where the first two inequalities are based on constraint (4), and the last inequality is derived from both the triangle inequality 7 and the fact that τ(⋅) is non-decreasing.
Appendix C: Proof of Property 4
Assume that τ(⋅) belongs to
ρ1
H . To prove that τ(⋅) is minimal 2-feasible is equivalent to showing that τ(⋅) is 2-feasible but not 1-feasible. Since it is trivial that τ(⋅) is both non-decreasing and not
z Case 1 (σ ≥||a||2≥ a1≥ a2 ≥0):
according to the arithmetic-geometric inequality, (C.2)
≥0.
[1] M. Bierling, “Displacement estimation by hierarchical block-matching,” in Proceedings of SPIE Conference on Visual Communications and Image Processing, vol. 1001, pp. 942-951, 1988.
[2] V. Barnett and T. Lewis, Outliers in Statistical Data. New York: John Wiley and Sons, 1994.
[3] M. J. Black and P. Anandan, “The robust estimation of multiple motions: parametric and piecewise-smooth flow fields,” Computer Vision and Image Understanding, vol. 63, pp. 75-104, 1996.
[4] M. J. Black and A. Rangarajan, “On the unification of line processes, outlier rejection, and robust statistics with applications in early vision,” International Journal of Computer Vision, vol. 19, pp. 57-91, 1996.
[5] A. Can, C. V. Stewart, and B. Roysam, “Robust hierarchical algorithm for constructing a mosaic from images of the curved human retina,” in Proceedings of IEEE Conference Computer Vision and Pattern Recognition, pp. 286-292, 1999.
[6] J.-H. Chen, C.-S. Chen, and Y.-S. Chen, “Fast method for robust template matching,” Technical Report TR-IIS-01-015, Institute of Information Science, Academia Sinica, Taipei, Taiwan, 2001.
[7] Y.-S. Chen, Y.-P. Hung, and C.-S. Fuh, “Fast block matching algorithm based on the winner-update strategy,” IEEE Transactions on Image Processing, vol. 10, pp. 1212-1222, 2001.
[8] K. H. K. Chow and M. L. Liou, “Genetic motion search algorithm for video compression,” IEEE Transactions on Circuit System and Video Technology, vol. 3, pp. 440-445, 1993.
[9] J. Flusser, “Refined moment calculation using image block representation,” IEEE Transactions on Image Processing, vol. 11, pp. 1977-1978, 2000.
[10] J. Flusser and B. Zitová, “Combined invariants to linear filtering and rotation,” International Journal of Pattern Recognition and Artificial Intelligence, vol. 13, pp. 1123-1136, 1999.
[11] H. Frigui and R. Krishnapuram, “A robust competitive clustering algorithm with applications in computer vision,” IEEE Transactions on Pattern Analysis and Machine Intelligence, to appear.
[12] S. Geman and D. E. McClure, “Statistical methods for tomographic image reconstruction,” Bulletin of the International Statistical Institute, vol. 52, pp.5-21, 1987.
[13] M. Ghanbari, “The cross-search algorithm for motion estimation,” IEEE Transactions on Communication, vol. 38, pp. 950-953, 1990.
[14] R. M. Haralick, H. Joo, C.-N. Lee, and et al., “Pose estimation from corresponding point data,” IEEE Transactions Systems, Man, and Cybernetics, vol. 19, pp. 1426-1446, 1989.
[15] H.-C. Huang, Y.-P Hung, and W.-L. Hwang, “Adaptive early jump-out technique for fast motion estimation in video coding,” CVGIP: Graphical Models and Image Processing, vol. 59, pp. 388-394, 1997.
[16] P. J. Huber, Robust Statistics. New York: John Wiley and Sons, 1981.
[17] R. A. Horn, and C. R. Johnson, Matrix Analysis. Cambridge University Press, 1985.
[18] K. Jonsson, J. Kittler, Y. P. Li, and J. Matas, “Support vector machines for face authentication,” Image and Vision Computing, to appear.
[19] K. Kawamura, K. Hasegawa, O. Yamashita, and et al., “Object recognition using local EGI and 3D models with M-estimators,” in Proceedings of IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems, pp. 80-86, 1999.
[20] R. Klette, K. Schlüns, and A. Koschan, Computer Vision Three-Dimensional Data from Images, Chapter 4, Springer, 1998.
[21] D. Keysers, W. Macherey, J. Dahmen, and H. Ney. “Learning of Variability for Invariant Statistical Pattern Recognition,” in Proceedings of 12th European Conference on Machine Learning, Freiburg, Germany, pp.
263-275, 2001.
[22] T. Koga, K. Iinuma, A. Hirano, and et al., “Motion compensated interframe coding for video conference,”
in Proceedings of National Telecommunications Conference, pp. G5.3.1-G5.3.5, 1981.
[23] S.-H. Lai, “Robust image matching under partial occlusion and spatially varying illumination change,”
Computer Vision and Image Understanding, vol. 78, pp. 84-98, 2000.
[24] S. Lawrence, C. L. Giles, A. -C. Tsoi, and A. Back, “Face recognition: a convolutional neural network approach,” IEEE Transactions on Neural Networks, vol. 8, pp. 98-113, 1997.
[25] C.-H. Lee and L.-H. Chen, “A fast motion estimation algorithm based on the block sum pyramid,” IEEE Transactions on Image Processing, vol. 6, pp. 1587-1591, 1997.
[26] W. Li and E. Salari, “Successive elimination algorithm for motion estimation,” IEEE Transactions on Image Processing, vol. 4, pp. 105-107, 1995.
[27] H.-Y. M. Liao, C.-C. Han, and G.-J. Yu, “Face+hair+shoulders+blackground≠face,” Technical Report TR-IIS-97-007, Institute of Information Science, Academia Sinica, Taipei, Taiwan, 1997.
[28] C.-H. Lin and J.-L. Wu, “A lightweight genetic block-matching algorithm for video coding,” IEEE Transactions on Circuit System Video Technology, vol. 8, pp. 386-392, 1998.
[29] L.-K. Liu and E. Feig, “A block-based gradient descent search algorithm for block motion estimation in video coding,” IEEE Transactions on Circuit System Video Technology, vol. 6, pp. 419-422, 1996.
[30] P. Meer, D. Mintz, A. Rosenfeld and D. Y. Kim, “Robust regression methods for computer vision: a review,” International Journal of Computer Vision, vol. 6, pp. 59-70, 1991.
[31] M. J. Mirza and K. L. Boyer, “Performance evaluation of a class of M-estimators for surface parameter estimation in noisy range data,” IEEE Transactions on Robotics and Automation, vol. 9, pp. 75-85, 1993.
[32] S. I. Olsen, “Epipolar line estimation,” in Proceedings of Second European Conference on Computer Vision, pp. 307-311, 1992.
[33] J. Park, B. Jiang, and U. Neumann, “Vision-based pose computation: robust and accurate augmented reality tracking,” in Proceedings of the IEEE International Workshop on Augmented Reality, pp. 3-12, 1999.
[34] S. Russel and P. Norvig, Artificial Intelligence, pp. 75, Prentice Hall, New Jersey, 1995.
[35] M.-C. Shie, W.-H. Fang, K.-J. Hung, and et al., “Fast, robust block motion estimation using simulated annealing,” IEICE Transactions on Fundamentals, vol. E83, pp. 121-127, 2000.
[36] D.-G. Sim, O.-K. Kwon, and R.-H. Park, “Object matching algorithms using robust hausdorff distance measures,” IEEE Transactions Image Processing, vol. 8, pp. 425-429, 1999.
[37] P. Simard, Y. Le Cun, J. Denker, “Efficient Pattern Recognition Using a New Transformation Distance”, in Neural Information Processing Systems, S. J. Hanson, J. D. Cowan, and C. L. Giles, Eds. 1993, vol. 5, pp.
50-58, Morgan Kaufmann.
[38] C. V. Stewart, “Robust parameter estimation in computer vision,” SIAM Review, vol. 41, pp. 513-537, 1999.
[39] S. Zhu and K.-K. Ma, “A new diamond search algorithm for fast block-matching motion estimation,” IEEE Transactions Circuit System and Video Technology, vol. 9, pp. 287-290, 2000.
[40] Z. Zhang, “Parameter estimation techniques: a tutorial with application to conic fitting,” Image and Vision Computing, Vol. 15, pp. 59-76, 1997.
Figure 1. This figure demonstrates the p-pyramid constructed from a 1-d signal where each element in higher levels is composed of its two son elements. Five p-pyramids are depicted in solid or dashed lines. The three black nodes are the ones that are shared between two pyramids.
Figure 2. Shapes of commonly used M-estimators with threshold σ=70. (a) The shape of ρ1. (b) The shape of ρ2. (c) The shape of ρ3. (d)The shape of ρ4. (e)The shape of ρ5. (f)The shape of ρ6.
A B C
I2 I1 I0
u u+1 u+2 u+3 u+4 u+5 u+6 u+7
……
Pyramid 1 Pyramid 2 Pyramid 3 Pyramid 4 Pyramid 5
…
………
……
…………
Figure 3. Illustration of the search strategies introduced in Section 3.A. The Li and Salari method [26] only searches the layer 0 and layer n of the tree in a depth-first order. The Lee and Chen method [25] searches the entire tree in a depth-first order. Both methods prune the search branches by comparing the current reference value with the error associated with the vertex. The Chen et al.
method [7] uses the uniform cost search [34] (the branch-and-bound strategy) for the entire tree to prune the unnecessary search branches.
Figure 4. (a) One of the synthetic input signals. (b),(c) and (d) The process of generating a test signal from an input signal.
It F-W,-W Fu,v FW,W
Layer n Layer 0
Search Tree
(c) Adding gaussian noise
(b) Extraction of a partial segment
(a) An input signal
(d) Adding impulse noises that serve as outliers. In this case, the outlier ratio is 0.05.
(a)
(b)
Figure 5. Comparisons between our method and the FS method for robust template matching in the signal matching experiment are shown. Note that simple truncation and the 1-pyramid are used in this experiment. (a) The operation count ratio vs. outlier ratio. (b) The time consumption ratio vs.
outlier ratio.
(a) (b)
Figure 6. (a) Part of a face-only database used in this paper, showing 100 images from 10 people with 10 images for each person (b) Contaminated images of a person with different outlier ratios.From left to right, top to bottom, the outlier ratios are set from 0 to 0.1.
(a) (b)
Figure 7. Comparisons between the SSD and SRD using Huber's estimator. (a) The hit ratio vs.outlier ratio. (b) The time consumption ratio vs. outlier ratio.
(a) (b)
Figure 8. Comparisons between the SSD and the SRD using Tukey's estimator. (a) The hit ratio vs. outlier ratio. (b) The time consumption ratio vs. outlier ratio.(a) (b)
Figure 9. Comparisons between the SSD and the SRD using Geman and McClure's estimator. (a) The hit ratio vs. outlier ratio. (b) The time consumption ratio vs. outlier ratio.(a) (b)
Figure 10. Comparisons between the SSD and the SRD using the trimmed mean M-estimator.(a) The hit ratio vs. outlier ratio. (b) The time consumption ratio vs. outlier ratio.