Documentation of local parametrization

[Kyle]

Hi everyone,

I have one question about local parametrizations: homogeneous vector and quaternion.

Ceres implement local parametrization for homogeneous vector [HarleyZisserman] (with correction) in the code.

but I see in Ceres' documentation that you show a different form.

Is there something wrong in the documentation? or Two forms are equivalent? and how?

For quaternion, I find a local parametrization in the article:

J. Schmidt and H. Niemann, "Using Quaternions for Parametrizing 3–D Rotations in Unconstrained Nonlinear Optimization",  VMV01.

It seems different from Ceres' documentation.

Can anyone tell me which article/book in the bibliography is implemented in Ceres?

Thanks for all your help.

[Mike Vitus]

Hi Kyle,

The homogeneous vector local parameterization should follow Harley And Zisserman's.  Can you explain where you think they differ?

In regards to the quaternion local parameterization. The one implemented in Ceres is based on the exponential map. You can think of the local parameter as an angle axis representation where x = theta * v with ||v|| = 1.  This is then converted to a quaternion through standard formulas and then used to update the quaternion through normal quaternion multiplication. Unfortunately, there aren't great references on this, but this might help.

cheers,

Mike

[Kyle]

Thanks for your rely,

 

The homogeneous vector local parameterization should follow Harley And Zisserman's.  Can you explain where you think they differ?

The local parametrization for homogeneous vectors in http://ceres-solver.org/nnls_modeling.html#_CPPv2N5ceres33HomogeneousVectorParameterizationE

and that in Ceres' code (Harley And Zisserman's) are different.

Can you confirm that there is still one error in Harley And Zisserman's correction (attached file), (page 5, last line)?

f(y) should be (sinc(||y||/2) y^T/2, cos(||y||/2))^T

 

In regards to the quaternion local parameterization. The one implemented in Ceres is based on the exponential map. You can think of the local parameter as an angle axis representation where x = theta * v with ||v|| = 1.  This is then converted to a quaternion through standard formulas and then used to update the quaternion through normal quaternion multiplication. Unfortunately, there aren't great references on this, but this might help.

Kyle 

[Kyle]

In regards to the quaternion local parameterization. The one implemented in Ceres is based on the exponential map. You can think of the local parameter as an angle axis representation where x = theta * v with ||v|| = 1.  This is then converted to a quaternion through standard formulas and then used to update the quaternion through normal quaternion multiplication. Unfortunately, there aren't great references on this, but this might help.

Quaternion parametrization (http://ceres-solver.org/nnls_modeling.html#_CPPv2N5ceres26QuaternionParameterizationE)

If we consider Deltax as an angle axis representation, then convert it to a quaternion, and then update through normal quaternion multiplication,

Eq.(3) should be x' = [cos(||Deltax||/2), sinc(||Deltax||/2)  Deltax/2] x     ?

 (follow Harley And Zisserman's, A.4.3.3 with correction)

Kyle

[Kyle]

 Nonlinear optimization moves x to a new point x' on the unit sphere.

The quaternion difference between these two quaternions is  Deltaq = x' x^{-1}

(multiplication between two quaternions)

Deltaq is a unit quaternion since x and x' are unit quaternions.

Thus, there exists a vector v \in R^3, ||v|| = 1 and theta \in R such that Deltaq = [cos(theta), sin(theta) v].

(Proposition 12, Erik B. Dam, Quaternions, Interpolation and Animation, 1998).

The vector Deltax is defined by Deltax = theta v.

Finally, the update operator is defined by the multiplication between two quaternions

x' = [cos(||Deltax||), sin(||Deltax||) Deltax / ||Deltax||] x 

as (http://ceres-solver.org/nnls_modeling.html#_CPPv2N5ceres26QuaternionParameterizationE).

Note that, Deltax is not angle-axis representation of quaternion Deltaq.

2 theta v is the angle-axis representation of Deltaq.

Topic closed.

Thanks for the excellent library.

Kyle