This Article 
 Bibliographic References 
 Add to: 
Direct Analytical Methods for Solving Poisson Equations in Computer Vision Problems
May 1990 (vol. 12 no. 5)
pp. 435-446

Direct analytical methods are discussed for solving Poisson equations of the general form Delta u=f on a rectangular domain. Some embedding techniques that may be useful when boundary conditions (obtained from stereo and occluding boundary) are defined on arbitrary contours are described. The suggested algorithms are computationally efficient owing to the use of fast orthogonal transforms. Applications to shape from shading, lightness and optical flow problems are also discussed. A proof for the existence and convergence of the flow estimates is given. Experiments using synthetic images indicate that results comparable to those using multigrid can be obtained in a very small number of iterations.

[1] B. K. P. Horn and M. J. Brooks, "The variational approach to shape from shading,"Comp. vision, Graphics, and Image Processing, vol. 33, no. 2, pp. 174-208, Feb. 1986.
[2] B. K. P. Horn, "Determining lightness from an image,"Comput. Graphics, Image Processing, vol. 3, pp. 277-299, Dec. 1974.
[3] D. Terzopoulos, "Image analysis using multigrid relaxation methods,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-8, pp. 129-139, Mar. 1986.
[4] B. K. P. Horn and B. Schunck, "Determining optical flow,"Artificial Intell., vol. 17, pp. 185-203, Aug. 1981.
[5] B. Buzbee, G. Golub, and C. Nielson, "On direct methods for solvine Poisson's equations,"SIAM J. Numer. Anal., vol. 7, pp. 627- 656, Dec. 1970.
[6] A. Blake, "On the geometric information obtainable from simultaneous observation of stereo contour and shading," Dep. Comput. Sci., Univ. Edinburgh, Rep. CSR 205-86, 1986.
[7] R. T. Frankot and R. Chellappa, "A method for enforcing integrability in shape from shading algorithms,"IEEE Trans. Pattern Anal. Machine Intell., vol. 10, pp. 439-451, July 1988.
[8] B. Buzbeeet al., "The direct solution of the discrete Poisson equation on irregular regions,"SIAM J. Numer. Anal., vol. 8, pp. 722- 736, Dec. 1971.
[9] D. Lee, "A provably convergent algorithm for shape from shading," inProc. DARPA Image Understanding Workshop, Miami Beach, FL, Dec. 1985, pp. 489-496.
[10] A. N. Tikhonov and V. A. Arsenin,Solutions of Ill-Posed Problems. Washington, DC: Winston, 1977.
[11] T. Poggio and V. Torre, "Ill-posed problems and regularization analysis in early vision," Memo 773, MIT AI Lab., Apr. 1984.
[12] A. Blake and A. Zisserman,Visual Reconstruction. Cambridge, MA: MIT Press, 1987.
[13] A. Bruss, "Is what you see what you get," inProc. Int. Joint Conf. Artificial Intelligence, Karlsruhe, Germany, Aug. 1983.
[14] K. Ikeuchi and B. K. P. Horn, "Numerical shape from shading and occluding boundaries,"Artificial Intell., vol. 17, pp. 141-185, Aug. 1981.
[15] M. J. Brooks and B. K. P. Horn, "Shape and source from shading," inProc. Int. Joint Conf. Artificial Intelligence, Los Angeles, CA, Aug. 1985, pp. 932-936.
[16] M. Strat, "A numerical method for shape from shading from a single image," M.S. thesis, Dep. Elec. Eng. Comput. Sci., Massachusetts Inst. Technol., Cambridge, 1979.
[17] K. Ikeuchi, "Constructing a depth map from images," Artificial Intell. Lab., Massachusetts Inst. Technol., Cambridge, Rep. A.I.M 744, Aug. 1983.
[18] G. Smith,Numerical Solution of Partial Differential Equations: Finite Difference Methods. London: Oxford, University Press, 1978.
[19] A. Rosenfeld and A. Kak,Digital Picture Processing, New York: Academic, 1976.
[20] G. Strang,Linear Algebra and Its Applications. New York: Academic, 1976.
[21] P. N. Swarztrauber, "The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle,"SIAM Rev., vol. 19, pp. 490-501, July 1977.
[22] M. Shao, T. Simchony, and R. Chellappa, "New algorithms for reconstruction of a 3-D depth map from one or more images," inProc. Comput. Vision Pattern Recognition(Ann Arbor, MI), June 1988, pp. 530-535.

Index Terms:
stereo boundary; Poisson equations; computer vision; boundary conditions; occluding boundary; contours; fast orthogonal transforms; shape from shading; lightness; optical flow; convergence; multigrid; computer vision; computerised picture processing; optical information processing
T. Simchony, R. Chellappa, M. Shao, "Direct Analytical Methods for Solving Poisson Equations in Computer Vision Problems," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 12, no. 5, pp. 435-446, May 1990, doi:10.1109/34.55103
Usage of this product signifies your acceptance of the Terms of Use.