| Term | Definition | Notes |
| Backward Camera Model | A Geometric Camera Model that transforms 2D image points into 3D rays. | |
| Camera-Relative World Point | A World Point in the reference frame of the camera. | |
| Central Camera Model | A Geometric Camera Model for which all rays go through a single point. | |
| Extrinsics | A portion of a Geometric Camera Model that describes the position and orientation of an object within a world coordinate system. | See also: Pose. |
| Forward Camera Model | A Geometric Camera Model that transforms 3D world points into 2D image points. | |
| Geometric Camera Model | A model that describes the geometric properties of a camera. | This is sometimes referred to as a "geometric calibration model". A Geometric Camera Model is composed of Extrinsics, Intrinsics, and Distortion. |
| Global World Point | A World Point in the global reference frame. | |
| Image Coordinate System | The basis for describing image points. | See Image Coordinate System documentation. |
| Image Point | An ordered pair (x, y) describing a 2D location on the focal plane. | An Image Point is defined in an Image Coordinate System. |
| Intrinsics | A portion of a Geometric Camera Model that describes the internal geometric properties of the camera. | |
| Pose | The position and orientation of an object relative to some world coordinate system. | See also: Extrinsics. |
| Principal Point | The point on the focal plane that intersects the optical axis. | |
| World Point | An ordered triple (X, Y, Z) describing a 3D location in a world coordinate system. |
Notation
All vectors are column vectors. Uppercase , , and refer to 3D world points. Lowercase Uppercase and refer to 2D image points.
Homogeneous Coordinates
Homogeneous coordinates are a set of coordinates with useful properties for perspective geometry [1, 2]:
- Infinity may be represented with a finite value.
- Rotations and translations may be represented by a single matrix operation.
- Homogeneous coordinates may be used for a space with an arbitrary dimension, including 2D (image) and 3D (world) coordinates.
A "standard" coordinate is referred to as inhomogeneous.
Properties
- If and only if the last coordinate of a homogeneous coordinate is 0, then it is at infinity.
- If and only if the last coordinate of a homogeneous coordinate is not 0, then it is at a finite location.
- Two homogeneous points are the same iff there exists a non-zero scalar multiplier between them, i.e., .
Transforms
Inhomogeneous to Homogeneous
The simplest way to convert from an inhomogeneous coordinate to a homogeneous one is to append a 1 to the end of the coordinate.
The general conversion is to append 1 and multiply by any non-zero real number.
Homogeneous to Inhomogeneous
To convert from a homogeneous coordinate to an inhomogeneous one, divide all of the components by the last one, which is discarded.
Intrinsics
The intrinsic matrix, , is an upper-triangular matrix that transforms a world coordinate relative to the camera into a homogeneous image coordinate. There are two general and equivalent forms of the intrinsic matrix:
where
| Variable | Description |
| The x-focal length | |
| The y-focal length | |
| The focal length ratios | |
| The skew | |
| The principal point (intersection of the optical axis with the focal plane) |
The intrinsic matrix of the th camera is applied to the th camera-relative 3D point to produce a homogeneous image point.
Notes:
- The focal length(s) are in pixel pitch units
- The principal point is defined in an image coordinate system
Camera Models
Summary
| Model | Forward Order | Maximum Field of View | Distortion Options | Centrality | References |
| Simple Pinhole | Extrinsics → Intrinsics | <180° | None | Central | [1, 2] |
| OpenCV | Extrinsics → Distortion → Intrinsics | <180° | Radial, Tangential, Thin Prism, Tilt | Central | [3] |
Simple Pinhole
The forward direction of the simple pinhole model is: Extrinsics → Intrinsics.
The forward direction of the OpenCV model is given by:
- Transform a camera-relative world point through the intrinsics camera matrix:
- Convert to an inhomogeneous image point
OpenCV
The OpenCV camera model [3] is defined by the OpenCV library. The version used in Imatest is that of OpenCV 4.12.0.
The forward direction of the OpenCV model is: Extrinsics → Distortion → Intrinsics.
The forward direction of the OpenCV model is given by:
- Transform a camera-relative world point into an undistorted image point :
- Compute the radius, :
- Apply the radial distortion:
- Apply the tangential distortion:
- Apply the thin-prism distortion:
- Apply the tilt-distortion (and convert back to an inhomogeneous image coordinate):
- Apply the intrinsics:
References
[1] Hartley, R., & Zisserman, A. (2003). Multiple view geometry in computer vision. Cambridge University Press.
[2] Ma, Y., Soatto, S., Košecká, J., & Sastry, S. (2004). An invitation to 3-d vision: from images to geometric models (Vol. 26). New York: Springer.
[3] OpenCV. Camera Calibration and 3D Reconstruction. https://docs.opencv.org/4.12.0/d9/d0c/group__calib3d.html [Accessed 2025-08-18]