Chapter 2: Camera Models & Calibration

In the pinhole model, light from a 3D point passes through a single point (the optical center) and lands on the image plane at distance $f$ (the focal length) behind it. By similar triangles, a point $(X, Y, Z)$ in camera coordinates projects to image coordinates $(fX/Z,\ fY/Z)$. This division by $Z$ is called perspective division and is responsible for the foreshortening effect: distant objects appear smaller. The optical axis is the line through the pinhole perpendicular to the image plane, and the principal point is where this axis intersects the image plane.

The focal length $f$ determines the camera's field of view (FOV). A short focal length produces a wide FOV (wide-angle lens), capturing more of the scene but with more perspective distortion. A long focal length produces a narrow FOV (telephoto lens), magnifying distant objects but capturing less of the scene. The relationship is $\text{FOV} = 2 \arctan\!\left(\frac{\text{sensor\_size}}{2f}\right)$. In the intrinsic matrix, focal length is expressed in pixels: $f_x = f_{mm} \cdot \frac{\text{image\_width\_px}}{\text{sensor\_width\_mm}}$.

The full pinhole projection equation combines intrinsic and extrinsic parameters into a single $3 \times 4$ projection matrix $P = K[R|t]$. Given a 3D world point $X_w$ in homogeneous coordinates, the pixel location is found by: $x = P X_w$, then dividing by the third component. The intrinsic matrix $K$ is $3 \times 3$ (focal lengths, principal point, skew), and $[R|t]$ is the $3 \times 4$ extrinsic matrix (rotation and translation from world to camera). This compact equation is the mathematical backbone of multi-view geometry.