Why EKF-SLAM uses EKF instead of linear Kalman Filter?

Question

I am writing a bachelor's diploma on vSLAM. I learned and programmed EKF-SLAM like it's written in MonoSLAM paper, and I was going to write, that I can't use KF and have to use EKF because of non-linearity of observation function, but wait, how is it possible, if everything is linear?!

I understand, that if I store direction of a camera in form of axis-angle vector or quaternion then there will be something non-linear, but what if I will store it directly as values of rotation matrix? Then my observation function is just going to be a multiplication of matrices, which is a linear operator, and therefore linear. Am I wrong?

Answer 1

Well in EKF SLAM, the underlying assumption is that all states are gaussian in nature and we can linearize the motion and observation models around the mean at each instant. This is done to write the equations in a form similar to what you see in the Kalman Filter.

Note: This linearization is just an approximation that we have made. It leads to error due to which nowadays we use other methods like Graph SLAM or particle filters.

Even if you represent the angles in form of a rotation matrix, the cosine and sine functions are non- linear in nature which we can't get rid of easily. The states contain angles and model is a function of states which is non- linear in nature.

You can store them in your state space however that just increases the dimension of your state space since instead of three euler angles, you now have 6 terms (cosine and sine value of each angle). This is really undesirable in nature as it unnecessarily increases the computation time of the over all algorithm (which is of the order $O(n^{2.4})$ in EKF SLAM.