Module 5: Evaluation Metrics
After the "playground" of alignment is over, how do we actually know if our algorithm did a good job? In a real-world scenario, we often have the Ground Truth. To see how close our prediction is to the ground truth, we use two main referees: RRE for the rotation and RTE for the movement.
5.1 Relative Rotation Error (RRE)
Think of RRE as a measure of how "tilted" or "twisted" your result is compared to the perfect answer. It calculates the angular difference in degrees. If the ground truth says the object should be rotated exactly \(90^\circ\), but your algorithm says \(88^\circ\), your RRE is \(2^\circ\). In 3D math, we calculate this by looking at the "distance" between two rotation matrices on a special mathematical surface called a manifold.
For a predicted rotation \(\mathbf{R}_p\) and ground-truth rotation \(\mathbf{R}_{gt}\), the RRE is:
5.2 Relative Translation Error (RTE)
RTE is much simpler to visualize. It measures the straight-line distance like Euclidean distance between where the object is and where it should be.
The Intuition: Imagine you are moving a chair to its correct spot. If the chair's center point is supposed to be at the coordinate \((0, 0, 0)\) but your algorithm puts it at \((0, 0, 0.1)\), your RTE is \(0.1\) units. It’s essentially the "gap" between the two positions.
Given the predicted translation \(\mathbf{t}_p\) and the ground-truth translation \(\mathbf{t}_{gt}\), we calculate the \(L_2\) norm: