question about a jacobian

[fushi219]

Hi! Thanks for your beautiful code and I am confused about a jacobian and wish to get your help.

I have no idea about jacobians from https://github.com/SpectacularAI/HybVIO/blob/main/src/odometry/triangulation.cpp#L270 to https://github.com/SpectacularAI/HybVIO/blob/main/src/odometry/triangulation.cpp#L322. Are these calculations used for jacobian of landmark reprojection errors to imu poses in ekf state, which will be used in later filter update? (in this way, we do not need to project H into nullspaces of landmark.) But how the jacobians are derivated?

I know its quite boring and troublesome to explain it, so a liitle explain will be grateful, thanks!

[oseiskar]

Hello. You are correct, those Jacobians are intermediary steps and the goal is computing the Jacobian of the reprojection errors accurately (by differentiating an entire Gauss-Newton iteration process) so the MSCKF nullspace projection trick is not needed. This is the main difference between PIVO and MSCKF. HybVIO is a derivative of the PIVO method in this sense.

What it actually differentiated in that code block are (related to) the left and right hand side of a Gauss-Newton update step in the triangulation process.

[fushi219]

Thanks a lot! I have understood the jacobians mentioned above, however, i can't figure out these:

HybVIO/src/odometry/triangulation.cpp

Lines 337 to 338 in e325353

	const Vector3d dX_Eerror = -X.solve(dETE.block<3, 3>(0, j * 3) * X.solve(Eerror));
	dpfi.col(j) += -X.solve(dEerror.col(j)) - dX_Eerror;

wish to get some help. Thanks!