---
title: Camera Mathematics
categories: lecture
---

# Key Concepts

- calibration
- perspective

# Lectore Notes

## Image Projection

### World frame and camera frame

+ Change of basis

$$\mathbf{X} = R\mathbf{X}_0 + T \in\mathbb{R}^3 $$

+ or with homogeneous co-ordinates

$$\mathbf{X} = \begin{bmatrix} R & T \\ 0 & 1 \end{bmatrix}\cdot \mathbf{X}_0$$

### Projection

+ Recall the ideal perspective

$$\mathbf{x} = \begin{bmatrix}x\\y\end{bmatrix}
= \frac fZ\begin{bmatrix}x\\y\end{bmatrix}$$

+ In homogeneous co-ordinates we have

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

+ $Z$ is typically unknown.
+ We write $\lambda (=Z)$ for this unknown constant.

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

### Ideal Camera Projection

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

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

+ Canonical projection matrix $\Pi_0$

### From meters to pixels

+ Same units for world frame and image frame, i.e. meter
+ Meaningless - images are measured in pixels

$$\begin{bmatrix} x_s\\y_s\end{bmatrix} =
\begin{bmatrix} s_x & 0\\ 0 & s_y\end{bmatrix} \cdot
\begin{bmatrix} x\\y\end{bmatrix} $$

+ Each pixel is $s_x\times s_y$ in metric units
+ Then we translate the origin

$$x' = x_s + o_x \quad y' = y_s + o_y $$

+ or in homogeneous co-ordinates

$$\begin{bmatrix} x'\\y'\end{bmatrix} =
\begin{bmatrix} s_x & 0 & o_x \\ 0 & s_y & o_y \\ 0 & 0 & 1 \end{bmatrix} \cdot
\begin{bmatrix} x\\y\\1\end{bmatrix} $$

+ Sometimes the pixels are not rectangular

$$\begin{bmatrix} x'\\y'\\1\end{bmatrix} =
\begin{bmatrix} s_x & s_\theta & o_x \\ 0 & s_y & o_y \\ 0 & 0 & 1 \end{bmatrix} \cdot
\begin{bmatrix} x\\y\\1\end{bmatrix} $$

## Summary

+ **Camera Intrinsic Parameters**

$$\lambda\begin{bmatrix} x'\\y'\\1\end{bmatrix} =
\begin{bmatrix} s_x & s_\theta & o_x \\ 0 & s_y & o_y \\ 0 & 0 & 1 \end{bmatrix} \cdot
\begin{bmatrix} f & 0 & 0 \\ 0 & f & 0 \\ 0 & 0 & 1 \end{bmatrix} \cdot
\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix} \cdot
\begin{bmatrix} X\\Y\\Z\\1\end{bmatrix} $$

## Distortion

### Calibration

+ Camera Calibration is a transformation in the *ideal* model
+ In addition we need calibration to compensate for distortion
    = imperfection in the camera
+ Two types
    + Radial Distortion - typical for wide angle lenses
    + Tangential Distortion - when image plane and lens are not parallel
      (ignored in the textbook)

### 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

$$
\begin{align}
  x_d &= x+ [2p_1xy+p_2(r^2+2x^2)]
  y_d &= y+[p_1(r^2+2y^2)+2p_2xy]
\end{align}
$$