Fig. 1. Detected optical flow for Translating Tree using frames 7, 8, 9, 10, and 11. (a)–(d) are an image from sequences of images, the ground truth of optical flow, optical flow detected without presmoothing, and optical flow detected with presmoothing, respectively.

Fig. 2. Detected optical flow for Divergent Tree using frames 7, 8, 9, 10, and 11. (a)–(d) are an image from sequences of images, the ground truth of optical flow, optical flow detected without presmoothing, and optical flow detected with presmoothing, respectively. The window size is 5×5.

Fig. 3. (a) and (b) shows optical flow for votings of 99% of 600 combinations and 90% of 600 combinations, respectively of Translating Tree. (c) and (d) show error histograms of results in (a) and (b), respectively. We have measured absolute errors less than 1 deg, 2 deg, and 3 deg, average and standard derivation σ.

Fig. 4. (a)–(d) shows the original image for frame 1, optical flow detected using frames 1 and 2, optical flow detected using frames 1 and 5, and optical flow detected using frames 1, 2, 3, 4, 5 and 7, and using weighted voting in the accumulator space for Hamburg Taxi.

Fig. 5. Using frames 1, 2, 3, 4, and 5 of Hamburg Taxi. We have evaluated the effect of the selection of thresholds. (a)–(d) are the original image, optical flow detected using all combinations of linear constraints, optical flow detected using randomly selected 30% of all combinations, and optical flow detected using randomly selected 10% of the total number of combinations for selection of a pair of equations.

Fig. 6. For Rubic Cube, optical flow computed from (a) frames 1, 2, 3, and 4, (b) frames 1, 3, 5 and 7, and (c) frames 1, 4 and 7. The motion of the cube is 0.4 pixel/frame in average and the motion of the turning table 1.3 pixel/frame in average. Here, the window size is 5×5.

Fig. 7. For Hamburg Taxi, (a) original image, (b) optical flow vectors estimated using Lucas-and-Kanade’s method and, (c) proposed method for 5×5 window using randomly selected 50% of the total number of combinations for selection of a pair of equations.

Fig. 8. For Rubic Cube, (a) original image, (b) optical flow estimated using Lucas-and-Kanade’s method and, (c) proposed method for 5×5 window using randomly selected 50% constrains.

Fig. 9. For Rubic Cube, The average of optical flow detected from frame 1, 2, and 3 (a) and the average of optical flow detected from frames 1, 2, and 4 (b).

Fig. 10. For SIR Tree, (a) original image, (b) optical flow estimated using Lucas-and-Kanade’s method and, (c) proposed method for 5×5 window using randomly selected 50% of the total number of combinations for selection of a pair of equations.

Table 1. Statistical data of some methods for the estimation of optical flow

In this table, for our method 1, we set threshold is 99% of 600 combinations, which is the total number of all combinations for pairs of equations, with presmoothing. For our method 2, we set threshold is 90% of 600 combinations, which is the total number of all combinations for pairs of equations, with presmoothing. For Horn-and-Schunck’s criterion we set λ²=0.5 with presmoothing in spatiotemporal domain. In Barron et al. (1992), λ for Horn-and-Schunck’s is 0.5; that is λ²=0.025 which is smaller than our selection. Lucas-and-Kanade’s method, we computed the optical flow when the smallest eigenvalue of matrix A^TW²A is larger than 0.5, which is used as the threshold (For matrix W, see Appendix B.) for the spatiotemporal-presommothed sequence. Results in Lucas-and-Kanade’s methods (2) and (3) are form Barron et al. (1992) for λ₂1.0 and λ₂5.0, respectively, where λ₂ is the smallest eigenvalue of A^TW²A. For Nagel’s and Anandan’s criterion, we quoted the figures from Barron et al. (1992).