vcglib/vcg/complex/algorithms/harmonic.h

/****************************************************************************
* 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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* GNU General Public License (http://www.gnu.org/licenses/gpl.txt)          *
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#ifndef __VCGLIB_HARMONIC_FIELD
#define __VCGLIB_HARMONIC_FIELD

#include <vcg/complex/complex.h>
#include <Eigen/Sparse>

namespace vcg {
namespace tri {

template <class MeshType, typename Scalar = double>
class Harmonic
{
public:
    typedef typename MeshType::VertexType VertexType;
    typedef typename MeshType::FaceType   FaceType;
    typedef typename MeshType::CoordType  CoordType;
    typedef typename MeshType::ScalarType ScalarType;

    typedef double CoeffScalar;

    typedef typename std::pair<VertexType *, Scalar> Constraint;
    typedef typename std::vector<Constraint>         ConstraintVec;
    typedef typename ConstraintVec::const_iterator   ConstraintIt;

    /**
     * @brief ComputeScalarField
     * Generates a scalar harmonic field over the mesh.
     * For more details see:\n Kai Xua, Hao Zhang, Daniel Cohen-Or, Yueshan Xionga,'Dynamic Harmonic Fields for Surface Processing'.\nin Computers & Graphics, 2009
     * @param m the mesh
     * @param constraints the Dirichlet boundary conditions in the form of vector of pairs <vertex pointer, value>.
     * @param field the accessor to use to write the computed per-vertex values (must have the [ ] operator).
     * @return true if the algorithm succeeds, false otherwise.
     * @note the algorithm has unexpected behavior if the mesh contains unreferenced vertices.
     */
    template <typename ACCESSOR>
    static bool ComputeScalarField(MeshType & m, const ConstraintVec & constraints, ACCESSOR field, bool biharmonic = false)
    {
        typedef Eigen::SparseMatrix<CoeffScalar> SpMat;  // sparse matrix type
        typedef Eigen::Triplet<CoeffScalar>      Triple; // triplet type to fill the matrix

        RequirePerVertexFlags(m);
        RequireCompactness(m);
        RequireFFAdjacency(m);
        MeshAssert<MeshType>::FFAdjacencyIsInitialized(m);
        MeshAssert<MeshType>::NoUnreferencedVertex(m);

        if (constraints.empty())
            return false;

        int n  = m.VN();

        // Generate coefficients
        std::vector<Triple>          coeffs;   // coefficients of the system
        std::map<size_t,CoeffScalar> sums;     // row sum of the coefficient matrix

        vcg::tri::UpdateFlags<MeshType>::FaceClearV(m);
        for (size_t i = 0; i < m.face.size(); ++i)
        {
            FaceType & f = m.face[i];
            assert(!f.IsV());

            f.SetV();

            // Generate coefficients for each edge
            for (int edge = 0; edge < 3; ++edge)
            {
                CoeffScalar weight;
                WeightInfo res = CotangentWeightIfNotVisited(f, edge, weight);

                if (res == EdgeAlreadyVisited) continue;
                assert(res == Success);

                // Add the weight to the coefficients vector for both the vertices of the considered edge
                size_t v0_idx = vcg::tri::Index(m, f.V0(edge));
                size_t v1_idx = vcg::tri::Index(m, f.V1(edge));

                coeffs.push_back(Triple(v0_idx, v1_idx, -weight));
                coeffs.push_back(Triple(v1_idx, v0_idx, -weight));

                // Add the weight to the row sum
                sums[v0_idx] += weight;
                sums[v1_idx] += weight;
            }
        }

        // Setup the system matrix
        SpMat laplaceMat;        // the system to be solved
        laplaceMat.resize(n, n); // eigen initializes it to zero
        laplaceMat.reserve(coeffs.size());
        for (std::map<size_t,CoeffScalar>::const_iterator it = sums.begin(); it != sums.end(); ++it)
        {
            coeffs.push_back(Triple(it->first, it->first, it->second));
        }
        laplaceMat.setFromTriplets(coeffs.begin(), coeffs.end());

        if (biharmonic)
        {
            SpMat lap_t = laplaceMat;
            lap_t.transpose();
            laplaceMat = lap_t * laplaceMat;
        }


        // Setting the constraints
        const CoeffScalar alpha = pow(10.0, 8.0); // penalty factor alpha
//        const CoeffScalar alpha = CoeffScalar(1e5); // penalty factor alpha

        Eigen::Matrix<CoeffScalar, Eigen::Dynamic, 1> b, x; // Unknown and known terms vectors
        b.setZero(n);

        for (ConstraintIt it=constraints.begin(); it!=constraints.end(); it++)
        {
            size_t v_idx = vcg::tri::Index(m, it->first);
            b(v_idx) = alpha * it->second;
            laplaceMat.coeffRef(v_idx, v_idx) += alpha;
        }

        // Perform matrix decomposition
        Eigen::SimplicialLDLT<SpMat> solver;
        solver.compute(laplaceMat);
        // TODO eventually use another solver (e.g. CHOLMOD for dynamic setups)
        if(solver.info() != Eigen::Success)
        {
            // decomposition failed
            switch (solver.info())
            {
            // possible errors
            case Eigen::NumericalIssue :
            case Eigen::NoConvergence :
            case Eigen::InvalidInput :
            default : return false;
            }
        }

        // Solve the system: laplacianMat x = b
        x = solver.solve(b);
        if(solver.info() != Eigen::Success)
        {
            return false;
        }

        // Set field value using the provided handle
        for (size_t i = 0; int(i) < n; ++i)
        {
            field[i] = Scalar(x[i]);
        }

        return true;
    }

    enum WeightInfo
    {
        Success            = 0,
        EdgeAlreadyVisited
    };


    /**
     * @brief CotangentWeightIfNotVisited computes the cotangent weighting for an edge
     * (if it has not be done yet).
     * This must be ensured setting the visited flag on the face once all edge weights have been computed.
     * @param f the face
     * @param edge the edge of the provided face for which we compute the weight
     * @param weight the computed weight (output)
     * @return Success if everything is fine, EdgeAlreadyVisited if the weight
     * for the considered edge has been already computed.
     * @note the mesh must have the face-face topology updated
     */
    template <typename ScalarT>
    static WeightInfo CotangentWeightIfNotVisited(const FaceType &f, int edge, ScalarT & weight)
    {
        const FaceType * fp = f.cFFp(edge);
        if (fp != NULL && fp != &f)
        {
            // not a border edge
            if (fp->IsV()) return EdgeAlreadyVisited;
        }

        weight = CotangentWeight<ScalarT>(f, edge);

        return Success;
    }

    /**
     * @brief ComputeWeight computes the cotangent weighting for an edge
     * @param f the face
     * @param edge the edge of the provided face for which we compute the weight
     * @return the computed weight
     * @note the mesh must have the face-face topology updated
     */
    template <typename ScalarT>
    static ScalarT CotangentWeight(const FaceType &f, int edge)
    {
        assert(edge >= 0 && edge < 3);

        // get the adjacent face
        const FaceType * fp = f.cFFp(edge);

        //       v0
        //      /|\
        //     / | \
        //    /  |  \
        //   /   |   \
        // va\   |   /vb
        //    \  |  /
        //     \ | /
        //      \|/
        //       v1

        ScalarT cotA = 0;
        ScalarT cotB = 0;

        // Get the edge (a pair of vertices)
        VertexType * v0 = f.cV(edge);
        VertexType * v1 = f.cV((edge+1)%f.VN());

        if (fp != NULL &&
            fp != &f)
        {
            // not a border edge
            VertexType * vb = fp->cV((f.cFFi(edge)+2)%fp->VN());
            ScalarT angleB = ComputeAngle<ScalarT>(v0, vb, v1);
            cotB = vcg::math::Cos(angleB) / vcg::math::Sin(angleB);
        }

        VertexType * va = f.cV((edge+2)%f.VN());
        ScalarT angleA = ComputeAngle<ScalarT>(v0, va, v1);
        cotA = vcg::math::Cos(angleA) / vcg::math::Sin(angleA);

        return (cotA + cotB) / 2;
    }

    template <typename ScalarT>
    static ScalarT ComputeAngle(const VertexType * a, const VertexType * b, const VertexType * c)
    {
        //       a
        //      /
        //     /
        //    /
        //   /  ___ compute the angle in b
        // b \
        //    \
        //     \
        //      \
        //       c
        assert(a != NULL && b != NULL && c != NULL);
        Point3<ScalarT> A,B,C;
        A.Import(a->P());
        B.Import(b->P());
        C.Import(c->P());
        ScalarT angle = vcg::Angle(A - B, C - B);
        return angle;
    }
};

}
}
#endif // __VCGLIB_HARMONIC_FIELD