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 ${\hat {{\mathbf {x_{i}}}}},{\hat {{\mathbf {x_{i}}}}}'$

References

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

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