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 X ^ {displaystyle {hat {mathbf {X} }}} recreates the point's true projection x {displaystyle mathbf {x} } . More precisely, let P {displaystyle mathbf {P} } be the projection matrix of a camera and x ^ {displaystyle {hat {mathbf {x} }}} be the image projection of X ^ {displaystyle {hat {mathbf {X} }}} , i.e. x ^ = P X ^ {displaystyle {hat {mathbf {x} }}=mathbf {P} ,{hat {mathbf {X} }}} . The reprojection error of X ^ {displaystyle {hat {mathbf {X} }}} is given by d ( x , x ^ ) {displaystyle d(mathbf {x} ,,{hat {mathbf {x} }})} , where d ( x , x ^ ) {displaystyle d(mathbf {x} ,,{hat {mathbf {x} }})} denotes the Euclidean distance between the image points represented by vectors x {displaystyle mathbf {x} } and x ^ {displaystyle {hat {mathbf {x} }}} . 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 X ^ {displaystyle {hat {mathbf {X} }}} recreates the point's true projection x {displaystyle mathbf {x} } . More precisely, let P {displaystyle mathbf {P} } be the projection matrix of a camera and x ^ {displaystyle {hat {mathbf {x} }}} be the image projection of X ^ {displaystyle {hat {mathbf {X} }}} , i.e. x ^ = P X ^ {displaystyle {hat {mathbf {x} }}=mathbf {P} ,{hat {mathbf {X} }}} . The reprojection error of X ^ {displaystyle {hat {mathbf {X} }}} is given by d ( x , x ^ ) {displaystyle d(mathbf {x} ,,{hat {mathbf {x} }})} , where d ( x , x ^ ) {displaystyle d(mathbf {x} ,,{hat {mathbf {x} }})} denotes the Euclidean distance between the image points represented by vectors x {displaystyle mathbf {x} } and x ^ {displaystyle {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 { x i ↔ x i ′ } {displaystyle {mathbf {x_{i}} leftrightarrow mathbf {x_{i}} '}} . We wish to find a homography H ^ {displaystyle {hat {mathbf {H} }}} and pairs of perfectly matched points x i ^ {displaystyle {hat {mathbf {x_{i}} }}} and x ^ i ′ {displaystyle {hat {mathbf {x} }}_{i}'} , i.e. points that satisfy x i ^ ′ = H ^ x ^ i {displaystyle {hat {mathbf {x_{i}} }}'={hat {H}}mathbf {{hat {x}}_{i}} } that minimize the reprojection error function given by 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 x i ^ , x i ^ ′ {displaystyle {hat {mathbf {x_{i}} }},{hat {mathbf {x_{i}} }}'}