diff --git a/modules/calib3d/include/opencv2/calib3d.hpp b/modules/calib3d/include/opencv2/calib3d.hpp index 8d6fedf1a2..7145aa9ac6 100644 --- a/modules/calib3d/include/opencv2/calib3d.hpp +++ b/modules/calib3d/include/opencv2/calib3d.hpp @@ -567,7 +567,97 @@ focal length. function requires exactly four object and image points. The function estimates the object pose given a set of object points, their corresponding image -projections, as well as the camera matrix and the distortion coefficients. +projections, as well as the camera matrix and the distortion coefficients, see the figure below +(more precisely, the X-axis of the camera frame is pointing to the right, the Y-axis downward +and the Z-axis forward). + +![](pnp.jpg) + +Points expressed in the world frame \f$ \bf{X}_w \f$ are projected into the image plane \f$ \left[ u, v \right] \f$ +using the perspective projection model \f$ \Pi \f$ and the camera intrinsic parameters matrix \f$ \bf{A} \f$: + +\f[ + \begin{align*} + \begin{bmatrix} + u \\ + v \\ + 1 + \end{bmatrix} &= + \bf{A} \hspace{0.1em} \Pi \hspace{0.2em} ^{c}\bf{M}_w + \begin{bmatrix} + X_{w} \\ + Y_{w} \\ + Z_{w} \\ + 1 + \end{bmatrix} \\ + \begin{bmatrix} + u \\ + v \\ + 1 + \end{bmatrix} &= + \begin{bmatrix} + f_x & 0 & c_x \\ + 0 & f_y & c_y \\ + 0 & 0 & 1 + \end{bmatrix} + \begin{bmatrix} + 1 & 0 & 0 & 0 \\ + 0 & 1 & 0 & 0 \\ + 0 & 0 & 1 & 0 + \end{bmatrix} + \begin{bmatrix} + r_{11} & r_{12} & r_{13} & t_x \\ + r_{21} & r_{22} & r_{23} & t_y \\ + r_{31} & r_{32} & r_{33} & t_z \\ + 0 & 0 & 0 & 1 + \end{bmatrix} + \begin{bmatrix} + X_{w} \\ + Y_{w} \\ + Z_{w} \\ + 1 + \end{bmatrix} + \end{align*} +\f] + +The estimated pose is thus the rotation (`rvec`) and the translation (`tvec`) vectors that allow to transform +a 3D point expressed in the world frame into the camera frame: + +\f[ + \begin{align*} + \begin{bmatrix} + X_c \\ + Y_c \\ + Z_c \\ + 1 + \end{bmatrix} &= + \hspace{0.2em} ^{c}\bf{M}_w + \begin{bmatrix} + X_{w} \\ + Y_{w} \\ + Z_{w} \\ + 1 + \end{bmatrix} \\ + \begin{bmatrix} + X_c \\ + Y_c \\ + Z_c \\ + 1 + \end{bmatrix} &= + \begin{bmatrix} + r_{11} & r_{12} & r_{13} & t_x \\ + r_{21} & r_{22} & r_{23} & t_y \\ + r_{31} & r_{32} & r_{33} & t_z \\ + 0 & 0 & 0 & 1 + \end{bmatrix} + \begin{bmatrix} + X_{w} \\ + Y_{w} \\ + Z_{w} \\ + 1 + \end{bmatrix} + \end{align*} +\f] @note - An example of how to use solvePnP for planar augmented reality can be found at