# American Institute of Mathematical Sciences

September  2007, 6(3): 853-871. doi: 10.3934/cpaa.2007.6.853

## The complete orthogonal V-system and its applications

 1 Faculty of Information Technology, Macau University of Science and Technology, Macau, Macau 2 College of Sciences, North China University of Technology, Beijing 100041, China 3 School of Mathematics and Computer Sciences, Harbin Normal University, Harbin 150080, China

Received  May 2006 Revised  February 2007 Published  June 2007

Based on the function generator on [0,1], a class of complete orthogonal function system called as the V-system is studied in this paper. The V-system is composed by piecewise polynomials, and is capable of exactly describing the geometric information expressed by the popularly and widely used polynomial spline curves and surfaces. The V-system has all of the beautiful properties of the U-system: continuity, discontinuity, orthogonal completeness and reproducibility. In addition, the V-system also has the concise structure, compactly local support and multi-resolution capability. The V-system is the generalization of the well-known Haar function system, and is also a new class of practical and flexible wavelet bases. By utilizing the concepts of the energy and the descriptor of the V-system, we study the degree of similarity of geometric models which can be used in image analysis and processing.
Citation: Hui Ma, Dongxu Qi, Ruixia Song, Tianjun Wang. The complete orthogonal V-system and its applications. Communications on Pure & Applied Analysis, 2007, 6 (3) : 853-871. doi: 10.3934/cpaa.2007.6.853
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