This Article 
 Bibliographic References 
 Add to: 
Statistical Analysis of Inherent Ambiguities in Recovering 3-D Motion from a Noisy Flow Field
October 1992 (vol. 14 no. 10)
pp. 995-1013

The inherent ambiguities in recovering 3-D motion information from a single optical flow field are studied using a statistical model. The ambiguities are quantified using the Cramer-Rao lower bound. As a special case, the performance bound for the motion of 3-D rigid planar surfaces is studied in detail. The dependence of the bound on factors such as the underlying motion, surface position, surface orientation, field of view, and density of available pixels are derived as closed-form expressions. A subset of the results support S. Adiv's (1989) analysis of the inherent ambiguities of motion parameters. For the general motion of an arbitrary surface. It is shown that the aperture problem in computing the optical flow restricts the nontrivial information about the 3-D motion to a sparse set of pixels at which both components of the flow velocity are observable. Computer simulations are used to study the dependence of the inherent ambiguities on the underlying motion, the field of view, and the number of feature points for the motion in front of a nonplanar environment.

[1] G. Adiv, "Determining three-dimensional motion and structure from optical flow generated by several moving objects,"IEEE Trans. Patt. Anal. Machine Intell., vol. PAMI-7, no. 4, pp. 384-401, July 1985.
[2] G. Adiv, "Inherent ambiguities in recovering 3-D motion and structure from a noisy flow field,"IEEE Trans. Patt. Anal. Machine Intell., vol. 11, no. 5, pp. 477-489, May 1989.
[3] J. K. Aggarwal, "Motion and time-varying imagery--An overview," inProc. IEEE Workshop Motion: Representation Anal.(Kiawah Island, SC), May 1986.
[4] J. Aloimonos and D. Shulman,Integration of Visual Modules: An Extension of the Marr Paradigm. Boston, MA: Academic, 1989.
[5] P. Anandan, "Computing dense displacement fields with confidence measures in scenes containing occlusion," inProc. DARPA Image Understanding Workshop, Oct. 1984, pp. 236-246.
[6] P. Anandan, "A unified perspective on computational techniques for the measurement of visual motion," inProc. First Int. Conf. Comput. Vision(London), June 1987, pp. 219-230.
[7] P. Anandan and R. Weiss, "Introducing a smoothness constraint in a matching approach for the computation of displacement fields," inProc. DARPA Image Understanding Workshop, Dec. 1985, pp. 186-196.
[8] T. J. Broida, S. Chandrashekhar, and R. Chellappa, "Recursive estimation of 3-D kinematics and structure from noisy monocular image sequences,"IEEE Trans. Aerosp. Electron. Syst., vol. AES-26, pp. 639-656, Aug. 1990.
[9] T. Broida and R. Chellappa, "Estimation of object motion parameters from noisy images,"IEEE Trans. Pattern Anal. Machine Intell, vol. PAMI-8, no. 1, Jan. 1986.
[10] T. J. Broida and R. Chellappa, "Performance bounds for estimating three-dimensional motion parameters from a sequence of noisy images,"J. Optical Soc. Amer. A, vol. 6, no. 6, pp. 879-889, June 1989.
[11] A. R. Bruss and B. K. P. Horn, "Passive navigation,"Comput. Vision Graphics Image Processing, vol. 21, no. 1, pp. 3-20, Jan. 1983.
[12] B. Cernuschi-Friaset al., "Toward a model-based bayesian theory for estimating and recognizing parameterized 3-d objects using two or more images taken from different positions,"IEEE Trans. Patt. Anal. Machine Intell., vol. 11, no. 10, pp. 1028-1052, Oct. 1989.
[13] E. Hildreth, "Computation underlying the measurement of visual motion,"Artificial Intell., vol. 23, pp. 309-354, 1984.
[14] E. Hildreth,The Measurement of Visual Motion, Cambridge, MA: MIT Press, 1983.
[15] B. K. P. Horn,Robot Vision. Cambridge, MA: M.I.T. Press, 1986.
[16] B. K. P. Horn, "Motion fields are hardly ever ambiguous,"Int. J. Comput. Vision, vol. 1, pp. 259-274, Oct. 1987.
[17] B. K. P. Horn, "Relative orientation,"Int. J. Computer Vision, vol. 4, no. 1, pp. 59-78, 1990.
[18] H. C. Longuet-Higgins, "A computer algorithm for reconstructing a scene from two projections,"Nature, vol. 293, pp. 133-135, Sept. 1981.
[19] K. Prazdny, "Egomotion and relative depth map from optical flow,"Biol. Cybern., vol. 36, pp. 87-102, Feb. 1980.
[20] H. W. Sorenson,Parameter Estimation, Principles and Problems. New York: Marcel Dekker, 1980.
[21] M. E. Spetsakis and J. Aloimonos, "Optimal motion estimation," inProc. Workshop Visual Motion(Irvine, CA), Mar. 1989, pp. 229-237.
[22] G. Strang,Linear Algebra and Its Application. New York: Academic, 1980.
[23] R. Y. Tsai and T. S. Huang, "Uniqueness and estimation of three-dimensional motion parameters of rigid objects with curved surfaces,"IEEE Trans. Patt. Anal. Machine Intell., vol. PAMI-6, pp. 13-27, Jan. 1984.
[24] S. Ullman, "Analysis of visual motion by biological and computer systems,"Comput., vol. 14, pp. 57-69, Aug. 1981.
[25] A. M. Waxman, B. Kamgar-Parsi, and M. Subbarao, "Closed-form solutions to image flow equations for 3d structure and motion,"Int. J. Comput. Vision, vol. 1, pp. 239-258, Oct. 1987.
[26] J. Weng, T. S. Huang, and N. Ahuja, "Motion from images: Image matching, parameter estimation, and intrinsic stability," inProc. IEEE Workshop Visual Motion, (Irvine, CA), Mar. 20-22, 1989, pp. 359-366.
[27] G. S. Young and R. Chellappa, "3-d motion estimation using a sequence of noisy stereo images : Models, estimation, and uniqueness results,"IEEE Trans. Patt. Anal. Machine Intell., vol. 12, no. 8, pp. 735-759, Aug. 1990.

Index Terms:
3D motion recovery; image processing; inherent ambiguities; optical flow field; statistical model; Cramer-Rao lower bound; surface orientation; field of view; aperture; feature points; image processing; optical information processing; statistical analysis
G.S.J. Young, R. Chellappa, "Statistical Analysis of Inherent Ambiguities in Recovering 3-D Motion from a Noisy Flow Field," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 14, no. 10, pp. 995-1013, Oct. 1992, doi:10.1109/34.159903
Usage of this product signifies your acceptance of the Terms of Use.