--- title: Camera Mathematics categories: session --- # Key Concepts - calibration - perspective # Briefing ## Image Projection ### World frame and camera frame $$\mathbf{X} = R\mathbf{X}_0 + T\in\mathbb{R}^3$$ ### Projection $$ Z \begin{bmatrix}x\\y\\1\end{bmatrix} = \begin{bmatrix}f 0 0 0 \\0 f 0 0 \\0 0 1 0\end{bmatrix} \cdot \begin{bmatrix}X\\Y\\Z\\1\end{bmatrix}$$ Note that $Z$ is typically unknown. We write $\lambda (=Z)$ for this unknown constant. ### Ideal Camera Projection $$ \lambda \begin{bmatrix}x\\y\\1\end{bmatrix} = \begin{bmatrix}f 0 0 \\0 f 0 \\0 0 1 \end{bmatrix} \cdot \begin{bmatrix}1 0 0 0 \\0 1 0 0 \\0 0 1 0\end{bmatrix} \cdot \begin{bmatrix}R T \\0 1\end{bmatrix} \cdot \begin{bmatrix}X\\Y\\Z\\1\end{bmatrix}$$ $$ \lambda \textbf{x} = K_f\Pi_0\mathbf{X} = K_f\Pi_0g\mathbf{X}_0$$ ### From meters to pixels + Same units for world frame and image frame, i.e. meter + Meaningless - images are measured in pixels **TODO** complete ## Distortion ### Radial Distortion + Artifact of wide angle lenses (wide field of view). + Simplest effective model: $$ \begin{align} x &= x_d(1 + a_1r^2 + a_2r^4) \\ y &= y_d(1 + a_1r^2 + a_2r^4) \\ r &= x_d^2+y_d^2 \end{align} $$ + $(x,y)$ are the true co-ordinate of some point + $(x_d,y_d)$ are the co-ordinates in the distorted image + This is easy to automate, so we can disregard distortion for analysis + OpenCV uses at sixth order model $$ \begin{align} x_d &= x(1 + a_1r^2 + a_2r^4 + a_3r^6) \\ y_d &= y(1 + a_1r^2 + a_2r^4 + a_3r^6) \end{align} $$ ## Tangential Distortion Not discussed in the text book + Occurs when the lens and image plane are not parallel $$ \begin{align} x_d &= x+ [2p_1xy+p_2(r^2+2x^2)] y_d &= y+[p_1(r^2+2y^2)+2p_2xy] \end{align} $$ # Exercises # Debrief