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

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 $\{{𝐱}_{𝐢}\leftrightarrow {{𝐱}_{𝐢}}^{\prime }\}$. We wish to find a homography $\widehat{𝐇}$ and pairs of perfectly matched points $\widehat{{𝐱}_{𝐢}}$ and ${{\widehat{𝐱}}_i}^{\prime }$, i.e. points that satisfy ${\widehat{{𝐱}_{𝐢}}}^{\prime }=\hat{H}{\widehat{𝐱}}_{𝐢}$ that minimize the reprojection error function given by

$\sum _{i}d({𝐱}_{𝐢},\widehat{{𝐱}_{𝐢}}{)}^2+d({{𝐱}_{𝐢}}^{\prime },{\widehat{{𝐱}_{𝐢}}}^{\prime }{)}^2$

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 $\widehat{{𝐱}_{𝐢}},{\widehat{{𝐱}_{𝐢}}}^{\prime }$

References

• Richard Hartley and Andrew Zisserman (2003). Multiple View Geometry in computer vision. Cambridge University Press. ISBN 0-521-54051-8.