Zhang's camera calibration

• Camera calibration from a single plane at few (at least three, two skew is ignored) orientations.
• Without loss of generality, assume that the model plane is on $Z = 0$ .
• Then, for points on the model plane
  $$s \left[ \begin{array}{c} u \\ v \\ 1 \end{array}\right] = {\bf C... ...f t} \end{array}\right] \left[ \begin{array}{c} X \\ Y \\ 1 \end{array}\right]$$

• Thus, a model point ${\bf M}$ and its image ${\bf m}$ are related by a homography ${\bf H}$ , where
  $$s {\bf m} = {\bf H} {\bf M}$$
  with
  $${\bf H} = {\bf C} \left[ \begin{array}{ccc} {\bf r_1} & {\bf r_2} & {\bf t} \end{array}\right]$$

• From
  $${\bf H} = \left[ \begin{array}{ccc} {\bf h}_1 & {\bf h}_2 & {\bf ... ...} \left[ \begin{array}{ccc} {\bf r_1} & {\bf r_2} & {\bf t} \end{array}\right]$$
  using the fact that ${\bf r_1}$ and ${\bf r_2}$ are orthonormal, we obtain the relationships
  $$\begin{array}{ccc} {\bf h}_1^t {\bf C}^{-t}{\bf C}^{-1} {\bf h}_2... ...-1} {\bf h}_1 & = & {\bf h}_2^t {\bf C}^{-t}{\bf C}^{-1} {\bf h}_2 \end{array}$$

• Each such homography provides two constraints on the camera intrinsics (image of the absolute conic). Three independent orientations are sufficient to solve for camera internals linearly. Two are sufficient if the skew is ignored.
• Once the camera internals matrix ${\bf C}$ is known, the externals can be readily obtained.
  $$\begin{array}{ccc} {\bf r}_1 & = & \lambda {\bf C}^{-1} {\bf h}_1... ...times {\bf r}_2\\ {\bf t} & = & \lambda {\bf C}^{-1} {\bf h}_3 \\ \end{array}$$
  with $\lambda = 1/\vert\vert{\bf C}^{-1}{\bf h}_1\vert\vert = 1/ \vert\vert{\bf C}^{-1}{\bf h}_2\vert\vert$ . Of course, the computed matrix ${\bf R} = \left[ {\bf r}_1\;\; {\bf r}_2 \;\; {\bf r}_3 \right]$ does not, in general, satisfy the properties of a rotation matrix. The orthonormality properties can be enforced in manned similar to the one described in Tsai's.
• The method can be extended to also obtain the radial lens distortion parameters.