When are simple LS estimators enough? An empirical study of LS, TLS and GTLS

A variety of least-squares estimators of significantly different complexity and generality are available to solve over-constrained linear systems. The most theoretically general may not necessarily be the best choice in practice; problem conditions may be such that simpler and faster algorithms, if theoretically inferior, would yield acceptable errors. We investigate when this may happen using homography estimation as the reference problem. We study the errors of LS, TLS, equilibrated TLS and GTLS algorithms with different noise types and varying intensity and correlation levels. To allow direct comparisons with algorithms from the applied mathematics and computer vision communities, we consider both inhomogeneous and homogeneous systems. We add noise to image co-ordinates and system matrix entries in separate experiments, to take into account the effect on noise properties (heteroscedasticity) of pre-processing data transformations. We find that the theoretically most general algorithms may not always be worth their higher complexity; comparable results are obtained with moderate levels of noise intensity and correlation. We identify such levels quantitatively for the reference problem, thus suggesting when simpler algorithms can be applied with limited errors in spite of their restrictive assumptions.