Extrinsic Parameters

Transformation

Transformation from "world space" to "camera space": 6 degree of freedom.

Notation

$^AP$ is the coordinates of $P$ in frame $A$.

Given some origin $^A O$, $$^A P = \begin{bmatrix} ^Ax \\ ^Ay \\ ^Az \end{bmatrix} \Leftrightarrow \overrightarrow{OP} = (^Ax \cdot i_A)(^Ay \cdot j_A)(^Az \cdot k_A)$$

Translation

$$^BP = ^AP + ^B(O_A)$$ Using homogeneous coordinates, translation can be expressed as a matrix multiplication. $$\begin{bmatrix} ^Bp \\ 1 \end{bmatrix} = \begin{bmatrix} I & ^BO_A \\ 0^T & 1 \end{bmatrix} \begin{bmatrix} ^AP \\ 1 \end{bmatrix}$$ Where $I$ is 3x3 identity.

Translation is commutative.

Rotation

$$^PB = (^B_AR) (^AP)$$ $^B_AR$ is the rotation matrix, which means describing frame A in the coordinates system of frame B $$^B_AR = \begin{bmatrix} i_A \cdot i_B & j_A \cdot i_B & k_A \cdot i_B \\ i_A \cdot j_B & j_A \cdot j_B & k_A \cdot j_B \\ i_A \cdot k_B & j_A \cdot k_B & k_A \cdot k_B \\ \end{bmatrix} \\ = \begin{bmatrix} ^Bi_A & ^Bj_A & ^Bk_A \end{bmatrix} \\ = \begin{bmatrix} {^Ai_B}^T \\ {^Aj_B}^T \\ {^Ak_B}^T \end{bmatrix} \\ $$

Example: Rotation about z axis

$$R_z(\theta) = \begin{bmatrix} \cos(\theta) & -\sin(\theta) & 0 \\ \sin(\theta) & \cos(\theta) & 0 \\ 0 & 0 &1 \\ \end{bmatrix}$$

Rotation with Homogeneous Coordinates

$$\begin{bmatrix} ^Bp \\ 1 \end{bmatrix} = \begin{bmatrix} ^B_AR & 0 \\ 0^T & 1 \end{bmatrix} \begin{bmatrix} ^AP \\ 1 \end{bmatrix}$$

Rotation is not commutative

Total Rigid Transformation

$$^PB = (^B_AR) (^BP) + ^BO_A$$

Using homogeneous coordinates: $$\begin{bmatrix} ^Bp \\ 1 \end{bmatrix} = \begin{bmatrix} I & ^BO_A \\ 0^T & 1 \end{bmatrix} \begin{bmatrix} ^B_AR & 0 \\ 0^T & 1 \end{bmatrix} \begin{bmatrix} ^AP \\ 1 \end{bmatrix} \\ = \begin{bmatrix} ^R_AB & ^BO_A \\ 0^T & 1 \end{bmatrix} \begin{bmatrix} ^AP \\ 1 \end{bmatrix} \\ = ^B_AT \begin{bmatrix} ^AP \\ 1 \end{bmatrix}$$ So $$\begin{bmatrix} ^AP \\ 1 \end{bmatrix} = ^A_BT \begin{bmatrix} ^BP \\ 1 \end{bmatrix} = (^B_AT)^{-1} \begin{bmatrix} ^AP \\ 1 \end{bmatrix}$$ Homogeneous coordinates:

Homogeneous coordinates allows us to write coordinate transforms as a single matrix! $$^B \overrightarrow{p} = (^B_AT) \space (^A \overrightarrow{p})$$

$$^B \overrightarrow{p}_{(4 \times 1)} = \begin{bmatrix} \begin{bmatrix} & & \\ & ^B_AR & \\ & & \end{bmatrix} \begin{bmatrix} | \\ ^B_A \overrightarrow{t} \\ | \end{bmatrix}\\ \begin{matrix} 0 & 0 &0 \end{matrix} \begin{matrix} &1 \end{matrix} \end{bmatrix}_{(4 \times 4)} \begin{bmatrix} x \\ y \\ z \\ 1 \end{bmatrix}_{(4 \times 1)}$$

From World to Camera

From world to camera is the extrinsic parameter matrix (4 x 4)