template<class TT>
class sophus::InverseDepthPoint3< TT >
Inverse depth point representation.
(a, b) := (x/z, y/z) and psi := 1/z
following https://ethaneade.com/thesis_revised.pdf, pp. 79
Let us assume we have Euclidean 3d point (x,y,z) in a local reference frame (e.g. camera origin). One can construct an inverse depth point (in the same local reference frame) as follows:
First we project the point (x,y,z) through the origin (0,0,0) onto the z=1 plane. We call the projection (a, b) := (x/z, y/z).
In other words, (a, b) is the intersection of the line through (0, 0, 0) to (x, y, z) and the 2d Euclidean plane z=1.
Now, we can describe almost any 3d point in our local reference frame as a point (a',b') in the Euclidean plane z=1 and the inverse depth psi := 1/z.
For example, the Euclidean point (2, 0, 8) is represented as (a, b) = (2/8, 0/8) = (1/4, 0) and inverse depth psi = 1/8.
The only Euclidean 3d point we cannot describe is the origin (0,0,0) (since there are infinitely many lines through the origin which intersect with the plane z=1).
The advantage of using an inverse depth representation over Euclidean representation is that we can also represent points at infinity. Let (a,b) a 2d Euclidean point on our reference plane z=1; nothing stops us from choosing a psi=0, which corresponds to a "z = 1/0 = infinity".
In summary, using this representation, we can represent
- points at infinity: psi == 1/z == 0
- points close to +infinity: psi == 1/z == +e
- points close to -infinity: psi == 1/z == -e
- points one unit in front: psi == 1/z == +1
- points one unit behind: psi == 1/z == -1
- points close to zero, in front: psi == 1/z == +999999
- points close to zero, behind: psi == 1/z == -999999
1.8.17