WhereTh is the number of trajectories in sequence Sh.
The third row of fig. 6.2 shows the 2D segmentation of our technique for the example se- quences in the figure. The misclassified trajectories are outlined with white boxes. The per- centage misclassification rate for these sequences are Mh = 0.4%,0.0%,6.1% from left to right.
The fourth row shows the results provided by one of the 3D segmentation algorithms. Again misclassified trajectories are outlined with white boxes. The percentage misclassification rate for the example sequences are Mh= 0.6%,0.0%,6.1% from left to right respectively.
This section has illustrated a few results and how they are measured. We now go on to present a complete analysis comparing with five 3D segmentation techniques and the technique of Fradet [46].
6.1.1 Misclassification Results
We have compared the proposed algorithm with the five 3D segmentation approaches. These approaches are titled Multistage Learning (MSL) [59],Generalized Principal Component Analysis (GPCA) [118–121], Random Sample Consensus(RANSAC) [42, 115], Local Subspace Affinity (LSA) [132] and Agglomerative Lossy Compression (ALC) [97, 133].
In general most 3D segmentation algorithms do not perform well when the number of moving object Nh in a sequence is more than two. Therefore the misclassification rates are reported
for sequences with two and three moving objects separately. Table 6.1 shows the number of sequences (Nh), the mean number of trajectories ( ¯Th) and frames ( ¯Fh) for the sequences with
two or three moving objects. There are 120 and 35 sequences with two and three moving objects respectively.
Recall that the sequences in the Hopkins dataset are categorized into checkerboard, traffic, and articulated/non-rigid sequences. Each category has different motion sequence characteris- tics. The objects in the checkerboard sequences are physically controlled to have either rota- tional, translational or stationary motion. However for the traffic and articulated sequences the motions of the objects are not controlled.
Since the three categories in the Hopkins dataset all have different motion characteristics, the misclassification rates are reported for each category. For our technique the number of
6.1. Quantitative Analysis on the Hopkins Dataset 109 Nh = 2 Nh = 3 #Seq. ¯Th F¯h #Seq. ¯Th F¯h Checkerboard 78 291 28 26 437 28 Traffic 31 241 30 7 332 31 Articulated 11 155 40 2 122 31 All 120 266 30 35 398 29
Table 6.1: The number of sequences (#Seq.), the mean number of trajectories (T¯h) and frames
¯
Fhfor the three categories in the Hopkins dataset. These statistics are shown for sequences with two (Nh= 2) and three (Nh= 3) moving objects.
1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 0.5 1 1.5 2 2.5 3 3.5 ¯ B ¯M
Figure 6.4: The mean bundle ratios B¯ versus the mean misclassification rates M¯ for the 7 experiments outlined in table 6.2.
bundlesbh produced per moving object for sequenceSh influences the misclassification rateMh.
Consider that Bh is the ratio of the number of bundlesbh to the number of moving objectsNh
defined as follows.
Bh =
bh
Nh
(6.2) It is reasonable to expect that increasing the ratio Bh (more bundles per object) will cause
the misclassification rate Mh to decrease. Therefore we conducted 8 experiments on all 155 sequences where the parameters in the refinement stage of our technique were varied to control the ratio Bh of bundles per moving object.
Initialization Refinement Affine Mer. MS Band. Separation Mer.
Experiment sn,m k1, k2, k3, k4 dτ, Dτ ςτ, Rτ B¯ M¯ 0 0.001 0.5,0.5,0.5,0.5 0,∞ 0,∞ 5.65 0.69 1 0.001 0.5,0.5,0.5,0.5 0.5,0.80 1.0,0.70 3.78 0.68 2 0.001 0.5,0.5,0.5,0.5 1.0,0.70 2.0,0.50 3.00 0.93 3 0.001 0.5,0.5,0.5,0.5 1.0,0.60 2.0,0.40 2.48 1.20 4 0.001 0.5,0.5,0.5,0.5 1.0,0.60 3.0,0.40 2.29 1.29 5 0.001 0.5,0.5,0.5,0.5 1.0,0.50 3.0,0.30 2.19 1.44 6 0.001 0.5,0.5,0.5,0.5 1.0,0.40 3.0,0.20 2.07 1.63 7 0.001 0.5,0.5,0.5,0.5 1.2,0.35 5.0,0.15 1.81 3.46
Table 6.2: The parameters for 8 experiments conducted for the proposed algorithm on the 155 sequences in the Hopkins dataset. The parameters for the Affine merging sn,m (Affine Mer.) and the Mean shift clustering bandwidth k1, k2, k3, k4 (MS Band.) in the initialization stage
were kept constant over all experiments. The trajectory separation merging parametersdτ,Dτ,
ςτ and Rτ (Separation Mer.) in the refinement stage were varied so the bundle ratio Bh (6th
column)decreases monotonically from experiment 1-7. In experiment 0 the trajectory bundles are not merged in therefinementstage. The mean misclassification rateM¯ for each experiment is shown in the end column. The results Experiment 1 are used for comparison with other segmentation methods. The row for this experiment is in bold text.
mean bundle ratios ¯B and mean misclassification rates ¯M over all 155 sequences. Note that experiment 0 acts as a control where no merging of the trajectory bundles is allowed in the refinement stage. The parameters for the Affine mergingsn,m (Affine Mer.) and the Mean shift clustering bandwidth k1, k2, k3, k4 (MS Band.) in the initialization stage were kept constant
over all experiments. However the trajectory separation merging parameters dτ,Dτ,ςτ and Rτ
(Separation Mer.) in therefinement stage were varied to control the bundle ratio Bh.
For experiments 1-7 the trajectory separation merging thresholds dτ, Dτ, ςτ and Rτ are relaxed monotonically. That is, experiment 1 has the ‘tightest’ thresholds (0.5,0.8,1.0,0.70) to deter the merging of the trajectory bundles, so this experiment has the highest bundle ratio ¯B = 3.78. The 7th experiment however has the ‘loosest’ thresholds (1.2,0.35,5.0,0.15) to encourage more merging of the trajectory bundles. Hence this experiment has the lowest bundle ratio
¯
B = 1.81.
The mean bundle ratios ¯B versus the mean misclassification rates ¯M for the 8 experiments are plotted in fig.6.4. As expected the mean misclassification rate ¯M decreases as the mean number of bundles per moving object ( ¯B) increases. The plot in fig. 6.4 suggests that we can not expect much more improvement in the mean misclassification rate ¯M by increasing the
6.1. Quantitative Analysis on the Hopkins Dataset 111 Checkerboard Proposed MSL GPCA RANSAC LSA ALC
Average 0.56% 2.13% 5.80% 6.05% 2.14% 2.24%
Median 0.00% 0.00% 1.03% 1.84% 0.27% 0.00%
Traffic Proposed MSL GPCA RANSAC LSA ALC
Average 0.05% 0.74% 8.01% 8.15% 9.51% 1.04%
Median 0.00% 0.00% 0.80% 0.52% 5.66% 0.00%
Articulated Proposed MSL GPCA RANSAC LSA ALC
Average 4.10% 1.90% 9.78% 10.17% 8.77% 6.69%
Median 0.00% 0.00% 5.56% 6.35% 5.33% 0.00%
All Proposed MSL GPCA RANSAC LSA ALC
Average 0.75% 1.75% 6.74% 6.97% 4.65% 2.34%
Median 0.00% 0.00% 1.32% 1.84% 0.89% 0.00%
Table 6.3: Misclassification rates for sequences with two object motions Nh = 2.
bundle ratio ¯B beyond 3.78. Therefore the results of experiment 1 are used later in this chapter for the comparison with the other segmentation approaches. This experiment has the lowest mean misclassification rate ¯M = 0.68, and is highlighted in bold text in table 6.2.
The next section reports the misclassification rates for the sequences with two moving objects.