Pinhole Projection

The pinhole model is the simplest and most widely used camera projection. It models a camera as an ideal point through which all light rays pass — no lens, no aperture, just a geometric center of projection.

Mathematical Definition

Given a 3D point ${\mathbf{P}}_c=[X,Y,Z{]}^T$ in the camera frame, the pinhole projection maps it to normalized image coordinates:

$$\mathbf{n}=\Pi ({\mathbf{P}}_c)=\left[\begin{array}{c}X/Z\\ Y/Z\end{array}\right]$$

This is perspective division: the point is projected onto the $Z=1$ plane. The projection is defined only for points in front of the camera ($Z>0$).

The inverse (back-projection) maps normalized coordinates to a 3D direction:

$${\Pi }^{-1}(\mathbf{n})=\left[\begin{array}{c}{n}_x\\ n_y\\ 1\end{array}\right]$$

This defines a ray from the camera center through the image point.

Camera Frame Convention

calibration-rs uses a right-handed camera frame:

• $X$: right
• $Y$: down
• $Z$: forward (optical axis)

This matches the convention used by OpenCV and most computer vision libraries. A point with positive $Z$ is in front of the camera; negative $Z$ is behind it.

The ProjectionModel Trait

#![allow(unused)]
fn main() {
pub trait ProjectionModel<T: RealField> {
    fn in_domain(&self, p_c: &Vector3<T>) -> bool;
    fn project(&self, p_c: &Vector3<T>) -> Point2<T>;
    fn unproject(&self, n: &Point2<T>) -> Ray<T>;
    fn is_central(&self) -> bool;
}
}

The projection takes the camera-frame point, and back-projection returns a Ray (origin and direction). For Pinhole, in_domain is z > 0, project is (x / z, y / z) and unproject returns the direction (n_x, n_y, 1) from the camera centre. project is total: callers that need validity check in_domain first.

Note the use of .clone() — this is required for compatibility with dual numbers used in automatic differentiation (see Autodiff and Generic Residual Functions).

Limitations

The pinhole model assumes:

• No lens distortion — addressed by the distortion stage
• No sensor tilt — addressed by the sensor stage
• Central projection — all rays pass through a single point

For wide-angle lenses the distortion model corrects for lens effects after pinhole projection to normalized coordinates. A fisheye lens that sees near or beyond 90° from the axis needs a different projection: see Fisheye Projection.