Reprojection error

2023-12-18T14:06:09Z

92.76.185.7: Reprojection error has nothing to do with old phones, old cameras or image artifacts, this was either vandalism or a case of mistaken identity.

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

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 <math>\{\mathbf{x_i} \leftrightarrow \mathbf{x_i}'\}</math>. We wish to find a [[homography (computer vision)|homography]] <math>\hat{\mathbf{H}}</math> and pairs of perfectly matched points <math>\hat{\mathbf{x_i}}</math> and <math>\hat{\mathbf{x}}_i'</math>, i.e. points that satisfy <math>\hat{\mathbf{x_i}}' = \hat{H}\mathbf{\hat{x}_i}</math> that minimize the reprojection error function given by
:<math> \sum_i d(\mathbf{x_i}, \hat{\mathbf{x_i}})^2 + d(\mathbf{x_i}', \hat{\mathbf{x_i}}')^2</math>
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 <math>\hat{\mathbf{x_i}}, \hat{\mathbf{x_i}}'</math>

==References==

*{{cite book |
author=Richard Hartley and Andrew Zisserman |
title=Multiple View Geometry in computer vision |
publisher=Cambridge University Press|
year=2003 |
isbn=0-521-54051-8}}

[[Category:Geometry in computer vision]]

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Reprojection error