Intrinsic Camera Calibration

Intrinsic Parameters

From the 3D coordinates in the camera frame to the 2D image plane via projection.

Ideal Perspective Projection

$$u = f \frac{x}{z}$$

$$v = f \frac{y}{z}$$

Real Intrinsic parameters

1. "Pixels" are in some arbitrary spatial units, maybe pixels are note square $$u = \alpha \frac{x}{z} \\ v = \beta \frac{y}{z}$$

1. We don't know the origin of our camera pixel coordinates $$u = \alpha \frac{x}{z} + u_0 \\ v = \beta \frac{y}{z} + v_0$$

2. Maybe skew between camera pixel axes $$v' \sin(\theta) = v \\ u' = u - \cos(\theta) v' = u - \cot(\theta) v$$

Intrinsic Parameters, Homogeneous coords

$$\overrightarrow{p'}= K \space ^C \overrightarrow{p}$$

$$K = \begin{bmatrix} f & s & c_x \\ 0 & af & c_y \\ 0 & 0 & 1 \end{bmatrix}$$

$\overrightarrow{p'}$: Pixels, $\overrightarrow{p}$: Camera 3D coordinates

5 DOF

where

$f$ - focal length, $s$ - skew, $a$ - aspect ratio, $c_x, c_y$ - offset

If square pixels, no skew, and optical center is in the center

$$K = \begin{bmatrix} f & 0 & 0 \\ 0 & f & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

Only 1 DOF