Next: 5 Discussion Up: Finding the Epipole from Previous: 3 Computing the epipole

4 Experimental results

 

In this section we present initial results on synthetic and real image sequences to check the consistency and robustness of the proposed method.

4.1 Synthetic data

In order to check the robustness of the method, we ran a series of tests on synthetic optical flows. The flows were produced by the rigid motion of two planar surfaces, and corrupted by varying amounts of noise. Figure 1 shows an example of ideal and noisy optical flow, respectively.

 
Figure 1:   An ideal (left) and noise-corrupted (right) optical flow produced by two planar surfaces (see text). The dots identify the epipole estimated by the proposed method.

In each experiment we generated randomly a the 3-D motion, location and orientation of the two planar surfaces. In order to simulate ecologically plausible motions, the parameters were obtained by drawing and from the interval [-0.5,0.5] cm/frame, from [-2,2] cm/frame, and , , and from [-1,1] degree/frame. One plane formed an angle of 45 with the image plane lying at a distance of 200 cm from the origin; the other plane was parallel to the image plane and lying at a distance of 150 cm. These orientations were perturbed by noise to obtain the orientations actually used in the tests.

We added increasing amounts of noise to each of the two components of the motion field at a certain fraction of the image locations, selected at random. At each randomly selected location, and given a maximum percentage of noise, say k %, the motion field components were perturbed by an additive, random amount in the intervals and , respectively. In the experiments reported here we perturbed 10%, 30%, and 50% of the image locations with uniform noise of maximum percentages 5%, 10% and 20%. The number of flow estimates used varied between 6 to 16384. Notice that simply adding Gaussian or other noise to all synthetic flow vectors does not seem a plausible simulation of the flows generated by reliable algorithms on real sequences; for instance, flow vectors at adjacent locations are generally not independent; practically correct estimates can be expected at many locations, but significantly wrong estimates are normally obtained at others. It seemed therefore reasonable to apply noise to varying fractions of flow vectors only.

Table 1 summarizes the results of our tests on the accuracy of epipole localization. Every test consisted of ten different trials. It can be seen that the method performs rather accurately and tolerates noise very well. As can be seen from the rightmost column of Table 1 , even better results can be obtained by discarding flow estimates at which the residual is larger than a fixed threshold.

 

Fract k n LS < thr
10% 5% 256 1.20 0.02
10% 20% 4096 2.32 0.46
30% 5% 4096 2.03 0.84
30% 10% 4096 1.88 0.30
30% 20% 16384 7.23 1.98
Table 1:   Summary of the performance of the proposed method in synthetic tests. The first three columns give, respectively, the fraction of flow points corrupted by noise, the maximum percentage of noise (see text), and the number of flow estimates used. The two rightmost columns report the average error (in pixels) on the localization of the true epipole obtained as the minimum of the residual surface considering all the points (fourth column) and only the points at which the relative residual was smaller than 2% (fifth column).

4.2 Real image sequences

We also applied the proposed method to real images. Since no exact ground truth was available in these experiments, we concentrated on verifying the consistency and reliability of the method in terms of stability of the estimated epipole's location, in sequences where the instantaneous epipole was approximately stationary. An example is given in Figure 2 , which shows four frames of a sequence of 30 images ( ) taken by a camera mounted on a vehicle. Figure 3 shows two of the optical flows computed through a block-matching algorithm with subpixel precision. No flow estimates were generated at locations where the residual of the block-matching was not sufficiently different from its minimum value in a neighborhood of the minimum. The large dots denote the estimated location of the epipole. In this sequence the vehicle (and also the vehicles traveling in the opposite direction) was turning on a turn of approximately constant curvature; therefore the epipole can be expected to be roughly in the same location throughout the entire sequence. Consistently, the standard deviation of the epipole location estimated by our algorithm was less than 3 pixels. Very similar results (not reported here) have been obtained on two other real image sequences.

 
Figure 2:   Four frames of a long image sequence (from upper left to lower right).

 
Figure 3:   Optical flows relative to the sequence of Figure 2 computed by a block-matching technique.



Next: 5 Discussion Up: Finding the Epipole from Previous: 3 Computing the epipole

Adrian F Clark
Wed Jul 9 12:25:56 BST 1997