Abstract
We present a new mathematical formulation of some curve and surface reconstruction algorithms by the introduction of auxiliary variables. For deformable models and templates, the extraction of a shape is obtained through the minimization of an energy composed of an internal regularization term (not necessary in the case of parametric models) and an external attraction potential. Two-step iterative algorithms have been often used where, at each iteration, the model is first locally deformed according to the potential data attraction and then globally smoothed (or fitted in the parametric case).
We show how these approaches can be interpreted as the introduction of auxiliary variables and the minimization of a two-variables energy. The first variable corresponds to the original model we are looking for, while the second variable represents an auxiliary shape close to the first one. This permits to transform an implicit data constraint defined by a non convex potential into an explicit convex reconstruction problem. This approach is much simpler since each iteration is composed of two simple to solve steps. Our formulation permits a more precise setting of parameters in the iterative scheme to ensure convergence to a minimum.
We show some mathematical properties and results on this new auxiliary problem, in particular when the potential is a function of the distance to the closest feature point. We then illustrate our approach for some deformable models and templates.
Similar content being viewed by others
References
W.E.L. Grimson, From Images to Surfaces: A Computational Study of the Human Early Vision System, The MIT Press, 1981.
Demetri Terzopoulos, “The computation of visible-surface representations,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. PAMI-10, No. 4, pp. 417–438, 1988.
Michael Kass, Andrew Witkin, and Demetri Terzopoulos, “Snakes: Active contour models,” International Journal of Computer Vision, Vol. 1, No. 4, pp. 321–331, 1988.
Laurent D. Cohen and Isaac Cohen, “Finite element methods for active contour models and balloons for 2-D and 3-D images,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. PAMI-15, No. 11, pp. 1131–1147, 1993.
F. Leitner, I. Marque, S. Lavallée, and P. Cinquin, “Dynamic segmentation: finding the edge with snake-splines,” in Proceedings of International Conference on Curves and Surfaces, Academic Press: Chamonix, France, June 1990, pp. 1–4.
D. Mumford and J. Shah, “Boundary detection by minimizing functionals,” in Proceedings of Computer Vision and Pattern Recognition, June 1985, San Francisco.
D. Geman and C. Yang, “Nonlinear image recovery with halfquadratic regularization,” in IEEE Transactions on Image Processing, Vol. 4, No. 7, pp. 932–946, July 1995.
A.L. Yuille, P.W. Hallinan, and D.S. Cohen, “Feature extraction from faces using deformable templates,” International Journal of Computer Vision, Vol. 8, No. 3, 1993.
Isaac Weiss, “Shape reconstruction on a varying mesh,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. PAMI-12, No. 4, 1990.
A.N. Tikhonov and V.Y. Arsenin, Solutions of Ill-Posed Problems, Winston and Sons, 1977.
H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1983.
T. Poggio, H. Voohrees, and A. Yuille, “A regularized solution to edge detection,” Journal of Complexity, Vol. 4, pp. 106–123, 1988.
Laurent D. Cohen, “On active contour models and balloons,” Computer Vision, Graphics, and Image Processing: Image Understanding, Vol. 53, No. 2, pp. 211–218, 1991.
Isaac Cohen and Laurent D. Cohen, “A hybrid hyperquadric model for 2-D and 3-D data fitting,” in Proceedings of the 12th IEEE International Conference on Pattern Recognition (ICPR'94), Jerusalem, 1994, pp. B-403–405, Part of Inria TR 2188, to appear in Computer Vision, Graphics, and Image Processing: Image Understanding.
Eric Bardinet, Laurent Cohen, and Nicholas Ayache, “Fitting 3D data using superquadrics and free-form deformations,” in Proceedings of the 12th IEEE International Conference on Pattern Recognition (ICPR'94), Jerusalem, October 1994, pp. A-79–83.
D. Suter, “Constraint networks in vision,” IEEE Transactions on Computers, Vol. 40, No. 12, pp. 1359–1367, 1991.
Andrew Blake and Andrew Zisserman, Visual Reconstruction, The MIT Press, 1987.
D. Geman and Reynolds, “Constrained restoration and the recovery of discontinuities,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 14, pp. 367–383, 1992.
G. Aubert, M. Barlaud, L. Blanc-Féraud, and P. Charbonnier, ”A deteministic algorithm for edge-preserving computed imaging using Legendre transform,” in Proc. 12th International Conference on Pattern Recognition, Jerusalem, Israel, October 1994, pp. C-188–191.
P.G. Ciarlet, Introduction à l'Analyse Numérique Matricielle et à l'Optimisation, Masson, Paris, 1985.
I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North Holland: Amsterdam, 1976.
Gunilla Borgefors, “Distance transformations in arbitrary dimensions,” Computer Vision, Graphics, and Image Processing, Vol. 27, pp. 321–345, 1984.
P.E. Danielsson, “Euclidean distance mapping,” Computer Vision, Graphics, and Image Processing, Vol. 14, pp. 227–248, 1980.
P.J. Huber, Robust Statistics, Wiley, 1981.
D. Geiger and A.L. Yuille, “A common framework for image segmentation,” International Journal of Computer Vision, Vol. 6, No. 3, pp. 227–234, 1991.
N. Kiryati and A.M. Bruckstein, “What's in a set of points,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. PAMI-14, No. 4, pp. 496–500, 1992.
A.P. Pentland, “Recognition by parts,” in Proc. First International Conference on Computer Vision, pp. 612–620, 1987.
Song Han, Dmitry B. Goldgof, and Kevin W. Bowyer, “Using hyperquadrics for shape recovery from range data,” in Proc. Fourth International Conference on Computer Vision, Berlin, June 1993, pp. 492–496. IEEE.
Paul Besl and Neil McKay, “A method for registration of 3-D shapes,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 14, No. 2, pp. 239–256, 1992.
Z. Zhang, “Iterative point matching for registration of free-form curves and surfaces,” International Journal of Computer Vision, Vol. 13, No. 2, pp. 119–152, 1994.
J. Feldmar and N. Ayache, “Locally affine registration of free-form surfaces,” in IEEE Proceedings of Computer Vision and Pattern Recognition 1994 (CVPR'94), Seattle, USA, June 1994. To appear in International Journal of Computer Vision.
Isaac Cohen, Laurent D. Cohen, and Nicholas Ayache, “Using deformable surfaces to segment 3-D images and infer differential structures,” Computer Vision, Graphics, and Image Processing: Image Understanding, Vol. 56, No. 2, pp. 242–263, 1992.
W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, Numerical Recipes in C. The Art of Scientific Programming, Cambridge University Press, 1990.
Pietro Perona and Jitendra Malik, “Scale space and edge detection using anisotropic diffusion,” in Proc. IEEE Workshop on Computer Vision, Miami, FL, 1987, pp. 16–22.
Laurent Cohen, “Filtres adaptatifs et contours,” Technical Report, Schlumberger Montrouge Recherche, Octobre 1989.
L.I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” in Ecoles CEA-EDF-INRIA; Problèmes Non Linéaires Appliqués: Modélisations Mathématiques Pour le Traitement d'Images, Rocquencourt, France, March 1992, pp. 149–179.
Isaac Cohen, “Nonlinear variational method for optical flow computation,” in Proceedings of the 8th Scandinavian Conference on Image Analysis, Tromso, Norway, June 1993, pp. 523–530. IAPR.
S. Kumar and D. Goldgof, “A robust technique for the estimation of the deformable hyperquadrics from images,” in Proceedings of the 12th IEEE International Conference on Pattern Recognition (ICPR'94), Jerusalem, October 1994, pp-74–78.
A.P. Dempster, N.M. Laird, and D.B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” Journal of the Royal Statistical Society B, Vol. 39, No. 1, pp. 1–38, 1977.
Reginald L. Lagendijk and Jan Biemond, Iterative Identification and Restoration of Images, Kluwer Academic Press, 1991.
B. Chalmond, F. Coldefy, and B. Lavayssiere, “3D curve reconstruction from degraded projections,” in L. Schumaker P.J. Laurent and A. Le Mehauté (Eds.), Wavelets, Images, and Surface Fitting. Proceedings of the Conference on Curves and Surfaces, pp. 113–119, 1994.
Laurent D. Cohen and Isaac Cohen, “A finite element method applied to new active contour models and 3D reconstruction from cross sections,” in Proc. Third International Conference on Computer Vision, Osaka, Japan, December 1990, pp. 587–591.
Sylvie Menet, Philippe Saint-Marc, and Gerard Medioni, “B-snakes: Implementation and application to stereo,” in Proceedings of the Seventh Israeli Conference on Artificial Intelligence and Computer Vision, Tel Aviv, Israel, December 1990, pp. 227–240.
S. Sinha and B. Schunk, “Surface approximation using weighted splines,” in Proc. 1991 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Maui, Hawai, June 1991.
André Guéziec, “Large deformable splines, crest lines and matching,” in Proc. Fourth International Conference on Computer Vision, Berlin, May 1993.
N. Ayache, P. Cinquin, I. Cohen, L. Cohen, F. Leitner, and O. Monga, “Segmentation of complex 3D medical objects: a challenge and a requirement for computer assisted surgery planning and performing,” in Computer Integrated Surgery, R. Taylor and S. Lavallee (Eds.), MIT Press, 1994.
T.A. Foley, “Interpolation with interval and point tension controls using cubic weighted ν-Splines,” ACM Transactions on Mathematical Software, Vol. 13, No. 1, pp. 68–96, 1987.
B.A. Barsky, “Exponential and polynomial methods for applying tension to an interpolating spline curve,” Computer Vision, Graphics, and Image Processing, Vol. 27, pp. 1–18, 1984.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cohen, L.D. Auxiliary variables and two-step iterative algorithms in computer vision problems. J Math Imaging Vis 6, 59–83 (1996). https://doi.org/10.1007/BF00127375
Issue Date:
DOI: https://doi.org/10.1007/BF00127375