Revisiting Sampson Approximations for Geometric Estimation Problems
Felix Rydell Angélica Torres Viktor Larsson
Many problems in computer vision can be formulated as geometric estimation
problems, i.e. given a collection of measurements (e.g. point correspondences)
we wish to fit a model (e.g. an essential matrix) that agrees with our
observations. This necessitates some measure of how much an observation
``agrees" with a given model. A natural choice is to consider the smallest
perturbation that makes the observation exactly satisfy the constraints.
However, for many problems, this metric is expensive or otherwise intractable
to compute. The so-called Sampson error approximates this geometric error
through a linearization scheme. For epipolar geometry, the Sampson error is a
popular choice and in practice known to yield very tight approximations of the
corresponding geometric residual (the reprojection error).
  In this paper we revisit the Sampson approximation and provide new
theoretical insights as to why and when this approximation works, as well as
provide explicit bounds on the tightness under some mild assumptions. Our
theoretical results are validated in several experiments on real data and in
the context of different geometric estimation tasks.