Analytical Derivatives with Local Parameterization

[Fernando Nobre]

Hello, 

I am having trouble getting a Cost Function (analytical derivatives) to work with a local parameterization. I have a basic example to illustrate:

I have a SE3 pose I wish to minimize with regards to another SE3 pose. Each pose has 7 parameters (3 for translation and 4 for the rotation quaternion).

The residual is taken in tangent space, so it has dimension of 6 (3 for translation, 3 for rotation)

I know I have to provide a 6x7 jacobian in the cost function. Since the 7x6 local param jacobian will convert that jacobian into the final 6x6 form.

I am analytically calculating the 6x7 jacobian, and associating a local parameterization to the parameter block, which implements ComputeJacobian analytically and returns the 7x6 matrix.

This same local parameterization works just fine with an autodiff cost function, and i've verified that the derivatives i'm providing are correct.

Is there possibly some problem in the way I'm formulating the problem? The number if unsuccesfull steps leads me to believe the derivatives w.r.t. the global parameters are incorrect but I have verified those. 

Here is a summary:

                                     Original                  Reduced

Parameter blocks                            1                        1

Parameters                                  7                        7

Effective parameters                        6                        6

Residual blocks                             1                        1

Residual                                    6                        6

Minimizer                        TRUST_REGION

Dense linear algebra library            EIGEN

Trust region strategy     LEVENBERG_MARQUARDT

                                        Given                     Used

Linear solver                     DENSE_SCHUR              DENSE_SCHUR

Threads                                     1                        1

Linear solver threads                       1                        1

Linear solver ordering              AUTOMATIC                        1

Cost:

Initial                          8.149899e+02

Final                            8.467852e+01

Change                           7.303114e+02

Minimizer iterations                       25

Successful steps                            9

Unsuccessful steps                         16

Time (in seconds):

Preprocessor                           0.0001

  Residual evaluation                  0.0000

  Jacobian evaluation                  0.0000

  Linear solver                        0.0002

Minimizer                              0.0006

Postprocessor                          0.0000

Total                                  0.0007

Termination:                      CONVERGENCE (Function tolerance reached. |cost_change|/cost: 9.896785e-07 <= 1.000000e-06)

final norm: 

13.0137

[Keir Mierle]

Hi Nombre,

I'm not sure exactly what is going on in your case, but consider trying the two methods we have for checking derivatives:

Easiest:

Turn on check_gradients

https://ceres-solver.googlesource.com/ceres-solver/+/master/include/ceres/solver.h:697

More difficult but more informative:

https://ceres-solver.googlesource.com/ceres-solver/+/master/include/ceres/gradient_checker.h

Did you check that the analytic and autodiff versions are identical?

Thanks,

Keir

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[Fernando Nobre]

Hi Keir,

Thanks for that suggestion, it helped me figure out what was wrong: My state parameter was stored in memory as quaternion in the first four positions and then translation in the last three. 

The jacobian i was providing had the derivatives w.r.t. the translation in the first three columns, which was being interpreted by ceres as the derivative w.r.t. the first three parameters of the quaternion, so it was obviously totally incorrect.

Now I am able to get the correct solution using analytical cost functions with a local parameterization. Thanks!