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Constructing Intrinsic Parameters with Active Models for Invariant Surface Reconstruction
July 1993 (vol. 15 no. 7)
pp. 668-681

A technique for constructing a canonical surface parameterization in terms of lines of curvature is presented. Two methods of computing the canonical invariant representation are also presented. In the first method, a static instance of the controlled continuity spline is used for the stabilizer. Ways to modify it to reflect a change of parameters to the lines of curvature are described. In the second method, the dynamic instance of the controlled continuity spline called the deformable model is used. A force field defined in terms of the principal vectors is synthesized and applied to the parameter curves of the deformable model to coerce them along the lines of curvature. In essence, any transformation of parameters requires a modification of the stabilizer in the first method, whereas in the second method, it is tantamount to synthesizing a new force field. Experimental results with real and synthetic range data are included.

[1] A. Barr, "Superquadrics and angle preserving transformations,"IEEE Comput. Graphics Applications, pp. 11-23, Jan. 1981.
[2] A. Blake, "Reconstructing a visible surface," inProc. Nat. Conf. Artificial Intell. AAAI-84, Aug. 1984, pp. 23-26.
[3] A. Blake and A. Zisserman,Visual Reconstruction. Cambridge, MA: MIT Press, 1987.
[4] B. Bhanu, "Representation and shape matching of 3D objects,"IEEE Trans. Patt. Anal. Machine Intell., vol. PAMI-6, pp. 340-351, 1984.
[5] R. M. Bolle and B. C. Vemuri, "On three-dimensional surface reconstruction methods,"IEEE Trans. Patt. Anal. Machine Intell., vol. PAMI-13, no. 1, pp. 1-13, 1991.
[6] W. M. Boothby,An Introduction to Differentiable Manifolds and Riemanian Geometry. New York: Academic, 1986.
[7] T. E. Boult and J. R. Kender, "Visual surface reconstruction using sparse depth data," inProc. IEEE Conf. Comput. Vision Patt. Recogn., June 1986, pp. 68-76.
[8] M. Brady, J. Ponce, A. Yuille, and H. Asada, "Describing surfaces,"Comput. Vision Graphics Image Processing, vol. 32, pp. 1-28, 1985.
[9] F. H. Clarke,Optimization and Nonsmooth Analysis. New York: Wiley-Interscience, 1983.
[10] G. Dahlquist andÅ. Björk,Numerical Methods. Englewood Cliffs, NJ: Prentice-Hall, 1974.
[11] H. Delingette, M. Hebert, and K. Ikeuchi, "Shape representation and image segmentation using deformable surfaces," inProc. IEEE Conf. Comput. Vision Pattern Recogn., June 1991.
[12] M. P. Do Carmo,Differential Geometry of Curves and Surfaces. Englewood Cliffs, NJ: Prentice-Hall, 1976.
[13] J. Duchon, "Interpolation de functions de deux variables suivant le principe de la flexion des plaques minces,"Rev. Francaise d'Automatique Informatique et Recherche Operationelle, vol. 5, no. 12, Dec. 1976.
[14] T. Hughes,The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Englewood Cliffs, NJ: Prentice-Hall, 1987.
[15] W. Hurewicz,Lectures on Ordinary Differential Equations. Cambridge, MA: MIT Press, 1958.
[16] S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, "Optimization by simulated annealing,"Sci., vol. 220, pp. 671-680, 1983.
[17] D. Lee and T. Pavlidis, "One-dimensional regularization with discontinuities,"IEEE Trans. Patt. Anal. Machine Intell., vol. PAMI-10, pp. 822-829, 1986.
[18] J. Marroquin, S. Mitter, and T. Poggio, "Probabilistic solution of ill-posed problems in computational vision,"J. Amer. Stat. Assoc., vol. 82, no. 397, 1987.
[19] R. Malladi, "Deformable models: Canonical parameters for surface representation and multiple view integration," M.S. thesis, Dept. of CIS, Univ. of Florida, Gainesville, May 1991.
[20] J. Meinguet, "Multivariate interpolation at arbitrary points made simple,"J. Applied Math. Phys., vol. 30, pp. 292-304, 1979.
[21] M. Potmesil, "Generating models for solid objects by matching 3D surface segments," inProc. IEEE Int. Joint Conf. Artificial Intell.(Karlsruhe, Germany), Aug. 1983.
[22] R. T. Rockafellar,The Theory of Subgradients and its Applications to Problems of Optimization: Convex and Nonconvex Functions. Berlin: Helderman Verlag, 1981.
[23] P. Sander and S. W. Zucker, "Inferring differential structure from 3D images: Smooth cross section of fiber bundles,"IEEE Trans. Patt. Anal Machine InteIl., vol. PAMI-12, pp. 833-854, 1990.
[24] L. L. Schumaker, "Fitting surfaces to scattered data," inApproximation Theory II(G. G. Lorentz, C. K. Chui, and L. L. Schumaker, Eds.). New York: Academic, 1976, pp. 203-267.
[25] S. S. Sinha and P. J. Besl, "Principal patches: A viewpoint invariant surface description," inProc. IEEE Int. Conf. Robotics Automat., 1990.
[26] R. L. Stevenson and E. J. Delp, "Invariant reconstruction of visual surfaces," inProc. IEEE Workshop Interpretation 3D Scenes(Austin, TX), Nov. 27-29, 1989, pp. 131-137.
[27] R. Szeliski,Bayesian Modeling of Uncertainty in Low-Level Vision, Boston: Kluwer, 1989.
[28] D. Terzopoulos, "Matching deformable models to images: Direct and iterative solutions," inTopical Mtg. Machine Vision. Washington, DC: Opt. Soc. Amer, 1987, pp. 164-167, vol. 12.
[29] D. Terzopoulos, "The computation of visible surface representations,"IEEE Trans. Patt. Anal. Machine Intell., vol. 10, pp. 417-438, 1988.
[30] D. Terzopoulos, A. Witkin, and M. Kass, "Constraints on deformable models: Recovering 3D shape and nonrigid motion,"Artificial Intell., vol. 36, pp. 91-123, 1988.
[31] J. F. Thompson, Z. U. A. Warsi, and C. W. Mastin,Numerical Grid Generation: Foundations and Applications. New York: Elsevier, 1985.
[32] B.C. Vemuri, A. Mitiche, and J.K. Aggarwal, "Curvature-based representation of objects from range data,"Image and Vision Comput., vol. 4, no. 2, pp. 107-114, May 1986.
[33] B. C. Vemuri and J. K. Aggarwal, "Representation and recognition of objects from dense range maps,"IEEE Trans. Circuits Syst., vol. CAS-34, no. 11, pp. 1351-1363, Nov. 1987.
[34] B. C. Vemuri, D. Terzopoulos, and P. J. Lewicki, "Canonical parameters for invariant surface representation," inProc. SPIE Symp. Advances Intelligent Robotic Syst.(Philadelphia, PA), Nov. 1989, pp. 75-86, vol. 119.
[35] B. C. Vemuri and G. Skofteland, "Motion estimation from multi-sensor data for tele-robotics," inProc. IEEE Int. Workshop Intelligent Motion Contr.(Istanbul, Turkey), Aug. 1990, pp. 361-367.
[36] B. C. Vemuri and R. Malladi, "Deformable models: Canonical parameters for surface representation and multiple view integration," inProc. IEEE Conf. Comput. Vision Patt. Recogn., June 1991, pp. 724-725.
[37] Y. F. Wang and J. F. Wang, "Surface reconstruction using deformable models with interior and boundary constraints,"Proc. ICCV, pp. 300-303, 1990.
[38] D. M. Young,Iterative Solution of Large Linear Systems. New York, 1971.

Index Terms:
curvature lines; active models; invariant surface reconstruction; canonical invariant representation; controlled continuity spline; deformable model; force field; principal vectors; image reconstruction; invariance; splines (mathematics)
B.C. Vemuri, R. Malladi, "Constructing Intrinsic Parameters with Active Models for Invariant Surface Reconstruction," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 15, no. 7, pp. 668-681, July 1993, doi:10.1109/34.221168
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