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<S: RealField> {
    fn project_dir(&self, dir_c: &Vector3<S>) -> Option<Point2<S>>;
    fn unproject_dir(&self, n: &Point2<S>) -> Vector3<S>;
}
}

The Pinhole struct implements this trait:

#![allow(unused)]
fn main() {
pub struct Pinhole;

impl<S: RealField> ProjectionModel<S> for Pinhole {
    fn project_dir(&self, dir_c: &Vector3<S>) -> Option<Point2<S>> {
        if dir_c.z > S::zero() {
            Some(Point2::new(
                dir_c.x.clone() / dir_c.z.clone(),
                dir_c.y.clone() / dir_c.z.clone(),
            ))
        } else {
            None
        }
    }

    fn unproject_dir(&self, n: &Point2<S>) -> Vector3<S> {
        Vector3::new(n.x.clone(), n.y.clone(), S::one())
    }
}
}

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 cameras with significant distortion (wide-angle, fisheye), the distortion model corrects for lens effects after pinhole projection to normalized coordinates.