Collinearity equation

Let x, y, and z refer to a coordinate system with the x- and y-axis in the sensor plane. Denote the coordinates of the point P on the object by ${\displaystyle x_{P},y_{P},z_{P}}$, the coordinates of the image point of P on the sensor plane by x and y and the coordinates of the projection (optical) centre by ${\displaystyle x_{0},y_{0},z_{0}}$. As a consequence of the projection method there is the same fixed ratio ${\displaystyle \lambda }$ between ${\displaystyle x-x_{0}}$ and ${\displaystyle x_{0}-x_{P}}$, ${\displaystyle y-y_{0}}$ and ${\displaystyle y_{0}-y_{P}}$, and the distance of the projection centre to the sensor plane ${\displaystyle z_{0}=c}$ and ${\displaystyle z_{P}-z_{0}}$. Hence:

${\displaystyle x-x_{0}=-\lambda (x_{P}-x_{0})}$

${\displaystyle y-y_{0}=-\lambda (y_{P}-y_{0})}$

${\displaystyle c=\lambda (z_{P}-z_{0}),}$

Solving for ${\displaystyle \lambda }$ in the last equation and entering it in the others yields:

${\displaystyle x-x_{0}=-c\ {\frac {x_{P}-x_{0}}{z_{P}-z_{0}}}}$

${\displaystyle y-y_{0}=-c\ {\frac {y_{P}-y_{0}}{z_{P}-z_{0}}}}$

The point P is normally given in some coordinate system "outside" the camera by the coordinates X, Y and Z, and the projection centre by ${\displaystyle X_{0},Y_{0},Z_{0}}$. These coordinates may be transformed through a rotation and a translation to the system on the camera. The translation doesn't influence the differences of the coordinates, and the rotation, often called camera transform, is given by a 3×3-matrix R, transforming ${\displaystyle (X-X_{0},Y-Y_{0},Z-Z_{0})}$ into:

${\displaystyle x_{P}-x_{0}=R_{11}(X-X_{0})+R_{21}(Y-Y_{0})+R_{31}(Z-Z_{0})}$

${\displaystyle y_{P}-y_{0}=R_{12}(X-X_{0})+R_{22}(Y-Y_{0})+R_{32}(Z-Z_{0})}$

and

${\displaystyle z_{P}-z_{0}=R_{13}(X-X_{0})+R_{23}(Y-Y_{0})+R_{33}(Z-Z_{0})}$

Substitution of these expressions, leads to a set of two equations, known as the collinearity equations:

${\displaystyle x-x_{0}=-c\ {\frac {R_{11}(X-X_{0})+R_{21}(Y-Y_{0})+R_{31}(Z-Z_{0})}{R_{13}(X-X_{0})+R_{23}(Y-Y_{0})+R_{33}(Z-Z_{0})}}}$

${\displaystyle y-y_{0}=-c\ {\frac {R_{12}(X-X_{0})+R_{22}(Y-Y_{0})+R_{32}(Z-Z_{0})}{R_{13}(X-X_{0})+R_{23}(Y-Y_{0})+R_{33}(Z-Z_{0})}}}$

The most obvious use of these equations is for images recorded by a camera. In this case the equation describes transformations from object space (X, Y, Z) to image coordinates (x, y). It forms the basis for the equations used in bundle adjustment. They indicate that the image point (on the sensor plate of the camera), the observed point (on the object) and the projection center of the camera were aligned when the picture was taken.

• 3D projection
• Epipolar geometry
• Pinhole camera model