Hi

I've been working feverishly to straighten up the Structure from Motion Toy Library, and make it more robust. During my experiments with different methods I wanted to test out a different method for decomposing the Essential matrix to rotation R and translation t, other than that of Hartley and Zisserman using SVD. That's when I came upon this paper: here by Berthold Horn from 1990, that traces the steps of Longuet-Higgins who came up with the derivation for the Essential matrix. It has a closed form solution that works pretty well, and here it is implemented with the Eigen math library (a very good library to get to know).

void DecomposeEssentialUsingHorn90(double _E[9], double _R1[9], double _R2[9], double _t1[3], double _t2[3]) { //from : http://people.csail.mit.edu/bkph/articles/Essential.pdf using namespace Eigen; Matrix3d E = Map<Matrix<double,3,3,RowMajor> >(_E); Matrix3d EEt = E * E.transpose(); Vector3d e0e1 = E.col(0).cross(E.col(1)),e1e2 = E.col(1).cross(E.col(2)),e2e0 = E.col(2).cross(E.col(2)); Vector3d b1,b2; #if 1 //Method 1 Matrix3d bbt = 0.5 * EEt.trace() * Matrix3d::Identity() - EEt; //Horn90 (12) Vector3d bbt_diag = bbt.diagonal(); if (bbt_diag(0) > bbt_diag(1) && bbt_diag(0) > bbt_diag(2)) { b1 = bbt.row(0) / sqrt(bbt_diag(0)); b2 = -b1; } else if (bbt_diag(1) > bbt_diag(0) && bbt_diag(1) > bbt_diag(2)) { b1 = bbt.row(1) / sqrt(bbt_diag(1)); b2 = -b1; } else { b1 = bbt.row(2) / sqrt(bbt_diag(2)); b2 = -b1; } #else //Method 2 if (e0e1.norm() > e1e2.norm() && e0e1.norm() > e2e0.norm()) { b1 = e0e1.normalized() * sqrt(0.5 * EEt.trace()); //Horn90 (18) b2 = -b1; } else if (e1e2.norm() > e0e1.norm() && e1e2.norm() > e2e0.norm()) { b1 = e1e2.normalized() * sqrt(0.5 * EEt.trace()); //Horn90 (18) b2 = -b1; } else { b1 = e2e0.normalized() * sqrt(0.5 * EEt.trace()); //Horn90 (18) b2 = -b1; } #endif //Horn90 (19) Matrix3d cofactors; cofactors.col(0) = e1e2; cofactors.col(1) = e2e0; cofactors.col(2) = e0e1; cofactors.transposeInPlace(); //B = [b]_x , see Horn90 (6) and http://en.wikipedia.org/wiki/Cross_product#Conversion_to_matrix_multiplication Matrix3d B1; B1 << 0,-b1(2),b1(1), b1(2),0,-b1(0), -b1(1),b1(0),0; Matrix3d B2; B2 << 0,-b2(2),b2(1), b2(2),0,-b2(0), -b2(1),b2(0),0; Map<Matrix<double,3,3,RowMajor> > R1(_R1),R2(_R2); //Horn90 (24) R2 = (cofactors.transpose() - B1*E) / b1.dot(b1); R1 = (cofactors.transpose() - B2*E) / b2.dot(b2); Map<Vector3d> t1(_t1),t2(_t2); t1 = b2; t2 = b1; cout << "Horn90 provided " << endl << R1 << endl << "and" << endl << R2 << endl; }

Once I get the rest of the Structure from Motion library up to a certain level, I will make a more thorough post.

Enjoy

Roy.