Inverse depth parametrization

In computer vision, the inverse depth parametrization is a parametrization used in methods for 3D reconstruction from multiple images such as simultaneous localization and mapping (SLAM). Given a point formula_1 in 3D space observed by a monocular pinhole camera from multiple views, the inverse depth parametrization of the point's position is a 6D vector that encodes the optical centre of the camera formula_2 when in first observed the point, and the position of the point along the ray passing through formula_1 and formula_2.
Inverse depth parametrization generally improves numerical stability and allows to represent points with zero parallax. Moreover, the error associated to the observation of the point's position can be modelled with a Gaussian distribution when expressed in inverse depth. This is an important property required to apply methods, such as Kalman filters, that assume normality of the measurement error distribution. The major drawback is the larger memory consumption, since the dimensionality of the point's representation is doubled.
Definition.
Given 3D point formula_5 with world coordinates in a reference frame formula_6, observed from different views, the inverse depth parametrization formula_7 of formula_1 is given by:
where the first five components encode the camera pose in the first observation of the point, being formula_10 the optical centre, formula_11 the azimuth, formula_12 the elevation angle, and formula_13 the inverse depth of formula_14 at the first observation.