Reprojection error information

This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (July 2014) (Learn how and when to remove this message)

The reprojection error is a geometric error corresponding to the image distance between a projected point and a measured one. It is used to quantify how closely an estimate of a 3D point ${\displaystyle {\hat {\mathbf {X} }}}$ recreates the point's true projection ${\displaystyle \mathbf {x} }$. More precisely, let ${\displaystyle \mathbf {P} }$ be the projection matrix of a camera and ${\displaystyle {\hat {\mathbf {x} }}}$ be the image projection of ${\displaystyle {\hat {\mathbf {X} }}}$, i.e. ${\displaystyle {\hat {\mathbf {x} }}=\mathbf {P} \,{\hat {\mathbf {X} }}}$. The reprojection error of ${\displaystyle {\hat {\mathbf {X} }}}$ is given by ${\displaystyle d(\mathbf {x} ,\,{\hat {\mathbf {x} }})}$, where ${\displaystyle d(\mathbf {x} ,\,{\hat {\mathbf {x} }})}$ denotes the Euclidean distance between the image points represented by vectors ${\displaystyle \mathbf {x} }$ and ${\displaystyle {\hat {\mathbf {x} }}}$.

Minimizing the reprojection error can be used for estimating the error from point correspondences between two images. Suppose we are given 2D to 2D point imperfect correspondences ${\displaystyle \{\mathbf {x_{i}} \leftrightarrow \mathbf {x_{i}} '\}}$. We wish to find a homography ${\displaystyle {\hat {\mathbf {H} }}}$ and pairs of perfectly matched points ${\displaystyle {\hat {\mathbf {x_{i}} }}}$ and ${\displaystyle {\hat {\mathbf {x} }}_{i}'}$, i.e. points that satisfy ${\displaystyle {\hat {\mathbf {x_{i}} }}'={\hat {H}}\mathbf {{\hat {x}}_{i}} }$ that minimize the reprojection error function given by

${\displaystyle \sum _{i}d(\mathbf {x_{i}} ,{\hat {\mathbf {x_{i}} }})^{2}+d(\mathbf {x_{i}} ',{\hat {\mathbf {x_{i}} }}')^{2}}$

So the correspondences can be interpreted as imperfect images of a world point and the reprojection error quantifies their deviation from the true image projections ${\displaystyle {\hat {\mathbf {x_{i}} }},{\hat {\mathbf {x_{i}} }}'}$