TTC_Lidar_Camera.md

The Math Behind Time-to-Collision (TTC)

The velocity $v_0$ or $v_r$ (reference velocity) in the first column is computed from two successive Lidar measurements as follows:

Up to 7th frame it's obvious that the vehicle has a negative acceleration and after that it is positive. This is also noticeable in the TTC camera calculations

The math behind the TTC Camera detection

A short overview can be found on these pictures regarding the TTC calculation.

There is no distance calculation but the ratio of the height on the image plane or the descriptors. The ratio of all relative distances between each other can be used to compute a reliable TTC estimate by replacing the height ratio $h_1/h_0$ with the mean or median of all distance ratios $d_k/d_k'$

Performance Evaluation (TTC Lidar)

Ref Velocity	TTC Lidar	Computation Method
0.640001	12.5156	Median point method
0.609999	12.9722	Minimum point method
0.630002	12.6142	Median point method
0.640001	12.264	Minimum point method
0.560002	14.091	Median point method
0.559998	13.9161	Minimum point method
0.469999	16.6894	Median point method
1.08	7.11572	Minimum point method
0.494998	15.7465	Median point method
0.469999	16.2511	Minimum point method
0.604999	12.7835	Median point method
0.609999	12.4213	Minimum point method
0.640001	11.9844	Median point method
0.220003	34.3404	Minimum point method
0.580001	13.1241	Median point method
0.799999	9.34376	Minimum point method
0.580001	13.0241	Median point method
0.409999	18.1318	Minimum point method
0.669999	11.1746	Median point method
0.409999	18.0318	Minimum point method
0.580001	12.8086	Median point method
1.88	3.83244	Minimum point method
0.819998	8.95978	Median point method
-0.669999	-10.8537	Minimum point method
0.73	9.96439	Median point method
0.780001	9.22307	Minimum point method
0.750003	9.59863	Median point method
0.649996	10.9678	Minimum point method
0.834999	8.52157	Median point method
0.870004	8.09422	Minimum point method
0.740001	9.51552	Median point method
2.15	3.17535	Minimum point method
0.725	9.61241	Median point method
-0.689998	-9.99424	Minimum point method
0.819998	8.3988	Median point method
0.819998	8.30978	Minimum point method

I calculated the TTC first based on the constant velocity model with the minimum lidar points detected and then proceeded with the median points in the respective bounding boxes. As the bold values indicate, especially the minus ones, the collision should have already happened in the past, so it's not a feasible solution to consider the minimum values but the median ones.

Performance Evaluation (TTC Camera)

Among the detectors/descriptor pairs the best performance is achieved using the AKAZE/FREAK.

It's known to yield the best result in high frame rate or high resolution images. Here an indirect supportive article can be found: http://tulipp.eu/wp-content/uploads/2019/03/2017_TUD_HEART_kalms.pdf

As seen above, which is also noticeable in the reference velocity calculations from the lidar measurements, up to 7th frame the vehicle decelerates and then accelerates again. So the constant velocity model is not capable of predicting the outcome of future frames.