Pinhole camera model

A camera could be approximated by a projective model, often called pinhole projection. The simplest representation of a camera is a light sensible surface (sensor): an image plane, a lens (projective projection) at a given position and orientation in space.

.. figure:: pinholeCamera.png :align: center

The pinhole camera model. An oriented central projective camera.

The pinhole camera geometry models the projective camera with two sub-parametrizations, intrinsic and extrinsic parameters. Intrinsic parameters model the optic component (without distortion) and extrinsic model the camera position and orientation in space. This projection of the camera is described as:

.. math:: P_{3 \times 4} = K [R|t] = \begin{bmatrix} fk_u & & c_u \ & fk_v & c_v \ & & 1 \end{bmatrix} \begin{bmatrix} & & & t_x \ & R_{3 \times 3} & & t_y \ & & & t_z \ \end{bmatrix}

Intrinsic parameters :math:[f; cu; cv]:
- :math:ku, kv : scale factor relating pixels to distance (often equal to 1),
- :math:f : the focal distance (distance between focal and image plane),
- :math:cu, cv : the principal point, which would be ideally in the centre of the image.
Extrinsic parameters :math:[R|t] = [R| - RC]:
- :math:R : the rotation of the camera to the world frame,
- :math:t : the translation of the camera. t is not the position of the camera. It is the position of the origin of the world coordinate system expressed in coordinates of the camera-centred coordinate system. The position, :math:C, of the camera expressed in world coordinates is :math:C=-R^{-1} t = -R^T t (since R is a rotation matrix).

A 3D point is projected in a image with the following formula (homogeneous coordinates):

.. math:: x_i = PX_i = K[R|t]X_i

.. math:: \begin{bmatrix} u_i\ v_i\ w_i\ \end{bmatrix} = \begin{bmatrix} fk_u & & c_u \ & fk_v & c_v \ & & 1\ \end{bmatrix} \begin{bmatrix} & & & t_x \ & R_{3 \times 3} & & t_y \ & & & t_z \ \end{bmatrix} \begin{bmatrix} X_i\ Y_i\ Z_i\ W_i\ \end{bmatrix}

OpenMVG Pinhole camera models

Pinhole intrinsic
- :class:Pinhole_Intrinsic : public IntrinsicBase
  - classic pinhole camera (Focal + principal point and image size).
- :class:Pinhole_Intrinsic_Radial_K1 : public Pinhole_Intrinsic
  - classic pinhole camera (Focal + principal point and image size) + radial distortion defined by one factor.
  - can add and remove distortion
- :class:Pinhole_Intrinsic_Radial_K3 : public Pinhole_Intrinsic
  - classic pinhole camera (Focal + principal point and image size) + radial distortion by three factors.
  - can add and remove distortion
- :class:Pinhole_Intrinsic_Brown_T2 : public Pinhole_Intrinsic
  - classic pinhole camera (Focal + principal point and image size) + radial distortion by three factors + tangential distortion by two factors.
  - can add and remove distortion
- :class:Pinhole_Intrinsic_Fisheye : public Pinhole_Intrinsic
  - classic pinhole camera (Focal + principal point and image size) + fish-eye distortion by four factors.
  - can add and remove distortion
Simple pinhole camera models (intrinsic + extrinsic(pose))

.. code-block:: c++

// Setup a simple pinhole camera at origin // Pinhole camera P = K[R|t], t = -RC Mat3 K; K << 1000, 0, 500, 0, 1000, 500, 0, 0, 1; PinholeCamera cam(K, Mat3::Identity(), Vec3::Zero());

OpenMVG Spherical camera models

:class:Intrinsic_Spherical : public IntrinsicBase
- a camera model for equirectangular panorama (360x180 degree). No focal is required, the camera is modelized as a unit sphere.