We may rewrite coplanarity equations by using the following symbols
In this case, this equation in a simplified form is
This equation can be written in the following vector form based on the unknown coefficients
where,
If we collect all y ̃ in one matrix, then we will have the following matrix form
This means that e is a solution of the above homogeneous linear equations.
Spectral analysis method using singular values
If we consider singular value decomposition of matrix A, we will have
In this case, the values on the main diameter of the diagonal matrix Σ will be single values. In an optimistic case, when A is non-singular, there will be non-zero value equal to minimum number of dimensions. A non-zero value σ will be a singular value for the matrix A if and only if there exists a vector u_(m×1) and a vector v_(n×1) such that
In this case, the vectors u and v are called the left and right singlular vectors for the singlular value of σ, respectively. The left and right singlular vectors of matrix A can be extracted from the columns of matrices U and V.
A Numerical solution for Essential matrix
If we return to the equation e^T.Y=0, using the definition stated above, the e matrix is a right singular matrix for zero singular value of the Y matrix.
In the case where the Y matrix is constructed using more than 8 points, there is a possibility that none of the singular values will be zero, this happens especially when there is noise in the observations, in this case the smallest singular value is considered in the Y matrix. Finally, by arranging the matrix e, the Essential matrix is formed.
Another side-effect of using the above equations happens when the Essential matrix is obtained, which does not have the form of an ideal Essential matrix where there are two non-zero singular values, and the third singular value is equal to zero. In this case, we use the following method to find the matrix E’ that meets the above conditions and has the closest distance to the estimated basic matrix (E_est).
In S matrix, we expect one of the diagonal values to be much smaller compared to the other two values. To apply the above constraint, we construct the S’ as
that s_1 and s_2 are the largest singular values of the E_est matrix, respectively. Using the above matrix, we form the E’ matrix as follows
In this way, the matrix E’ is a relatively acceptable solution for our problem.