# American Institute of Mathematical Sciences

November  2011, 16(4): 1055-1069. doi: 10.3934/dcdsb.2011.16.1055

## Feature extraction of the patterned textile with deformations via optimal control theory

 1 College of Mathematics and Computer Science, Chongqing Normal University, Chongqing, China 2 Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong 3 Department of Industrial and Manufacturing Systems Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong

Received  September 2010 Revised  April 2011 Published  August 2011

In handling textile materials, deformation is very common and is unavoidable. When the fabrics are dispatched for further feature extractions, it's necessary to recover the original shape for comparison with a standard template. This recovery problem is investigated in this paper. By introducing a set of recovered functions, the problem is formulated as a combined optimal control and optimal parameter selection problem, governed by the dynamical system of a set of two-dimensional control functions. After parameterization of the control functions, the problem is transformed into a nonlinear optimization problem, where gradient based optimization methods can be applied. We also analyze the convergence of the parameterization method. Several numerical examples are used to demonstrate the method.
Citation: Zhi Guo Feng, K. F. Cedric Yiu, K.L. Mak. Feature extraction of the patterned textile with deformations via optimal control theory. Discrete and Continuous Dynamical Systems - B, 2011, 16 (4) : 1055-1069. doi: 10.3934/dcdsb.2011.16.1055
##### References:
 [1] A. Bodnarova, M. Bennamoun and K. K. Kubik, Suitability analysis of techniques for flaw detection in textiles using texture analysis, Pattern Analysis and Applications, 3 (2000), 254-266. doi: 10.1007/s100440070010. [2] A. Bodnarova, M. Bennamoun and S. Latham, Optimal Gabor filters for textile flaw detection, Pattern Recognition, 35 (2002), 2973-2991. doi: 10.1016/S0031-3203(02)00017-1. [3] D. Chetverikov and A. Hanbury, Finding defects in texture using regularity and local orientation, Pattern Recognition, 35 (2002), 2165-2180. doi: 10.1016/S0031-3203(01)00188-1. [4] C. H. Chan and G. Pang, Fabric defect detection by Fourier analysis, IEEE Trans. Ind. Application, 36 (2000), 1267-1276. doi: 10.1109/28.871274. [5] Z. G. Feng and K. L. Teo, Optimal feedback control for stochastic impulsive linear systems subject to Poisson processes, in "Optimization and Optimal Control," Springer Optimization and Its Applications, 39, Springer, New York, (2010), 241-258. [6] Z. G. Feng, K. L. Teo and V. Rehbock, Hybrid method for a general optimal sensor scheduling problem in discrete time, Automatica J. IFAC, 44 (2008), 1295-1303. doi: 10.1016/j.automatica.2007.09.024. [7] W. G. Litvinov, Optimal control of electrorheological clutch described by nonlinear parabolic equation with nonlocal boundary conditions, Journal of Industrial and Management Optimization, 7 (2011), 291-315. [8] S. V. Lomov, G. Huysmans, Y. Luo, R. S. Parnas, A. Prodromou, I. Verpoest and F. R. Phelan, Textile composites: Modelling strategies, Composites Part A: Applied Science and Manufacturing, 32 (2001), 1379-1394. doi: 10.1016/S1359-835X(01)00038-0. [9] R. Li, K. L. Teo, K. H. Wong and G. R. Duan, Control parameterization enhancing transform for optimal control of switched systems, Mathematical and Computer Modelling, 43 (2006), 1393-1403. doi: 10.1016/j.mcm.2005.08.012. [10] K. L. Mak, P. Peng and K. F. C. Yiu, Fabric defect detection using morphological filters, Image and Vision Computing, 27 (2009), 1585-1592. doi: 10.1016/j.imavis.2009.03.007. [11] K. L. Mak and P. Peng, An automated inspection system for textile fabrics based on Gabor filters, Robotics and Computer-Integrated Manufacturing, 24 (2008), 359-369. doi: 10.1016/j.rcim.2007.02.019. [12] M. J. D. Powell, A fast algorithm for nonlinearly constrained optimization calculations, in "Numerical Analysis" (ed. G. A. Watson) (Proc. 7th Biennial Conf., Univ. Dundee, Dundee, 1977), Lecture Notes in Mathematics, 630, Springer, Berlin, (1978), 144-157. [13] Pablo Rodriguez-Ramirez and Michael Basin, An optimal impulsive control regulator for linear systems, Numerical Algebra, Control and Optimization, 1 (2011), 275-282. [14] M. Tarfaoui and S. Akesbi, A finite element model of mechanical properties of plain weave, Colloids and Surfaces A: Physicochemical and Engineering Aspects, 187-188 (2001), 439-448. doi: 10.1016/S0927-7757(01)00611-2. [15] K. L. Teo, C. J. Goh and K. H. Wong, "A Unified Computational Approach to Optimal Control Problems," Pitman Monographs and Surveys in Pure and Applied Mathematics, 55, Longman Scientific & Technical, Harlow, copublished in the United States with John Wiley & Sons, Inc., New York, 1991. [16] K. F. C. Yiu, K. L. Mak and K. L. Teo, Airfoil design via optimal control theory, Journal of Industrial and Management Optimization, 1 (2005), 133-148. doi: 10.3934/jimo.2005.1.133. [17] D. J. Yao, H. L. Yang and R. M. Wang, Optimal financing and dividend strategies in a dual model with proportional costs, Journal of Industrial and Management Optimization, 6 (2010), 761-777.

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##### References:
 [1] A. Bodnarova, M. Bennamoun and K. K. Kubik, Suitability analysis of techniques for flaw detection in textiles using texture analysis, Pattern Analysis and Applications, 3 (2000), 254-266. doi: 10.1007/s100440070010. [2] A. Bodnarova, M. Bennamoun and S. Latham, Optimal Gabor filters for textile flaw detection, Pattern Recognition, 35 (2002), 2973-2991. doi: 10.1016/S0031-3203(02)00017-1. [3] D. Chetverikov and A. Hanbury, Finding defects in texture using regularity and local orientation, Pattern Recognition, 35 (2002), 2165-2180. doi: 10.1016/S0031-3203(01)00188-1. [4] C. H. Chan and G. Pang, Fabric defect detection by Fourier analysis, IEEE Trans. Ind. Application, 36 (2000), 1267-1276. doi: 10.1109/28.871274. [5] Z. G. Feng and K. L. Teo, Optimal feedback control for stochastic impulsive linear systems subject to Poisson processes, in "Optimization and Optimal Control," Springer Optimization and Its Applications, 39, Springer, New York, (2010), 241-258. [6] Z. G. Feng, K. L. Teo and V. Rehbock, Hybrid method for a general optimal sensor scheduling problem in discrete time, Automatica J. IFAC, 44 (2008), 1295-1303. doi: 10.1016/j.automatica.2007.09.024. [7] W. G. Litvinov, Optimal control of electrorheological clutch described by nonlinear parabolic equation with nonlocal boundary conditions, Journal of Industrial and Management Optimization, 7 (2011), 291-315. [8] S. V. Lomov, G. Huysmans, Y. Luo, R. S. Parnas, A. Prodromou, I. Verpoest and F. R. Phelan, Textile composites: Modelling strategies, Composites Part A: Applied Science and Manufacturing, 32 (2001), 1379-1394. doi: 10.1016/S1359-835X(01)00038-0. [9] R. Li, K. L. Teo, K. H. Wong and G. R. Duan, Control parameterization enhancing transform for optimal control of switched systems, Mathematical and Computer Modelling, 43 (2006), 1393-1403. doi: 10.1016/j.mcm.2005.08.012. [10] K. L. Mak, P. Peng and K. F. C. Yiu, Fabric defect detection using morphological filters, Image and Vision Computing, 27 (2009), 1585-1592. doi: 10.1016/j.imavis.2009.03.007. [11] K. L. Mak and P. Peng, An automated inspection system for textile fabrics based on Gabor filters, Robotics and Computer-Integrated Manufacturing, 24 (2008), 359-369. doi: 10.1016/j.rcim.2007.02.019. [12] M. J. D. Powell, A fast algorithm for nonlinearly constrained optimization calculations, in "Numerical Analysis" (ed. G. A. Watson) (Proc. 7th Biennial Conf., Univ. Dundee, Dundee, 1977), Lecture Notes in Mathematics, 630, Springer, Berlin, (1978), 144-157. [13] Pablo Rodriguez-Ramirez and Michael Basin, An optimal impulsive control regulator for linear systems, Numerical Algebra, Control and Optimization, 1 (2011), 275-282. [14] M. Tarfaoui and S. Akesbi, A finite element model of mechanical properties of plain weave, Colloids and Surfaces A: Physicochemical and Engineering Aspects, 187-188 (2001), 439-448. doi: 10.1016/S0927-7757(01)00611-2. [15] K. L. Teo, C. J. Goh and K. H. Wong, "A Unified Computational Approach to Optimal Control Problems," Pitman Monographs and Surveys in Pure and Applied Mathematics, 55, Longman Scientific & Technical, Harlow, copublished in the United States with John Wiley & Sons, Inc., New York, 1991. [16] K. F. C. Yiu, K. L. Mak and K. L. Teo, Airfoil design via optimal control theory, Journal of Industrial and Management Optimization, 1 (2005), 133-148. doi: 10.3934/jimo.2005.1.133. [17] D. J. Yao, H. L. Yang and R. M. Wang, Optimal financing and dividend strategies in a dual model with proportional costs, Journal of Industrial and Management Optimization, 6 (2010), 761-777.
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