Reason for 0.5 factor in the error

[nkhedekar]

In the IntegratedMatchingCostFactor (and others), the error is evaluated as const double error = 0.5 * residual.transpose() * residual; . This error is then used to create the HessianFactor as new gtsam::HessianFactor(keys()[0], H_source, -b_source, error).

As per the documentation of the HessianFactor, the constant term that needs to be provided to the constructor is $f = b^Tb$ where $b = \Sigma^{-1/2} (z - h(x_0))$ is the whitened negative error at the linearization point $x_0$.

Maybe I am misunderstanding something but I don't follow the reason for the factor of 0.5 in the code?

[koide3]

Hmmm, maybe I added that constant for some reasons (consistency with GTSAM factors?), but I'm not very confident now. While the error value itself does not affect the optimization as long as H and b are correct, I'll re-check whether 0.5 is necessary.

[koide3]

Sorry, it took a long time to come back to this issue. I validated the linearized factor by comparing its error values with nonlinear ICP error values under small displacements. Without multiplying 0.5, the linear factor nicely approximates the nonlinear factor, and thus it seems 0.5 is not necessary. I'll soon add a patch to remove it from matching cost factors.

w/ 0.5

w/o 0.5