opencv/sources/modules/calib3d/src/usac/pnp_solver.cpp

// This file is part of OpenCV project.
// It is subject to the license terms in the LICENSE file found in the top-level directory
// of this distribution and at http://opencv.org/license.html.

#include "../precomp.hpp"
#include "../usac.hpp"
#include "../polynom_solver.h"
#if defined(HAVE_EIGEN)
#include <Eigen/Eigen>
#include <Eigen/QR>
#elif defined(HAVE_LAPACK)
#include "opencv_lapack.h"
#endif

namespace cv { namespace usac {
class PnPMinimalSolver6PtsImpl : public PnPMinimalSolver6Pts {
private:
    Mat points_mat;
public:
    // linear 6 points required (11 equations)
    int getSampleSize() const override { return 6; }
    int getMaxNumberOfSolutions () const override { return 1; }

    explicit PnPMinimalSolver6PtsImpl (const Mat &points_) :
        points_mat(points_)
    {
        CV_DbgAssert(!points_mat.empty() && points_mat.isContinuous());
    }
    /*
        DLT:
        d x = P X, x = (u, v, 1), X = (X, Y, Z, 1), P = K[R t]
        is 3x4 projection matrix with rows p1, p2, p3. d is depth

        u = p1^T X / p3^T X
        v = p2^T X / p3^T X

        (p1^T - u p3^T) X = 0
        (p2^T - v p3^T) X = 0

        (p11 - u p31) X + (p12 - u p32) Y + (p13 - u p33) Z + (p14 - u p34) = 0
        (p12 - v p31) X + (p22 - v p32) Y + (p23 - v p33) Z + (p24 - v p34) = 0

        [X, Y, Z, 1, 0, 0, 0, 0, -u X, -u Y, -u Z, -u] [p11]    [0]
        [0, 0, 0, 0, X, Y, Z, 1, -v X, -v Y, -v Z, -v] [p12]    [0]
        .                                                     = [0]
        .
        .                                              [p34]    [0]

        minimum 11 equations, each point gives 2 equation, so at least 6 points are required.

        @points is array Nx5
        u1 v1 X1 Y1 Z1
        ...
        uN vN XN YN ZN
        @P is output projection matrix

        A1 =
        [X1, Y1, Z1, 1, 0, 0, 0, 0, -u1 X1, -u1 Y1, -u1 Z1, -u1] [p11]    [0]
        [X2, Y2, Z2, 1, 0, 0, 0, 0, -u2 X2, -u2 Y2, -u2 Z2, -u2] [p12]    [0]
        [X3, Y3, Z3, 1, 0, 0, 0, 0, -u3 X3, -u3 Y3, -u3 Z3, -u3] [p13]    [0]
        [X4, Y4, Z4, 1, 0, 0, 0, 0, -u4 X4, -u4 Y4, -u4 Z4, -u4] [p14]    [0]
        [X5, Y5, Z5, 1, 0, 0, 0, 0, -u5 X5, -u5 Y5, -u5 Z5, -u5] [p21]    [0]
                                                                 [p22]
        A2 = (without first 4 columns)
        [0, 0, 0, 0, X1, Y1, Z1, 1, -v1 X1, -v1 Y1, -v1 Z1, -v1] [p23]  = [0]
        [0, 0, 0, 0, X2, Y2, Z2, 1, -v2 X2, -v2 Y2, -v2 Z2, -v2] [p24]    [0]
        [0, 0, 0, 0, X3, Y3, Z3, 1, -v3 X3, -v3 Y3, -v3 Z3, -v3] [p31]    [0]
        [0, 0, 0, 0, X4, Y4, Z4, 1, -v4 X4, -v4 Y4, -v4 Z4, -v4] [p32]    [0]
        [0, 0, 0, 0, X5, Y5, Z5, 1, -v5 X5, -v5 Y5, -v5 Z5, -v5] [p33]    [0]
        [0, 0, 0, 0, X6, Y6, Z6, 1, -v6 X6, -v6 Y6, -v6 Z6, -v6] [p34=1]  [0]

        P = null A; dim null A = n - rank(A) = 12 - 11 = 1
    */

    int estimate (const std::vector<int> &sample, std::vector<Mat> &models) const override {
        std::vector<double> A1 (60, 0), A2(56, 0); // 5x12, 7x8
        const float * points = points_mat.ptr<float>();
        // std::vector<double> A_all(11*12, 0);
        // int cnt3 = 0;

        int cnt1 = 0, cnt2 = 0;
        for (int i = 0; i < 6; i++) {
            const int smpl = 5 * sample[i];
            const double u = points[smpl    ], v = points[smpl + 1];
            const double X = points[smpl + 2], Y = points[smpl + 3], Z = points[smpl + 4];

            if (i != 5) {
                A1[cnt1++] = X;
                A1[cnt1++] = Y;
                A1[cnt1++] = Z;
                A1[cnt1++] = 1;
                cnt1 += 4; // skip zeros
                A1[cnt1++] = -u * X;
                A1[cnt1++] = -u * Y;
                A1[cnt1++] = -u * Z;
                A1[cnt1++] = -u;
            }

            A2[cnt2++] = X;
            A2[cnt2++] = Y;
            A2[cnt2++] = Z;
            A2[cnt2++] = 1;
            A2[cnt2++] = -v * X;
            A2[cnt2++] = -v * Y;
            A2[cnt2++] = -v * Z;
            A2[cnt2++] = -v;
        }

        Math::eliminateUpperTriangular(A1, 5, 12);


        int offset = 4*12;
        // add last eliminated row of A1
        for (int i = 0; i < 8; i++)
            A2[cnt2++] = A1[offset + i + 4/* skip 4 first cols*/];


        // must be full-rank
        if (!Math::eliminateUpperTriangular(A2, 7, 8))
            return 0;
        // fixed scale to 1. In general the projection matrix is up-to-scale.
        // P = alpha * P^, alpha = 1 / P^_[3,4]

        Mat P = Mat_<double>(3,4);
        auto * p = (double *) P.data;
        p[11] = 1;

        // start from the last row
        for (int i = 6; i >= 0; i--) {
            double acc = 0;
            for (int j = i+1; j < 8; j++)
                acc -= A2[i*8+j]*p[j+4];

            p[i+4] = acc / A2[i*8+i];
            // due to numerical errors return 0 solutions
            if (std::isnan(p[i+4]))
                return 0;
        }

        for (int i = 3; i >= 0; i--) {
            double acc = 0;
            for (int j = i+1; j < 12; j++)
                acc -= A1[i*12+j]*p[j];

            p[i] = acc / A1[i*12+i];
            if (std::isnan(p[i]))
                return 0;
        }
        models = std::vector<Mat>{P};
        return 1;
    }
};
Ptr<PnPMinimalSolver6Pts> PnPMinimalSolver6Pts::create(const Mat &points_) {
    return makePtr<PnPMinimalSolver6PtsImpl>(points_);
}

class PnPNonMinimalSolverImpl : public PnPNonMinimalSolver {
private:
    Mat points_mat;
public:
    explicit PnPNonMinimalSolverImpl (const Mat &points_) :
        points_mat(points_)
    {
        CV_DbgAssert(!points_mat.empty() && points_mat.isContinuous());
    }

    int estimate (const std::vector<int> &sample, int sample_size,
                  std::vector<Mat> &models, const std::vector<double> &weights) const override {
        const float * points = points_mat.ptr<float>();
        if (sample_size < 6)
            return 0;

        double AtA [144] = {0}; // 12x12
        double a1[12] = {0, 0, 0, -1, 0, 0, 0,  0, 0, 0, 0, 0},
               a2[12] = {0, 0, 0,  0, 0, 0, 0, -1, 0, 0, 0, 0};

        if (weights.empty())
            for (int i = 0; i < sample_size; i++) {
                const int smpl = 5 * sample[i];
                const double u = points[smpl], v = points[smpl + 1];
                const double X = points[smpl + 2], Y = points[smpl + 3], Z = points[smpl + 4];

                a1[0 ] = -X;
                a1[1 ] = -Y;
                a1[2 ] = -Z;
                a1[8 ] = u * X;
                a1[9 ] = u * Y;
                a1[10] = u * Z;
                a1[11] = u;

                a2[4 ] = -X;
                a2[5 ] = -Y;
                a2[6 ] = -Z;
                a2[8 ] = v * X;
                a2[9 ] = v * Y;
                a2[10] = v * Z;
                a2[11] = v;

                // fill covariance matrix
                for (int j = 0; j < 12; j++)
                    for (int z = j; z < 12; z++)
                        AtA[j * 12 + z] += a1[j] * a1[z] + a2[j] * a2[z];
            }
        else
            for (int i = 0; i < sample_size; i++) {
                const int smpl = 5 * sample[i];
                const double weight = weights[i], u = points[smpl], v = points[smpl + 1];
                const double weight_X = weight * points[smpl + 2],
                             weight_Y = weight * points[smpl + 3],
                             weight_Z = weight * points[smpl + 4];

                a1[0 ] = -weight_X;
                a1[1 ] = -weight_Y;
                a1[2 ] = -weight_Z;
                a1[3 ] = -weight;
                a1[8 ] = u * weight_X;
                a1[9 ] = u * weight_Y;
                a1[10] = u * weight_Z;
                a1[11] = u * weight;

                a2[4 ] = -weight_X;
                a2[5 ] = -weight_Y;
                a2[6 ] = -weight_Z;
                a2[7 ] = -weight;
                a2[8 ] = v * weight_X;
                a2[9 ] = v * weight_Y;
                a2[10] = v * weight_Z;
                a2[11] = v * weight;

                // fill covariance matrix
                for (int j = 0; j < 12; j++)
                    for (int z = j; z < 12; z++)
                        AtA[j * 12 + z] += a1[j] * a1[z] + a2[j] * a2[z];
            }

        // copy symmetric part of covariance matrix
        for (int j = 1; j < 12; j++)
            for (int z = 0; z < j; z++)
                AtA[j*12+z] = AtA[z*12+j];

#ifdef HAVE_EIGEN
        models = std::vector<Mat>{ Mat_<double>(3,4) };
        Eigen::HouseholderQR<Eigen::Matrix<double, 12, 12>> qr((Eigen::Matrix<double, 12, 12>(AtA)));
        const Eigen::Matrix<double, 12, 12> &Q = qr.householderQ();
        // extract the last null-vector
        Eigen::Map<Eigen::Matrix<double, 12, 1>>((double *)models[0].data) = Q.col(11);
#else
        Matx<double, 12, 12> Vt;
        Vec<double, 12> D;
        if (! eigen(Matx<double, 12, 12>(AtA), D, Vt)) return 0;
        models = std::vector<Mat>{ Mat(Vt.row(11).reshape<3,4>()) };
#endif
        return 1;
    }
    int estimate (const std::vector<bool> &/*mask*/, std::vector<Mat> &/*models*/,
            const std::vector<double> &/*weights*/) override {
        return 0;
    }
    void enforceRankConstraint (bool /*enforce*/) override {}
    int getMinimumRequiredSampleSize() const override { return 6; }
    int getMaxNumberOfSolutions () const override { return 1; }
};
Ptr<PnPNonMinimalSolver> PnPNonMinimalSolver::create(const Mat &points) {
    return makePtr<PnPNonMinimalSolverImpl>(points);
}

class PnPSVDSolverImpl : public PnPSVDSolver {
private:
    std::vector<double> w;
    Ptr<PnPNonMinimalSolver> pnp_solver;
public:
    int getSampleSize() const override { return 6; }
    int getMaxNumberOfSolutions () const override { return 1; }

    explicit PnPSVDSolverImpl (const Mat &points_) {
        pnp_solver = PnPNonMinimalSolver::create(points_);
    }
    int estimate (const std::vector<int> &sample, std::vector<Mat> &models) const override {
        return pnp_solver->estimate(sample, 6, models, w);
    }
};
Ptr<PnPSVDSolver> PnPSVDSolver::create(const Mat &points_) {
    return makePtr<PnPSVDSolverImpl>(points_);
}

class P3PSolverImpl : public P3PSolver {
private:
    /*
     * calibrated normalized points
     * K^-1 [u v 1]^T / ||K^-1 [u v 1]^T||
     */
    const Mat points_mat, calib_norm_points_mat;
    const Matx33d K;
    const double VAL_THR = 1e-4;
public:
    /*
     * @points_ is matrix N x 5
     * u v x y z. (u,v) is image point, (x y z) is world point
     */
    P3PSolverImpl (const Mat &points_, const Mat &calib_norm_points_, const Mat &K_) :
        points_mat(points_), calib_norm_points_mat(calib_norm_points_), K(K_)
    {
        CV_DbgAssert(!points_mat.empty() && points_mat.isContinuous());
        CV_DbgAssert(!calib_norm_points_mat.empty() && calib_norm_points_mat.isContinuous());
    }

    int estimate (const std::vector<int> &sample, std::vector<Mat> &models) const override {
        /*
         * The description of this solution can be found here:
         * http://cmp.felk.cvut.cz/~pajdla/gvg/GVG-2016-Lecture.pdf
         * pages: 51-59
         */
        const float * points = points_mat.ptr<float>();
        const float * calib_norm_points = calib_norm_points_mat.ptr<float>();

        const int   idx1 = 5*sample[0],   idx2 = 5*sample[1],   idx3 = 5*sample[2];
        const Vec3d X1 (points[idx1+2], points[idx1+3], points[idx1+4]);
        const Vec3d X2 (points[idx2+2], points[idx2+3], points[idx2+4]);
        const Vec3d X3 (points[idx3+2], points[idx3+3], points[idx3+4]);

        // find distance between world points d_ij = ||Xi - Xj||
        const double d12 = norm(X1 - X2);
        const double d23 = norm(X2 - X3);
        const double d31 = norm(X3 - X1);

        if (d12 < VAL_THR || d23 < VAL_THR || d31 < VAL_THR)
            return 0;

        const int c_idx1 = 3*sample[0], c_idx2 = 3*sample[1], c_idx3 = 3*sample[2];
        const Vec3d cx1 (calib_norm_points[c_idx1], calib_norm_points[c_idx1+1], calib_norm_points[c_idx1+2]);
        const Vec3d cx2 (calib_norm_points[c_idx2], calib_norm_points[c_idx2+1], calib_norm_points[c_idx2+2]);
        const Vec3d cx3 (calib_norm_points[c_idx3], calib_norm_points[c_idx3+1], calib_norm_points[c_idx3+2]);

        // find cosine angles, cos(x1,x2) = K^-1 x1.dot(K^-1 x2) / (||K^-1 x1|| * ||K^-1 x2||)
        // calib_norm_points are already K^-1 x / ||K^-1 x||, so we perform only dot product
        const double c12 = cx1(0)*cx2(0) + cx1(1)*cx2(1) + cx1(2)*cx2(2);
        const double c23 = cx2(0)*cx3(0) + cx2(1)*cx3(1) + cx2(2)*cx3(2);
        const double c31 = cx3(0)*cx1(0) + cx3(1)*cx1(1) + cx3(2)*cx1(2);

        Matx33d Z, Zw;
        auto * z = Z.val, * zw = Zw.val;

        // find coefficients of polynomial a4 x^4 + ... + a0 = 0
        const double c12_p2 = c12*c12, c23_p2 = c23*c23, c31_p2 = c31*c31;
        const double d12_p2 = d12*d12, d12_p4 = d12_p2*d12_p2;
        const double d23_p2 = d23*d23, d23_p4 = d23_p2*d23_p2, d23_p6 = d23_p2*d23_p4, d23_p8 = d23_p4*d23_p4;
        const double d31_p2 = d31*d31, d31_p4 = d31_p2*d31_p2;
        const double a4 = -4*d23_p4*d12_p2*d31_p2*c23_p2+d23_p8-2*d23_p6*d12_p2-2*d23_p6*d31_p2+d23_p4*d12_p4+2*d23_p4*d12_p2*d31_p2+d23_p4*d31_p4;
        const double a3 = 8*d23_p4*d12_p2*d31_p2*c12*c23_p2+4*d23_p6*d12_p2*c31*c23-4*d23_p4*d12_p4*c31*c23+4*d23_p4*d12_p2*d31_p2*c31*c23-4*d23_p8*c12+4*d23_p6*d12_p2*c12+8*d23_p6*d31_p2*c12-4*d23_p4*d12_p2*d31_p2*c12-4*d23_p4*d31_p4*c12;
        const double a2 = -8*d23_p6*d12_p2*c31*c12*c23-8*d23_p4*d12_p2*d31_p2*c31*c12*c23+4*d23_p8*c12_p2-4*d23_p6*d12_p2*c31_p2-8*d23_p6*d31_p2*c12_p2+4*d23_p4*d12_p4*c31_p2+4*d23_p4*d12_p4*c23_p2-4*d23_p4*d12_p2*d31_p2*c23_p2+4*d23_p4*d31_p4*c12_p2+2*d23_p8-4*d23_p6*d31_p2-2*d23_p4*d12_p4+2*d23_p4*d31_p4;
        const double a1 = 8*d23_p6*d12_p2*c31_p2*c12+4*d23_p6*d12_p2*c31*c23-4*d23_p4*d12_p4*c31*c23+4*d23_p4*d12_p2*d31_p2*c31*c23-4*d23_p8*c12-4*d23_p6*d12_p2*c12+8*d23_p6*d31_p2*c12+4*d23_p4*d12_p2*d31_p2*c12-4*d23_p4*d31_p4*c12;
        const double a0 = -4*d23_p6*d12_p2*c31_p2+d23_p8-2*d23_p4*d12_p2*d31_p2+2*d23_p6*d12_p2+d23_p4*d31_p4+d23_p4*d12_p4-2*d23_p6*d31_p2;

        double roots[4] = {0};
        int num_roots = solve_deg4(a4, a3, a2, a1, a0, roots[0], roots[1], roots[2], roots[3]);

        models = std::vector<Mat>(); models.reserve(num_roots);
        for (double root : roots) {
            if (root <= 0) continue;

            const double n12 = root, n12_p2 = n12 * n12;
            const double n13 = (d12_p2*(d23_p2-d31_p2*n12_p2)+(d23_p2-d31_p2)*(d23_p2*(1+n12_p2-2*n12*c12)-d12_p2*n12_p2))
                                / (2*d12_p2*(d23_p2*c31 - d31_p2*c23*n12) + 2*(d31_p2-d23_p2)*d12_p2*c23*n12);
            const double n1 = d12 / sqrt(1 + n12_p2 - 2*n12*c12); // 1+n12^2-2n12c12 is always > 0
            const double n2 = n1 * n12;
            const double n3 = n1 * n13;

            if (n1 <= 0 || n2 <= 0 || n3 <= 0)
                continue;
            // compute and check errors
            if (fabs((sqrt(n1*n1 + n2*n2 - 2*n1*n2*c12) - d12) / d12) > VAL_THR ||
                fabs((sqrt(n2*n2 + n3*n3 - 2*n2*n3*c23) - d23) / d23) > VAL_THR ||
                fabs((sqrt(n3*n3 + n1*n1 - 2*n3*n1*c31) - d31) / d31) > VAL_THR)
                continue;

            const Vec3d nX1 = n1 * cx1;
            Vec3d Z2 = n2 * cx2 - nX1; Z2 /= norm(Z2);
            Vec3d Z3 = n3 * cx3 - nX1; Z3 /= norm(Z3);
            Vec3d Z1 = Z2.cross(Z3);   Z1 /= norm(Z1);
            const Vec3d Z3crZ1 = Z3.cross(Z1);

            z[0] = Z1(0);     z[3] = Z1(1);     z[6] = Z1(2);
            z[1] = Z2(0);     z[4] = Z2(1);     z[7] = Z2(2);
            z[2] = Z3crZ1(0); z[5] = Z3crZ1(1); z[8] = Z3crZ1(2);

            Vec3d Zw2 = (X2 - X1) / d12;
            Vec3d Zw3 = (X3 - X1) / d31;
            Vec3d Zw1 = Zw2.cross(Zw3); Zw1 /= norm(Zw1);
            const Vec3d Z3crZ1w = Zw3.cross(Zw1);

            zw[0] = Zw1(0);     zw[3] = Zw1(1);     zw[6] = Zw1(2);
            zw[1] = Zw2(0);     zw[4] = Zw2(1);     zw[7] = Zw2(2);
            zw[2] = Z3crZ1w(0); zw[5] = Z3crZ1w(1); zw[8] = Z3crZ1w(2);

            const Matx33d R = Math::rotVec2RotMat(Math::rotMat2RotVec(Z * Zw.inv()));
            Matx33d KR = K * R;
            Matx34d P;
            hconcat(KR, -KR * (X1 - R.t() * nX1), P);
            models.emplace_back(P);
        }
        return static_cast<int>(models.size());
    }
    int getSampleSize() const override { return 3; }
    int getMaxNumberOfSolutions () const override { return 4; }
};
Ptr<P3PSolver> P3PSolver::create(const Mat &points_, const Mat &calib_norm_pts, const Mat &K) {
    return makePtr<P3PSolverImpl>(points_, calib_norm_pts, K);
}
}}