Error in calculating the best fitting planar equation using SVD

Question

I am trying to find the best fitting plane for a set of points, and I use SVD to calculate the equation of the plane given by ax+by+cz+d=0.

I have implemented SVD and successfully able to get the normal to the plane, but i'm unable to calculate d.

After some digging, I substituted the centroid calculated back in the equation to calculate d, but I am getting an incorrect value. I'm sure this is a incorrect value because I'm comparing this with the RANSAC method.

The implementation of my code is as follows

    pcl::ModelCoefficients normal_extractor::plane_est_svd(pcl::PointCloud<pcl::PointXYZ>::ConstPtr point_cloud)
{
    Eigen::MatrixXd points_3D(3,point_cloud->width);
    //assigning the points from point cloud to matrix
    for (int i=0;i<point_cloud->width;i++)
    {
        points_3D(0,i) = point_cloud->at(i).x;
        points_3D(1,i) = point_cloud->at(i).y;
        points_3D(2,i) = point_cloud->at(i).z;
    }
    // calcaulating the centroid of the pointcloud
    Eigen::MatrixXd centroid = points_3D.rowwise().mean();
    //std::cout<<"The centroid of the pointclouds is given by:\t"<<centroid<<std::endl;
    //subtract the centroid from points
    points_3D.row(0).array() -= centroid(0);
    points_3D.row(1).array() -= centroid(1);
    points_3D.row(2).array() -= centroid(2);
    //calculate the SVD of points_3D matrix
    Eigen::JacobiSVD<Eigen::MatrixXd> svd(points_3D,Eigen::ComputeFullU);
    Eigen::MatrixXd U_MAT = svd.matrixU();
    //std::cout<<"U matrix transpose is:"<<U_MAT<<std::endl<<std::endl<<"U matrix is:"<<svd.matrixU()<<std::endl;
    /*********************************************************************************************
     * caculating d by sybstituting the centroid back in the quation
     *      aCx+bCy+cCz = -d
     ********************************************************************************************/
    //double d = -((U_MAT(0,2)*points_3D(0,1))+ (U_MAT(1,2)*points_3D(1,1)) + (U_MAT(1,2)*points_3D(1,2)));
    double d = -((U_MAT(0,2)*centroid(0))+ (U_MAT(1,2)*centroid(1)) + (U_MAT(1,2)*centroid(2)));

    pcl::ModelCoefficients normals;
    normals.values.push_back(U_MAT(0,2));
    normals.values.push_back(U_MAT(1,2));
    normals.values.push_back(U_MAT(2,2));
    normals.values.push_back(d);
    return(normals);

}

The results I am getting are

RANSAC method:

    a = -0.0584306  b = 0.0358117   c = 0.997649    d = -0.161604

SVD method:

    a = 0.0584302   b = -0.0357721  c = -0.99765    d = 0.00466139

From the results, I believe the normals are calculated just fine (although the direction is reversed), but the value of d is incorrect. I'm not sure where I am going wrong. Any help is really appreciated.

Thanks in Advance..

Answer 1

1201ProgramAlarm is right, there is a typo in U_MAT(1,2)*centroid(2).

To avoid such typo better write:

d = -centroid.dot(U_MAT).col(2);

you can also simplify:

points_3D.colwise() -= centroid;

For future reference, here is a self-contained example:

#include <iostream>
#include <Eigen/Dense>
using namespace Eigen;
using namespace std;

int main()
{
  int n = 10;
  // generate n points in the plane centered in p and spanned bu the u,v vectors.
  MatrixXd points_3D(3,n);
  Vector3d u = Vector3d::Random().normalized();
  Vector3d v = Vector3d::Random().normalized();
  Vector3d p = Vector3d::Random();
  points_3D = p.rowwise().replicate(n) + u*VectorXd::Random(n).transpose() + v*VectorXd::Random(n).transpose();
  MatrixXd initial_points = points_3D;

  Vector3d centroid = points_3D.rowwise().mean();
  points_3D.colwise()-=centroid;
  JacobiSVD<MatrixXd> svd(points_3D,ComputeFullU);
  Vector3d normal = svd.matrixU().col(2);
  double d = -normal.dot(centroid);

  cout << "Plane equation: " << normal.transpose() << " " << d << endl;
  cout << "Distances: " << (normal.transpose() * initial_points).array() + d << endl;
}