Intrinsics Matrix

The intrinsics matrix $K$ maps from the sensor coordinate system (normalized or sensor-transformed coordinates) to pixel coordinates. It encodes the camera's internal geometric properties: focal length, principal point, and optional skew.

Definition

$$K=\left[\begin{array}{ccc}{f}_x& s& c_x\\ 0& f_y& c_y\\ 0& 0& 1\end{array}\right]$$

where:

• $f_x,f_y$ — focal lengths in pixels (horizontal and vertical). These combine the physical focal length with the pixel pitch: $f_x=f/{\Delta }_x$ where $f$ is the focal length in mm and ${\Delta }_x$ is the pixel width in mm.
• $c_x,c_y$ — principal point in pixels. The projection of the optical axis onto the image plane. Typically near the image center but not exactly at it.
• $s$ — skew coefficient. Non-zero only if pixel axes are not perpendicular. Effectively zero for all modern sensors; calibration-rs defaults to zero_skew: true in most configurations.

Forward and Inverse Transform

Sensor to pixel (forward):

$$\left[\begin{array}{c}u\\ v\end{array}\right]=\left[\begin{array}{c}{f}_x\cdot s_x+s\cdot s_y+c_x\\ f_y\cdot s_y+c_y\end{array}\right]$$

where $[s_x,s_y{]}^T$ are sensor coordinates (output of the distortion + sensor stages).

Pixel to sensor (inverse):

$$s_y=\frac{v-c_y}{f_y},s_x=\frac{u-c_x-s\cdot s_y}{f_x}$$

The IntrinsicsModel Trait

#![allow(unused)]
fn main() {
pub trait IntrinsicsModel<S: RealField> {
    fn sensor_to_pixel(&self, sensor: &Point2<S>) -> Point2<S>;
    fn pixel_to_sensor(&self, pixel: &Point2<S>) -> Point2<S>;
}
}

The FxFyCxCySkew<S> struct implements this trait:

#![allow(unused)]
fn main() {
pub struct FxFyCxCySkew<S: RealField> {
    pub fx: S, pub fy: S,
    pub cx: S, pub cy: S,
    pub skew: S,
}
}

Coordinate Utilities

The coordinate_utils module provides convenience functions that combine intrinsics with distortion:

• pixel_to_normalized(pixel, K) — apply $K^{-1}$ to get normalized coordinates on the $Z=1$ plane
• normalized_to_pixel(normalized, K) — apply $K$ to get pixel coordinates
• undistort_pixel(pixel, K, distortion) — pixel → normalized → undistort → return normalized
• distort_to_pixel(normalized, K, distortion) — distort → apply $K$ → return pixel

These functions are used extensively in the linear initialization algorithms, which need to convert between pixel and normalized coordinate spaces.

Aspect Ratio

For most cameras, $f_x\ne f_y$ because pixels are not perfectly square. The ratio $f_x/f_y$ reflects the pixel aspect ratio. Typical industrial cameras have aspect ratios very close to 1.0 (within 1-3%).

OpenCV equivalence: cv::initCameraMatrix2D provides initial intrinsics estimates. The $K$ matrix format is identical to OpenCV's cameraMatrix.