rpnp.cpp

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/* +------------------------------------------------------------------------+
   |                     Mobile Robot Programming Toolkit (MRPT)            |
   |                          http://www.mrpt.org/                          |
   |                                                                        |
   | Copyright (c) 2005-2017, Individual contributors, see AUTHORS file     |
   | See: http://www.mrpt.org/Authors - All rights reserved.                |
   | Released under BSD License. See details in http://www.mrpt.org/License |
   +------------------------------------------------------------------------+ */

#include "vision-precomp.h"  // Precompiled headers
#include <iostream>
#include <vector>
#include <cmath>

#include <mrpt/utils/types_math.h>  // Eigen must be included first via MRPT to enable the plugin system
#include <Eigen/Dense>
#include <Eigen/SVD>
#include <Eigen/StdVector>
#include <unsupported/Eigen/Polynomials>

#include "rpnp.h"

mrpt::vision::pnp::rpnp::rpnp(
        Eigen::MatrixXd obj_pts_, Eigen::MatrixXd img_pts_, Eigen::MatrixXd cam_,
        int n0)
{
        obj_pts = obj_pts_;
        img_pts = img_pts_;
        cam_intrinsic = cam_;
        n = n0;

        // Store obj_pts as 3XN and img_projections as 2XN matrices
        P = obj_pts.transpose();
        Q = img_pts.transpose();

        for (int i = 0; i < n; i++) Q.col(i) = Q.col(i) / Q.col(i).norm();

        R.setZero();
        t.setZero();
}

bool mrpt::vision::pnp::rpnp::compute_pose(
        Eigen::Ref<Eigen::Matrix3d> R_, Eigen::Ref<Eigen::Vector3d> t_)
{
        // selecting an edge $P_{ i1 }P_{ i2 }$ by n random sampling
        int i1 = 0, i2 = 1;
        double lmin =
                Q(0, i1) * Q(0, i2) + Q(1, i1) * Q(1, i2) + Q(2, i1) * Q(2, i2);

        Eigen::MatrixXi rij(n, 2);

        R_ = Eigen::MatrixXd::Identity(3, 3);
        t_ = Eigen::Vector3d::Zero();

        for (int i = 0; i < n; i++)
                for (int j = 0; j < 2; j++) rij(i, j) = rand() % n;

        for (int ii = 0; ii < n; ii++)
        {
                int i = rij(ii, 0), j = rij(ii, 1);

                if (i == j) continue;

                double l = Q(0, i) * Q(0, j) + Q(1, i) * Q(1, j) + Q(2, i) * Q(2, j);

                if (l < lmin)
                {
                        i1 = i;
                        i2 = j;
                        lmin = l;
                }
        }

        // calculating the rotation matrix of $O_aX_aY_aZ_a$.
        Eigen::Vector3d p1, p2, p0, x, y, z, dum_vec;

        p1 = P.col(i1);
        p2 = P.col(i2);
        p0 = (p1 + p2) / 2;

        x = p2 - p0;
        x /= x.norm();

        if (std::abs(x(1)) < std::abs(x(2)))
        {
                dum_vec << 0, 1, 0;
                z = x.cross(dum_vec);
                z /= z.norm();
                y = z.cross(x);
                y /= y.norm();
        }
        else
        {
                dum_vec << 0, 0, 1;
                y = dum_vec.cross(x);
                y /= y.norm();
                z = x.cross(y);
                x /= x.norm();
        }

        Eigen::Matrix3d R0;

        R0.col(0) = x;
        R0.col(1) = y;
        R0.col(2) = z;

        for (int i = 0; i < n; i++) P.col(i) = R0.transpose() * (P.col(i) - p0);

        // Dividing the n - point set into(n - 2) 3 - point subsets
        // and setting up the P3P equations

        Eigen::Vector3d v1 = Q.col(i1), v2 = Q.col(i2);
        double cg1 = v1.dot(v2);
        double sg1 = sqrt(1 - cg1 * cg1);
        double D1 = (P.col(i1) - P.col(i2)).norm();
        Eigen::MatrixXd D4(n - 2, 5);

        int j = 0;
        Eigen::Vector3d vi;
        Eigen::VectorXd rowvec(5);
        for (int i = 0; i < n; i++)
        {
                if (i == i1 || i == i2) continue;

                vi = Q.col(i);
                double cg2 = v1.dot(vi);
                double cg3 = v2.dot(vi);
                double sg2 = sqrt(1 - cg2 * cg2);
                double D2 = (P.col(i1) - P.col(i)).norm();
                double D3 = (P.col(i) - P.col(i2)).norm();

                // get the coefficients of the P3P equation from each subset.

                rowvec = getp3p(cg1, cg2, cg3, sg1, sg2, D1, D2, D3);
                D4.row(j) = rowvec;
                j += 1;

                if (j > n - 3) break;
        }

        Eigen::VectorXd D7(8), dumvec(8), dumvec1(5);
        D7.setZero();

        for (int i = 0; i < n - 2; i++)
        {
                dumvec1 = D4.row(i);
                dumvec = getpoly7(dumvec1);
                D7 += dumvec;
        }

        Eigen::PolynomialSolver<double, 7> psolve(D7.reverse());
        Eigen::VectorXcd comp_roots = psolve.roots().transpose();
        Eigen::VectorXd real_comp, imag_comp;
        real_comp = comp_roots.real();
        imag_comp = comp_roots.imag();

        Eigen::VectorXd::Index max_index;

        double max_real = real_comp.cwiseAbs().maxCoeff(&max_index);

        std::vector<double> act_roots_;

        int cnt = 0;

        for (int i = 0; i < imag_comp.size(); i++)
        {
                if (std::abs(imag_comp(i)) / max_real < 0.001)
                {
                        act_roots_.push_back(real_comp(i));
                        cnt++;
                }
        }

        double* ptr = &act_roots_[0];
        Eigen::Map<Eigen::VectorXd> act_roots(ptr, cnt);

        if (cnt == 0)
        {
                return false;
        }

        Eigen::VectorXd act_roots1(cnt);
        act_roots1 << act_roots.segment(0, cnt);

        std::vector<Eigen::Matrix3d> R_cum(cnt);
        std::vector<Eigen::Vector3d> t_cum(cnt);
        std::vector<double> err_cum(cnt);

        for (int i = 0; i < cnt; i++)
        {
                double root = act_roots(i);

                // Compute the rotation matrix

                double d2 = cg1 + root;

                Eigen::Vector3d unitx, unity, unitz;
                unitx << 1, 0, 0;
                unity << 0, 1, 0;
                unitz << 0, 0, 1;
                x = v2 * d2 - v1;
                x /= x.norm();
                if (std::abs(unity.dot(x)) < std::abs(unitz.dot(x)))
                {
                        z = x.cross(unity);
                        z /= z.norm();
                        y = z.cross(x);
                        y / y.norm();
                }
                else
                {
                        y = unitz.cross(x);
                        y /= y.norm();
                        z = x.cross(y);
                        z /= z.norm();
                }
                R.col(0) = x;
                R.col(1) = y;
                R.col(2) = z;

                // calculating c, s, tx, ty, tz

                Eigen::MatrixXd D(2 * n, 6);
                D.setZero();

                R0 = R.transpose();
                Eigen::VectorXd r(
                        Eigen::Map<Eigen::VectorXd>(R0.data(), R0.cols() * R0.rows()));

                for (int j = 0; j < n; j++)
                {
                        double ui = img_pts(j, 0), vi = img_pts(j, 1), xi = P(0, j),
                                   yi = P(1, j), zi = P(2, j);
                        D.row(2 * j) << -r(1) * yi + ui * (r(7) * yi + r(8) * zi) -
                                                                r(2) * zi,
                                -r(2) * yi + ui * (r(8) * yi - r(7) * zi) + r(1) * zi, -1, 0,
                                ui, ui * r(6) * xi - r(0) * xi;

                        D.row(2 * j + 1)
                                << -r(4) * yi + vi * (r(7) * yi + r(8) * zi) - r(5) * zi,
                                -r(5) * yi + vi * (r(8) * yi - r(7) * zi) + r(4) * zi, 0, -1,
                                vi, vi * r(6) * xi - r(3) * xi;
                }

                Eigen::MatrixXd DTD = D.transpose() * D;

                Eigen::EigenSolver<Eigen::MatrixXd> es(DTD);

                Eigen::VectorXd Diag = es.pseudoEigenvalueMatrix().diagonal();

                Eigen::MatrixXd V_mat = es.pseudoEigenvectors();

                Eigen::MatrixXd::Index min_index;

                Diag.minCoeff(&min_index);

                Eigen::VectorXd V = V_mat.col(min_index);

                V /= V(5);

                double c = V(0), s = V(1);
                t << V(2), V(3), V(4);

                // calculating the camera pose by 3d alignment
                Eigen::VectorXd xi, yi, zi;
                xi = P.row(0);
                yi = P.row(1);
                zi = P.row(2);

                Eigen::MatrixXd XXcs(3, n), XXc(3, n);
                XXc.setZero();

                XXcs.row(0) = r(0) * xi + (r(1) * c + r(2) * s) * yi +
                                          (-r(1) * s + r(2) * c) * zi +
                                          t(0) * Eigen::VectorXd::Ones(n);
                XXcs.row(1) = r(3) * xi + (r(4) * c + r(5) * s) * yi +
                                          (-r(4) * s + r(5) * c) * zi +
                                          t(1) * Eigen::VectorXd::Ones(n);
                XXcs.row(2) = r(6) * xi + (r(7) * c + r(8) * s) * yi +
                                          (-r(7) * s + r(8) * c) * zi +
                                          t(2) * Eigen::VectorXd::Ones(n);

                for (int ii = 0; ii < n; ii++)
                        XXc.col(ii) = Q.col(ii) * XXcs.col(ii).norm();

                Eigen::Matrix3d R2;
                Eigen::Vector3d t2;

                Eigen::MatrixXd XXw = obj_pts.transpose();

                calcampose(XXc, XXw, R2, t2);

                R_cum[i] = R2;
                t_cum[i] = t2;

                for (int k = 0; k < n; k++) XXc.col(k) = R2 * XXw.col(k) + t2;

                Eigen::MatrixXd xxc(2, n);

                xxc.row(0) = XXc.row(0).array() / XXc.row(2).array();
                xxc.row(1) = XXc.row(1).array() / XXc.row(2).array();

                double res = ((xxc.row(0) - img_pts.col(0).transpose()).norm() +
                                          (xxc.row(1) - img_pts.col(1).transpose()).norm()) /
                                         2;

                err_cum[i] = res;
        }

        int pos_cum =
                std::min_element(err_cum.begin(), err_cum.end()) - err_cum.begin();

        R_ = R_cum[pos_cum];
        t_ = t_cum[pos_cum];

        return true;
}

void mrpt::vision::pnp::rpnp::calcampose(
        Eigen::MatrixXd& XXc, Eigen::MatrixXd& XXw, Eigen::Matrix3d& R2,
        Eigen::Vector3d& t2)
{
        Eigen::MatrixXd X = XXc;
        Eigen::MatrixXd Y = XXw;
        Eigen::MatrixXd K =
                Eigen::MatrixXd::Identity(n, n) - Eigen::MatrixXd::Ones(n, n) * 1 / n;
        Eigen::VectorXd ux, uy;
        uy = X.rowwise().mean();
        ux = Y.rowwise().mean();

        // Need to verify sigmax2
        double sigmax2 =
                (((X * K).array() * (X * K).array()).colwise().sum()).mean();

        Eigen::MatrixXd SXY = Y * K * (X.transpose()) / n;

        Eigen::JacobiSVD<Eigen::MatrixXd> svd(
                SXY, Eigen::ComputeThinU | Eigen::ComputeThinV);

        Eigen::Matrix3d S = Eigen::MatrixXd::Identity(3, 3);
        if (SXY.determinant() < 0) S(2, 2) = -1;

        R2 = svd.matrixV() * S * svd.matrixU().transpose();

        double c2 = (svd.singularValues().asDiagonal() * S).trace() / sigmax2;
        t2 = uy - c2 * R2 * ux;

        Eigen::Vector3d x, y, z;
        x = R2.col(0);
        y = R2.col(1);
        z = R2.col(2);

        if ((x.cross(y) - z).norm() > 0.02) R2.col(2) = -R2.col(2);
}

Eigen::VectorXd mrpt::vision::pnp::rpnp::getpoly7(const Eigen::VectorXd& vin)
{
        Eigen::VectorXd vout(8);
        vout << 4 * pow(vin(0), 2), 7 * vin(1) * vin(0),
                6 * vin(2) * vin(0) + 3 * pow(vin(1), 2),
                5 * vin(3) * vin(0) + 5 * vin(2) * vin(1),
                4 * vin(4) * vin(0) + 4 * vin(3) * vin(1) + 2 * pow(vin(2), 2),
                3 * vin(4) * vin(1) + 3 * vin(3) * vin(2),
                2 * vin(4) * vin(2) + pow(vin(3), 2), vin(4) * vin(3);
        return vout;
}

Eigen::VectorXd mrpt::vision::pnp::rpnp::getp3p(
        double l1, double l2, double A5, double C1, double C2, double D1, double D2,
        double D3)
{
        double A1 = (D2 / D1) * (D2 / D1);
        double A2 = A1 * pow(C1, 2) - pow(C2, 2);
        double A3 = l2 * A5 - l1;
        double A4 = l1 * A5 - l2;
        double A6 = (pow(D3, 2) - pow(D1, 2) - pow(D2, 2)) / (2 * pow(D1, 2));
        double A7 = 1 - pow(l1, 2) - pow(l2, 2) + l1 * l2 * A5 + A6 * pow(C1, 2);

        Eigen::VectorXd vec(5);

        vec << pow(A6, 2) - A1 * pow(A5, 2), 2 * (A3 * A6 - A1 * A4 * A5),
                pow(A3, 2) + 2 * A6 * A7 - A1 * pow(A4, 2) - A2 * pow(A5, 2),
                2 * (A3 * A7 - A2 * A4 * A5), pow(A7, 2) - A2 * pow(A4, 2);

        return vec;
}