martiege / lab_03

Lab 3 - An introduction to Lie theory

Welcome to Lab 3! We will here play around with some of the tools we get from Lie theory in practice.

I have made the following scripts:

• ex1_interpolation.py
• ex2_mean_pose.py
• ex3_noise_propagation.py

Your task this week is to finish these scripts, and then experiment with them to get a better understanding of how Lie theory can be used to work with poses on the manifold. See the sections below for details.

The exercises are based on visgeom, and pylie. You can install all dependencies using pip:

pip install -r requirements.txt

Exercise 1 - Linear interpolation of poses on the manifold

Using pylie, we can quite easily implement linear interpolation for poses on the manifold. For hints, see Example 4.8 in the compendium.

• Open ex1_interpolation.py.
• Implement linear interpolation on the manifold (see TODO 1)
• Play around with different poses and see how the interpolation is performed
• Try extrapolating
• Instead of interpolating on the pose manifold, try instead to interpolate the rotation and translation parts separately. How does this procedure compare to the full pose interpolation?

Exercise 2 - Estimate the mean pose from a set of random poses

We will here draw poses from a distribution, and try to recover the mean pose by estimating it on the manifold. For hints, see Examples 6.17 and 7.20 in the compendium.

• Open ex2_mean_pose.py
• Finish draw_random_poses() (see TODO 1)
• Study algorithm 2 in Example 7.20 in the compendium.
• Finish compute_mean_pose() (see TODOs 2 and 3)
• Play around with different distribution parameters. Try for example to increase the uncertainty in the rotation around the y-axis.

Exercise 3 - Propagating uncertainty in backprojection

We will here estimate the distribution of an observed 3D point in the world, based on uncertainties in measured camera pose, pixel position, and depth. For hints, see Examples 6.19 in the compendium.

• Open ex3_noise_propagation.py.
• Finish backproject() (see TODO 1).
• Implement uncertainty propagation (see TODO 2). Some of the Jacobians involved are implemented in pylie.
• Play around with different distribution parameters. Try for example to increase the uncertainty in the camra rotation around the y-axis to a large value (0.2 for example).
• When is this first order approximation to the distribution a good approximation? When is it bad?