# Reprojection error

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 ${\hat {\mathbf {X} }}$ recreates the point's true projection $\mathbf {x}$ . More precisely, let $\mathbf {P}$ be the projection matrix of a camera and ${\hat {\mathbf {x} }}$ be the image projection of ${\hat {\mathbf {X} }}$ , i.e. ${\hat {\mathbf {x} }}=\mathbf {P} \,{\hat {\mathbf {X} }}$ . The reprojection error of ${\hat {\mathbf {X} }}$ is given by $d(\mathbf {x} ,\,{\hat {\mathbf {x} }})$ , where $d(\mathbf {x} ,\,{\hat {\mathbf {x} }})$ denotes the Euclidean distance between the image points represented by vectors $\mathbf {x}$ and ${\hat {\mathbf {x} }}$ .
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 $\{\mathbf {x_{i}} \leftrightarrow \mathbf {x_{i}} '\}$ . We wish to find a homography ${\hat {\mathbf {H} }}$ and pairs of perfectly matched points ${\hat {\mathbf {x_{i}} }}$ and ${\hat {\mathbf {x} }}_{i}'$ , i.e. points that satisfy ${\hat {\mathbf {x_{i}} }}'={\hat {H}}\mathbf {{\hat {x}}_{i}}$ that minimize the reprojection error function given by
$\sum _{i}d(\mathbf {x_{i}} ,{\hat {\mathbf {x_{i}} }})^{2}+d(\mathbf {x_{i}} ',{\hat {\mathbf {x_{i}} }}')^{2}$ 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 ${\hat {\mathbf {x_{i}} }},{\hat {\mathbf {x_{i}} }}'$ 