To the Graduate Council: I am submitting herewith a dissertation ...

Chapter 3: A Feature Saliency Descriptor for Registration and Pose Recovery 28The common approach to computing the motion parameters ( R , t)uses a two-stagetechnique: starting with a linear method, followed by a non-linear refinement step[Faugeras93][Higgins81]. In the linear method the coplanarity constraint (3.3) isrewritten as:m T Em 0(3.4)1 2=where E is a 3x3 matrix, known as the Essential Matrix, that embeds the motionparameters:E×= [ t]R , with[ t]×⎡ 0⎢= ⎢ tz⎢⎣−ty− tt0xzty⎤⎥− tx ⎥ the anti-symmetric matrix0 ⎥⎦representing the cross product in a linear fashion. A system of homogeneous equationsin the entries of E is then set from the point matches. Commonly more than 8 matchesare used resulting in an over-constrained system, which is solved using singular valuedecomposition (SVD). The actual motion parameters are easily computed from theessential matrix. The scale of the translation, and hence of the reconstruction, isinherently ambiguous in the SFM problem, therefore we have only five independentparameters to recover: ϕ , ϕ , ϕ , t , t ) , the three rotation angles and two translations.(x y z y zUnless some knowledge about the actual dimensions of the scene is available[Morris01], the scale will be arbitrary.For the non-linear refinement step several formulations were proposed, where therotation matrix was parametrized as an orthonormal matrix or using unit quaternions[Horn90]. A classic nonlinear method is based on the least squares minimization ofthe Longuet-Higgins cost function [Faugeras93]:minR,ti=1...∑N matchesi1i22m .( t × Rm )(3.5)