Eigen: convert rotation matrix to quaternion then back getting completely different matrices - matrix

Question

Can anyone help me out with Eigen? I tried to convert quaternion to matrix and then back and got completely different matrices. I am not able to trust quaternion before understanding this issue.
Here is the code:
#include <Eigen/Geometry>
#include <iostream>
void Print_Quaternion(Eigen::Quaterniond &q){
std::cout<<"["<<q.w()<<" "<<q.x()<<" "<<q.y()<<" "<<q.z()<<"]"<<std::endl;
}
void Verify_Orthogonal_Matrix(Eigen::Matrix3d &m)
{
std::cout<<"|c0|="<<m.col(0).norm()<<",|c1|="<<m.col(1).norm()<<",|c2|="<<m.col(2).norm()<<std::endl;
std::cout<<"c0c1="<<m.col(0).dot(m.col(1))<<",c1c2="<<m.col(1).dot(m.col(2))<<",c0c2="<<m.col(0).dot(m.col(2))<<std::endl;
}
int main()
{
Eigen::Matrix3d m; m<<0.991601,0.102421,-0.078975,0.125398,-0.611876,0.78095,-0.0316631,0.784294,0.619581;
std::cout<<"Input matrix:"<<std::endl<<m<<std::endl;
std::cout<<"Verify_Orthogonal_Matrix:"<<std::endl;
Verify_Orthogonal_Matrix(m);
std::cout<<"Convert to quaternion q:"<<std::endl;
Eigen::Quaterniond q(m);
Print_Quaternion(q);
std::cout<<"Convert back to rotation matrix m1="<<std::endl;
Eigen::Matrix3d m1=q.normalized().toRotationMatrix();
std::cout<<m1<<std::endl;
std::cout<<"Verify_Orthogonal_Matrix:"<<std::endl;
Verify_Orthogonal_Matrix(m1);
std::cout<<"Convert again to quaternion q1="<<std::endl;
Eigen::Quaterniond q1(m1);
Print_Quaternion(q1);
}
Here is the result I got:
Input matrix:
0.991601 0.102421 -0.078975
0.125398 -0.611876 0.78095
-0.0316631 0.784294 0.619581
Verify_Orthogonal_Matrix:
|c0|=1,|c1|=1,|c2|=1
c0c1=-4.39978e-07,c1c2=4.00139e-07,c0c2=2.39639e-08
Convert to quaternion q:
[0.706984 0.00118249 -0.0167302 0.00812501]
Convert back to rotation matrix m1=
0.998617 -0.0230481 -0.047257
0.0228899 0.99973 -0.00388638
0.0473339 0.0027993 0.998875
Verify_Orthogonal_Matrix:
|c0|=1,|c1|=1,|c2|=1
c0c1=1.73472e-18,c1c2=-4.33681e-19,c0c2=6.93889e-18
Convert again to quaternion q1=
[0.999653 0.001672 -0.0236559 0.0114885]
Did I do something wrong here? I feel that this should be a well-known problem but I got stuck here. Can someone help me out?

Answer 1

Input matrix is not a rotation matrix, it contains mirroring. Its determinant == -1, but rotation should have +1.
check the code for orthogonalization, and look and the signs of last col
m.col(0).normalize();
m.col(1).normalize();
m.col(2) = m.col(0).cross(m.col(1));
m.col(2).normalize();
m.col(0) = m.col(1).cross(m.col(2));
m.col(0).normalize();
std::cout << "orthogonal matrix:" << std::endl << m << std::endl;
Input matrix:
0.991601 0.102421 -0.078975
0.125398 -0.611876 0.78095
-0.0316631 0.784294 0.619581
orthogonal matrix:
0.991601 0.102421 0.078975
0.125398 -0.611876 -0.78095
-0.0316628 0.784294 -0.619581