The linear solution will not find an optimal solution for the projection matrices as it is dependent on the initial projection matrix P' . The non-linear approach solves this problem by optimising the values of P' so that the algebraic error Hartley98 along the whole image sequence is minimised.
An initial solution is computed using the linear method, and the twelve entries of the last camera matrix are adjusted, such that the algebraic error of the whole sequence is minimised. To ensure the minimisation process does not converge on the trivial solution (where all cameras are at the same point) the constraint ||t||=1 must be enforced. This is implemented using Levenberg-Marquardt minimisation.
This algorithm is an extension of the work by Hartley Hartley98 from the three view trifocal tensor to long image sequences. Hartley solved for the trifocal tensor by optimising the epipoles so that the algebraic error of the camera matrix entries was minimised. In this work one matrix (P') is fixed and the projection matrices of the image sequence are computed.