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A Projection Operator for the Restoration of Divergence-Free Vector Fields
March 1988 (vol. 10 no. 2)
pp. 248-256

The theory of image restoration by projection onto convex sets can also be applied to the restoration of vector fields. These can have properties that restrict them to lie in well-defined closed convex sets. One of the properties, divergence freedom, is considered, and the theory and numerical implementation of its projection operator are presented. The performance of the operator is illustrated by restoring, from partial information, two simulated divergence-free vector fields. This projection operator finds an important application in the restoration of velocity fields or optical flows computed from an image sequence when the real velocity field is known, a priori to be divergence-free.

[1] D. C. Youla and H. Webb, "Image restoration by the method of convex projections: Part I-Theory,"IEEE Trans. Med. Imaging, vol. MI-1, pp. 81-94, Oct. 1982.
[2] M. I. Sezan and H. Stark, "Image restoration by the method of convex projections: Part II-Application and numerical results,"IEEE Trans. Med. Imaging, vol. MI-1, pp. 95-191, Oct. 1982.
[3] M. I. Sezan and H. Stark, "Tomographic image reconstruction from incomplete view data by convex projection and direct Fourier inversion,"IEEE Trans. Med. Imaging, vol. MI-3, pp. 91-98, June. 1984.
[4] R. T. Chin, C.-L. Yeh, and W. S. Olson, "Restoration of multi-channel microwave radiometric images,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-7, pp. 475-484, July 1985.
[5] B. K. P. Horn and B. G. Schnunck, "Determining optical flow,"Artificial Intell., vol. 17, pp. 185-203, Aug. 1981.
[6] E. Hildreth, "Computation underlying the measurement of visual motion,"Artificial Intell., vol. 23, pp. 309-354, 1984.
[7] D. R. Wilkie,Muscle, London and Southampton: Camelot, 1968.
[8] V. Girault and P.-A. Raviart,Finite Element Approximation of the Navier-Stokes Equutions. Berlin, Heidelberg, New York: Springer-Verlag. 1979.
[9] R. Teman,Navier Stokes Equation., Amsterdam, New York, Oxford: North-Holland, 1979, p. 17.
[10] R. A. Adams,Sobolev Spaces. New York, San Francisco, and London: Academic. 1975.
[11] G. Dahlquist,Numerical Methods. Englcwoods Cliffs, NJ: Prentice-Hall, 1969.

Index Terms:
picture processing; projection operator; divergence-free vector fields; image restoration; closed convex sets; optical flows; image sequence; picture processing; vectors
Citation:
P.Y. Simard, G.E. Mailloux, "A Projection Operator for the Restoration of Divergence-Free Vector Fields," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 10, no. 2, pp. 248-256, March 1988, doi:10.1109/34.3886
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