Why Plus in QuaternionParameterization not half angle axis

[拂石]

Hi, all.

I have two questions and wish to get your help.

First, question about Plus function in QuaternionParameterization and EigenQuaternionParametetization. 

When the ceres calculate the delta step of so3( delta = theta * Phi, where Phi  the axis and theta the angle), the corresponding delta quaternion should be as [cos(theta / 2), sin(theta  / 2) * Phi], why the Plus function use the format[cos(theta), sin(theta) * Phi] ?

Second, problem of analytical jacobian of  QuaternionParameterizaion quaternion.

I have a residual of N dimension, and i have derivated the analytical jacobian of resiaudl to QuaternionParameterization quaternion, as J with format N * 4.

Should i have to multiply or divide it by 2  to fit the missing 2 in my first qusetion?

Thanks a lot!

[Frank Neuhaus]

Hi,

about your first question: I have asked this in Github already. Basically, the decision on whether or not to add / 2 is arbitrary and does not affect optimization. It does however affect covariance estimation, so keep that in mind. Adding / 2 would probably be more common, but it was found not to be worth changing, as it could potentially break people's existing code. (See here: https://github.com/ceres-solver/ceres-solver/issues/311 )

As for your second question: There is no need to adjust your analytic Jacobian with regards to the QuaternionManifold / Parameterization as Ceres will do that for you.

Basically, say your cost function is:

f(q) -> R^2 where q is a quaternion.

What Ceres does internally is: Compute the Jacobian of f(q [+] delta) as d f(q [+] delta)  / d delta = (d f / d q)   *    (d q [+] (delta+h) / d h) | delta=0

The only thing you provide as a user is d f / d q, which is a 2x4 Jacobian. The term (d q [+] (delta+h) / d h) | delta=0 is computed by the QuaternionParameterization / QuaternionManifold, which will include the correct scale factor.

Providing the Jacobian of f(q [+] delta) directly is not possible in Ceres, unfortunately.

Hope this helps

   Frank

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[拂石]

Hi, Frank. Thanks for your reply. After considering your answer for a long time, i think i finally understand it, and hope you to see if it is correct. In a word, in QuaternionParameterization, ceres use 1/2 AngleAxis to represent so3 lie algebra, so the Plus function does not have the '1/ 2' scale. Then, the final jacobian of residual to rotation calculated by chain role is "residual to 1/2 AngleAxis", so if we wish to use J^TJ to calculate covariance, the result is aboout "half angleAxis".

[Frank Neuhaus]

Hi,

yes, I think you have understood it correctly.

Best,

   Frank

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