Optimizing the coordinate points of an image using the "Polar Geometry constraint"

0. background

The two corresponding points of two pictures are known, and fundamental Matrix F, how to optimize the coordinates of two points using the polar constraint?

1. Concept

Here we introduce Sampson approximation: Sampson approximation (first-order geometry correction)

First, we introduce the Sampson modifier:

After introducing this error function, we can get the coordinates of the corrected point:

of which, J is the Jacobian,

Error Term:

2. the Jacobian solution for polar geometry constraints

3. The above deduction requires attention to several places :

(1) theorem of vector differential

(2) Denominator note plus transpose

Here, the denominator has transpose, because in solving the Jacobian, and the denominator is a vector, the general is 1xN, and the x vector is generally Nx1, so here we want to add a transpose.

We give a description:

4. when the above-mentioned Jacobian solution is complete, we can get two-dimensional point coordinates on the revised image :

5. Reference Documents :

"Multiple View Geometry in computer Vision" P314

12.4 Sampson Approximation (first-order geometric correction)

6. Acknowledgements

Thank you very much for the detailed derivation of Huangshan Mountain, as well as a lot of proof content!

Optimizing the coordinate points of an image using the "Polar Geometry constraint"