A<=std::numeric_limits<ScalarType>::epsilon())
{
KH[(*vi)] = 0;
KG[(*vi)] = 0;
}
else
{
KH[(*vi)] = (((TDContr)[*vi].dot((*vi).cN())>0)?1.0:-1.0)*((TDContr)[*vi] / (TDAreaPtr) [*vi].A).Norm();
KG[(*vi)] /= (TDAreaPtr)[*vi].A;
}
}
}
/// \brief Update the mean and the gaussian curvature of a vertex.
/**
The function uses the VF adiacency to walk around the vertex.
Based on the paper <a href="http://www2.in.tu-clausthal.de/~hormann/papers/Dyn.2001.OTU.pdf">
<em> "Optimizing 3d triangulations using discrete curvature analysis" </em> </a>
it compute an approximation of the gaussian and of the absolute mean curvature
*/
static void PerVertexAbsoluteMeanAndGaussian(MeshType & m)
{
tri::RequireVFAdjacency(m);
tri::RequireCompactness(m);
const bool areaNormalize = true;
const bool barycentricArea=false;
auto KH = vcg::tri::Allocator<MeshType>:: template GetPerVertexAttribute<ScalarType> (m, std::string("KH"));
auto KG = vcg::tri::Allocator<MeshType>:: template GetPerVertexAttribute<ScalarType> (m, std::string("KG"));
int faceCnt=0;
for(VertexIterator vi = m.vert.begin(); vi != m.vert.end(); ++vi)
{
VertexPointer v=&*vi;
VFIteratorType vfi(v);
ScalarType A = 0;
KH[v] = 0;
ScalarType AngleDefect = (ScalarType)(2.0 * M_PI);;
while (!vfi.End()) {
faceCnt++;
FacePointer f = vfi.F();
CoordType nf = TriangleNormal(*f);
int i = vfi.I();
VertexPointer v0 = f->V0(i), v1 = f->V1(i), v2 = f->V2(i);
assert (v==v0);
ScalarType ang0 = math::Abs(Angle(v1->P() - v0->P(), v2->P() - v0->P() ));
ScalarType ang1 = math::Abs(Angle(v0->P() - v1->P(), v2->P() - v1->P() ));
ScalarType ang2 = M_PI - ang0 - ang1;
ScalarType s01 = SquaredDistance(v1->P(), v0->P());
ScalarType s02 = SquaredDistance(v2->P(), v0->P());
// voronoi cell of current vertex
if(barycentricArea)
A+=vcg::DoubleArea(*f)/6.0;
else
{
if (ang0 >= M_PI/2)
A += (0.5f * DoubleArea(*f) - (s01 * tan(ang1) + s02 * tan(ang2)) / 8.0 );
else if (ang1 >= M_PI/2)
A += (s01 * tan(ang0)) / 8.0;
else if (ang2 >= M_PI/2)
A += (s02 * tan(ang0)) / 8.0;
else // non obctuse triangle
A += ((s02 / tan(ang1)) + (s01 / tan(ang2))) / 8.0;
}
// gaussian curvature update
AngleDefect -= ang0;
// mean curvature update
// Note that the standard abs mean curvature approximation would require
// to sum all the edges*diehedralAngle. Here with just VF adjacency
// we make a rough approximation that 1/2 of the edge len plus something
// that is half of the diedral angle
ang1 = math::Abs(Angle(nf, v1->N()+v0->N()));
ang2 = math::Abs(Angle(nf, v2->N()+v0->N()));
KH[v] += math::Sqrt(s01)*ang1 + math::Sqrt(s02)*ang2 ;
++vfi;
}
KH[v] /= 4.0;
if(areaNormalize) {
if(A <= std::numeric_limits<float>::epsilon()) {
KH[v] = 0;
KG[v] = 0;
}
else {
KH[v] /= A;
KG[v] = AngleDefect / A;
}
}
}
}
/*
Compute principal curvature directions and value with normal cycle:
@inproceedings{CohMor03,
author = {Cohen-Steiner, David and Morvan, Jean-Marie },
booktitle = {SCG '03: Proceedings of the nineteenth annual symposium on Computational geometry},
title - {Restricted delaunay triangulations and normal cycle}
year = {2003}
}
*/
static void PrincipalDirectionsNormalCycle(MeshType & m){
tri::RequireVFAdjacency(m);
tri::RequireFFAdjacency(m);
typename MeshType::VertexIterator vi;
for(vi = m.vert.begin(); vi != m.vert.end(); ++vi)
if(!((*vi).IsD())){
vcg::Matrix33<ScalarType> m33;m33.SetZero();
face::JumpingPos<typename MeshType::FaceType> p((*vi).VFp(),&(*vi));
p.FlipE();
typename MeshType::VertexType * firstv = p.VFlip();
assert(p.F()->V(p.VInd())==&(*vi));
do{
if( p.F() != p.FFlip()){
Point3<ScalarType> normalized_edge = p.F()->V(p.F()->Next(p.VInd()))->cP() - (*vi).P();
ScalarType edge_length = normalized_edge.Norm();
normalized_edge/=edge_length;
Point3<ScalarType> n1 = NormalizedTriangleNormal(*(p.F()));
Point3<ScalarType> n2 = NormalizedTriangleNormal(*(p.FFlip()));
ScalarType n1n2 = (n1 ^ n2).dot(normalized_edge);
n1n2 = std::max(std::min( ScalarType(1.0),n1n2),ScalarType(-1.0));
ScalarType beta = math::Asin(n1n2);
m33[0][0] += beta*edge_length*normalized_edge[0]*normalized_edge[0];
m33[0][1] += beta*edge_length*normalized_edge[1]*normalized_edge[0];
m33[1][1] += beta*edge_length*normalized_edge[1]*normalized_edge[1];
m33[0][2] += beta*edge_length*normalized_edge[2]*normalized_edge[0];
m33[1][2] += beta*edge_length*normalized_edge[2]*normalized_edge[1];
m33[2][2] += beta*edge_length*normalized_edge[2]*normalized_edge[2];
}
p.NextFE();
}while(firstv != p.VFlip());
if(m33.Determinant()==0.0){ // degenerate case
(*vi).K1() = (*vi).K2() = 0.0; continue;}
m33[1][0] = m33[0][1];
m33[2][0] = m33[0][2];
m33[2][1] = m33[1][2];
Eigen::Matrix3d it;
m33.ToEigenMatrix(it);
Eigen::SelfAdjointEigenSolver<Eigen::Matrix3d> eig(it);
Eigen::Vector3d c_val = eig.eigenvalues();
Eigen::Matrix3d c_vec = eig.eigenvectors();
Point3<ScalarType> lambda;
Matrix33<ScalarType> vect;
vect.FromEigenMatrix(c_vec);
lambda.FromEigenVector(c_val);
ScalarType bestNormal = 0;
int bestNormalIndex = -1;
for(int i = 0; i < 3; ++i)
{
float agreeWithNormal = fabs((*vi).N().Normalize().dot(vect.GetColumn(i)));
if( agreeWithNormal > bestNormal )
{
bestNormal= agreeWithNormal;
bestNormalIndex = i;
}
}
int maxI = (bestNormalIndex+2)%3;
int minI = (bestNormalIndex+1)%3;
if(fabs(lambda[maxI]) < fabs(lambda[minI])) std::swap(maxI,minI);
(*vi).PD1().Import(vect.GetColumn(maxI));
(*vi).PD2().Import(vect.GetColumn(minI));
(*vi).K1() = lambda[2];
(*vi).K2() = lambda[1];
}
}
static void PerVertexBasicRadialCrossField(MeshType &m, float anisotropyRatio = 1.0 )
{
tri::RequirePerVertexCurvatureDir(m);
CoordType c=m.bbox.Center();
float maxRad = m.bbox.Diag()/2.0f;
for(size_t i=0;i<m.vert.size();++i) {
CoordType dd = m.vert[i].P()-c;
dd.Normalize();
m.vert[i].PD1().Import(dd^m.vert[i].N());
m.vert[i].PD1().Normalize();
m.vert[i].PD2().Import(m.vert[i].N()^CoordType::Construct(m.vert[i].PD1()));
m.vert[i].PD2().Normalize();
// Now the anisotropy
// the idea is that the ratio between the two direction is at most <anisotropyRatio>
// eg |PD1|/|PD2| < ratio
// and |PD1|^2 + |PD2|^2 == 1
float q =Distance(m.vert[i].P(),c) / maxRad; // it is in the 0..1 range
const float minRatio = 1.0f/anisotropyRatio;
const float maxRatio = anisotropyRatio;
const float curRatio = minRatio + (maxRatio-minRatio)*q;
float pd1Len = sqrt(1.0/(1+curRatio*curRatio));
float pd2Len = curRatio * pd1Len;
// assert(fabs(curRatio - pd2Len/pd1Len)<0.0000001);
// assert(fabs(pd1Len*pd1Len + pd2Len*pd2Len - 1.0f)<0.0001);
m.vert[i].PD1() *= pd1Len;
m.vert[i].PD2() *= pd2Len;
}
}
};
} // end namespace tri
} // end namespace vcg
#endif