##
**Integral equation-wavelet collocation method for geometric transformation and application to image processing.**
*(English)*
Zbl 1474.65459

Summary: Geometric (or shape) distortion may occur in the data acquisition phase in information systems, and it can be characterized by geometric transformation model. Once the distorted image is approximated by a certain geometric transformation model, we can apply its inverse transformation to remove the distortion for the geometric restoration. Consequently, finding a mathematical form to approximate the distorted image plays a key role in the restoration. A harmonic transformation cannot be described by any fixed functions in mathematics. In fact, it is represented by partial differential equation (PDE) with boundary conditions. Therefore, to develop an efficient method to solve such a PDE is extremely significant in the geometric restoration. In this paper, a novel wavelet-based method is presented, which consists of three phases. In phase 1, the partial differential equation is converted into boundary integral equation and representation by an indirect method. In phase 2, the boundary integral
equation and representation are changed to plane integral equation and representation by boundary measure formula. In phase 3, the plane integral equation and representation are then solved by a method we call wavelet collocation. The performance of our method is evaluated by numerical experiments.

### MSC:

65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |

94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |

42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |

65T60 | Numerical methods for wavelets |

PDF
BibTeX
XML
Cite

\textit{L. Yang} et al., Abstr. Appl. Anal. 2014, Article ID 798080, 17 p. (2014; Zbl 1474.65459)

Full Text:
DOI

### References:

[1] | Li, Z. C.; Bui, T. D.; Tang, Y. Y.; Suen, C. Y., Computer Transformation of Digital images and Patterns, 17 (1989), Singapore: World Scientific, Singapore · Zbl 0744.68135 |

[2] | Tang, Y. Y.; Suen, C. Y., Image transformation approach to nonlinear shape restoration, IEEE Transactions on Systems, Man and Cybernetics, 23, 1, 155-171 (1993) · Zbl 0777.68101 |

[3] | Cheng, H. D.; Tang, Y. Y.; Suen, C. Y., Parallel image transformation and its VLSI implementation, Pattern Recognition, 23, 10, 1113-1129 (1990) |

[4] | Wolberg, G., Digital Image Warping (1990), Los Alamitos, CA, USA: IEEE Computer Society Press, Los Alamitos, CA, USA |

[5] | Zienkiewics, O. C., The Finite Element Method (1977), London, UK: McGraw-Hill, London, UK |

[6] | Mitchell, A. R., The Finite Difference Method in Partial Differential Equations (1980), New York, NY, USA: Addision-Wesley, New York, NY, USA · Zbl 0417.65048 |

[7] | Atkinson, K. E., The Numerical Solution of Integral Equations of the Second Kind, 4 (1997), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0899.65077 |

[8] | Hsiao, G. C.; Wendland, W. L., A finite element method for some integral equations of the first kind, Journal of Mathematical Analysis and Applications, 58, 3, 449-481 (1977) · Zbl 0352.45016 |

[9] | Donoho, D. L., Orthonormal ridgelets and linear singularities, SIAM Journal on Mathematical Analysis, 31, 5, 1062-1099 (2000) · Zbl 0952.42020 |

[10] | Hsiao, G.; MacCamy, R. C., Solution of boundary value problems by integral equations of the first kind, SIAM Review, 15, 4, 77-93 (1973) · Zbl 0235.45006 |

[11] | Antoniadis, A.; Brossat, X.; Cugliari, J.; Poggi, J.-M., Clustering functional data using wavelets, International Journal of Wavelets, Multiresolution and Information Processing, 11, 1 (2013) · Zbl 1271.62131 |

[12] | Bhatnagar, G.; Wu, Q. M. J., An image fusion framework based on human visual system in framelet domain, International Journal of Wavelets, Multiresolution and Information Processing, 10, 1 (2012) |

[13] | Chen, G.; Qian, S.-E.; Ardouin, J.-P.; Xie, W., Super-resolution of hyperspectral imagery using complex Ridgelet transform, International Journal of Wavelets, Multiresolution and Information Processing, 10, 3 (2012) · Zbl 1248.65131 |

[14] | Dahmen, W.; Prossdorf, S.; Schneider, R., Multiscale Methods for Pseudodifferential Equations. Equations, Recent Advances in Wavelet Analysis (1993) |

[15] | Wagner, R. L.; Chew, W. C., Study of wavelets for the solution of electromagnetic integral equations, IEEE Transactions on Antennas and Propagation, 43, 8, 802-810 (1995) |

[16] | Xiang, Z.; Lu, Y., An effective wavelet matrix transform approach for efficient solutions of electromagnetic integral equations, IEEE Transactions on Antennas and Propagation, 45, 8, 1205-1213 (1997) · Zbl 0945.78013 |

[17] | Wells,, R. O.; Zhou, X., Wavelet solutions for the Dirichlet problem, Numerische Mathematik, 70, 3, 379-396 (1995) · Zbl 0824.65108 |

[18] | Resnikoff, H. L.; Wells, R. O., Wavelet Analysis-The Scalable Structure of Information (1998), New York, NY, USA: Springer, New York, NY, USA · Zbl 0922.42020 |

[19] | Kumar, S.; Kumar, S.; Raman, B.; Sukavanam, N., Image disparity estimation based on fractional dual-tree complex wavelet transform: a multi-scale approach, International Journal of Wavelets, Multiresolution and Information Processing, 11, 1 (2013) · Zbl 1266.42086 |

[20] | Maity, S. P.; Kundu, M. K., Performance improvement in spread spectrum image watermarking using wavelets, International Journal of Wavelets, Multiresolution and Information Processing, 9, 1, 1-33 (2011) · Zbl 1208.94060 |

[21] | Maity, S. P.; Phadikar, A.; Kundu, M. K., Image error concealment based on QIM data hiding in dual-tree complex wavelets, International Journal of Wavelets, Multiresolution and Information Processing, 10, 2 (2012) · Zbl 1248.68536 |

[22] | Zaied, M.; Said, S.; Jemai, O.; Ben Amar, C., A novel approach for face recognition based on fast learning algorithm and wavelet network theory, International Journal of Wavelets, Multiresolution and Information Processing, 9, 6, 923-945 (2011) · Zbl 1242.68278 |

[23] | Delves, L. M.; Mohamed, J. L., Computational Methods for Integral Equations (1985), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0592.65093 |

[24] | Dahmen, W., Wavelet and multiscale methods for operator equations, ACTA Numerica, 6, 55-228 (1997) · Zbl 0884.65106 |

[25] | Zhou, Y. H.; Wang, J. Z., A generalized Gaussian integral method for the calculation of scaling function transforms of wavelets and its applications, Acta Mathematica Scientia A: Shuxue Wuli Xuebao, 19, 3, 293-300 (1999) · Zbl 0941.65015 |

[26] | Cohen, A.; Daubechies, I.; Vial, P., Wavelets on the interval and fast wavelet transforms, Applied and Computational Harmonic Analysis: Time-Frequency and Time-Scale Analysis, Wavelets, Numerical Algorithms, and Applications, 1, 1, 54-81 (1993) · Zbl 0795.42018 |

[27] | Umbaugh, S. E., Computer Vision and Image Processing (1998), New York, NY, USA: Prentice-Hall, New York, NY, USA |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.