Setting the equations
Camera model and frame of reference
In this model, we distinguish two kinds of parameters:
- Modal parameters: and
- Intrinsic parameters: u0, v0 and
The modal parameters determine the model of projection:
| | |
Pure perspective projection | 1 | 1 |
Weak perspective projection (i.e. affine or orthographic) | 0 | 0 |
Para-perspective projection | 1 | 0 |
Representation of rigid displacement
We consider motion of rigid objects and the ego-motion of the camera.
It means that the tokens in the scene are undergoing a rigid
displacement parameterized by a rotation matrix R, which can be expressed as:
where r=[r0 r1 r2]T
and a translation vector t = [t0 t1 t2]T
Equations in the general case
Considering two consecutive frames:
which can be also written as:
and interpreted as a bilinear form in m and m', so that we can express this result in a more familiar form, using a generalization of the fundamental matrix:
This is easily generalizable to N views.
The particular cases
We will focus, in this paper, on the cases concerning the model of camera and the displacements. The cases concerning the particularities of the scene will not be examined here, except the planar case; we will suppose the points in a general configuration (coplanar or non-coplanar).
Particular cases in term of model of camera
We consider here modal and intrinsic parameters.
Modal parameters induce three cases, as seen previously.
Concerning intrinsic parameters, several cases have to be considered:
- which allows to represent the fact that the retina might not be orthogonal with respect to the optical axis and can be approximated to zero or considered as a fixed parameter
- and the focal lengths:
- and are fixed
- and are fixed and known
- the ratio between by is fixed and precalibrated
- and approximed to their 1st order
- and the coordinates of the principal point c:
- and fixed but unknowns
- and fixed but knowns (for example, fixed at the center of the image)
- and approximed to their 1st order
Particular cases in term of displacement of the camera (or objects in the scene)
Let us enumerate the different motion constraints:
Particular case: the case of a pure rotation and the planar case
In the
We introduce in this case the normale to the plane n = [n0 n1 n2]T and associate particular cases:
- one or more component of n vanishes:
(n0 = 0) and / or (n1 = 0) and / or (n2 = 0)
- relations with translation or rotation vectors:
- n // t
- n // r
- n t
- n r
Constraints: compatibility - redondance - consistence
The set of all constraints is made of atomic constraints, as seen previously, and molecular constraints, logical "AND" combinations of atomic constraints, excepted the non-possibles constraints.
We have to note that some combinations (by example, non-zero translation, parallelism and orthogonality) are in contradiction.
In other hand, if we compare:
(r0 = 0) (t r) and (t1 = 0) (t2 = 0) (t r)
we observe that we obtain the same constraint. This shows the redondance of molecular constraints.
The particular cases are the molecular constraints which verify the following properties, aiming to have a physical meaning:
- an unique model of camera have to be chosen
- an unique mode of rotation have to be chosen
- the translation must not be zero (we fixed one component to 1)
- the principal point is in the image
In other hand, the constraints give us not necessary enough information. In this case, we can focus only on what is determinable or add enough other constraints to solve the whole problem.
Using particular cases
- to determine the model of projection of the camera
- to determine the calibration parameters (if they are constants)
- to determine the type of displacement and, if possible, the characteristics (direction of translation, axis of rotation, angle of rotation)
- eventually, to segment the points in sets of rigidly bind points