Camera Geometry

Projective Mapping from Euclidean Space to the Image Plane

Vol. I · Part 03

The Pinhole Paradigm

In computer vision, the transformation of the three-dimensional world onto a two-dimensional manifold is classically modeled via the Pinhole Camera Model. This mathematical abstraction decomposes the projection into two essential components:

  • Extrinsics:Representing the rigid body transformation (R,t)(R, t) from world coordinates to the camera frame.
  • Intrinsics:Defining the internal projection KK that maps 3D rays onto discrete pixel coordinates.
Annotation // Instructional Scope
This module explores the relationship between focal length ff and perspectival distortion, while visualizing the relative motion (R,t)(R, t) of the optical center.

Intrinsic Calibration (K)

The intrinsic matrix K encapsulates the internal geometry of the optical system. Through the interactive plate below, observe how focal length and principal point offsets distort the projection.

Fig 3.B — Projective Manifold (Image Plane)

Matrix K

K=[400.00.0320.00400.0240.0001]K = \begin{bmatrix} \color{#1E3A8A}{400.0} & 0.0 & 320.0 \\ 0 & \color{#1E3A8A}{400.0} & 240.0 \\ 0 & 0 & 1 \end{bmatrix}

Projection Control

400.0
400.0
320.0
240.0
0.00

Geometric Note

The intrinsic matrix defines the mapping between camera-centric rays and the discretized pixel grid.

Plate 3.1 — Variations in Intrinsic Scale and Perspective

Fig. 3.1 — Focal length and principal point under the intrinsic matrix K

Annotation // Optical Magnification
Increasing the focal parameters (fx,fy)(f_x, f_y) effectively narrows the viewing frustum, resulting in a linear magnification of the observed features.

Extrinsic Orientation [R∣t][R | t]

The extrinsic parameters define the camera's pose relative to a global coordinate system, enabling the transformation of world points into the local optical reference.

Fig. 3.A — Volumetric Cartesian Projection

Euclidean Space // View A

Fig 3.B — Projective Manifold (Image Plane)

Image Plane // View B

Translation Vector

t=[0.00.03.0]t = \begin{bmatrix} \color{#1E3A8A}{0.0} \\ \color{#1E3A8A}{0.0} \\ \color{#1E3A8A}{3.0} \end{bmatrix}

Rotation [R]

0.50
0.50
0.00

Translation [t]

0.00
0.00
3.00

Plate 3.2 — Spatial Rigid Body Transformation and Euclidean Projections

Fig. 3.2 — Camera pose [R | t] relative to the world frame

Annotation // The Optical Axis
In our convention, the Z-axis projects forward into the scene. For an object to remain within the visible frustum, its translation TzT_z must remain positive relative to the camera origin.