reprojection error

{{Inline|date=July 2014}}

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 $\backslash hat\{\backslash mathbf\{X\}\}$ recreates the point's true projection $\backslash mathbf\{x\}$. More precisely, let $\backslash mathbf\{P\}$ be the projection matrix of a camera and $\backslash hat\{\backslash mathbf\{x\}\}$ be the image projection of $\backslash hat\{\backslash mathbf\{X\}\}$, i.e. $\backslash hat\{\backslash mathbf\{x\}\}=\backslash mathbf\{P\}\; \backslash ,\; \backslash hat\{\backslash mathbf\{X\}\}$. The reprojection error of $\backslash hat\{\backslash mathbf\{X\}\}$ is given by $d(\backslash mathbf\{x\},\; \backslash ,\; \backslash hat\{\backslash mathbf\{x\}\})$, where $d(\backslash mathbf\{x\},\; \backslash ,\; \backslash hat\{\backslash mathbf\{x\}\})$ denotes the Euclidean distance between the image points represented by vectors $\backslash mathbf\{x\}$ and $\backslash hat\{\backslash mathbf\{x\}\}$.

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 $\backslash \{\backslash mathbf\{x\_i\}\; \backslash leftrightarrow\; \backslash mathbf\{x\_i\}\text{'}\backslash \}$. We wish to find a homography $\backslash hat\{\backslash mathbf\{H\}\}$ and pairs of perfectly matched points $\backslash hat\{\backslash mathbf\{x\_i\}\}$ and $\backslash hat\{\backslash mathbf\{x\}\}\_i\text{'}$, i.e. points that satisfy $\backslash hat\{\backslash mathbf\{x\_i\}\}\text{'}\; =\; \backslash hat\{H\}\backslash mathbf\{\backslash hat\{x\}\_i\}$ that minimize the reprojection error function given by

:$\backslash sum\_i\; d(\backslash mathbf\{x\_i\},\; \backslash hat\{\backslash mathbf\{x\_i\}\})^2\; +\; d(\backslash mathbf\{x\_i\}\text{'},\; \backslash hat\{\backslash mathbf\{x\_i\}\}\text{'})^2$

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 $\backslash hat\{\backslash mathbf\{x\_i\}\},\; \backslash hat\{\backslash mathbf\{x\_i\}\}\text{'}$