The Community for Technology Leaders
RSS Icon
Issue No.11 - November (2011 vol.33)
pp: 2229-2244
Dimitris Arabadjis , National Techncal University of Athens, Athens
Panayiotis Rousopoulos , National Techncal University of Athens, Athens
Constantin Papaodysseus , National Techncal University of Athens, Athens
Michalis Exarhos , National Techncal University of Athens, Athens
Michail Panagopoulos , Ionian University, Corfu
Lena Papazoglou-Manioudaki , National Archaeological Museum of Greece, Athens
In this paper, a general methodology is introduced for the determination of potential prototype curves used for the drawing of prehistoric wall paintings. The approach includes 1) preprocessing of the wall-paintings contours to properly partition them, according to their curvature, 2) choice of prototype curves families, 3) analysis and optimization in 4-manifold for a first estimation of the form of these prototypes, 4) clustering of the contour parts and the prototypes to determine a minimal number of potential guides, and 5) further optimization in 4-manifold, applied to each cluster separately, in order to determine the exact functional form of the potential guides, together with the corresponding drawn contour parts. The methodology introduced simultaneously deals with two problems: 1) the arbitrariness in data-points orientation and 2) the determination of one proper form for a prototype curve that optimally fits the corresponding contour data. Arbitrariness in orientation has been dealt with a novel curvature based error, while the proper forms of curve prototypes have been exhaustively determined by embedding curvature deformations of the prototypes into 4--manifolds. Application of this methodology to celebrated wall paintings excavated at Tyrins, Greece, and the Greek island of Thera manifests that it is highly probable that these wall paintings were drawn by means of geometric guides that correspond to linear spirals and hyperbolae. These geometric forms fit the drawings' lines with an exceptionally low average error, less than 0.39 mm. Hence, the approach suggests the existence of accurate realizations of complicated geometric entities more than 1,000 years before their axiomatic formulation in the Classical Ages.
Rotation and translation invariant curve fitting, pattern recognition in paintings, optimization in differentiable manifolds, geometric guides in prehistoric wall paintings, minimal parameters set for curve description, fitting prototype curves to drawn borders.
Dimitris Arabadjis, Panayiotis Rousopoulos, Constantin Papaodysseus, Michalis Exarhos, Michail Panagopoulos, Lena Papazoglou-Manioudaki, "Optimization in Differentiable Manifolds in Order to Determine the Method of Construction of Prehistoric Wall Paintings", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.33, no. 11, pp. 2229-2244, November 2011, doi:10.1109/TPAMI.2011.65
[1] S. Ahn, W. Rauh, H. Cho, and H. Warnecke, "Orthogonal Distance Fitting of Implicit Curves and Surfaces," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 24, no. 5, pp. 620-638, May 2002.
[2] S.J. Ahn, W. Rauh, and H.-J. Warnecke, "Least-Squares Orthogonal Distances Fitting of Circle, Sphere, Ellipse, Hyperbola, and Parabola," Elsevier Pattern Recognition, vol. 34, no. 12, pp. 2283-2203, 2001.
[3] K. Katani, "Cramer-Rao Lower Bounds for Curve Fitting," Elsevier Graphical Models and Image Processing, vol. 60, no. 2, pp. 93-99, 1998.
[4] C. Papaodysseus, D.K. Fragoulis, M. Panagopoulos, T. Panagopoulos, P. Rousopoulos, M. Exarhos, and A. Skembris, "Determination of the Method of Construction of 1650 B.C. Wall Paintings," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 28, no. 9, pp. 1-11, Sept. 2006.
[5] S. Sener and M. Unel, "A New Affine Invariant Fitting Algorithm for Algebraic Curves," Image Analysis and Recognition, pp. 344-351, Springer, 2004.
[6] S. Belongie, J. Malik, and J. Puzicha, "Shape Matching and Object Recognition Using Shape Contexts," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 24, no. 4, pp. 509-522, Apr. 2002.
[7] S. Flöry and M. Hofer, "Constrained Curve Fitting on Manifolds," Computer-Aided Design, vol. 40, no. 1, pp. 25-34, 2008.
[8] A. Spira and R. Kimmel, "Enhancing Images Painted on Manifolds," Scale Space and PDE Methods in Computer Vision, pp. 492-502, Springer, 2005.
[9] G. Taubin, F. Cukierman, S. Sullivan, J. Ponce, and D. Kriegman, "Parameterized Families of Polynomials for Bounded Algebraic Curve and Surface Fitting," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 16, no. 3, pp. 287-303, Mar. 1994.
[10] M.M. Blane, Z. Lei, H. Ivi, and D.B. Cooper, "The 3l Algorithm for Fitting Implicit Polynomial Curves and Surfaces to Data," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 22, no. 3, pp. 298-313, Mar. 2000.
[11] G. Taubin, "Estimation of Planar Curves, Surfaces, and Nonplanar Space Curves Defined by Implicit Equations with Applications to Edge and Range Image Segmentation," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 13, no. 11, pp. 1115-1138, Nov. 1991.
[12] N. Chernov and C. Lesort, "Statistical Efficiency of Curve Fitting Algorithms," Computational Statistics and Data Analysis, vol. 47, no. 4, pp. 713-728, 2004.
[13] A. Spira and R. Kimmel, "Pictorial Analysis of Line-Drawings," Computational Aesthetics in Graphics, Visualization, and Imaging, pp. 123-130, Springer, 2008.
[14] V.J. Katz, "The 3l Algorithm for Fitting Implicit Polynomial Curves and Surfaces to Data," Math. Magazine, vol. 52, no. 3, pp. 146-156, 1997.
[15] T. Kanungo, D.M. Mount, N.S. Netanyahu, C.D. Piatko, R. Silverman, and A.Y. Wu, "An Efficient K-Means Clustering Algorithm: Analysis and Implementation," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 24, no. 7, pp. 881-892, July 2002.
[16] S. Immerwahr, Aegean Painting in the Bronze Age. Pennsylvania State Univ. Press, 1990.
[17] K. Birtacha and M. Zacharioudakis, "Sterotypes in Theran Wall Painting: Modules and Patterns in the Procedure of Painting," Proc. First Int'l Symp. the Wall Paintings of Thera, pp. 159-172, 1997.
[18] C. Doumas, The Wall-Paintings of Thera. Thera Foundation, 1992.
[19] C. Papaodysseus, M. Exarhos, T. Panagopoulos, C. Triantafillou, G. Roussopoulos, A. Pantazi, V. Loumos, D. Fragoulis, and C. Doumas, "Identification of Geometrical Shapes in Paintings and Its Application to Demonstrate the Foundations of Geometry in 1650 B.C.," IEEE Trans. Image Processing, vol. 14, no. 7, pp. 862-873, July 2005.
14 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool