Reprojection error

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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[edit]

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