This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
On the Mathematical Foundations of Smoothness Constraints for the Determination of Optical Flow and for Surface Reconstruction
November 1991 (vol. 13 no. 11)
pp. 1105-1114

Gradient-based approaches to the computation of optical flow often use a minimization technique incorporating a smoothness constraint on the optical flow field. The author derives the most general form of such a smoothness constraint that is quadratic in first derivatives of the grey-level image intensity function based on three simple assumptions about the smoothness constraint: (1) it must be expressed in a form that is independent of the choice of Cartesian coordinate system in the image: (2) it must be positive definite; and (3) it must not couple different component of the optical flow. It is shown that there are essentially only four such constraints; any smoothness constraint satisfying (1), (2), or (3) must be a linear combination of these four, possibly multiplied by certain quantities invariant under a change in the Cartesian coordinate system. Beginning with the three assumptions mentioned above, the author mathematically demonstrates that all best-known smoothness constraints appearing in the literature are special cases of this general form, and, in particular, that the 'weight matrix' introduced by H.H. Nagel is essentially (modulo invariant quantities) the only physically plausible such constraint.

[1] P. Anandan, "A unified perspective on computational techniques for the measurement of visual motion," inProc. 1st Int. Conf. Comput. Vision, (London, England), June 1987.
[2] P. Anandan and R. Weiss, "Introducing a smoothness constraint in a matching approach for the computation of displacement fields," COINS Tech. Rep. 85-38, Univ. Mass., Amherst, Dec. 1985.
[3] M. Brady and B.K.P. Horn, "Rotationally symmetric operators for surface interpolation,"CVGIP, vol. 22, pp. 70-94, 1983.
[4] M. P. do Carmo,Differential Geometry of Curves and surfaces. Englewood Cliffs, NJ: Prentice-Hall 1976.
[5] E. Hildreth,The Measurement of Visual Motion, Cambridge, MA: MIT Press, 1983.
[6] W. E. L. Grimson,From Images to Surfaces: A Computational Study of the Human Early visual System. Cambridge, MA: MIT Press, 1981.
[7] B. K. P. Horn and E. J. Weldon, Jr. "Robust direct methods for recovering motion," Univ. Hawaii, Manoa, Tech. Rep., Apr. 1987.
[8] B. K. P. Horn and B. G. Schunck, "Determining optical flow,"Artificial Intell., vol. 23, pp. 185-203, 1981.
[9] D. Marr and E. Hildreth, "Theory of edge detection," Phil. Trans. Roy. Soc. London, vol. B207, pp. 187-217, 1980.
[10] H.-H. Nagel, "Constraints for the estimation of displacement vector fields from image sequences," inProc. IJCAI83, (Karlsruhe, W. Germany), 1983, pp. 945-951.
[11] H.-H. Nagel and W. Enkelmann, "An investigation of smoothness constraints for the estimation of displacement vector fields from image sequences,"IEEE Trans. Patt. Anal. Machine Intell., vol. PAMI-8, no. 5, pp. 565-593, Sept. 1986.
[12] Y.C. Lee et al., "Internal Thermal Resistance of a Multi-Chip Packaging Design for VLSI Based Systems,"IEEE Trans. Components, Hybrids, and Manufacturing Technology, Vol. 12, No. 2, June 1989, pp. 163- 169.
[13] H.-H. Nagel, "Image sequences-ten (Octal) years-from phenomenology towards a theoretical foundation,"Int. J. Patt. Recognition Artificial Intell., vol. 2, pp. 459-483, 1988.
[14] B. G. Schunck, "The motion constraint equation and surface structure,"Nat. Conf. Artificial Intell., (Austin, TX), 1984.
[15] B. G. Schunck, "The motion constraint equation for optical flow," in Proc. Int. Conf. Patt. Recognition (Montreal, Canada), 1984.
[16] M. A. Snyder, "On the mathematical foundations of smoothness constraints for the determination of optical flow and for surface reconstruction," COINS Tech. Rep. 89-05, Univ. Mass., Amherst, 1989.
[17] M. A. Snyder, "On the mathematical foundations of smoothness constraints for the determination of optical flow and for surface reconstruction," COINS Tech. Rep. 89-05, Univ. Mass., Amherst, 1989.
[18] G. Strang,Introduction to Applied Mathematics, Wellesley, MA: Wellesley-Cambridge Press, 1986.
[19] J. A. Thorpe,Elementary Topics in Differential Geometry. New York: Springer Verlag, New York, 1979.
[20] P. Anandan, personal communication.

Index Terms:
computer vision; gradient-based methods; smoothness constraints; optical flow; surface reconstruction; minimization technique; grey-level image intensity function; positive definite; Cartesian coordinate system; weight matrix; computer vision; minimisation
Citation:
M.A. Snyder, "On the Mathematical Foundations of Smoothness Constraints for the Determination of Optical Flow and for Surface Reconstruction," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 13, no. 11, pp. 1105-1114, Nov. 1991, doi:10.1109/34.103272
Usage of this product signifies your acceptance of the Terms of Use.