Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Multiresolution Analysis Adapted to Irregularly Spaced Data | Hindawi Publishing Corporation EURASIP Journal on Advances in Signal Processing Volume 2009 Article ID 293952 19 pages doi 2009 293952 Research Article Multiresolution Analysis Adapted to Irregularly Spaced Data Anissa Mokraoui1 and Pierre Duhamel EURASIP Member 2 1L2TI Institut Galilee Universite Paris 13 99 Avenue Jean Baptiste Clement 93 430 Villetaneuse France 2LSS CNRS SUPELEC Plateau deMoulon 91 192 Gifsur Yvette France Correspondence should be addressed to Anissa Mokraoui Received 26 February 2009 Revised 18 July 2009 Accepted 18 September 2009 Recommended by Enrico Capobianco This paper investigates the mathematical background of multiresolution analysis in the specific context where the signal is represented by irregularly sampled data at known locations. The study is related to the construction of nested piecewise polynomial multiresolution spaces represented by their corresponding orthonormal bases. Using simple spline basis orthonormalization procedures involves the construction of a large family of orthonormal spline scaling bases defined on consecutive bounded intervals. However if no more additional conditions than those coming from multiresolution are imposed on each bounded interval the orthonormal basis is represented by a set of discontinuous scaling functions. The spline wavelet basis also has the same problem. Moreover the dimension of the corresponding wavelet basis increases with the spline degree. An appropriate orthonormalization procedure of the basic spline space basis whatever the degree of the spline allows us to i provide continuous scaling and wavelet functions ii reduce the number of wavelets to only one and iii reduce the complexity of the filter bank. Examples of the multiresolution implementations illustrate that the main important features of the traditional multiresolution are also satisfied. Copyright 2009 A. Mokraoui and P. Duhamel. This is an open access article distributed under the .