---
title: Camera Mathematics
categories: session
---

# Key Concepts

- calibration
- perspective

# Briefing

## 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.

$$ \lambda \begin{bmatrix}x\\y\\1\end{bmatrix} = 
\begin{bmatrix}f  0 0 \\0 f   0 \\0 0 1 \end{bmatrix} \cdot
\begin{bmatrix}1 0 0 0 \\0 1  0 0 \\0 0 1 0\end{bmatrix} \cdot
\begin{bmatrix}R T \\0 1\end{bmatrix} \cdot
\begin{bmatrix}X\\Y\\Z\\1\end{bmatrix}$$

$$ \lambda \textbf{x} = K_f\Pi_0\mathbf{X} = K_f\Pi_0g\mathbf{X}_0$$

## Ideal Camera Projection

# Exercises

# Debrief