/****************************************************************************
* VCGLib                                                            o o     *
* Visual and Computer Graphics Library                            o     o   *
*                                                                _   O  _   *
* Copyright(C) 2004-2016                                           \/)\/    *
* Visual Computing Lab                                            /\/|      *
* ISTI - Italian National Research Council                           |      *
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* All rights reserved.                                                      *
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* (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.                                                         *
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****************************************************************************/
/****************************************************************************

****************************************************************************/

#ifndef VCGLIB_UPDATE_CURVATURE_FITTING
#define VCGLIB_UPDATE_CURVATURE_FITTING

#include <vcg/space/index/grid_static_ptr.h>
#include <vcg/simplex/face/topology.h>
#include <vcg/simplex/face/pos.h>
#include <vcg/simplex/face/jumping_pos.h>
#include <vcg/container/simple_temporary_data.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 <vcg/complex/algorithms/nring.h>

#include <Eigen/Core>
#include <Eigen/QR>
#include <Eigen/LU>
#include <Eigen/SVD>
#include <Eigen/Eigenvalues>


namespace vcg {
namespace tri {

/// \ingroup trimesh

/// \headerfile curvature_fitting.h vcg/complex/algorithms/update/curvature_fitting.h

/// \brief Computation of per-vertex directions and values of curvature.
/**
This class is used to compute the per-vertex directions and values of curvature using a quadric fitting method.
*/

template <class MeshType>
class UpdateCurvatureFitting
{

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 typename MeshType::VertexPointer VertexTypeP;
    typedef vcg::face::VFIterator<FaceType> VFIteratorType;
    typedef typename MeshType::CoordType CoordType;
    typedef typename CoordType::ScalarType ScalarType;

class Quadric
{
    public:

    Quadric(double av, double bv, double cv, double dv, double ev)
    {
        a() = av;
        b() = bv;
        c() = cv;
        d() = dv;
        e() = ev;
    }

    double& a() { return data[0];}
    double& b() { return data[1];}
    double& c() { return data[2];}
    double& d() { return data[3];}
    double& e() { return data[4];}

    double data[5];

    double evaluate(double u, double v)
    {
        return a()*u*u + b()*u*v + c()*v*v + d()*u + e()*v;
    }

    double du(double u, double v)
    {
        return 2.0*a()*u + b()*v + d();
    }

    double dv(double u, double v)
    {
        return 2.0*c()*v + b()*u + e();
    }

    double duv(double /*u*/, double /*v*/)
    {
        return b();
    }

    double duu(double /*u*/, double /*v*/)
    {
        return 2.0*a();
    }

    double dvv(double /*u*/, double /*v*/)
    {
        return 2.0*c();
    }

    static Quadric fit(std::vector<CoordType> VV)
    {
        assert(VV.size() >= 5);
        Eigen::MatrixXf A(VV.size(),5);
        Eigen::MatrixXf b(VV.size(),1);
        Eigen::MatrixXf sol(VV.size(),1);

        for(unsigned int c=0; c < VV.size(); ++c)
        {
            double u = VV[c].X();
            double v = VV[c].Y();
            double n = VV[c].Z();

            A(c,0) = u*u;
            A(c,1) = u*v;
            A(c,2) = v*v;
            A(c,3) = u;
            A(c,4) = v;

            b(c,0) = n;
        }

        sol = ((A.transpose()*A).inverse()*A.transpose())*b;
        return Quadric(sol(0,0),sol(1,0),sol(2,0),sol(3,0),sol(4,0));
    }
};

    static CoordType project(VertexType* v, VertexType* vp)
    {
      return vp->P() - (v->N() * ((vp->P() - v->P()) * v->N()));
    }


    static std::vector<CoordType> computeReferenceFrames(VertexTypeP vi)
    {
      vcg::face::VFIterator<FaceType> vfi(vi);

      int i = (vfi.I()+1)%3;
      VertexTypeP vp = vfi.F()->V(i);

      CoordType x = (project(&*vi,vp) - vi->P()).Normalize();
      //assert(fabs(x * vi->N()) < 0.1);

      std::vector<CoordType> res(3);

      res[0] = x;
      res[1] = (vi->N() ^ res[0]).Normalize();
      res[2] = (vi->N())/(vi->N()).Norm();

      return res;
    }

    static std::set<CoordType> getSecondRing(VertexTypeP v)
    {
        std::set<VertexTypeP> ris;
        std::set<CoordType> coords;

        vcg::face::VFIterator<FaceType> vvi(v);
        for(;!vvi.End();++vvi)
        {
            vcg::face::VFIterator<FaceType> vvi2(vvi.F()->V((vvi.I()+1)%3));
            for(;!vvi2.End();++vvi2)
            {
                ris.insert(vvi2.F()->V((vvi2.I()+1)%3));
            }
        }
                typename std::set<VertexTypeP>::iterator it;
        for(it = ris.begin(); it != ris.end(); ++it)
            coords.insert((*it)->P());

        return coords;
    }

    static Quadric fitQuadric(VertexTypeP v, std::vector<CoordType>& ref)
    {
        std::set<CoordType> ring = getSecondRing(v);
        if (ring.size() < 5)
            return Quadric(1,1,1,1,1);
        std::vector<CoordType> points;

        typename std::set<CoordType>::iterator b = ring.begin();
        typename std::set<CoordType>::iterator e = ring.end();

        while(b != e)
        {
            // vtang non e` il v tangente!!!
            CoordType vTang = *b - v->P();

            double x = vTang * ref[0];
            double y = vTang * ref[1];
            double z = vTang * ref[2];
            points.push_back(CoordType(x,y,z));
            ++b;
        }
        return Quadric::fit(points);
    }


    static void computeCurvature(MeshType & m)
    {
      Allocator<MeshType>::CompactVertexVector(m);
      tri::RequireCompactness(m);
      tri::RequireVFAdjacency(m);

        vcg::tri::UpdateTopology<MeshType>::VertexFace(m);

        vcg::tri::UpdateNormal<MeshType>::PerVertexAngleWeighted(m);
        vcg::tri::UpdateNormal<MeshType>::NormalizePerVertex(m);


        VertexIterator vi;
        for(vi = m.vert.begin(); vi!=m.vert.end(); ++vi )
        {
            std::vector<CoordType> ref = computeReferenceFrames(&*vi);

            Quadric q = fitQuadric(&*vi,ref);
            double a = q.a();
            double b = q.b();
            double c = q.c();
            double d = q.d();
            double e = q.e();

            double E = 1.0 + d*d;
            double F = d*e;
            double G = 1.0 + e*e;

            CoordType n = CoordType(-d,-e,1.0).Normalize();

            vi->N() = ref[0] * n[0] + ref[1] * n[1] + ref[2] * n[2];

            double L = 2.0 * a * n.Z();
            double M = b * n.Z();
            double N = 2 * c * n.Z();

            // ----------------- Eigen stuff
            Eigen::Matrix2d m;
            m << L*G - M*F, M*E-L*F, M*E-L*F, N*E-M*F;
            m = m / (E*G-F*F);
            Eigen::SelfAdjointEigenSolver<Eigen::Matrix2d> eig(m);

            Eigen::Vector2d c_val = eig.eigenvalues();
            Eigen::Matrix2d c_vec = eig.eigenvectors();

            c_val = -c_val;

            CoordType v1, v2;
            v1[0] = c_vec(0,0);
            v1[1] = c_vec(0,1);
            v1[2] = 0;

            v2[0] = c_vec(1,0);
            v2[1] = c_vec(1,1);
            v2[2] = 0;

            v1 = v1.Normalize();
            v2 = v2.Normalize();

            v1 = v1 * c_val[0];
            v2 = v2 * c_val[1];

            CoordType v1global = ref[0] * v1[0] + ref[1] * v1[1] + ref[2] * v1[2];
            CoordType v2global = ref[0] * v2[0] + ref[1] * v2[1] + ref[2] * v2[2];

            v1global.Normalize();
            v2global.Normalize();

            if (c_val[0] > c_val[1])
            {
                (*vi).PD1().Import(v1global);
                (*vi).PD2().Import(v2global);
                (*vi).K1()  = c_val[0];
                (*vi).K2()  = c_val[1];
            }
            else
            {
                (*vi).PD1().Import(v2global);
                (*vi).PD2().Import(v1global);
                (*vi).K1()  = c_val[1];
                (*vi).K2()  = c_val[0];
            }
            // ---- end Eigen stuff
        }
    }

    // GG LOCAL CURVATURE

    class QuadricLocal
    {
    public:

        QuadricLocal ()
        {
            a() = b() = c() = d() = e() = 1.0;
        }

        QuadricLocal (double av, double bv, double cv, double dv, double ev)
        {
            a() = av;
            b() = bv;
            c() = cv;
            d() = dv;
            e() = ev;
        }

        double& a() { return data[0];}
        double& b() { return data[1];}
        double& c() { return data[2];}
        double& d() { return data[3];}
        double& e() { return data[4];}

        double data[5];

        double evaluate(double u, double v)
        {
            return a()*u*u + b()*u*v + c()*v*v + d()*u + e()*v;
        }

        double du(double u, double v)
        {
            return 2.0*a()*u + b()*v + d();
        }

        double dv(double u, double v)
        {
            return 2.0*c()*v + b()*u + e();
        }

        double duv(double /*u*/, double /*v*/)
        {
            return b();
        }

        double duu(double /*u*/, double /*v*/)
        {
            return 2.0*a();
        }

        double dvv(double /*u*/, double /*v*/)
        {
            return 2.0*c();
        }


        static QuadricLocal fit(std::vector<CoordType> &VV, bool svdRes, bool detCheck)
        {
            assert(VV.size() >= 5);
            Eigen::MatrixXd A(VV.size(),5);
            Eigen::MatrixXd b(VV.size(),1);
            Eigen::MatrixXd sol(5,1);

            for(unsigned int c=0; c < VV.size(); ++c)
            {
                double u = VV[c].X();
                double v = VV[c].Y();
                double n = VV[c].Z();

                A(c,0) = u*u;
                A(c,1) = u*v;
                A(c,2) = v*v;
                A(c,3) = u;
                A(c,4) = v;

                b[c] = n;
            }


            static int count = 0, index = 0;
            double min = 0.000000000001; //1.0e-12
            /*
            if (!count)
                printf("GNE %e\n", min);
             */

            if (detCheck && ((A.transpose()*A).determinant() < min && (A.transpose()*A).determinant() > -min))
            {
                //A.svd().solve(b, &sol); A.svd().solve(b, &sol);
                //cout << sol << endl;
                printf("Quadric: unsolvable vertex %d %d\n", count, ++index);
                //return Quadric (1, 1, 1, 1, 1);
//                A.svd().solve(b, &sol);
                Eigen::JacobiSVD<Eigen::MatrixXd> svd(A);
                sol=svd.solve(b);
                return QuadricLocal(sol[0],sol[1],sol[2],sol[3],sol[4]);
            }
            count++;

            //for (int i = 0; i < 100; i++)
            {
                if (svdRes)
                {
                  Eigen::JacobiSVD<Eigen::MatrixXd> svd(A);
                  sol=svd.solve(b);
                  //A.svd().solve(b, &sol);
                }
                else
                    sol = ((A.transpose()*A).inverse()*A.transpose())*b;

            }

            return QuadricLocal(sol[0],sol[1],sol[2],sol[3],sol[4]);
        }
    };

    static void expandMaxLocal (MeshType & mesh, VertexType *v, int max, std::vector<VertexType*> *vv)
    {
    Nring<MeshType> rw = Nring<MeshType> (v, &mesh);
    do rw.expand (); while (rw.allV.size() < max+1);
    if (rw.allV[0] != v)
        printf ("rw.allV[0] != *v\n");
    vv->reserve ((size_t)max);
    for (int i = 1; i < max+1; i++)
        vv->push_back(rw.allV[i]);

    rw.clear();
    }


    static void expandSphereLocal (MeshType & mesh, VertexType *v, float r, int min, std::vector<VertexType*> *vv)
    {
    Nring<MeshType> rw = Nring<MeshType> (v, &mesh);

    bool isInside = true;
    while (isInside)
    {
        rw.expand();
        vv->reserve(rw.allV.size());

        typename std::vector<VertexType*>::iterator b = rw.lastV.begin();
        typename std::vector<VertexType*>::iterator e = rw.lastV.end();
        isInside = false;
        while(b != e)
        {
        if (((*b)->P() - v->P()).Norm() < r)
        {
            vv->push_back(*b);;
            isInside = true;
        }
        ++b;
        }
    }
    //printf ("%d\n", vv->size());
    rw.clear();

    if (vv->size() < min)
    {
        vv->clear();
        expandMaxLocal (mesh, v, min, vv);
    }
    }


    static void getAverageNormal (VertexType *vp, std::vector<VertexType*> &vv, CoordType *ppn)
    {
    *ppn = CoordType (0,0,0);
    for (typename std::vector<VertexType*>::iterator vpi = vv.begin(); vpi != vv.end(); ++vpi)
        *ppn += (*vpi)->N();
    *ppn += (*vp).N();
    *ppn /= vv.size() + 1;
    ppn->Normalize();
    }


    static void applyProjOnPlane (CoordType ppn, std::vector<VertexType*> &vin, std::vector<VertexType*> *vout)
    {
    for (typename std::vector<VertexType*>::iterator vpi = vin.begin(); vpi != vin.end(); ++vpi)
        if ((*vpi)->N() * ppn > 0.0f)
        vout->push_back (*vpi);
    }

    static CoordType projectLocal(VertexType* v, VertexType* vp, CoordType ppn)
    {
        return vp->P() - (ppn * ((vp->P() - v->P()) * ppn));
    }


    static void computeReferenceFramesLocal (VertexType *v, CoordType ppn, std::vector<CoordType> *ref)
    {
    vcg::face::VFIterator<FaceType> vfi (v);

    int i = (vfi.I() + 1) % 3;
    VertexTypeP vp = vfi.F()->V(i);

    CoordType x = (projectLocal (v, vp, ppn) - v->P()).Normalize();

    assert(fabs(x * ppn) < 0.1);

    *ref = std::vector<CoordType>(3);

    (*ref)[0] = x;
    (*ref)[1] = (ppn ^ (*ref)[0]).Normalize();
    (*ref)[2] = ppn.Normalize(); //ppn / ppn.Norm();
    }


    static void fitQuadricLocal (VertexType *v, std::vector<CoordType> ref, std::vector<VertexType*> &vv, QuadricLocal *q)
    {
    bool svdResolution = false;
    bool zeroDeterminantCheck = false;

    std::vector<CoordType> points;
    points.reserve (vv.size());

    typename std::vector<VertexType*>::iterator b = vv.begin();
    typename std::vector<VertexType*>::iterator e = vv.end();

    while(b != e)
    {
        CoordType cp = (*b)->P();

        // vtang non e` il v tangente!!!
        CoordType vTang = cp - v->P();

        double x = vTang * ref[0];
        double y = vTang * ref[1];
        double z = vTang * ref[2];
        points.push_back(CoordType(x,y,z));
        ++b;
    }

    *q = QuadricLocal::fit (points, svdResolution, zeroDeterminantCheck);
    }


    static void finalEigenStuff (VertexType *v, std::vector<CoordType> ref, QuadricLocal q)
    {
    double a = q.a();
    double b = q.b();
    double c = q.c();
    double d = q.d();
    double e = q.e();

    double E = 1.0 + d*d;
    double F = d*e;
    double G = 1.0 + e*e;

    CoordType n = CoordType(-d,-e,1.0).Normalize();

    v->N() = ref[0] * n[0] + ref[1] * n[1] + ref[2] * n[2];

    double L = 2.0 * a * n.Z();
    double M = b * n.Z();
    double N = 2 * c * n.Z();

    // ----------------- Eigen stuff
    Eigen::Matrix2d m;
    m << L*G - M*F, M*E-L*F, M*E-L*F, N*E-M*F;
    m = m / (E*G-F*F);
    Eigen::SelfAdjointEigenSolver<Eigen::Matrix2d> eig(m);

    Eigen::Vector2d c_val = eig.eigenvalues();
    Eigen::Matrix2d c_vec = eig.eigenvectors();

    c_val = -c_val;

    CoordType v1, v2;
    v1[0] = c_vec[0];
    v1[1] = c_vec[1];
    v1[2] = d * v1[0] + e * v1[1];

    v2[0] = c_vec[2];
    v2[1] = c_vec[3];
    v2[2] = d * v2[0] + e * v2[1];

    v1 = v1.Normalize();
    v2 = v2.Normalize();

    CoordType v1global = ref[0] * v1[0] + ref[1] * v1[1] + ref[2] * v1[2];
    CoordType v2global = ref[0] * v2[0] + ref[1] * v2[1] + ref[2] * v2[2];

    v1global.Normalize();
    v2global.Normalize();

    v1global *= c_val[0];
    v2global *= c_val[1];

    if (c_val[0] > c_val[1])
    {
        (*v).PD1() = v1global;
        (*v).PD2() = v2global;
        (*v).K1()  = c_val[0];
        (*v).K2()  = c_val[1];
    }
    else
    {
        (*v).PD1() = v2global;
        (*v).PD2() = v1global;
        (*v).K1()  = c_val[1];
        (*v).K2()  = c_val[0];
    }
    // ---- end Eigen stuff
    }



    static void updateCurvatureLocal (MeshType & mesh, float radiusSphere)
    {
    bool verbose = false;
    bool projectionPlaneCheck = true;
    int vertexesPerFit = 0;

    int i = 0;
    VertexIterator vi;
    for(vi = mesh.vert.begin(); vi != mesh.vert.end(); ++vi, i++)
    {
        std::vector<VertexType*> vv;
        std::vector<VertexType*> vvtmp;

        int count;
        if (verbose && !((count = (vi - mesh.vert.begin())) % 1000))
        printf ("vertex %d of %d\n",count,mesh.vert.size());

        // if (kRing != 0)
        // 	expandRing (&*vi, kRing, 5, &vv);
        // else
        expandSphereLocal (mesh, &*vi, radiusSphere, 5, &vv);

        assert (vv.size() >= 5);

        CoordType ppn;
        // if (averageNormalMode)
        // 	//ppn = (*vi).N();
        getAverageNormal (&*vi, vv, &ppn);
        // else
        // 	getProjPlaneNormal (&*vi, vv, &ppn);

        if (projectionPlaneCheck)
        {
        vvtmp.reserve (vv.size ());
        applyProjOnPlane (ppn, vv, &vvtmp);
        if (vvtmp.size() >= 5)
            vv = vvtmp;
        }

        vvtmp.clear();

        // if (montecarloMaxVertexNum)
        // {
        // 	//printf ("P: %d\n", vv.size());
        // 	vvtmp.reserve (vv.size ());
        // 	//printf ("TP: %d\n", vvtmp.size());
        // 	applyMontecarlo (montecarloMaxVertexNum, vv, &vvtmp);
        // 	//printf ("TD: %d\n", vvtmp.size());
        // 	vv = vvtmp;
        // 	//printf ("D: %d\n", vv.size());
        // 	//printf ("\n");
        // }

        assert (vv.size() >= 5);

        std::vector<CoordType> ref;
        computeReferenceFramesLocal (&*vi, ppn, &ref);

        /*
          printf ("%lf %lf %lf - %lf %lf %lf - %lf %lf %lf\n",
          ref[0][0], ref[0][1], ref[0][2],
          ref[1][0], ref[1][1], ref[1][2],
          ref[2][0], ref[2][1], ref[2][2]);
        */

        vertexesPerFit += vv.size();
        //printf ("size: %d\n", vv.size());

        QuadricLocal q;
        fitQuadricLocal (&*vi, ref, vv, &q);

        finalEigenStuff (&*vi, ref, q);

    }

    //if (verbose)
        //printf ("average vertex num in each fit: %f, total %d, vn %d\n", ((float) vertexesPerFit) / mesh.vn, vertexesPerFit, mesh.vn);
    if (verbose)
        printf ("average vertex num in each fit: %f\n", ((float) vertexesPerFit) / mesh.vn);
    }

};

}
}
#endif
