The Solution of Nonlinear Equations via the Method of Hurwitz-Radon Matrices

Dariusz Jacek Jakóbczak

Department of Electronics and Computer Science, Technical University of Koszalin, Koszalin, Poland.

DOI: 10.4236/jcc.2014.210002                

Abstract

Image analysis and computer vision are interested in suitable methods to solve the nonlinear equations. Coordinate x  for f (x) = 0 is crucial because each equation can be transformed into f (x) = 0. A novel method of Hurwitz-Radon Matrices (MHR) can be used in approximation of a root of function in the plane. The paper contains a way of data approximation via MHR method to solve any equation. Proposed method is based on the family of Hurwitz-Radon (HR) matrices. The matrices are skew-symmetric and possess columns composed of orthogonal vectors. The operator of Hurwitz-Radon (OHR), built from these matrices, is described. Two-dimensional data are represented by discrete set of curve  f points. It is shown how to create the orthogonal OHR operator and how to use it in a process of data interpolation. MHR method is interpolating the curve point by point without using any formula or function.

Keywords

Image Analysis, Nonlinear Equation, Root of Function, Curve Interpolation, Hurwitz-Radon

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Ballard, D.H. (1982) Computer Vision. Prentice Hall, New York.
[2]	Yang, W.Y., Cao, W., Chung, T.-S. and Morris, J. (2005) Applied Numerical Methods Using MATLAB®. John Wiley & Sons, New York. http://dx.doi.org/10.1002/0471705195
[3]	Khalil, H.K. (2001) Nonlinear Systems. Prentice Hall, New York.
[4]	Kelley, C.T. (1995) Iterative Methods for Linear and Nonlinear Equations. Society for Industrial and Applied Mathematics, Philadelphia. http://dx.doi.org/10.1137/1.9781611970944
[5]	Markman, A.B. (1998) Knowledge Representation. Lawrence Erlbaum Associates, Mahwah.
[6]	Sowa, J.F. (2000) Knowledge Representation: Logical, Philosophical and Computational Foundations. Brooks/Cole, New York.
[7]	Soussen, C. and Mohammad-Djafari, A. (2004) Polygonal and Polyhedral Contour Reconstruction in Computed Tomography. IEEE Transactions on Image Processing, 11, 1507-1523. http://dx.doi.org/10.1109/TIP.2004.836159
[8]	Tang, K. (2005) Geometric Optimization Algorithms in Manufacturing. Computer-Aided Design & Applications, 2, 747-757. http://dx.doi.org/10.1080/16864360.2005.10738338
[9]	Kozera, R. (2004) Curve Modeling via Interpolation Based on Multidimensional Reduced Data. Silesian University of Technology Press, Gliwice.
[10]	Dahlquist, G. and Bjoerck, A. (1974) Numerical Methods. Prentice Hall, New York.
[11]	Ralston, A. (1965) A First Course in Numerical Analysis. McGraw-Hill Book Company, New York.
[12]	Eckmann, B. (1999) Topology, Algebra, Analysis-Relations and Missing Links. Notices of the American Mathematical Society, 5, 520-527.
[13]	Citko, W., Jakóbczak, D. and Sieńko, W. (2005) On Hurwitz-Radon Matrices Based Signal Processing. Workshop Signal Processing at Poznan University of Technology, Poznan.
[14]	Tarokh, V., Jafarkhani, H. and Calderbank, R. (1999) Space-Time Block Codes from Orthogonal Designs. IEEE Transactions on Information Theory, 5, 1456-1467. http://dx.doi.org/10.1109/18.771146
[15]	Sieńko, W., Citko, W. and Wilamowski, B. (2002) Hamiltonian Neural Nets as a Universal Signal Processor. 28th Annual Conference of the IEEE Industrial Electronics Society IECON, Sevilla.
[16]	Sieńko, W. and Citko, W. (2002) Hamiltonian Neural Net Based Signal Processing. The International Conference on Signal and Electronic System ICSES, Wroclaw.
[17]	Jakóbczak, D. (2007) 2D and 3D Image Modeling Using Hurwitz-Radon Matrices. Polish Journal of Environmental Studies, 16, 104-107.
[18]	Jakóbczak, D. (2010) Shape Representation and Shape Coefficients via Method of Hurwitz-Radon Matrices. In: Computer Vision and Graphics, in: Lecture Notes in Computer Science, 6374, 411-419.
[19]	Jakóbczak, D. (2009) Curve Interpolation Using Hurwitz-Radon Matrices. Polish Journal of Environmental Studies, 18, 126-130.
[20]	Jakóbczak, D. (2010) Application of Hurwitz-Radon Matrices in Shape Representation. In: Banaszak, Z. and Swic, A., Eds., Applied Computer Science: Modelling of Production Processes, Lublin University of Technology Press, Lublin, 63-74.
[21]	Jakóbczak, D. (2010) Object Modeling Using Method of Hurwitz-Radon Matrices of Rank k. In: Wolski, W. and Borawski, M., Eds., Computer Graphics: Selected Issues, University of Szczecin Press, Szczecin, 79-90.
[22]	Jakóbczak, D. (2010) Implementation of Hurwitz-Radon Matrices in Shape Representation. In: Choras, R.S., Ed., Advances in Intelligent and Soft Computing, Vol. 84, Image Processing and Communications Challenges, 39-50.
[23]	Jakóbczak, D. (2011) Object Recognition via Contour Points Reconstruction Using Hurwitz-Radon Matrices. In: Józefczyk, J. and Orski, D., Eds., Knowledge-Based Intelligent System Advancements: Systemic and Cybernetic Approaches, IGI Global, Hershey, 87-107.
[24]	Jakóbczak, D. (2011) Data Extrapolation and Decision Making via Method of Hurwitz-Radon Matrices. In: Jedrzejowicz, P., Thanh Nguyen, N. and Hoang, K., Eds., Computational Collective Intelligence, Technologies and Applications, Lecture Notes in Computer Science, Vol. 6922, 173-182.
[25]	Jakóbczak, D. (2011) Curve Parameterization and Curvature via Method of Hurwitz-Radon Matrices. Image Processing & Communications, 16, 49-56.